# CAPACITORS, TANKS, SPRINGS AND THE LIKE: A MULTIMEDIA TUTORIAL

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1 FFP Udine November 2011 CAPACITORS, TANKS, SPRINGS AND THE LIKE: A MULTIMEDIA TUTORIAL Assunta Bonanno, Michele Camarca, Peppino Sapia, P.E.R. Group, Physics Department University of Calabria Rende (CS) - Italy Abstract In this work we present an interactive multimedia tutorial allowing to comparatively explore the dynamical behavior of the Two-capacitor system and of others systems, quite different, but showing a very similar energetic behavior: i) two communicating tanks; ii) two coupled lossy springs sharing elongation ; iii) plastic collision between two material points; iv) two coaxial, rotationally coupled, disks spinning around an axis. The aim of the work is to give learners useful insights on the fundamental subject of energy transformation. The tutorial, appropriate for high school and first year university students, is implemented in a form that makes it also suitable for classroom use with an Interactive White Board. 1. Introduction There is a well-known textbook problem (Halliday et Al., 1997; Lorrain et al., 1988) that every once in a while, since forty years, appears in the literature (Cuvaj, 1968; Powell, 1979; Sciamanda, 1996; Mita and Boufaida, 1999; Al-Jaber and Salih, 2000; Boykin et al., 2002): the so called Two Capacitor Problem, also present in on-line forums as The Capacitor Conundrum (Sciamanda, 1996). The problem concerns two capacitors, one of which ( C 1 ) charged, the other ( C 2 ) uncharged, which are connected at time t=0. Simple calculations show that in the process a fraction C2 ( C1 + C2 ) of the initial electrostatic energy disappears, apparently giving rise to a paradox. The behavior of this system asks some questions usually worrying learners (first of all: where does lost electrostatic energy go?). The search for answering these questions allows, from a didactical point of view, to highlight many subtleties about energy transformation processes. In this regard, it is very useful comparing the two capacitors system with others systems, quite different, but showing a very similar energetic behavior: i) two communicating tanks containing a fluid, initially located in one of the two; ii) two coupled lossy springs sharing elongation ; iii) plastic collision between two material points; iv) two coaxial, rotationally coupled, disks spinning around an axis. In this connection we present here an interactive multimedia tutorial allowing to comparatively explore the dynamical behavior of described systems. The tutorial (implemented in a form that makes it also suitable for classroom use with an Interactive White Board) is appropriate for high school and first year university students and has manifold goals; among others: i) to give learners useful insights on the fundamental subject of energy transformation; ii) to shed light on the role of dissipative phenomena in the achievement of equilibrium; iii) to illustrate the role of analogies in the study of physical phenomena, in the aim to allow students to get used to a proper identification of similarities between apparently different scientific subjects; iv) to pave the way for further quantitative treatment of the problem. In the present paper, after summarizing in section 2 the two-capacitor-problem family, we will describe, in section 3, the tutorial. In section 4, eventually, we draw some interesting conclusion. Figure 1: The two capacitor problem. A capacitor is initially charged, the other uncharged (left). Then capacitors are connected so to share the electric charge (right). In this process half the electrostatic energy disappears.

2 FFP Udine November The two-capacitor(-like) problem(s) A capacitor is charged (Fig. 1, left) by a given potential difference (p.d.) V 1; then it is disconnected from the e.m.f. source and connected to an uncharged capacitor, with which it shares its electric charge (Fig. 1, right). To determine the final state one must impose the conservation of charge and the requirement of a common final p.d. through the two capacitors. Simple calculations show that the initial stored electrostatic energy (U i =1/2CV 2 1 ) halves in the final equilibrium state (U f =1/4CV 2 1 ); so that a definite fraction of the initial energy disappears. For sake of simplicity we consider the case in which capacitors have equal capacities, so that this fraction is ½. More generally, one can easily show that the disappearing fraction of the initial energy is given by: C2 C + C where the subscripts 1 and 2 refer, respectively, to the initially charged or uncharged capacitor. 1 2 A very interesting consideration, on a pedagogical point of view, is the following: there are different mechanical systems (apparently dissimilar from each other, and even more so than the system of two capacitors) that show exactly the same kind of energetic behavior (Fig. 2). In fact, Simple calculations allow to verify that for each of the systems represented in Figure 2 the energy loss for the attainment of the final state of equilibrium is equal to half the initial energy (Table 1). (1) Figure 2: Mechanical systems exhibiting the same behavior as the two capacitor system: a) two tanks sharing an amount of liquid; b) two coaxial rotating disks sharing angular momentum; c) two connected springs sharing elongation; d) two plastically colliding masses sharing momentum. In each case the equilibrium is driven by equating an intensive variable: a) liquid height; b) angular velocity; c) elastic force; d) velocity.

3 FFP Udine November 2011 System System parameter Capacitors capacity (C) System variables (equilibrium condition) V, V : potential 1 2 difference across capacitors V = V ) ( 1 2 Energy before Lost fraction of initial energy CV Tanks sectional area (S) h, 1 2 h : heights of the liquid in the tanks h = h ) ( ρ gsh 1 2 Moment of inertia of the disks (I) ω, ω : Angular 1 2 velocities of the disks ω = ω ) ( Iω Elastic constant of the springs (k) Springs elongations δ = δ ) ( kδ Masses of colliding material points (m) Speeds of the material points v = v ) ( mv Table 1: Two-capacitors and related systems. This table establishes a formal correspondence among the systems by comparing: characteristic parameters (second column); system variables and related equilibrium condition (third column); expression of the initial energy in terms of system variables (fourth column); fraction of the initial energy lost after equilibrium has been reached. A closer comparison among described systems permits to highlight the following similarities: i) they are two-component systems; ii) their energy depends quadratically on two symmetric intensive variables; iii) their approach to equilibrium is driven by the tendency to equalize these two variables; iv) they all need some form of dissipation to reach equilibrium, and lose half the energy in reaching equilibrium. These analogies, and their conceptual implications are explored in a paper we have recently written (Bonanno et al 2012). A useful visual tool in order to understand the energetic behavior of two-capacitor-like systems is a graphical representation (Sciamanda 1996) of the system state in a Cartesian plane whose coordinates are, for each one of the systems in Table 1, the corresponding couple of variables in the third column. As an example, Figure 3 shows the representation for the two-tanks case. A little algebra allows to see that the fixed energy 1 states of each system lie on an ellipse in that Cartesian plane; while equilibrium states between the two subsystems are represented by points belonging to the bisector of the first and third quadrants. Reaching the equilibrium state, starting from an imbalance between the two subsystems, requires a jump from an outer ellipse to the intersection point between the bisector and an inner ellipse: the jump requires the dissipation of a given amount of the system s energy: this amount is always one half, in terms of the energy corresponding to the starting ellipse. 1 With reference to Table 1, the various kind of energy are, respectively: electrostatic, gravitational potential, rotational kinetic, elastic potential, translational kinetic.

4 FFP Udine November 2011 Figure 3: A useful graphical representation of the energetic behavior of the two-tanks system. A given state is represented in a Cartesian plane whose coordinates are the liquid s levels (h 1 and h 2 ) in the two subsystems. Fixed-energy states lie on an ellipse, while equilibrium states belong to the bisector of the first and third quadrants. The example depicted in the figure refers to an initial state (point A) in which the liquid is entirely contained in the right tank, and a final (equilibrium) state in which liquid s heights in the two subsystems are equal. Figure 4: A screenshot from the tutorial, illustrating by means of an animation the distribution of the electrostatic energy between the two subsystems (single capacitors) as the whole system approaches equilibrium.

5 FFP Udine November The multimedia tutorial The interactive tutorial, implemented as a web-browsable hypertext, is intended to introduce in a visually appealing way the two-capacitor-like systems, in the aim to discuss the subtleties regarding the transfer of energy between systems. In particular it helps shedding light on the role of dissipative phenomena in the achievement of equilibrium, suitably for high school (and first year university) students. On a more strictly pedagogical point of view, the proposed multimedia permits learners to acknowledge the same role of different variables in unlike systems, so to enlighten the power of analogies in the study of physical phenomena. Finally, it paves the way for further quantitative treatment of the problem, examples of which can be found in (Bonanno et al 2012). Figures 4 and 5 show some screenshot of the tutorial, which can be freely accessed on the web at the following URL: Figure 5: A screenshot from the tutorial, illustrating the mandated energy loss in a system constituted by two springs sharing elongation. References Al-Jaber S M, and Salih S K, Energy consideration in the two-capacitor problem, Eur. J. Phys. 21, (2000). Boykin T B, Hite D, and Singh N, The two-capacitor problem with radiation, Am. J. Phys. 70 (4), (2002). Bonanno A, Camarca M, and Sapia P, Reaching the equilibrium: the role of dissipation in analogous systems. Eur. J. Phys. Submitted (2012). Cuvaj C, On conservation of energy in electric circuits, Am. J. Phys. 36, (1968). Halliday D, Resnick R, and Walker J, Fundamentals of Physics (Wiley, New York, 1997), 5th ed. Lorrain P, Corson D, and Lorrain F, Electromagnetic Fields and Waves (Freeman, New York, 1988), 3rd ed., p Mita K, and Boufaida M, Ideal capacitor circuits and energy conservation, Am. J. Phys. 67 (3), (1999). Powell R A, Two-capacitor problem: a more realistic view, Am. J. Phys. 47 (5), (1979). Sciamanda R J, Mandated energy dissipation- e pluribus unum, Am. J. Phys. 64 (10), (1996).

6 STUDY OF STABILITY MATTER PROBLEM IN MICROPOLAR GENERALISED THERMOELASTIC arminder singh, kounsil, c.t. institute of technology, jalandhar, india dr. gurpinder singh, samra, Lyallpur khalsa college, jalandhar, india Abstract The theory of micropolar thermoelasticity has many applications. One form of the recent years concerning the problem of propagation of thermal waves at finite speed and the possibility of second sound effects established a new thermo mechanical theory of deformable media that uses a general entropy balance as postulated and the theory is illustrated in detail in the context of flow of heat in a rigid solid, with particular reference to the propagation of thermal waves at finite speed. Then theory of thermoelasticity for non-polar bodies, based on the new procedures, was discussed and employed the eigen value approach to study the effect of rotation and relaxation time in two dimensional problem of generalized thermoelasticity. Recently investigation shows the dynamic response of a homogeneous, isotropic, generalized thermoelastic halfspace with voids subjected to normal, tangential force and thermal stress. In this paper we introduce the eigen value approach, following Laplace and Fourier transformation has been employed to find the general solution of the field equation in a micropolar generalized thermoelastic medium for plane strain problem. An application of an infinite space with an impulsive mechanical source has been taken to illustrate the utility of the approach. The integral transformation has been inverted by using a numerical inversion technique to get result in physical domain. The result in the form of normal displacement, normal force stress, tangential force stress, tangential couple stress and temperature field components have been obtained numerically and illustrated graphically. Special case of a thermoelastic solid has also been deduced. Keywords: Eigen value; Mechanical sources ; Laplace and Fourier transform ; Concentrated force; Romberg s integration 1. Introduction The theory of micropolarthermoelasticity has been a subject of intensive study. A comprehensive review of works on the subject was given by Eringen (1970) and Nowacki (1966). There has been very much written in recent years concerning the problem of propagation of thermal waves at finite speed. A generalized theory of linear micropolarthermoelasticity that admits the possibility of second sound effects was established by Boschi (1993). Recently, Green and Naghdi (1991) established a new thermomechanical theory of deformable media that uses a general entropy balance as postulated by Green and Naghdi (1977). The theory is illustrated in detail in the context of flow of heat in a rigid solid, with particular reference to the propagation of thermal waves at finite speed. A theory of thermoelasticity for non-polar bodies, based on the new procedures, was discussed by Green and Naghdi (1993).Bakshi, Bera and Debnath (2004) employed the eigen value approach to study the effect of rotation and relaxation time in two dimensional problem of generalized thermoelasticity. Kumar and Rani (2004) studied the deformation due to mechanical and thermal sources in generalized orthorhomticthermoelastic material. Kumar and Rani (2005) investigated the dynamic response of a homogeneous, isotropic, generalized thermoelastic halfspace with voids subjected to normal, tangential force and thermal stress. The micropolar theory was extended to include thermal effects by Nowacki (1966) and Eringen (1970).

7 Kumar and Chadha (1986) derived the expressions for displacements, microrotation, force stress, couple stress and first moment for a half - space subjected to an arbitrary temperature field and a particular case of line heat source has been discussed in detail. The uniqueness of the solution of some boundary value problems of the linear micropolarthermoelasticity was investigated by Cracium (1990). Passarella (1996) solved the initial-boundary value problem for micropolarthermoelasticity and proved a uniqueness theorem for the problem. Mahalanabis and Manna (1997) discussed eigen value approach to linear micropolarthermoelasticity by arranging basic equations of elasticity in the form of matrix deferential equation in the Hankel transform and extended the approach to linear thermoelasticity. Marin and Lupu (1998) investigated harmonic vibrations in thermoelasticity of micropolar bodies. Kumar and Deswal (2001) discussed the disturbance due to mechanical and thermal sources in homogeneous isotropic micropolar generalized thermoelastic half-space. 2. Formulation and solution of the Problem We consider a homogeneous, isotropic, micropolar generalized thermoelastic solid in an undisturbed state and initially at uniform temperature. We take a cartesian system (x,y,z) and z- axis pointing vertically into the medium. Following Eringen (1968), Lord and Shulman (1967) and Green and Lindsay (1972), the field equations and the constitutive relations in micropolar generalized thermoelastic solid without body forces, body couples and heat sources can be written as ν1+ T=ρ K=ρj =!" # + T \$ %+ν \$# +Ξ \$ % mij = αφ δ + β φ + γφ r, r ij i, j j, i, t ij = λu + T ( u + u ) + K ( u ε φ )- ν T + τ, r, rδ ij µ i, j j, i j, i ijr r 1 δ ij t For the L-S (Lord Shulman) theoryτ 1 = 0, Ξ = 1and for G - L (Green Lindsay) theoryτ 1 = 0, Ξ = 0, The thermal relaxations τ 0 and τ 1 satisfy the inequality τ1 τ 0 > 0 for the G-L theory only. However, it has been proved by Sturnin (2001) that the inequalities are not mandatory for τ 0 and τ 1 to follow. For two dimensional plane strain problem parallel toxz-plane, we assume =(,0,( +,,=0,-,0

8 The displacement components 1 3, u u and microrotation component - depend upon x, z and t and are independent of co-ordinate y, so that 0 y. With these considerations and using (2.6) and introducing the non-dimensional quantities as, C x x 1 = ω C z z, 1 = ω, T T T 0 =, T C u u ν ρω =, m T C m ν ω = Where ( ), K C + = µ λ ω C λ µ ρ + = Now applying Laplace and Fourier transform defined by ( ) ( ), exp(-pt)dt t z x f p z x f 0 =,,,, ( ) ( ), ) e(-,,,, ~ - dx x p z x f p z f ιξ ξ = on the set of equations (2.1)-(2.3), after suppressing primes, we get ( ) ( ) T ~ 1 ~ m ~ m ~ p m m 1 ~ = p dz d dz du u dz u d τ ιξ φ ξ ι ξ ( ) T ~ ~ m ~ p m ~ m m 1 ~ = dz d u dz du dz u d ξ φ ι ξ ξ ι ( ) ~ p m 2m ~ m ~ m ~ φ ξ ξ ι φ = u dz u d dz d

9 2 ~ d T d u~ = ε p dz dz Where ~ { T } ( ) ~ τ pξ ι ξ u + + ξ + p( 1 τ p) 0 λ + 2µ + K = ρ C m1 2 1, m λ + µ =, 2 2 ρ C1 m = K 4 2 ρ C1, m KC ρω =, 2 m µ 7 =, m 2 8 = 2 ρ C1 ρ C1 λ, 2 T 0 β1 ε = ρ K ω. Equations (2.9) - (2.12) can be written in the vector matrix differential equation form as d W dz Where ( ξ, z, p) = A( ξ, p) W ( ξ, z, p) W U O I =, DU A = A2 A 1 A 0 f12 f13 0 f 0 0 f, = f f A 2 g11 0 = 0 g41 g g g g g g and O is the Null matrix of order 4 with f lξ m =, f13 = m3 m3 f m, 31 5 = f = ε p( + τ p Ξ) m 42 0, 1,

10 g 22 = 2 2 ( 3ξ + ) m p, m 1 g 22 l m = m 1 4 ξ, 2 = + Ξ g = ξ + p( + τ p) ( τ ) g lε ξ p 1 p, 41 0 To solve the equation (2.14), we take W( ξ, z, p) X (, p) e 1, 44 0 q z = ξ for some q, So we obtain A( ξ, p)w( ξ, z, p)= q W( ξ, z, p), This leads to Eigen value problem. The characteristic equation corresponding to the matrix A is given by which on expansion provides us q σ q + σ q σ q + σ = 0 1 Where /0 1Ι=0 σ = g + g + g + g + f f + f f + f f, σ = g g + g g + g g + g g + g g + g g + f f g + f f g + f f g f f f f g g + f f g + f f g f f g + f f g + f f g , g g g f f σ = g g g + g g g + g g g + g g g g g g g g g + f f g g + σ f f g g + f f g g + f f g g f f g g f f g g f f g g f13 f - 31g22g44 g14g41g 22- g14g41g33,. 4 = g 11g22g33g 44 g23g32g11g 44 + g14g41g22g33 + g23g32g14g 41 The eigen values of the matrix A are the characteristic roots of the equation (2.19). The vectors (, p) X ξ corresponding to the eigen values qs can be determined by solving the homogeneous equations [ qi] X ( ξ, p) A s = 0 The set of eigen vectors X (,p) s ξ ; s =1, 2, 3,..., 8 may be defined as

11 p), ( X p), ( X p)=, ( X s2 s1 s ξ ξ ξ Where s1 s s s s a q b X (, p), - c ξ ξ = 2 s s s s s2 s s s a q b q X (, p), - q c q ξ ξ = s s s l1 s -a q b X (, p), - c ξ ξ = 2 s s s s l2 s s s a q -b q X (, p), q -c q ξ ξ = ( ) { } ( ) ( )( )( ) p p q m p m p q m p m m m m q p p m a s s s s 3 s }] { 1 [ Ξ + τ + τ + + ξ ε ξ + + τ + ξ ξ = ( ) { } ( )( ) ( )( )( ) ] } 2 { 1 [ m s 3 Ξ = p p q m p m p m q p m q m p m m q m q p p b s s s s s s τ τ ξ ξ ε ξ ξ τ ξ ι ( )( ) ( ) { } [ ] p p q b a p pq c s s s s s τ ξ ιξ τ ε Ξ + = ( ) { } ( ) ( )( )( ) q p p p m q p m m q q p p m m s s s s 3 5 s ] 1 1 } { 1 [ ξ + τ Ξ + τ ε + + ξ + τ + ξ = Thus solution of equation (2.14) is as given by Sharma and Chand.(1992) ( ) ( ) [ ] e p X E e p X E p) z,, W( s z s -q s s z s q s s = + + ξ + ξ = ξ ,, E, E, E, E, E, E, E and 8 E are eight arbitrary constants. The equation (2.32) represents a general solution of the plane strain problem for isotropic, micropolar generalized thermoelastic solid and gives the displacement, microrotation and temperature field in the transformed domain. 3.Applications

12 Mechanical Source We consider an infinite micropolar generalized thermoelastic space in which a concentrated z force where F 0 is the magnitude of the force, F = F0 δ (x) δ (t) acting in the direction of the z- axis at the origin of the Cartesian co-ordinate system as shown in fig.1. The boundary condition for present problem on the plane z=0 are MICROPOLAR THERMOELASTIC MEDIUM-II z< 0 F 0 O x ν z> 0 MICROPOLAR THERMOELASTIC MEDIUM-I + + u1(x, 0, t) - u1(x, 0, t) = 0, u 3 (x, 0, t) - u 3 (x, 0, t) = 0, (x, + φ 2 0,t) - φ2(x,0,t)= 0, T(x,0 +,t) -T(x,0,t)= 0, T + T + ( x, 0, t) ( x, 0, t) = 0, t31 ( x, 0, t) t31 ( x, 0, t) = 0, z z t + + ( 0, t) t ( x, 0, t) = F δ (x) δ (t), m ( x, 0, t) m ( x, 0, t) 33 x, = F0 Making use of equations (2.6)-(2.7) and F 0 = in equations (2.4)-(2.5), we get the stresses in K the non-dimensional form with primes. After suppressing the primes, we apply Laplace and Fourier transforms defined by equations (2.8) on the resulting equations and from equation (3.1), we get transformed components of displacement, microrotation, temperature field, tangential force stress, normal force stress and tangential couple stress for z > 0 are given by 3 { q } 1z q2z q e e e z e q4z u % ( ξ, z, p) = a q E + a q E + a q E + a q E, u % ( ξ, z, p) = b E e + b E e + b E e + b E e, q 1 z q 2 z q3 z q 4 z { q } 1z q2z q e e e z e q4z % ϕ ( ξ, z, p) = -ξ E + E + E + E, T% ( ξ, z, p) = c E e + c E e + c E e + c E e, q 1 z q 2 z q3z q 4 z

13 2 q z 2 ( ) e ( ) 2 q z 2 ( + ιξ + ξ ) e + ( + ιξ + ξ ) % t ( ξ, z, p) = m a q + ιξ b s + ξ m E + m a q + ιξ b m + ξ m E e + 1 q2z m a q b m m E m a q b m m E e, 3 q4z ( ( p) ) ιξ ( τ ) ιξ ( τ ) ιξ m a + + ( + τ p) % t ( ξ, z, p) = [ ιξm a q + m b q + c 1 + τ E e q1z q2z ( m 8a2 q 2 m1b2q 2 c2 1 1 p ) E 6e q3z ( m8a3q 3 m1b3q3 c3 1 1 p ) E7e q4z ( 8 4q 4 m1b4q 4 c4 1 1 ) E 8e, { q } 1z q2z q z q4z m% ( ξ, z, p) = ξ s q E e + q E e + q E e + q E e, for z<0, the above expressions get suitably modified, e.g. u % ( ξ, z, p) = a q E e + a q E e + a q E e + a q E e, q 1 z q 2 z q3 z q 4 z Making use of the transformed displacements, microrotation, microstretch and stresses given by (3.6)-(3.12) in the transformed boundary conditions, we obtain eight linear relations between the E i s, which on solving gives F E = E = c a a + c a a + c a a ( ) ( ) ( ) q 1 1 F E = E = c a a + c a a + c a a, ( ) ( ) ( ) q2 1 F E = E = c a a + c a a + c a a ( ) ( ) ( ) q3 1 F E = E = c a a + c a a + c a a ( ) ( ) ( ) q4 1 Where,,, {( ) ( ) ( )} c2 {( a3b1 a1b3 ) ( a1b4 a4b1 ) ( a4b3 a3b4 )} c3 {( a1b2 a2b1 ) ( a4b1 a1b4 ) ( a2b4 a4b2 )} c ( a b a b ) ( a b a b ) ( a b a b ) = m [ c a b a b + a b a b + a b a b { } Thus functions u% 1, u% 3, % φ 2, T %, t% 31, t% 33 and m% 32 have been determined in the transformed domain and these enable us to find the displacements, microrotation, temperature field and stresses. ],

14 Case I : For L-S theory, a s, b s and c s in the expressions (3.5)-(3.12) take the form ξ a = [ + p 1 + p q { m m + m 2m + + p m q { ξ ( τ 0 ) } ( 5 ξ 6 ) s s s m3 s ( s )( + 0 ) ε p 2m ξ p m q 1 τ p } ], ι b = [ + p 1 + p q { m m q + 2 m + + p m q m + p m q } { ξ ( τ 0 ) } 4 5 ( 5 ξ 6 )( 1ξ 3 ) s s s s s m3 s ( s )( + 0 ) ε p ξ 2m ξ p m q 1 τ p ], c s ( ιξ + ) ( 1 τ ) ε pqs as bs =, 2 2 qs { ξ + p + 0 p } Where m s [ { ξ p( 1 τ 0 p) qs } { m2qs ( m1ξ p qs m3 ) } m3 = ( 1 )( s ) ε p 1 + τ p q ξ ] ; s = 1,2,3,4 and ± qs (s =1, 2, 3, 4) are roots of the equation (2.19) in which σ1, σ 2, σ3 and σ 4 are obtained respectively from expressions (2.20)-(2.23) by taking τ 1 = 0, Ξ = 1. Case II : For G-L theory, as b s and c's in the expressions (3.5)-(3.12) take the form ξ a = [ + p 1 + p q { m m + m 2m + + p m q { ξ ( τ 0 ) } ( 5 ξ 6 ) s s s m3 s ( s )( + 1 ) ε p 2m ξ p m q 1 τ p } ], ι b = [ + p 1 + p q { m m q + 2 m + + p m q m + p m q } { ξ ( τ 0 ) } 4 5 ( 5 ξ 6 )( 1ξ 3 ) s s s s s m3 s ( s )( + 1 ) ε p ξ 2m ξ p m q 1 τ p ], c s ( ιξ + ) ( 1 τ ) ε pqs as bs =, 2 2 qs { ξ + p + 0 p } Where

15 m s [ { ξ p( 1 τ 0 p) qs } { m2qs ( m1ξ p qs m3 ) } m3 = and 2 2 ( 1 )( s ) ε p 1 + τ p q ξ ] ; s = 1,2,3,4 ± qs (s =1, 2, 3, 4) are roots of the equation (2.19) in which σ1, σ 2, σ3 and 4 respectively from expressions (2.20) - (2.23) by taking Ξ = 0 σ are obtained Case III : For Green and Naghdi theory (G-N), equations (2.1), (2.3) and (2.4) can be written as ν T=ρ t K T=ρC T t +4. \$ t t i j = λ u r, r δ + µ i j ( u + u ) + K ( u ε φ ) i, j j,i j,i i j r r νtδ and K is not the usual thermal conductivity but a material characteristics constant in G - N & theory and is given K C = ( λ + 2µ ) 4 With the help of equations (3.26)-(3.28) and following the procedure of the previous sections, we get the expressions for displacements, microrotation, temperature, field, force stresses and couple stress by taking in equations (3.5)-(3.11). ( 1+ τ p) = 1, ( τ p) 1 0 i j 1+ = 4 p, Particular Case I : Neglecting micropolarity effect i.e. α = β = γ = K = j = 0 in equations (3.5)- (3.12), the expressions for displacement components, force stresses and temperature field are obtained in a thermoelastic medium as q1 z q2z q3z { e e e 4 } u % ( ξ, z, p) = a q E + a q E + a q E, q z q 1 z 2 3 ξ = q z 3 u % (, z, p) b E e b E e b E e, q z q q 1 z z 2 3 { } T% ( ξ, z, p) = ξ E e + E e + c E e, 2 q z q q 1 2 z ( ) e ( ) e ( ) e z % t ( ξ, z, p) = m { a q + ιξ b E + a q + ιξ b E + a q + ιξ b E },

16 { ( p) } { ιξ ξ ( τ1 p) } { + ιξ 8 ξ ( + τ1 p) } q z % 1 t ( ξ, z, p) = [ m q b + ιξm a q ξ 1+ τ E e Where ( a ) F a E = E = 1 2q E E , q z 2 m q b + m a q 1+ E e + q z 3 m q b m a q 1 E e ], ( ) F a a = E = 2q ( ) F a a = E = q3 1,, {( ) ( ) ( )} = + + m a b a b a b a b a b2 a b, ξ 2 2 a = [m2 { ξ + p ( 1 + τ 0 p) q } + ε p ( 1+ τ 0 pξ )( 1 + τ1 p) }], s s m 3 3 s b = ι [{ ξ p ( 1 τ 0 p) q }( ) ( )( ) 1 3 } ], s s m ξ p m qs ε p ξ τ p τ p m Ξ + s ( 1+ τ pξ) 3 {( m ξ p q ) 1 s ξ m 3 2} 2 ε pqs = + m, s ξ m and q ( s 1,2,3 ) 6 ± = are the roots of the equation s q σ 1q + σ 2q σ 3 = m1ξ + p m3ξ + p 2 ε p ξ m2 σ1 = { ξ p ( 1 τ 0 p) } ( 1 τ 0 p )( 1 τ1 p), Ξ + m m m m m

17 m1ξ + p m3ξ + p m3ξ + p 2 σ 2 = { ξ p( 1 τ 0 p )} + m m m m ξ + p ε p m ξ + p Ξ m m m { ξ p( 1 τ 0 p) } ( 1 τ 0 p )( 1 τ1 p) ξ m2 2 2ε pξ m2 ε pξ { ξ + p( 1+ τ 0 p) } 1+ τ ( 0 pξ )( 1+ τ1 p) + ( 1+ τ 0 pξ )( 1 + τ1 p), m m m m m With λ + 2µ m =, ρc µ m3 = ρ C 2 1 i. For L-S theory : Taking τ 1 = 0, Ξ = 1in expression given by (3.29)-(3.33) of particular case I, we obtain expressions for displacement components, temperature field and force stresses. ii. For G-L theory : Taking Ξ = 0 in expressions given by (3.29)-(3.33)of particular case I, we obtain expressions for displacement components, temperature field and force stresses iii. For G-N theory : Neglecting micropolarity effect i.e. ( α = β = γ = K = j = 0 ) in subcase III of case I, we get the expressions for displacement components, temperature field and force stresses are obtained in a thermoelastic medium by taking 1 1, + τ1 p = τ 0 ε 1+ p = 4 p, 1 + τ0 pξ = p, 1 C ε =, ω = 1, 4 h 2 T 0 ν ε1 = ρ K in equations (3.29)-(3.44) as q1 q2 q3 { 4 } 0 0 z z z e e e u % ( ξ, z, p) = a q E + a q E + a q E, q z q z ξ = q z 3 u % (, z, p) b E e b E e b E e, q z 1 0 q z q 2 0 z 3 { } T% ( ξ, z, p) = ξ E e + E e + c E e, q z 2 q z ( ) 4 ( 2 ) ( 3 3 ) % t (, z, p) m { a q b E a q b E a q b E }, q z 3 31 ξ = 7 + ιξ e ιξ 2 5e ιξ 6e q z { ( )} ( ) q z 3 + { m q b + ιξm8a q ξ ( 1+ τ1 p) } E e ], { } % t (, z, p) [ m q b m a q 1 E m q b m a q 1 E q z 2 33 ξ = + ιξ ξ + τ p e + + ιξ ξ + τ1 p e

18 Where E 0 0 ( a ) F a = E 1 4 =, 0 0 E E q ( ) F a a = E = 2q 0 1 3, ( ) F a a = E 3 6 = 0 0 2q3 1, {( ) ( ) ( )} = m a b a b a b a b a b2 a b, { ( ) s } 0 ξ a = [m 1, s 0 2 ξ + p + τ 0 p q + ε1 p m 3 s { ( )}( ) ε 2 1 ξ ξ s ξ 1 3 s ι b = [ p p q m + p m q } s 0 m 0 s ε1 p q = ξ m s 2 0 s 0 and q ( s 1,2,3 ) s {( m ξ + p m q ) 2} 2 s ξ m, ε1 = 4ε, 1 ± = are the roots of the equation σ1 + σ 2 σ 3 = 0, q q q σ m ξ + p m ξ + p ε p ξ m ( ξ 4 p ), m m m m m = m1ξ + p m3ξ + p m3ξ + p 2 2 σ 2 = ( ξ 4 p ) + m m m m ξ + p ε p m ξ + p + + ( ξ 4 p ) m m m ξ m ε1 p ξ m2 ε1p ξ ( ξ + 4 p ) +, m m m m m

19 where f % e and f % 0 are even and odd parts of the functions f% ( ξ, z, p) respectively. Thus, expression (4.1) gives us the Laplace transform f ( x, z, p) of the function f (x, z, t). Following Honig and Hirdes (1984), the Laplace transform function f ( x, z, p) can be inverted to f (x, z, t). σ m ξ + p m ξ + p ε p ξ m ξ + p 4, = ( ξ p + ) + m m m m Thus, the expressions given by equations (3.5)-(3.12) with the help of (3.13)-(3.16) and (3.17) represent the solution of plane strain problem under consideration in the transformed domain using eigen value approach. 4. Inversion of the transforms To obtain the solution of the problem in the physical domain, we must invert the transforms for three theories that is L-S, G-L and G-N. These expressions are functions of z, the parameters of Laplace and Fourier transforms p and ξ respectively and hence are of the form f ( x, z, p ). To get the function f ( x, z, t) in the physical domain, first we invert the Fourier transform using f ( x, z, p) = ~ 1 exp( iξ x) f ( ξ, z, p) dξ = π 0 ~ { cos( ξ x) f + isin( ξ x) ~ f } The last step in the inversion process is to evaluate the integral in equation (4.1). This was done using Romberg s integration with adaptive step size. This method uses the results from successive refinements of the extended trapezoidal rule followed by extrapolation of the results to the limit when the step size tends to zero. The details can be found in Press et al. (1986). 5.Numerical Results and Discussion Following Eringen [1984], we take the following values of relevant parameters for the case of Magnesium crystal as e 0 dξ ρ= 1.74 gm µ = T = 23 C, 0 h = 1cm, 11 γ = cm 3, dyne, 4 dyne, j = K = ε = τ 0 z = = cm 2 dyne 13, cm sec, 2, λ= C K τ 1 11 dyne cm 0 = 0.23 Call gm C, = = cal / cm sec, 13 sec,

20 > U 3 (x,1) L-S(MTE) L-S(TE) G-L(MTE) G-L(TE) G-N(MTE) G-N(TE) > x Fig.2 Variation of normal displacement U 3 (x,1) -----> T 33 (x,1) L-S(MTE) L-S(TE) G-L(MTE) G-L(TE) G-N(MTE) G-N(TE) > x -4-6 Fig.3 Variation of normal force stress T 33 (x,1)

21 L-S(MTE) 1.5 G-L(MTE) G-N(MTE) -----> M 32 (x,1) > x Fig.4 Variation of tangential couple stress M 32 (x,1) 3 2 L-S(MTE) L-S(TE) G-L(MTE) G-L(TE) G-N(MTE) G-N(TE) -----> T * (x,1) > x Fig.5 Variation of temprature field T * (x,1) Discussion The variations of normal displacement U 3 with distance x for three different theories (L-S, G-L and G-N) in both media after multiplying the original values for G-N theory in MTE medium by

22 10 are shown in fig. 2 The values of normal displacement due to microrotation effect are less in MTE medium in comparison to TE medium in the 0 x 0.5 for all three theories, whereas the values of U 3 oscillate as x increases further in the rest of the range for both media. It is also evident that normal displacement decreases for both media for L-S and G-L theories, increases gradually in MTE medium for G-N theory and oscillate in TE medium for G-N theory. The values of normal force stress T33 in magnitude are more for three different theories in MTE medium in comparison to TE medium. It is also noticed that the values of normal force stress oscillate for L-S and G-L theories in MTE and TE media. The values of normal force stress also oscillate for G-N theory in TE medium, whereas these decrease gradually with increasing value of x in MTE medium. These variations of normal force stress have been shown in fig. 3 after dividing the original values by 10 in case of G-N theory in MTE medium. Fig. 4 depicts the variations of tangential couple stress M 32 for three different theories in MTE medium after dividing the original values for G-N theory by 10. The behaviour of tangential couple stress is oscillatory for three theories. It is noticed that the value of tangential couple stress for G-N theory are large in comparison to L-S and G-L theories in the range 0 x 2.5 and the values are small for the rest of the range. The range of values of temperature field in magnitude is large in case of three theories in MTE medium in comparison to TE medium. It is also observed that temperature field oscillate in TE medium for three different theories but in MTE medium for L-S and G-L theories, the temperature field oscillate. The values of temperature field for G-N theory decrease gradually with increasing value of x in MTE medium. These variations shown in fig. 5 after multiplying the original values in case of L-S and G-N theories by 10 2 and 10 2 respectively in MTE medium; the original values in case of G-N theory (TE medium) and also magnified by multiplying Conclusion From the above numerical results, we conclude that micropolarity has a significant effect on normal displacement, normal force stress and temperature field mechanicalsource for three theories.mcropolar effect is more appreciable for normal displacement and temperature field in, comparison to normal force stress. Application of the present paper may also be found in the field of steel and oil industries. The present Problem is also useful in the field of geomechanics, where, the interest is about the various phenomenon occurring in the earthquakes and measuring of displacements, stresses and temperature field due to the presence of certain sources. Nomenclature λ, µ = Lame's constants α, β,γ,k = Micropolar material constants α 0, λ 0, λ 1 stretch. = Material constants due to the presence of

23 λ I, µ I, K I, α I, ν, γ I, α 0I, λ 0I, λ 1I = Microstretch viscoelastic constants ρ = Density j = Micro-inertia u = Displacement vector, = Microrotation vector, = Scalar microstretch t ij m ij = Force stress tensor = Couple stress tensor λ l δ ij ε ijr ι = Iota = Microstress tensor = Kronecker delta = Alternating tensor = Gradient operator And dot denotes the partial derivative w.r.t. time. References : Boschi E. and Jesan D.(1993): A Generalized Theory of Linear MicropolarThermoelasticity,- Meccanica, Vol. VIII, pp Bakshi R., Bera R.K. and Debnath L.(2004): Eigen value approach to study the effect of rotation and relaxation time in two dimensional problems of generalized thermoelastic, Int. J. Engg. Sci., Vol. 42,pp Cracium L.(1990): On the uniqueness of the solution of some boundary value problems of some boundary value problems of the linear micropolarthermoelasticity, - Bull. Int. Politech. Iasi. Sect.,Vol.- 36, pp Eringen A.C.(1970):Foundations of MicropolarThermoelasticity,- International Center for Mechanical Science Courses and Lecturer. No.23. -Springer, Berlin. Eringen A.C.(1984): Plane waves in a non-local micropolar elasticity, -Int. J. Engg. Sci., Vol.22, pp Eringen A.C.(1968):Theory of micropolar elasticity in Fracture (chapter-7), Vol.II,Academic Press, New York, Ed. H. Leibowitz. Green A.E. and Lindsay K.A. (1972): Thermoelasticity, - J. Elasticity, Vol. 2, pp Green A.E. and Naghdi P.M. (1977): On thermodynamics and the nature of the second law, Proc. Ray Soc. London A., Vol.-357,pp

24 Green A.E. and Naghdi P.M. (1991): A Re-examination of the Basic Postulate of Thermomechanics,- Proc. Ray. Soc, London A, Vol pp Green A.E. and Naghdi P.M.(1993): Thermoelastic without energy dissipation, J. Elasticity, Vol. 31, pp Green A E and Lindsay K. A.(1972):Thermoelasticity, -J. Elasticity, Vol. 2, pp Honig G. and Hirdes U. (1984): A method for the numerical inversion of the Laplace transforms, - J. Comp. Appl. Math, Vol.10, pp Lord H. W. and Shulman Y.(1967): A generalized dynamical theory of thermoelasticity,- J. Mech. Phys. Solids, Vol.15, pp Kumar R. and ChadhaT.K. (1986): On torsional loading in an axismmetricmicropolar elastic medium, - Proc. Indian Acad. Sci.(Maths Sci.),Vol. 95, pp Kumar R. and Deswal S. (2001): Mechanical and thermal sources in the micropolargeneralized thermoelastic medium, J. Sound and Vibration,Vol. 239, pp Kumar R. and Rani L. (2004): Deformation due to mechanical and thermal sources in the generalized orthorhombic thermoelastic material,-sadhana, Vol. 29, pp Kumar R. and RaniL.(2005): Deformation due to mechanical and thermal sources in the generalized half-space with void, -Journal of Thermal Stress,Vol. 28, pp Mahalanabis and Manna (1997): Eigen value approach to the problems of linear micropolarthermoelasticity,- J. Indian Acad. Math., Vol.-19,pp MarinM. andlupu M.(1998): On harmonic vibrations in thermoelasticity of micropolar bodies, - J. Vibration and Control, Vol.-4, pp Nowacki W. (1966): Couple stresses in the theory of thermoelasticity, - Proc. ITUAM symposia, Springer-Verlag, pp Press W. H., Teukolsky S.A., Vellerling W.T. and Flannery B. P.(1986): Numerical Recipes, - Cambridge University Press, Cambridge. Passarella F.(1996):Some results in micropolarthermoelasticity, - Mechanic ResearchCommunication, Vol. 23, pp Sturnin D.V. (2001):On characteristics times in generalized thermoelasticity, - J. Appl. Math., Vol.-68, pp Sharma J.N. and Chand D.(1992): On the axisymmetric and plane strain problems of generalized thermoelasticity,- Int. J. Engg.Sci., Vol.-33, pp

25 SEISMIC HAZARD ASSESMENT: PARAMETRIC STUDIES ON GRID INFRASTRUCTURES Andrea Magrin, Cristina La Mura, Franco Vaccari, Giuliano F. Panza, Dipartimento di Geoscienze, Università di Trieste Alexander A. Gusev Institute of Volcanology and Seismology, Russian Academy of Sciences, Kamchatka Branch, Geophysical Service, Russian Academy of Sciences Iztok Gregori, Stefano Cozzini, CNR/IOM c/o Sissa, Trieste Abstract Seismic hazard assessment can be performed following a neo-deterministic approach (NDSHA), which allows to give a realistic description of the seismic ground motion due to an earthquake of given distance and magnitude. The approach is based on modelling techniques that have been developed from a detailed knowledge of both the seismic source process and the propagation of seismic waves. This permits us to define a set of earthquake scenarios and to simulate the associated synthetic signals without having to wait for a strong event to occur. NDSHA can be applied at the regional scale, computing seismograms at the nodes of a grid with the desired spacing, or at the local scale, taking into account the source characteristics, the path and local geological and geotechnical conditions. Synthetic signals can be produced in a short time and at a very low cost/benefit ratio. They can be used as seismic input in subsequent engineering analyses aimed at the computation of the full non-linear seismic response of the structure or simply the earthquake damaging potential. Massive parametric tests, to explore the influence not only of deterministic source parameters and structural models but also of random properties of the same source model, enable realistic estimate of seismic hazard and their uncertainty. This is particular true in those areas for which scarce (or no) historical or instrumental information is available. Here we describe the implementation of the seismological codes and the results of some parametric tests performed using the EU-India Grid infrastructure. 1 Neo-deterministic seismic hazard assessment The typical seismic hazard problem lies in the determination of the ground motion characteristics associated with future earthquakes, at both the regional and the local scale. Seismic hazard assessment can be performed in various ways, e.g. with a description of the groundshaking severity due to an earthquake of a given distance and magnitude ( groundshaking scenario ), or with probabilistic maps of relevant parameters describing ground motion. Seismic hazard assessment can be performed following a neo-deterministic approach (NDSHA), which allows to give a realistic description of the seismic ground motion due to an earthquake of given distance and magnitude. The procedure for neo-deterministic seismic zoning (PANZA et al., 2001) is based on the calculation of synthetic seismograms. It can be applied also to areas that have not yet been hit by a catastrophic event in historical times, but are potentially prone to it. The neodeterministic method allows to quantitatively model the effects of an earthquake which may happen in the future and therefore is a very effective technique in seismic hazard assessment, even in the regions with scarce or no historical or instrumental information available. Starting from the available information on the Earth s structure, seismic sources, and the level of seismicity of the investigated area, it is possible to compute complete synthetic seismograms and the related estimates on peak ground acceleration (PGA), velocity (PGV) and displacement (PGD) or any other parameter relevant to seismic engineering (such as design ground acceleration, DGA) which can be extracted from the computed theoretical signals.

26 NDSHA can be applied at the regional scale, computing seismograms at the nodes of a grid with the desired spacing, or at the local scale, taking into account the source characteristics, the path and local geological and geotechnical conditions. In the NDSHA approach, the definition of the space distribution of seismicity accounts only for the largest events reported in the earthquake catalogue at different sites, as follows. Earthquake epicenters reported in the catalogue are grouped into 0.2 x 0.2 cells, assigning to each cell the maxim um magnitude recorded within it. A smoothing procedure is then applied to account for spatial uncertainty and for source dimensions (PANZA et al., 2001). Only cells located within the seismogenic zones are retained. This procedure for the definition of earthquake locations and magnitudes for NDSHA makes the method pretty robust against uncertainties in the earthquake catalogue, which is not required to be complete for magnitudes lower than 5. A double-couple point source is placed at the center of each cell, with a focal mechanism consistent with the properties of the corresponding seismogenic zone and a depth, which is a function of magnitude. To define the physical properties of the source-site paths, the territory is divided into 1 x1 cells, each characterized by a structural mod el composed of flat, parallel anelastic layers that represent the average lithosphere properties at regional scale (Brandmayr et al., 2010). Synthetic seismograms are then computed by the modal summation technique for sites placed at the nodes of a grid with step 0.2 x 0.2 that covers the national territory, considering the average structural model associated to the regional polygon that includes the site. Seismograms are computed for an upper frequency content of 1 Hz, which is consistent with the level of detail of the regional structural models, and the point sources are scaled for their dimensions using the spectral scaling laws proposed by GUSEV (1983), as reported in AKI (1987). Figure 1: flow chart of national scale hazard package From the set of complete synthetic seismograms, various maps of seismic hazard describing the maximum ground shaking at the bedrock can be produced. The parameters representative of earthquake ground motion are maximum displacement, velocity, and acceleration. The acceleration parameter in the NDSHA is given by the design ground acceleration (DGA). This quantity is obtained by computing the response spectrum of each synthetic signal for periods of 1 s and longer (the periods considered in the generation of the synthetic seismograms) and extending the spectrum, at frequencies higher than 1 Hz, using the shape of the Italian design response spectrum for soil A (Norme Tecniche D.M. 14/09/ 2005), which defines the

27 normalized elastic acceleration response spectrum of the ground motion for 5% critical damping. For more details see PANZA et al. (1996). The neo-deterministic method has been recently adapted to account for the extended source process by including higher frequency content (up to 10 Hz) as well as the rupture process at the source and the consequent directivity effect (obtained by means of the PULSYN algorithm by GUSEV (2011)). The most common approach to the simulation of the earthquake source consists in adopting a kinematic model; the kinematic description is merely phenomenological but, also their simplest versions (e.g. Haskell, 1964) are able to describe the gross features of the rupture process by simply using five source parameters: the fault dimensions (length, L, and width, W), the amount of slip at any point of the fault, the rise-time, and the rupture velocity. In order to describe the source microstructure, i.e. its roughness, one can use a stochastic or a deterministic (or a combination of both) distribution of barriers and asperities on the fault surface resulting in a non-uniform distribution of slip. Actually dynamic considerations should underlie this choice, but a kinematical model simply takes into account the existence of fault irregularities. Thus, at a given site the motion is dependent on the size and the duration of each of the sub-sources, and on their distribution in space and in time. Furthermore, at epicentral distances comparable with the dimensions of the fault, the relative positions of the sub-sources with respect to a site receiver can play a fundamental role in the interference of the different wavetrains, resulting in the so-called directivity effect. For an earthquake with a given radiated energy, the decay of the source spectrum at high frequencies shows a strong azimuthal dependence, since the corner frequency at a given receiver in the near-field is a function of all the kinematic source parameters. The seismic waves due to an extended source are obtained by approximating the fault with a rectangular plane surface, on which the main rupture process is assumed to occur. The source is represented as a grid of point subsources, and their seismic moment rate functions are generated considering each of them as realizations (sample functions) of a non-stationary random process. Specifying in a realistic way the source length and width, as well as the rupture velocity, one can obtain realistic source time functions, valid in the far-field approximation. Furthermore, assuming a realistic kinematic description of the rupture process, the stochastic structure of the accelerograms can be reproduced, including the general envelope shape and peak factors. The extended earthquake source model allows us to generate a spectrum (amplitude and phase) of the source time function that takes into accounts both the rupture process and directivity effects, also in the near source region. Slip on the fault varies in space and increases weakly monotonously in time, so that the dislocation rate is non-negative. Its unit moment tensor (defined by slip direction and fault-normal direction) does not vary over this area or in time. Therefore, the description of the source in space-time is essentially scalar, in terms of the distribution over the fault area of the seismic moment (and its time rate). To specify temporal and spectral properties of the simulated sources, the equivalent point source (SSPS) moment rate time history &M 0 (t), its Fourier transform &M 0 ( f )and corresponding amplitude spectrum ( source spectrum ) &M 0 ( f ) have been widely used (e.g. Panza et al., 2001). 2 Uncertainties in hazard maps In NDSHA, the treatment of uncertainties is performed by sensitivity analyses for key modelling parameters. Fixing the uncertainty related to a particular input factor is an important component of the procedure. The input factors must account for the

28 imperfectness in the prediction of fault radiation, and for the use of Green functions, for a given medium, that are only imperfectly known. At present, the following factors are selected as assumedly dominating ones. - Fault radiation factors: - parameters related to the point source position/orientation representing the fault in world coordinates: centroid depth, strike/dip/rake. The point source horizontal position within the cell is not perturbed. - intrinsic fault parameters describing the extension of the fault, like fault length and width vs. Mw (moment magnitude) relationship, Brune s stress drop, faultaverage rupture velocity, directivity effects. - random seeds that define space-time structure of a particular realization of the random source, including random subsource time functions. -Path factors - parameters that specify uncertainties in the assumed 1D bedrock velocitydensity-q profiles. 3 Implementation of seismological codes on grid The use of the EU-India Grid infrastructure ( allows to conduct massive parametric tests, to explore the influence not only of deterministic source parameters and structural models but also of random properties of the same source model, to enable realistic estimate of seismic hazard and their uncertainty. The random properties of the source are specially important in the simulation of the high frequency part of the seismic ground motion. We have ported and tested seismological codes for national scale on the Grid infrastructure. The first step was the speed optimization of this package by the identification of the critical programs and of the hot spots within programs. The critical point in the algorithm was the computation of synthetic seismograms. The optimization was performed in two ways: first by removing of repeated formatted disk I/O, second by sorting of seismograms by source depth, to avoid the repeated computation of quantities that are depth-dependent. The second step was porting of the national scale hazard package on EU-IndiaGRID infrastructure. Two different types of parametric tests were developed: on the deterministic source parameters and on the random properties of the source model. The first experiment is performed by perturbing the properties of the seismic sources selected by the algorithm before the computation of synthetic seismograms. In the second test different sets of curves of source spectrum generated by a MonteCarlo simulation of the source model are used for scaling the seismograms. In both cases there are many independent runs to be executed, so a script for generation of the input and other scripts for checking the status of jobs, retrieving the results and relaunching aborted jobs were developed. 4 Preliminary results One preliminary test over deterministic source parameter for whole Italy ( persut Italy ), and two different tests over random properties ( seed1hz and seed10hz ) for the whole Italian territory, with different frequency content and different maximum distance for the computation of seismograms, were conducted. The performance of

29 the package over the grid in terms of computational time and number of successful jobs was tested, and submission of job and retrieval of its output were refined. The number of seismograms that must be computed determines the duration and the storage requirement of the run. This parameter seems critical for the success of the job. The test runs on the random component of the source gave an indication on the effective number of jobs that must be computed to have a good estimate of the distribution of the ground shaking peaks at each receiver. The first runs have provided a preliminary evaluation of the uncertainty of the hazard maps due to the random representation of the source and to the uncertainty on source parameter. Figure 3 shows an example of results of the test on the random component of the source model. The variability on the different random realizations of the source model (right) is shown in terms of ratio between standard deviation and average at each receiver. seed1hz persut Italy seed10hz type of submission direct to CE WMS WMS total n. of job % of successful job total time of computation of 948 h 1200 h 791 h successful job average time of computation for 17 h 3.5 h 1.6 h one job number of computed seismograms for one job Table 1: performance of the three test runs. Figure 2: maps of average of PGV (peak ground velocity) on different random realizations of source model (left) and variability of the PGV in terms of ratio beetwen standard deviation and average of the maximum peaks at each receiver.

30 5 Conclusions and perspectives We have ported and tested seismological codes for seismic hazard assessment at national scale on the Grid infrastructure. The use of the EU-India Grid infrastructure allows to conduct massive parametric tests for evaluating the uncertainties in the computed hazard maps. Two different types of parametric tests were developed: on the deterministic source parameters and on the random properties of the source model. The performance of the package over the grid in terms of computational time and number of successful jobs were tested and submission of job and retrieval of its output were refined. The tests on the random component of the source gave an indication on the effective number of jobs that must be executed to have a good estimate of the distribution of the ground shaking peaks at each site. At the same time, they provided a preliminary estimate of the uncertainty of the hazard maps due to the random representation of the source. The procedure followed for porting the package on the grid infrastructure can be implemented for other seismological programs as well. A Cooperation Project, aimed at the definition of seismic and tsunami hazard scenarios by means of indo-european e-infrastructures in the Gujarat region (India), has been recently funded by the Friuli Venezia Giulia Region. This two-years project, starting in November 2011, involves three Italian partners (DiGeo, University of Trieste; ICTP SAND Group; CNR/IOM uos Democritos) and two Indian partners (ISR, Gujarat; CSIR C-MMACS, Bangalore). The project aims to set up a system for the seismic characterization, integrated with the e-infrastructures distributed amongst India and Europe, to allow for the optimization of the computation of the ground shaking and tsunami scenarios. This goal will be attained thanks to the strict connection with the European project EU-IndiaGrid2, that will provide the necessary infrastructure. Thus, the project will permit developing an integrated system, with high scientific and technological content, for the definition of scenarios of ground shaking, providing in the same time to the local community (local authorities and engineers) advanced information for seismic and tsunami risk mitigation in the study region. References Aki K. Strong motion seismology (1987) In: Strong ground motion seismology, NATO ASI Series, Series C: Mathematical and Physical Sciences, vol. 204 (eds. ERDIK, M., and TOKSOZ, M.) (D. Reidel Publishing Company, Dordrecht), pp Brandmayr E, Raykova RB, Zuri M, Romanelli F, Doglioni C and Panza GF (2010) The lithosphere in Italy: structure and seismicity. In: Beltrando M, Peccerillo A, Mattei M, Conticelli S and Doglioni C (eds) The Geology of Italy Journal of the Virtual Explorer, Electronic edn, vol 36, paper 1. ISSN virtualexplorer.com.au/article/2009/224/lithospherestructure- seismicity Gusev A. A. (1983), Descriptive statistical model of earthquake source radiation and its application to an estimation of short period strong motion, Geophys J R Astron Soc, 74, Gusev A. A. (2011) Broadband kinematic stochastic simulation of an earthquake source: a refined procedure for application in seismic hazard studies. Pure Appl Geophys 168(1 2). doi: /s Haskell, N. A. (1964). Total energy and energy spectral density of elastic wave radiation from propagating faults, Bull Seismol Soc Am, 54, Norme Tecniche per le costruzioni (D.M. 14/09/2005), published on G.U. 23/09/2005 Panza G. F., Vaccari F., Costa G., Suhadolc P. and Fah D. (1996), Seismic input modelling for zoning and microzoning, Earthq Spectra 12, Panza G. F., Romanelli F. and Vaccari F. (2001), Seismic wave propagation in laterally heterogeneus anelastic media: theory and applications to seismic zonation, Adv Geophys 43, 1 95

31 Zuccolo E., Vaccari F., Peresan A. and Panza G.F. (2010) Neo-Deterministic and Probabilistic Seismic Hazard Assessments: a Comparison over the Italian Territory, Pageoph, DOI /s

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38 BASIC CONCEPT OF SUPERCONDUCTIVITY: A PATH FOR HIGH SCHOOL Marisa Michelini, Lorenzo Santi, Alberto Stefanel, Research Unit in Physics Education (URDF- UNIUD), University of Udine Abstract The need for renewed physics curricula, it suggests to include topics of modern physics. The phenomenology of superconductivity is interesting, because: macroscopic evidence of quantum processes; presentable with simple and motivating experiments; involving interesting technological applications. An approach designed to introduce superconductivity in the high school, based on experiments, aimed to recognize the change in electric and magnetic properties of an YBCO sample at phase transition is presented with some results about experimentations with students. 1. Introduction Teaching and learning Modern Physics is a challenge for Physics Education research involving multidimensional perspectives (PE 2000; AJP 2002, Meijer 2005; Johanssonm, Milstead 2008; Steinberg, Oberem 2000). The results of different international surveys show a diffuse illiteracy in the scientific areas and from different perspectives (PISA 2006), that motivated a diffuse reflection on how physics is taught, how can be proposed in a renewed ways (Euler 2004; Duit 2006). The curricular researches stress the need to reorganize classical physics as physics in context where the technological apparatuses used in everyday life become objects of study to develop concepts and theory, classical principles based techniques used in research offer to the pupils experiences of methodologies of physic (Battaglia et al. 2011; Zollman 2002; Cobai et al 2011). It seeks effective bridges from classical to quantum physics, to create continuity between what the students traditionally faced and the new topics (Cobai et al 1996; Zollman et al 2002; Johanssonm, Milstead 2008; Battaglia et al. 2011) also for education of Quantum Mechanics (PE 2000; AJP 2002). Research experimentations on modern physics evidence the feasibility of the introduction of Quantum Physics in Upper Secondary, particularly when the focus is on the basic concepts (Fischler, Lichtfeld 1992; Zollman ed 1999) or on the integration of physic and technology (Zollman 2002). Empirical studies showed some typical learning difficulties of students in the acquisition of a a quantum mechanical way of thinking and also made clear the effective learning produced by coherent research based teaching/learning sequences (Zollman eds 1999). In this perspective, superconductivity offers many opportunities to explore phenomena very interesting for students because challenges that stimulate the construction of models, activate a critical re-analysis of their knowledge on magnetic and electrical properties of materials, stimulate links between science and technology, thanks to really important applications (i.e Maglev Trains, supermagnets for physics research and MNR test) (Tasar 2009; Michelini, Viola 2011). We developed, in the collaboration of the European projects MOSEM 1-2 (Kedzierska 2010), an educational path for high school to introduce superconductivity, integrating it in the courses of electromagnetism. The educational path developed implement an IBL approach using a set of hands-on/minds-on apparatuses designed with simple materials and High Technology, YBCO samples, USB probe to explore resistivity versus temperature of solids (Gervasio, Michelini 2010). The present paper focus on the rational of the educational path, presenting its main steps aiming the following research questions: Which conceptual elements of superconductivity can be introduced from phenomenology? How to construct a coherent description of the magnetic and the electric properties of a superconductor? How to introduce the main aspects of the BCS theory? Some general results are also reported concerning the students learning, referring to a future work for a more extensive discussion of it. 2. Superconductivity and classical physics Superconduction is a quantum mechanical phenomenon to a large extent as implied in the Ginzburg-Landau theory (1950), in the BCS theory (1957), in the Josephson effect (1964). In spite of this, Essén and Fiolhais (2012) pointed that According to basic physics and a large number of independent investigators, the specific phenomenon of flux expulsion follows naturally from classical physics and the zero resistance property of the superconductor they are just perfect conductors. In particular they showed, following de Gennes (1999) and their own work (Fiolhais et al. 2011), that the Meissner effect (B=0 inside a superconductor) can be derived just from Maxwell equation applied for a R=0 conductor. At the same time they stressed on the possibility to use

39 classical electromagnetism (without ad hoc hypothesis) the derivate the London equations, usually framed coherently just in the BCS theory (1957). Badía-Majós (2006), aiming to understand stable levitation when a magnet is posed on a superconductor, proposed a treatment of superconductors of I and II type to give into account Meissner effect as well as pinning where the main concepts involved are electromagnetic energy, thermodynamic reversibility and irreversibility, Lenz- Faraday s law of induction, and the use of variational principles. These theoretical results ensure that the phenomenology of superconductive state, almost for what concern the Meissner effect, can be described in the framework of classical electromagnetim. From an educational perspective, the concept of electromagnetism (magnetic field, flux, magnetization) can be used as tools to explore the phenomenology of superconductivity. In this perspective the Meissner effect (B=0 inside a SC) can be explained as an em induction process occurring in an ideal conductor (R=0 Ω). The London equations constitute the theoretical (implicit) reference for the exploration and a possible intermediate goal of the educational path. The attempt to explain how the R=0 Ω condition can be created inside a superconductor bridges toward the quantum mechanical frame for the creation of the superconductivity state. 3. The rational of the path on Meissner effect for Upper Secondary School Students The educational path approaches the Meissner effect through an experimental exploration of the magnetic properties of a superconductor sample (SC a disc of YBCO with very weak pinning effect), aiming to the identification of the diamagnetic nature of a SC. An YBCO disc at room temperature (T e ) does not present magnetic properties. When it reaches the temperature of LN (T NL ), the evident levitation of a magnet on it poses the problem to individuate what kind of magnetic property has or acquire a SC when T is close to T NL. A systematic exploration of the interaction of the SC with different magnets and different objects (ferromagnetic objects in primis), with different configurations, put in evidence that a repulsive effect is ever observed when a magnet is posed close to the SC and no interaction is observed when a ferromagnetic object is putted close the SC alone. These exploration leads to the conclusion that: when T= T e, a SC does not evidence magnetic properties; when T T NL, suddenly, magnetic properties emerge (there is a phase transition?). Moreover the SC is not ferromagnetic, it do not transmit the magnetic field inside, it is not a permanent magnet, it do not behave like a magnet, it do not become like the mirror image of the levitated magnet. It always shows repulsive effects close to a magnet, like all the diamagnetic objects placed near a magnet. For that a SC at T T NL must be classified diamagnetic. The strength of the interaction between a SC and a magnet, several orders of magnitude greater than those observed with ordinary diamagnetic materials, suggest to give a more detailed characterization of the nature of the diamagnetism of the SC. Starting from the evidence that the SC shows magnetic interaction only when a magnet is close to it and that the SC do not interact with a ferromagnetic object, it is possible recognize that the interaction with a magnet do not depend on the pole put close to the surface of the magnet, the equilibrium position is always the same. Changing magnet (inside a range of course), the equilibrium position changes but is always the same, for the same magnet. Moreover when T NL <T<T e the B field can be different by 0 inside the YBCO sample (a magnet interact strongly with an iron ring also when a YBCO disc is putted in between), but when T T NL the magnetic field inside the SC sample must be very little (in this condition the magnet and the iron ring do not interact when the YBCO disc is in between). The magnetization of the SC is always adjusted to react to the external magnetic field, tending to preserve the initial situation. In particular if B=0 when the superconductivity state is created, the system tend to react to an external magnetic field creating a counter field that tend to maintain B=0 inside the SC (Meissner effect). It is possible to find a similar behavior in the case of eddy current produced in the electromagnetic induction. For instance a magnet is evidently braked when falling down on a thick copper layer or inside a copper tube. It is evident however that never can be stopped, as in the case of the SC. In fact If the magnet were to stop, the induced emf immediately ceases and, due to the Joule effect, also the induced current ceases immediately. The magnet would be stopped just falling over a conductor with R=0 Ω (an ideal perfect conductor!). Supposing that an electromagnetic induction process will be at the base of the superconductivity levitation, a link between the magnetic and electric property of the SC is activated. This suggest that the resistivity of the YBCO sample could be suddenly change when T T NL and the levitation phenomena is observed. The experimental measurement of the breakdown of the resistivity of a SC, give quantitative evidence to this link making explicit also

40 that a phase transition occurs when the YBCO sample changes from the ordinary conductor state, to the superconductor state. The phenomenological exploration leads to the conclusion that R=0 as well as B=0 inside the SC, aspects that characterize the Meissner effect. The levitation phenomena, that is a phenomenological evidence of this effect, can be described as a fall down of a magnet occurring at v=0 velocity. When the superconductive state is create with the magnet away from the SC, the magnetic flux changes from 0 to the value corresponding to B*S (B* is the mean value of the magnetic field produced by the magnet on the surface of the SC of area S). When the superconductive state is create with the magnet over the YBCO sample, the flux variation is due to the change in the magnetic properties of the SC. So also in this case, where apparently there isn t a flux variation, an electromagnetic induction process is at the base of the levitation phenomenon. To give into account how this superconductive state can be created, it is possible to start from the analysis of the energy of the electrons system inside of a crystal lattice. In the ordinary conduction state the electrons could be treated as not interacting particles, because of the negligible interaction occurring and due to two part: a repulsive part, due to the coulombian interaction essentially screened by the mean field produced by the lattice; an attractive part, due to an effective potential emerging as results of the interaction of the electron and the lattice. Only the second part is depending from the temperature, so when it is prevalent with respect to the coulombian part, the system of electrons become instable for creation on Cooper pairs, or in other words the formation of pair of correlated electrons is a process energetically favored. The collapse of these Cooper pairs on the same ground state is responsible of the R=0 property of an SC, that is recognized to be at the base of the superconduction behavior. The discussion of the creation of Cooper pairs ends the exploration of the Meissner effect that is the main focus of the present work. The educational path faces also the other effect evidenced usually by the II type of SC, as the persistent currents, the pinning effect and the correlated phenomenology. This effects are observed when the superconductivity state is created in presence of an external field. The pinning effect, due to the penetration of the magnetic field inside the SC sample inside the vortexes created by supercurrents, emerges as the anchoring of the magnet to the SC. This is at the base of the MAGLEV train. 4. From electron conduction to the superconduction of the Cooper pairs The Meissner effect is described characterizing the SC state, on the phenomenological level, with the two conditions B int =0 and R SC =0, but do not giving into account how the phase transition can occurs and the processes that produce the change from the initial situation, to the SC state. This change can be described in an educational perspective as follow. The energy of the electrons inside a solid, that can be written roughly as the following sum: E el = K e + V int = Σ (i) K i + ½ [Σ (i j) V coul - Σ (i) V 2ij ] where: - K e =Σ (i) K i =Σ (i) p 2 i /2m* is the kinetic energy of the electrons, p i being the momentum of the i-th electron and m* is the effective mass of the electron inside the lattice (including in a phenomenological way the interaction pf the main field produced by the lattice) - V coul =½ Σ (i j) V coul (r ij ) is the screened Coulomb energy (of the form: V coul (r ij )=V c exp(-βr ij ) (e 2 / r ij ) - V 2 = ½Σ (ij) V 2ij is an effective energy potential due to the interaction of electron and lattice (in the case of Sc of I type this can be written in a relative simple way as a phonon-electron interaction). In an ordinary conductor the electrons can be treated as a system of non-interacting particles, or weakly repulsing particles due to the predominant term V coul in the interaction part of energy V int. When the temperature increases V coul remains essentially the same, V 2 generally change. In the ordinary conductor V int remains greater than zero. Their state is characterized by the values of: energy, momentum, spin. (es. E, p l, ). Many energy level are possible and are one very close in energy to the other. Of course many electrons can possess the same energy but have different values of the momentum. For each momentum two state are possible (one with spin and one with spin ). The electrons, being spin ½ h particles, due to the Pauli principle, (at T=0 K) fill all the energy level till a maximum value named Fermi energy level. When T increases some electrons can usually acquire energy making a transition to excited state and leaving lower state partially empty. The presence of a band of energy level one very close to the other ensure that the system

41 of electron can exchange energy with the lattice. This exchange is at the base of the presence of a resistivity greater than zero in the ordinary conductors. In the SC this schema changes, because the attractive contribution V 2 can become greater than V coul and the interaction energy V int become negative (V int <0) and the electron are subjected to a net attractive energy. For anyhow smalli V int <0 the electrons system become instable for production of couple of electrons correlated by the long range interaction V 2. When one of this pair of electrons (Cooper pair) is created the energy of the system decrease of an amount of the order of the main value of <V 2 >. The pairs of electrons are (quasi) systems with integer spin. For energetic reasons (but not only) the coupling of electrons with opposite spin and opposite momentum is favored. Due to their integer spin, the electron pairs occupy all the same ground state, that is separated from the first excited state by an energy gap of an amount proportional to N<V 2 >, where N is the number of pairs created in the systems. The same process that produces the electron pairs, produces also the energy gap. The state created is not only relatively stable, but also ensures that the pairs of electrons cannot exchange energy with the crystal lattice. In other words the energy gap ensure that the resistivity of the system becomes zero. The creation of pairs of electrons is a crucial process in the creation of the superconductive state. In the case of SC of I type the schema described is framed in the BCS theory of superconductivity [ ], being the V2 term of interaction expressed as a potential energy interaction between two electrons mediated by the exchange of phonons. In the case of the SC of II type it is not clear what kind of interaction is responsible of the V2 energy potential. 6. Experimentation with students Explorative activities with students are carried out in informal learning setting, in four contexts (Udine, Pordenone, Frascati) with 685 students. These activities was important to test microsteps of the path, in particular for what concern the way in which students discuss the distinction between the levitation due to the Meissner effect and the levitation due to the pinning effect. The research experimentations of the educational path on Meissenr effect of SC was performed with students in eight contexts in Italy (Udine, Pordenone, Cosenza, Bari, Crotone, Siena) with 335 students of Italian upper secondary schools (17-19 aged). In these experimentation tutorial worksheets are used to monitor the students learning, analyzing their answers, drawings and schemas after the construction of typical categories. Few results concerning three main conceptual steps are given, referring to further work a more thorough details. The change in the properties of the YBCO disc when T T NL. When student answered to the question concerning the change of the properties of a YBCO sample when the temperature is T=T NL, sentencing: a) «the properties of the YBCO have changed», «the diminution of the temperature produced a change in the behavior of the YBCO» (40%); b) «The disc of YBCO changes his properties. It repels the magnetic field» (35%); «The YBCO disc is magnetized» (13%); Re-arrangement of the atoms (7%); other answers (5%). The change in the property of the superconductor is the main focus of about 90% of the students answers, explicitly referred to the magnetic properties in about half of the sample. When the students were requested to represents the magnetic field configuration that can give account of the stability of the levitation configuration, they drawn: field lines do not penetrating the YBCO if T<T NL (55%); field lines around the SC and aroud the magnet (37%); the lines of the magnet penetrate inside the SC at T>T NL (8%). The first two chategories include all the draw where the field do not entry in the SC. For what concern the intermediate way to resume the Meissner effect indicate that: It consists in canceling the magnetic field is part of a SC, At a certain temperature of the transition to the SC the Meissner effect means that the SC ceases to oppose any resistance to the passage of electricity and being all or almost all internal magnetic fields (22%); The Meissner effect makes levitating a magnet over a superconductor without any constraint (57%), an effect which changes the physical properties of objects subjected to cooling (14%); magnetic fields are arranged in such a manner (such as jammed) that the magnetysme in air because it has no possibility to deviate) then receives only fields opposite to the magnet. (7%). In all the students answers the Meissenr effect is identified phenomenologically. It Is also explicitly framed on the conceptual plane by the students of the first group. 7. Conclusion

42 An educational path on superconductivity for Upper Secondary School was designed. Students are engaged in an experimental exploration of the phenomenology of use the experiments developed in the context of the European projects MOSEM and MOSEM 2. The focus is on the analysis of the Meissner effect to characterize the perfect diamagnetism of the superconductors under the temperature of the phase transition. With an experimental exploration, carried out both with qualitative observations, both with measurements using on-line sensors, students gradually construct the conclusions that B=0 and r=0 inside a S under the critical temperature. The students use concepts of the electromagnetism (the field lines, the magnetization vector, the electromagnetic induction law) as tools to construct a link between magnetic and electric properties of a syperconductor, describing the phenomenology of the Meissner effect, according to the suggestion of many authors, that show the possibility to describe the Meissner effect in the framework of classical electromagnetism. In the phenomenological description of the superconduction the aim is the recognition of the role of the electromagnetic induction in the description of the Meissner effect. How this state is produced or the phase transition occurs, it is described for what concern the role of the creation of the Cooper pairs on the change in the energy of the electrons system of the superconductors. The focus on energy leave open the opportunity to go deep in the quantum description of the energy gap formation. From research experimentations carried out in eight different contexts with 335 students emerge that the majority of students recognize the change in the magnetic properties of the superconductor under a critical temperature, the B=0 condition, the different nature of the magnetic suspension and the levitation of a magnet on a superconductor sample. References AJP, (2002) Special Issues of Am. J. Phys. 70 (3) Badìa-Majòs A. (2006) Understanding stable levitation of superconductors from intermediate electromagnetics, Am. J. Phys. 74, Bardeen J., Cooper L.N., Schrieffer J. R. (1957) Theory of superconductivity, Phys. Rev. 108, Battaglia R O et al. (2011) in Rogers L. et al. Community and Cooperation, Vol II, Leicester: Leicester Univ. pp Cobai D., Michelini M., Pugliese S., Eds. (1996) Teaching the Science of Condensed Matter and New Materials, GIREP Book of selected papers, Udine: Forum. de Gennes P. G. (1999) Superconductivity of Metals and Alloys (Perseus Books, Reading, MA), pp Duit, R. (2006) Science Education Research An Indispensable Prerequisite for Improving Instructional Practice, ESERA Summer School, Essén H., Fiolhais N. (2012) A.J.P., 80 (2), Euler M. (2004) in Redish E. F., Vicentini M. (eds.), Proceedings of the International School of Physics Enrico Fermi, Varenna, Course Research on Physics Education July 2003, Italy, Amsterdam: IOS, pp ; Euler (2004) Quality development: challenges to Physics Education, In Quality Development in the Teacher Education and Training, M.Michelini ed., Udine: Girep, Forum. Farrell W. E. (1981) Classical derivation of the London equations, Phys. Rev. Lett. 47, Fiolhais M. C. N., Essén H., Providencia C., and Nordmark A. B. (2011) Magnetic field and current are zero inside ideal conductors, Prog. Electromagn. Res. B 27, Fischler, H., Lichtfeldt, M. (1992). IJSE, 14(2), Gervasio M, Michelini M (2010), Ginzburg V. L.and Landau L. D., (1950) On the theory of superconductivity, Zh. Eksp. Teor. Fiz. 20, English translation in Collected Papers by L. D. Landau edited by D. Ter Haar, (Pergamon, Oxford, 1965), pp Kedzierska E. et al. (2010), Il Nuovo Cimento, 33 (3) Johanssonm K E, Milstead D (2008) Phys. Educ. 43, Josephson B. D. (1964) Coupled superconductors, Rev. Mod. Phys. 36, Meijer F. (2005) in H. E. Fischer (Ed.), Developing standards in RSE, London: Taylor & Francis., Michelini M., Viola R. (2011) Il Nuovo Cimento, DOI /ncc/i PE (2000) Special Issues of Phys Educ.35 (6) PISA (2006), PISA results, Steinberg R. N., Oberem G. E. (2000) JCMST 19 (2) Taşar, M.F. (2009). In The International History, Philosophy, and ST Group Biennial Meeting, University of Notre Dame, South Bend, IN. Zollman D A et al. (2002) A.J.P. 70 (3) Zollmann D. Eds (1999) Research on Teaching and Learning Quantum Mechanics, at

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46 POPULARISATION OF PHYSICS IN THE WILD Beatrice Boccardi (Liceo Scientifico A. Labriola, Napoli, Esplica no-profit, Laboratorio per la Divulgazione Culturale e Scientifica nell Era Digitale, Villafranca Di Verona, VR, Michela Fragona, Giovanna Parolini (Esplica no-profit, Laboratorio per la Divulgazione Culturale e Scientifica nell Era Digitale, Villafranca Di Verona, VR, Abstract Science popularisation is important to inform tax payers on how their money is used and to inform the general public of the progresses of research. The web 2.0, comprising the 3D virtual worlds, offers science and physics popularisation new communication channels and tools constituting, as a whole, a large part of learning in the wild, with the consequent citizen science projects. Tags: physics popularisation, informal learning, web 2.0, virtual worlds, Second Life. 1 Introduction Science and physics popularisation is important as a means to promote further investments through the information of the tax payers on how their money has been used and on the improvements that scientific research can bring in their lives, so the communication of scientific knowledge to the general public is both a consequence and a precondition for the ongoing scientific-technological progress. Science popularisation gives the non-scientists an up-to-date knowledge, resulting in the constitution of a cultural environment where science is valued and supported. The web 2.0 has not just quickened or simplified the existing procedures, it has opened new domains to science popularisation and to the active involvement of the non-scientists. Seminars, lectures, science museums, etc., are the traditional formats for science popularisation, while the web 2.0 provides tools that make it accessible to everybody. The cultural revolution of the Internet and the web 2.0 is expressed in the triad of information-communication-sociality. The multimedia on the web, with music, videos and games has made it predominant for sociality, knowledge and learning. In this paper, we will describe in particular the experiences of physics popularisation in the virtual worlds. These are a specific instance of the cyberspace, true 3D social networks where the users interact directly through their graphical representation, the avatar, and build the world. Though they may appear like videogames, they constitute places where new forms of science popularisation are experimented. The definitions learning in the wild and citizen scientist not created, but made popular by Nature summarize the change in progress in learning and the popularisation of science: Much of what people know about science is learned informally (Learning in the wild, 2010) 1. In this context, wild stands for informal learning, without fixed schedules and programs. The result of this diffused learning mode is the citizen scientist, contributing to the citizen science projects with observations and data gathering, mostly thanks to the digital technologies. 2 Learning in the wild and citizen science Learning in the wild, made up of freely chosen experiences without an express formative aim, is often much more effective [than formal learning] at getting people excited about science (Learning in the wild 2010) and involves people of every age, so it can be the foundations of school learning and of lifelong learning. Forms of science popularisation and citizen science have existed since the XVIII century, but the web 2.0 has given the science enthusiasts the tools to communicate among themselves wherever they are based in the world, in a continuous junction between the physical world, where the facts 1 Learning in the wild, Nature, International weekly journal of science, Apr 8 th, 2010,

48 school website, an article in the local magazine, discussion of the results in a public Conference. In these projects the students learn through a hands-on approach, carry out true scientific project, build the measuring tools, study radioactivity in the field, apply the experimental method, evaluate the limits and validity of the measurements. The radon projects also improve the knowledge of the territory. These projects constitute physics laboratories and excellence courses and are the junction between the popularisation of physics and formal learning. While the schools need to change programs and methods, the general public is caught in a direct participation, through the installation of the radon measuring kit at home, or being invited to a conference. 3 Virtual worlds The virtual worlds, or metaverses, invented for playing, have turned out to be the most complete environments on the web to create and share culture. The features common to all of them are tridimensionality, the user present as an avatar, synchronous communication through text-chat and voice-based chat, but a few others are necessary to use them for the popularisation of physics: no pre-arranged storyline, no characters to interpret or levels to attain, user-created content. To the persistent base of the world designed by the developers, the users add regions (lands) designed by themselves - ancient Rome, Dracula s castle, islands devoted to culture, art, science - and create groups according to their specific interests, many of which active also on other social networks, websites and blogs. The virtual worlds are fundamental channels for the popularisation of science and physics, for the users attending them. The majority belong to an age level higher than in the real games, are educated, have various cultural interests. The virtual worlds are consistent with the emerging trends, the gender asymmetry in disfavour of women reported by other projects of dissemination is absent and they are fit for the differently abled, which they foster socially and physically, solving some motor or expression difficulties, in a place where it is easier to contact people and make friends. The social interaction with the avatars present in the same place, produces the sensation of feeling present, just like in the physical world (Baumgartner 2008) 15. The creativity prompted by building and scripting makes these worlds generative of new cultural contents. The user as the content creator and the social interaction in presence bear the immersivity of the environment, a positive emotion that prepares the avatar to a more active participation than in Real Life (Suler 1996) 16. Universities, non-profit organization, scientific groups give courses, make conferences and research. All this does not replace the activities in the physical world. Rather, they are expanded and given a new presentation. The playful dimension, even when it is not the immediate aim of communication, helps the transmission of knowledge and learning (Flanagan 2010) Science popularisation in the virtual worlds In the metaverses science popularisation may happen in forms different or impossible in the physical world, with objects where the game is implicit. Building and scripting enable you to represent physics concepts or events through interactive objects that are themselves a form of play - the avatar can get into a black hole or become a subatomic particle, a playful dimension that is part of the appeal of science popularisationin the metaverses. The science event within reach of teleport encourages the participation of those who up to that moment had left it out, for want of interest or prevented by everyday circumstances. The metaverses facilitate the discovery of 15 Baumgartner T, Speck D, Wettstein D, Masnari O, Beeli G, Jäncke L (2008). Feeling present in arousing virtual reality worlds: prefrontal brain regions differentially orchestrate presence experience in adults and children. Frontiers in Human Neuroscience, 2, 1-12, then click on fnhum _SUPPLEMENT.pdf for PDF 16 Suler J (1996) The Psychology of Cyberspace, revised Online treatise, 17 Flanagan T, Delphin G, Fargis M, Lexington C (2010), Arts, Mathematics and Physics in Second Life, paper for the Aplimat Conference 2010,

49 cultural fields so far ignored or considered intellectually distant. At the moment, the 3D world Second Life appears as the most complete and flexible for the popularisation of science. The group Real Life Education in Second Life has 4,800 members, educators in Real Life. Universities and institutes inworld are Stanford University, Media PlayLab (Singapore), Infolit ischool (UK), NOOA to mention but a few. Among the scientific groups, physical and astrophysical topics are prevailing. The NASA land in Second Life exhibits space and terrestrial environments that illustrate its fields of research - glaciers, underwater laboratories, volcanoes. Through its website, the NASA provides a guide for the games inworld, for students of different age groups: they deal with the building of space rockets and their importance in the development of space research, or with the students working at the organization of a mission on the Moon and the study of its habitat (NASA b) 18. Some of these games are suitable to promote a popular talk, as the playful dimension attracts also the adult receivers of science popularisation, who can play with the interactive objects or build some themselves. MICA-Meta Institute for Computational Astrophysics gathers international Real Life astronomers and astrophysicists to study the application of the virtual worlds and technologies to astronomical research. They give public lectures on Saturdays, which can be listened again from their website (MICA) 19. Science Friday presents a course on the basic concepts of astronomy and optics, with interactive models of astronomical objects. It broadcasts inworld the scientific programs of NPR-National Public Radio. The avatar audience can ask questions to the guests in the Real Life studio. The anglophone Kira Cafè is a bar to socialize and discuss interdisciplinary scientific subjects. Exploratorium is a science museum inspired to the Real Life homonymous museum in San Francisco, CA. It hosts a presentation of nuclear physics, experiments of physics, astronomy, on perception, and an interactive Sun: the avatar travels to the Earth riding a sun ray. An installation of the Earth-Moon system shows how a solar eclipse appears from the Moon. Second Life Physics Lab-SLPL, founded by a physics professor in Real Life, exhibits a cannon shooting cannon balls following two different trajectories, either according to Buridanian or Newtonian physics: this shows how the tools of the virtual world can be used for science popularisation. In the Italian community we can see the emergence of several groups that carry out extemporary activities or well established projects lasting for several years now. Italy has taken part in Avatar Project , to study the use of the virtual worlds as a classroom laboratory, co-financed by the European Commission. High school teachers and students experimented with the features of the virtual world: immersivity, creativity, learning made interactive and easier by the 3d ambient (Avatar) 20. Italian groups dealing with teacher formation are INDIRE (INDIRE) 21, with courses of building, maths and Italian l2. SecondAnitel, association of e-tutors, with courses on teaching with the digital technologies (ANITEL) 22. Imparafacile offers courses on e-learning and makes research on teaching methods in the virtual world (Imparafacile) 23. Immersiva.2life is engaged in the popularisation of culture and music through live concerts and forums on topical issues. EsplicaSL is the virtual representative of Esplica no-profit (Esplica) 24, to link the immersive creativity of the metaverse and the physical reality and to promote Second Life as an aggregation place for the diversely abled. In 2010 it realized Let s read in Second Life, a twinning with the ebookfest in Real Life (Cross-Universe) 25 and this year has presented its accomplishments at the exhibition for the 4th anniversary of Nonprofit Commons, a similar group in Second Life. Second Physics, group for the popularisation of physics and science, from 2008, is an example of the many activities possible in the metaverse in this field. Scienza on the Road, in collaboration with Immersiva2Life, is an itinerant series of one-hour conversations started in It has won 18 NASA b, and 19 MICA, and 20 Avatar, 21 INDIRE, 22 ANITEL, 23 Imparafacile, accessed 2011 December 24 Esplica, 25 Cross-Universe EBookFest,

50 fame by now, with more than 4,000 avatars attending the lectures in more than a hundred different Italian lands. At the moment Second Physics has more than 1,500 members one of the highest number among all the scientific groups comprising many non-italian-speaking avatars, who receive the English translation on a special board. So far, the conversations have dealt with Physics, Astronomy, Particle Physics, Mathematics, Acoustics, Optics, Medicine, Cybernetics, Literature for Science Popularisation. The audience can ask questions and discuss with the lecturers. The conversations are broadcast in streaming web and announced in blogs, online magazines and on FaceBook. Second Physics has taken part to the conference Virtual Worlds Best Practices in Education 2010, with school and university groups active in Real and Second Life, with a poster session and an article on the Journal of Virtual Studies (Boccardi 2011) 26. With SL Art, in 2010 Second Physics organized Scien&Art, a contest for the artists aiming at expressing their creativity through scientific concepts and the scientists trying to communicate scientific ideas in a creative way, which put into evidence the link between art and science. A 10-hour course Beyond the Third Dimension was held in 2009 and a course on the cosmic rays will be given in Second Campus, the project started this year with two 10-hour courses, on the literary fiction for science popularisation and on the graphics software Blender. The Science Café opened in 2010 at Second Physics, without a fixed schedule. On the occasion of scientific events in Real Life, an expert in the subject is invited to analyse the sense and the value of the piece of news. Besides the scheduled events, the land of Second Physics presents installations on the physicists researches and anecdotes, as prompts for quizzes that, while attracting the visitors attention, drive them to some googling to solve the enigmas. Other fixed installations are the optical illusions, and the presentations of past courses and exhibits. Finally, there are scientific sculptures, from Leonardo to the DNA helix to the cosmic rays. We end this description of the activities of Second Physics mentioning the support group to the free online course on AI by S. Thrun and P. Norvig with Stanford University, set up in Second Campus Hall to discuss the lessons weekly. Learning in the wild arises new cultural interests and the virtual world gives a space to share them. 5 Conclusions Learning in the wild is made up of freely chosen activities, without a direct formative aim. The web 2.0 has given the science enthusiasts the tools to contribute to scientific research through the citizen science projects making observations, collecting data, discussing with other citizen scientists online. The metaverses are showing great possibilities for science popularisation, thanks to the direct relationship between the divulger and the receiver through the avatar, both a playful and serious mode, that makes learning easier. This has changed the way we learn and formal learning, in its turn, will have to change programs, methods, evaluation systems, to keep up with tools that are not simply technologies, but represent a true cultural revolution where sharing, playing, free choice, are the keywords. 26 Boccardi B, Fabbri F L, Fragona M, Nardone A, Parolini G, Dos Santos R P (2011), Second Physics: Popularisation and Outreach of Science in the Italian Second Life Community,

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66 FFP Udine November 2011 GENERAL RELATIVISTIC QUANTUM THEORIES Foundations, the Leptons Masses Claudio Parmeggiani, University of Milan Abstract The space-time is represented by a usual, four-dimensional differential manifold, X. Then it is assumed that at every point x of X, we have a Hilbert space H(x) and a quantum description (states, observables, probabilities, expectations) based on H(x). The Riemannian structure of X induces a connection on the fiber bundle H and this assumption has many relevant consequences: the theory is regularized; the interaction energy is a well defined self-adjoint operator; finally, applying the theory to electro-weak interactions, we can obtain a specter of leptons masses (electron, muon, tau). Introduction. Summary We shall discuss (the foundations of) a General Relativistic Quantum Theory: now the Hilbert space of the quantum description is replaced by a fiber bundle (here called quantum bundle) based on a four-dimensional differential manifold (the space-time); the typical fiber is a complex, infinitedimensional Hilbert space; on every fiber there are states (vectors or rays), observables (operators), probabilities and expectations (Section 1), The ordinary derivative (in Schrödinger equations) is replaced by a sort of covariant derivative on the quantum bundle. This derivative, at a space-time point, depends on the value of the space-time metric tensor (the gravitational field) at the same point: this is the main hypothesis. The standard quantum field theories are formally recovered, for a constant metric tensor (that is when the gravitational field goes to zero); but, in fact, even for an isolated elementary particle we have to consider the gravitational field of the particle itself (Section 2). The quantum fields and the energy operator (the Hamiltonian ) are now well defined (eventually self-adjoint) operators; the Pauli-Jordan distributions are now true, differentiable functions (gravitation dependent, obviously); therefore the Theory is regularized: there are no more divergences (Section 3). The presence of our covariant derivative not only regularizes the theory but, as a consequence, the (ratios of the) masses of the leptons are, by some supplementary assumptions, theoretically predictable. In fact, imposing the constancy of the sum of the space-time dependent self mass (logarithmically diverging, with zero gravity) and of a space-time dependent counter term, we arrive at a second order partial differential equation; the self values of this equation are (proportional to) the leptons masses (Sections 4 and 5). Finally note that we have only to postulate the existence of three stable elementary leptons (electrons, positrons and electron neutrinos); the other ones (the µ and the τ particles, their neutrinos) and their masses are derived, so to speak, as excitations. 1. Fiber Bundles and Quantum Bundles A fiber bundle (see, for example, (Bourbaki 2007) or (Lang 1985)) is a tuple (F, X, π, F) where F and X (the base space) are manifolds; π is a continuous surjection from F to X; the F(x) := π 1 (x) (the fibers at x, a point of X) are all isomorphic to the typical fiber F. A (cross) section is a function Ψ from X to F such that: π(ψ(x)) = x (1) Locally (in an open subset U of X) F(x) can be identified to F; hence, locally, the sections can be expressed as (ordinary) maps from U to F; in the following we shall generally employ local expressions. A quantum bundle is defined as a tuple (H, X, π, H) where the base space X is a four-dimensional Riemannian manifold (g is its (+1, 1, 1, 1) metric tensor); the fibers at x, H(x), and the typical fiber, H, are infinite-dimensional, complex, isomorphic Hilbert spaces. The Schrödinger function is now a section of the quantum bundle, locally expressed by a map from X to H; the observables are

67 FFP Udine November 2011 expressed by operator-valued maps, assigned on X. For example, the average value of the observable ξ, the system being in the state Ψ, can be (locally) written as: (Ψ(x) ξ(x) Ψ(x)) H / (Ψ(x) Ψ(x)) H (2) where the inner product is taken on H. Obviously in a Schrödinger picture the section ξ will be constant (in a sense make precise in subsection 2.2). 2. Derivatives. Dynamics To relate quantum objects defined at different space-time points we need a way to connect them, at least when they are infinitely near ; thence we shall introduce a notion of covariant derivative Covariant Derivatives If Ψ is the local expression of a section of the Quantum Bundle and u is a 4-dimensional vector (space-like or time-like), the covariant derivative of Ψ, in the direction u, is defined as: D u Ψ(x) := u Ψ(x) + C u (x).ψ(x) (3) where u is the ordinary derivative (along u) and C u (x) is a space-time dependent operator on the Hilbert space H. And for an observable (an operator) ξ: D u ξ(x) = u ξ(x) + [C u (x), ξ(x)] (4) If γ is a smooth path on X, parameterized by the real variable s, a section Ψ along γ is said parallel if: 2.2. Schrödinger Equation The self-adjoint operator on H (u is again a 4-dimensional vector) D u Ψ(γ(s)) = 0 (u = γʹ(s)) (5) P u (x) = P 0 (x) u 0 + P 1 (x) u 1 + P 2 (x) u 2 + P 3 (x) u 3 (6) is the (local expression of the) energy-momentum observable, at space-time point x; it is an energy operator if u is time-like, a momentum operator when u is space-like. Typically P u (x) can be decomposed as: P u (x) = P u (FREE) (x) + P u (INT) (x) (7) where P u (INT) (x) = 0, for a space-like u. A section Ψ satisfies a Schrödinger equation (in a Schrödinger picture) if while, for all observables ξ: D u Ψ(x) = u Ψ(x) + C u (x).ψ(x) = i P u (x).ψ(x) D u ξ(x) = u ξ(x) + [C u (x), ξ(x)] = 0 Observe that, because P u (x) and P v (x) commute and D u P v (x) = 0 (for every couple of vectors, u and v), we have D u D v Ψ(x) = D v D u Ψ(x). In an interaction picture D u Ψ (INT) (x) = i P u (INT) (x).ψ (INT) (x) D u ξ (INT) (x) = i [P u (FREE) (x), ξ (INT) (x)] In the following we shall always use the interaction picture Explicit expression of the Covariant Derivative Having defined a vacuum state (at space-time point x), Φ 0 (x), others states (and observables) can be generated by means of the creation and annihilation operators, a* n,p (x) and a n,p (x) (see, for example, (Bogoliubov 1980) or (Weinberg 1995)). Here n is an index = 1, 2,, N and (8a) (8b) (9a) (9b) p = (ε, p), ε = (p 2 + M 2 ) (10)

68 FFP Udine November 2011 ε, p and M are the particles energy, momentum and (bare, unobservable) mass. Evidently we must have, for the vacuum, D u Φ 0 (x) = u Φ 0 (x) + C u (x).φ 0 (x) = 0 (and also u Φ 0 (x) = 0). In our interaction picture [P u (FREE) (x), a n,p (x)] = (p.u) a n,p (x) (11) so: D u a n,p (x) = i (p.u) a n,p (x). Afterward we shall assume (this is a fundamental postulate) that: [C u (x), a n,p (x)] = iκ c u (x,p) a n,p (x) (12) where c u is an ordinary numerical function dependent on x and p (but not on n) and, at least in a first approximation, that: c u (x,p) = αβ Γ u αβ (x). p α p β = ½ αβ u g αβ (x).p α p β (13) Here the g αβ are the contravariant components of the metric tensor g; the Γ are its Christoffel symbols; κ is a numerical, universal parameter. We are using absolute unit of measure, that is h PLANK = 2π and M PLANK = 1. Hence we are lead to the differential equation (c(x,p) := ½ αβ g αβ (x).p α p β ) which can be immediately integrated: i u a n,p (x) = ((p.u) + κ u c(x,p)) a n,p (x) (14) a n,p (x) = exp( ip(x x 0 )).exp( iκ(c(x,p) c(x 0,p))).a n,p (x 0 ) (15) x 0 is a fixed space-time point. Evidently for a constant, special relativistic g, we shall recover the usual definitions, relatively to the free creation and annihilation operators (see again (Bogoliubov 1980) or (Weinberg 1995)). For a near constant g (that is in a weak gravitational field approximation, see (Weinberg 1972)) c(x,p) = ½ M 2 + (M (GRAV) / r(x))(ε 2 + p 2 ) (16) M = (ε 2 p 2 ) is the bare mass; M (GRAV) is the gravitating mass; r(x) is the distance from x to the source of the gravitational field Commutations and Anti-commutations Relations At the same space-time point x 0 hence, for a real κ (positive or negative): [a m,p (x 0 ), a n,q *(x 0 )] ± = N p δ m,n δ p,q (17) [a m,p (x), a n,q *(y)] ± = N p exp(ip(y x)).exp(iκ(c(y,p) c(x,p))) δ m,n δ p,q (18) where x and y are different space-time points. The bracket [, ] ± refer to Fermi or Bose statistics; N p is a normalization factor, usually set equal to 1 (as in (Weinberg 1995)) or to (p 2 + M 2 )/M. Observe that, if κ is imaginary, κ = iκʹ, κʹ positive, we get a term like exp( κʹ(c(x,p) + c(y,p))). 3. Quantum Fields It is possible to define neutral and charged quantum fields φ and ψ (now well defined, operatorvalued functions assigned on X) as: φ(x) = (a p (x) + a p *(x)) dµ(p) (19) ψ(x) = a p (x) dµ(p) + b p *(x) dν(p) (20) dµ and dν are measure on R 3, eventually matrix-valued. Obviously, for κ=0, we shall obtain the usual, singular quantum fields. These are free fields (the P u (INT) (x) term is not considered), but the gravitational effects are fully included in the field operators. Afterward we can build the Pauli-Jordan functions (x, y): they depend on κ(c(y,p) c(x,p)) and are, generally, not translation invariant. But they are now true, differentiable functions (assigned on X 2 ) and, for κ=0, we shall obtain the usual distributions (singular functions), translation invariant. We have, for example, for a real κ:

69 FFP Udine November 2011 but for an imaginary κ = iκʹ, κʹ positive: (x, y) = [φ(x), φ(y)] = 2 sin(p(y x)+κ(c(y,p) c(x,p))) dµ(p) (21) (x, y) = [φ(x), φ(y)] = 2 sin(p(y x)).exp( κʹ(c(y,p)+c(x,p))) dµ(p) (22) that leads to an exact, canonical equal time commutation relation. 4. Interactions. Self-Energies 4.1. Interaction-Energy Operator Let us consider a tri-linear interaction-energy operator (the Hamiltonian, always in an interaction picture): H (INT) (x) = λ (ξ p *(x) A p,q (x) ξ q (x)) dµ(p,q) (23) here dµ is a measure on R 6 ; A p,q (x) is a self-adjoint operator (Bose statistics) related to the gauge fields which mediate the interaction; the operators ξ q (x) describe the interacting particles (Fermi statistics) and they are linear combinations of the a p (x), a p *(x), b p (x), b p *(x); λ is a coupling constant. Note that we are only considering the particles interactions and not the interactions between the gauge fields. When κ=0 the interaction-energy operator (formally) reduces to the standard expression (x = (t, r)): time dependent and needing a regularization. H (INT) (t) = λ (ξ p *(t) A p,q (t) ξ q (t)) dµ(p,q) (24) 4.2. Particles Self-Masses If we try to calculate the self-energies (or the self-masses) of our particles considering only the one-loop Feynman graphs, we arrive at a r (and g) dependent expression ((Weisskopf 1939), see also for a more modern, but essentially equivalent, presentation (Bogoliubov 1980) or (Weinberg 1995)): M (SELF) (r) = M.Λ.log(F g (r)) + const. + (25) logarithmic diverging when κ=0 (that is ignoring gravitation). Here M is again the bare mass and Λ a numerical, positive parameter, proportional to the squared coupling constant λ. In the weak gravitational field approximation (subsection 2.3): where M (GRAV) is the effective (inertial, gravitational) particle mass. F g (r) = M (GRAV) / r (26) 4.3. Counter-Terms Now we shall assume that the interaction-energy operator contains a mass-like counter-term and that the real function M (CTERM) (r) (> 0) is proportional to (ξ p *(x) M (CTERM) (x) ξ p (x)) dµ(p) (27) M.( k(r) / k(r)) (28) where M is the bare mass, the Laplace, second order, differential operator and k is a scalar, classical field. Alternatively we can include the counter-term in the free part of the energy operator so the bare mass will be a function of x. Presumably the counter-term is of geometrical origin, for example it is the Ricci scalar of a (modified) metric tensor g. In any case, imposing the constancy of the sum M (SELF) (r) + M (CTERM) (r) (29) (the effective mass cannot depends on r), we arrive at a R 3 differential equation (note that here the bare mass do not appears): k(r) / k(r) = Λʹ.log(F g (r)) + const. (30)

70 FFP Udine November 2011 For a radial symmetric k, rescaling the dependent variable and using the weak field expression of F g (r), M (GRAV) / r, Kʹʹ(r) + (2/r) Kʹ(r) = (log(r) w) K(r) (31) We are looking for a K(r) going to zero when r diverge, hence we are lead to a countable family of solutions of the above differential equation; the first self-values of w are: 1.044; 1.847; 2.290; 2.596; 2.830; 3.020; (32) Obviously we can only obtain in this way the masses ratios of the first, the second, the third,... variety (or generations) of particles. 5. Leptons Masses Consider now the case of the electro-weak interactions between leptons, mediate by the gauge fields A, Z, W, W*; M Z and M W are the Z-particle and W-particle masses. To calculate the selfmasses we need the three (one loop) Feynman graph (lept stands for a charged lepton): lept(lept-photon)lept + lept(lept-zparticle)lept + lept(neutrino-wparticle)lept (33) then in the differential equation of subsection 4.3 appears a sum of three mass-dependent terms: Λ A.log(M lept ) + Λ Z.log(M lept + M Z ) + Λ W.log(M neutrino + M W ) (34) The Λ A, Λ Z and Λ W are three positive parameters proportional to the squared electro-weak coupling constants; Λ A + Λ Z + Λ W = 1, as a consequence of the rescaling the dependent variable (subsection 4.3). Apparently the experimental data lead to Λ A /Λ W = and to Λ A /Λ Z = so: Finally, assuming that M neutrino << M W, we arrive at: Λ A = 0.151, Λ Z = 0.196, Λ W = (35) log(m µ /M electron ) = 5.32, log(m τ /M electron ) = 8.22 (36) in reasonable accord to the experimental values: and (we have introduced many approximations). But if, for example, we add to the self-masses a (log(r)) 2 term (coming from some two loops Feynman graphs), we obtain a quite better accord. It would be also of some interest to consider the not radial symmetric solutions of the k-field differential equation. Obviously we also obtain other, bigger mass values (corresponding, presumably, to highly instable particles): M 4 11 Gev, M 5 34 Gev, M 6 72 Gev, (37) If we try to calculate the neutrinos masses starting from the interaction-energy operator defined at the subsection 4.1, we arrive at a quite problematic result: the neutrinos predicted masses are very large, greater than the W-particle and Z-particle masses. To cure this problem it is apparently necessary to modify the structure of the energy operator. References Bogoliubov N N, Shirkov D V (1980), Introduction to the Theory of Quantized Fields, John Wiley and Sons Bourbaki N (2007), Eléments de Mathématique. Variétés différentielles et analytiques: Fascicule de résultats, Springer-Verlag Lang S (1985), Differential Manifolds, Springer-Verlag Weinberg S (1972), Gravitation and Cosmology: Principles and Applications of the General Theory of Relativity, John Wiley and Sons Weinberg S (1995), The Quantum Theory of Fields, Volume I, Cambridge UP Weisskopf V F (1939), On the Self-Energy and the Electromagnetic Field of the Electron, Physical Review, (56), 72

71 FFP Udine November 2011 A MODEL OF CONCEPT LEARNING IN PHYSICS Wagner Clemens, Vaterlaus Andreas, Solid State Dynamics and Education, ETH Zurich, Schafmattstr. 16, 8093 Zurich, Switzerland Abstract Learning concepts in physics is difficult due to the often simultaneous presence of misconceptions. We have developed a dynamic model of concept learning, which includes the dynamics of misconceptions. In its simplest form the model simulates the learning of the concept and the unlearning of the misconception with a two-dimensional differential equation system. The major conclusion from our model simulations is that while teaching a concept, misconceptions should be intensively addressed. Rapid decay of concept knowledge observed in experiments after concept teaching is explained in our model with the persistent presence of a high level of misconception. 1. Introduction Recently, exploring the time dependence of learning physics concepts has gathered increased attention (Bao 2008, Sayre 2009, Heckler 2010). The data about concept learning shows convincingly that the acquisition of concepts is a dynamic process. As a consequence the interpretation of quantities like the normalized gain, for example, requires great care since the results strongly depend on the time points when the data are recorded. Therefore a more thorough understanding of the dynamic process of learning is required. Heckler et al. (Heckler 2010) investigated concept learning of an introductory physics course at the university level. Data were collected from more than 1600 students. Every few days or every week 12 students (in the average) were asked to solve a set of multiple-choice questions concerning physics concepts. Students were asked before, while and after the topic was taught. The proportion of correct answers was then plotted against time. It turned out that the most effective learning event was the homework activity since the homework due time showed the highest level of concept knowledge. The goal of this work is to construct a model, which is able to simulate the time course of concept learning. The model we present is based on previous work done by Lei Bao (Bao 2008). The major difference is that our model consists of two variables, one for the concept c and one for the misconception m. It distinguishes between not knowing the concept (1 - c) and the level of misconception m. The differential equation system we present to simulate concept learning has two stable fixed points, a concept fixed point and a misconception fixed point. Learning of concepts means in terms of our model, to move the system away from the misconception fixed point so that it relaxes on the concept fixed point after the learning phase. The major conclusion from our model is that if misconceptions are not properly addressed in class, resulting in a reduced misconception level, teaching of concepts in physics will be almost futile. Student s concept knowledge rapidly decays, after the learning phase. Thus an initial unlearning phase of misconceptions is necessary (if misconceptions are initially present) to settle concept knowledge on the long run. 2. The Model Our model of concept learning consists of two coupled differential equations, one for the concept c and one for the misconception m dc/dt = α c(1-c) (1 - m) - γ c m, dm/dt = -β m (1-m) c + µ (1-m).

72 FFP Udine November 2011 The parameters α and γ are the coefficients for learning and unlearning of concepts. The misconception is learned with rate coefficient µ and unlearned with rate coefficient β. The levels of the concept and the misconception are in the interval [0, 1]. The variable c is defined as probability to solve a concept question correctly at the given time point. Although we call the variable m misconception level it summarizes all detrimental effects on the concept knowledge. Therefore m hampers the learning and promotes the unlearning of concepts. We assume that the learning of concepts is hard at the beginning and at the end. This justifies that the learning rate is proportional to c(1-c). The most progress is made for concept learning of c = 0.5. The same argument also holds for the unlearning of misconceptions (unlearning is proportional to m(1-m). However, learning of concepts does not only depend on the concept c but also on the level of misconception, m. Thus learning of concepts is easier the larger the factor (1-m) is. The stronger the misconceptions are the slower is the learning of concepts. The unlearning of concepts is proportional to the product of the level of the concept and the misconception. If there is no concept there is no unlearning and the stronger the misconception the stronger the unlearning. The unlearning of misconception is amplified by the presence of a high concept level. Therefore we multiplied the misconception unlearning term -βm(1-m) by c. Finally, misconceptions have a permanent learning term, (1-m), due to everyday experience (for example: there is no real world experiment without friction) or due to intuitive perception (for example: adding a resistor to a circuit should increase the equivalent resistance). 3. Results In order to find reasonable values for the learning and unlearning parameters we used the data provided by Heckler and Sayre (Heckler 2010). They applied multiple-choice questions to students in order to test their understanding of physics concepts in Mechanics and Electricity and Magnetism. A data point corresponds to the normalized answer of 12 students (average). At every time point a different representative sample of students was asked the same questions. The data points of the electric circuit problem are shown in Fig. 1. Figure 1: Fit of the model to the data provided by Heckler and Sayre (Heckler 2010) (electric circuit problem). The full line represents the concept knowledge and the dashed line the misconception level. If a topic is taught directly by lectures, labwork or homework we increase α and assume an enhanced learning ability α enh. To obtain the fit curve we used the procedure (fminsearch) provided by Matlab. The procedure was allowed to adjust the following parameters providing the results α = 0.4, α enh = 3 44, β = 0.24, γ = 0.043, µ = and the initial value of the misconception level m init = 0.8. The time points for the onset and the end of the enhanced learning phase were kept constant at t 1 = 28 days and t 2 = 39 days. These time points correspond to the onset and offset of the teaching period. We selected for the initial condition of the concept the first point of the measurement c init = c 5 = 0.47 (circuit question). The fitted misconception level at the beginning is approximately m init = 0.8 and seems to be rather high. The rate constant for enhanced learning is about 10 times higher α enh = 3 44 than the basic coefficient α = 0.4. It is also remarkable that the rate for unlearning misconceptions is of one magnitude smaller than the rate of concept learning.

73 FFP Udine November 2011 As a consequence, the unlearning of misconception has to be done in an efficient way in order to successfully teach concepts. If the misconception level is not substantially reduced (when starting at m init 0.8) during the learning phase, the concept level decays quite rapidly after the enhanced learning phase (see Fig. 1). Finally students end up in the misconception fixed point and all efforts to understand the concept were in vain. The long-term behavior of the system is shown in Fig. 2 (thin full line: concept; thin dashed line: misconception). It emphasis our point of view that we not only have to teach concepts, but we also have to unteach misconceptions. What happens to the student knowledge if misconceptions are unlearned at the same time as the concept is taught is shown in Fig. 2. In order to do the corresponding simulations we introduced an enhanced unlearning coefficient for the misconception β enh = 0.37 during the concept teaching period. Between days 28 and 39 we changed the misconception unlearning rate from β to β enh. During this time period the misconception level drops markedly so that after the learning phase the system drives towards the concept fixed point. Figure 2: Simulation of unlearning misconceptions. The thick full line represents the concept knowledge and the thick dashed line is the level of misconception. The thin lines represent the simulation of the concept (full) and the misconception (dashed) due to the data shown in Fig. 1 for comparison. 4. Discussion In order to provide a better understanding of the evolution of concept knowledge we have developed a new model, which allows for simulating the interaction between the concept and the misconception. Using the model we fit the data about concept knowledge recorded by Heckler and Sayre (Heckler 2010). In our model students start at a medium level of concept knowledge and at a strong misconception level. For the first 27 days the model moves towards the misconception fixed point. Between day 28 and 39 the learning activities take place leading to a change of the learning coefficient from α to α enh. After day 39 the topic is assumed to change and without unteaching misconceptions the system moves back to the misconception fixed point. In contrast, if the misconceptions are properly addressed during the learning phase the misconception level drops substantially so that after the learning phase the system ends up in the concept fixed point on the long run. References Bao L (2008) arxiv: v2 physics-ed-ph. Heckler A F and Sayre E C (2010) Am. J. Phys. 78, Sayre E C and Heckler A F (2009) Phys. Rev. STPER 5,

74 FFP Udine November 2011 Learning scenario for a 3D virtual environment: the case of Special Relativity de Hosson Cécile, Kermen Isabelle, Maisch Clémént, LDAR, Université Paris Diderot-Paris 7 (France) Parizot Etienne, APC, Université Paris Diderot-Paris 7 (France) Doat Tony, Vézien Jean-Marc, LIMSI CNRS/Orsay (France) Abstract Special Relativity, as introduced by Einstein, is regarded as one of the most important revolutions in the history of physics. Nevertheless, the observation of direct outcomes of this theory on mundane objects is impossible because they can only be witnessed when travelling at relative speeds approaching the speed of light c. These effects are so counterintuitive and contradicting with our daily understanding of space and time that physics students find it hard to learn special relativity beyond mathematical equations and to understand the deep implications of the theory. Although we cannot travel at the speed of light, Virtual Reality (VR) makes it possible to experiment the effects of relativity in a 3D immersive environment (a CAVE: Cave Automatic Virtual Environment). The use of the immersive environment is underpinned by the development of dedicated learning scenarios created through a dialectic between VR-related computational constraints and cognitive constraints that include students difficulties. 1.. Introduction This research takes place within the context of the EVEILS research project (French acronym for Virtual Spaces for the Education and Illustration of Science) 1. This project aims at exploring the innovating potential of Virtual Reality (VR) in several areas of science through an interdisciplinary approach involving physicists, VR specialists and physics education researchers. The project exploits advanced interfaces in order to confront a student with unusual phenomena otherwise inaccessible to human experience. The exploration of the cognitive modifications and pedagogical advantages associated with the immersion is part of the main goals of EVEILS. This educational aspect makes EVEILS quite specific among the research programs devoted to computer simulations associated with VR (Savage et al. 2007). Special Relativity, as introduced by Einstein, is regarded as one of the most important revolutions in the history of physics. Nevertheless, the observation of direct outcomes of this theory on mundane objects is impossible because they can only be witnessed when travelling at relative speeds approaching the speed of light c. The theory of Special Relativity teaches us that space and time are neither absolute, that is, independent of the observer (or the reference frame associated with the observer), nor independent from one another. Instead, they make up a global geometric structure with 4 dimensions, called space-time, whose time and space components depend on the reference frame used to describe physical bodies and events in terms of positions and instants. In particular, the length of a given object, as well as the duration of a phenomenon (between two well-defined events) will be not only appear different for two observers moving with respect to one another. These effects are so counterintuitive and contradicting with our daily understanding of space and time that physics students find it hard to learn relativity beyond mathematical equations and to understand the deep implications of the theory. Although macroscopic objects can not travel at the speed of light, Virtual Reality (VR) makes it possible to experiment the effects of relativity in a 3D immersive environment (a CAVE: Cave Automatic Virtual Environment, see fig.1) 2 where the speed of the light is simulated to a reduced value. The EVEILS project is a 1 EVEIL is supported by the French National Research Agency (ANR) and conducted under the responsibility of Pr E. Parizot (APC Université Paris Diderot-Paris 7). 2 A CAVE is a surround-screen, surround-sound, projection-based virtual reality system. The illusion of immersion is created by projecting 3D computer graphics into a cube composed of display screens that completely surround the viewer. It is

75 FFP Udine November 2011 framework designed to merge advanced 3D graphics with Virtual Reality interfaces in order to create an appropriate environment to study and learn relativity as well as to develop some intuition of the relativistic effects and the quadri-dimensional reality of space-time. Of course, very early, mathematics were used to predict what objects would look like in relativistic motions and this have been applied to VR computing. Nevertheless, the specificity of the EVEILS project is to make physics education research actively involved in both conception and evaluation of the scenarios to be implemented into the CAVE. Fig.1: Picture of the CAVE used for the EVEILS project (LIMSI, CNRS/Orsay, France) 2. Overview of the research The use of the immersive environment is underpinned by the development of dedicated learning scenarios created through a dialectic between VR-related computational constraints and cognitive constraints that include students difficulties. Investigating students understanding of relativistic situations (that involve speeds closed to c) led to the typifying of a cognitive profile that orientated the situations to be implemented into the CAVE and the associated learning scenarios (see fig.2). These scenarios aim at approaching the consequences of the invariance of the speed of light and more specifically the relativity of the simultaneity but also a deeper understanding of the concepts of reference frame and event (in physics). Here we will present the results of the characterization of the cognitive profile and its consequences on the development of the scenarios. Fig.2: Diagram of the research process coupled with head and hand tracking systems to produce the correct stereo perspective and to isolate the position and orientation of a 3D input device. The viewer explores the virtual world by moving around inside the cube and grabbing objects with a appropriate device.

76 FFP Udine November Students difficulties in special relativity: elements of the state of the art The transition from classical to relativistic kinematics requires a radical change in the conceptual framework. In the theory of special relativity, c is a constant that connects space and time in the unified structure of space-time. The speed of light is equal to that constant and thus is invariant with respect to any inertial reference frame. Besides, the simultaneity of two events is not absolute (two events at different locations that occur at the same time in a given reference frame are not simultaneous in all other reference frames). Assuming this change in the conceptual framework requires a sound knowledge of the concepts of reference frame and event that underpin the laws of classical kinematics. Studies conducted in order to characterize student s difficulties in special relativity are not very numerous. Nevertheless, from what have been explored we can detain that students use spontaneous kinematics lines of reasoning (such as absolute motion, distances and velocities) to explain mechanical phenomena in both classical and special relativity frameworks (Saltiel & Malgrange 1980, Villani & Pacca 1987). Students think that simultaneity is absolute and independent of relative motions (Villani & Pacca 1987, Scherr 2001). Students fail in understanding the concept of reference frame confusing reference frame and point of view. Thus, each observer constitutes a distinct reference frame (Scherr 2001). Moreover, they fail in defining and using the concept of event and thus confuse the instant of an event and the instant of the perception of that event by an observer (Scherr et al. 2001, de Hosson et al. 2010). By connecting students with visual consequences of movements whose speed appears close to the speed of light we also hope we can reach a better understanding of concepts involved in the classical kinematics framework. This echoes various researches that pointed out the benefits of introducing modern physics in high school in order to improve fundamental concepts comprehension even in classical physics (Biseci & Michelini, 2008). This could also contribute to update the physics content taught in the secondary school context. 4. Confronting students to the 3D virtual environment Considering both VR-related computational and cognitive constraints we designed learning scenarios to be implemented into the CAVE. These scenarios aim at giving direct access to: lengths and durations are not invariant and depend on the relative velocity between the objects and the reference frames involved. Thus, there is a priori a conflict between the intrinsic definition of the objects in their own reference frame and their actual occurrence in other reference frames, with respect to which they are moving. More precisely, Special Relativity teaches us that, in these other reference frames, the (instantaneous) lengths between two given points of the object are generally not the same. For instance, a billiard puck that is intrinsically a sphere, is no longer a sphere when described in the rest frame of the billiard board (see figure III), with respect to which it is moving. This calls for a consistent description of the objects in any reference frame, i.e. in the 4D space-time reality itself. the speed of light c is finite (and invariant), so we do not see the objects where they are now, but where they were when they emitted the photons that we perceive now. The determination of what a given observer effectively sees at a given location at a given time (i.e. at a given point in space-time), requires a framework in which the whole history containing the past positions of the various objects of the scene is accessible to find the emission event.

77 FFP Udine November 2011 Fig.3: The rendering of a relativistic scene involves a search in the history of each vertex (point) of the billiard pucks. Each time an image is generated for the user, we store the position of each vertex at the current simulation time, after applying Lorentz contraction to the intrinsic definition of the objects. The history table built in this way can then be deep-searched through by dichotomy to find the emission event associated with each vertex, at any later observation event (Doat et al. 2011). The general idea of the scenarios is to confront users immersed into the CAVE with objects (billiard pucks) moving on a billiard table (without frictions) at a speed approaching that of the light (figure 4). Interaction with the simulation is made possible by applying impulsions to the pucks, to observe the effect of the limited speed of light and the Lorentzian contraction of length in the carom billiard 3. Indeed, our application allows observing some subtle consequences of the theory of Special Relativity which are particularly important for physics education: The changes in the puck shape The apparent non-simultaneity of the bounces The apparent acceleration and deceleration of the pucks The aberration of the light Fig.4: The carom billiard running in the EVE CAVE (CNRS/LIMSI, Orsay, France) With respect to students spontaneous lines of reasoning (see above) we focused on the concept of event as defined by its coordinates in space and time in a given reference frame. Since the perception of an event (e.g. emission of light) by an observer depends on the time taken by the light to travel from the location of the event to the eyes of the observer, the signal travel time of light will impact the observations of the users immersed in the simulator. In everyday life the light signal travel time does not matter because it is close to zero whatever the location of the emission of the event. But when the students take it into 3 To avoid overloading the simulation and affecting the understanding of the scene by superimposing effects of very different nature, we have not implemented the Doppler effect nor the effects of changes in light intensity. Indeed, we initially limited the rendering to purely geometrical effects (space-time and related concepts). The Doppler effect is certainly an effect of space-time, but its manifestation depends on the physical nature of light (electromagnetic wave which actually has a frequency...). In our approach, for now, we do not question the luminous phenomenon itself but the space-time architecture of the physical reality.

78 FFP Udine November 2011 account, they consider that the order in which two events occur is a consequence of the order in which these two events are perceived as if causality would apply from future to past (de Hosson et al., 2010). In our environment the signal travel time cannot be neglected (c is fixed to 1m/s), thus mundane objects are not seen as in everyday life. The simulator we designed takes into account the relativistic effects (simulation algorithms based on Lorentz transform, Doat et al., 2011) and also those due to the light propagation. Then we make the hypothesis that the users of our simulator will take into account the time delay between emission and perception in a relevant way. According to our first scenarios, users (who also are observers) are asked about changes in the shape of the pucks and about the changes in their velocity. We also question them about the instant of the contact of the pucks with the billiard table. The corrected proper time of each puck is visible. It represents the time measured by a clock located in the puck itself. The delay of reception of the photons by the observer (who is actually not located where the time is measured) is taken into account. Thus, according to the movement of the puck, the perceived time seems to pass faster or slower. Moreover, the movement of the puck can be freeze so that the puck is seen using the Matrix effect: a camera turns around the puck showing each part of it without changing the initial point of view which is the observer s one. Fig.5: On the left screen-capture the apparent collision delay is observed; one the right screen-capture the observer can see the object deformation as an effect of the consideration of the finite value of c. The photons that arrive into the eye of the observer have not been emitted at the same time. The image seen is not the result of the emission of photons emitted simultaneously. The same scene can be replayed but as seen by an observer shifted on the left (or on the right). Then, two pucks (of two different colors) are in movement perpendicularly to the billiard table (fig.5). The observer can be located at equal distances between the red puck and the orange one. He can also moves to the left (or to the right) breaking the symmetry of the distances. All the effects observed are discussed. 5. Conclusion and perspectives Five users were immersed in the simulator and had to explain what they saw when confronted to the learning pathway. We undertook a lexical analysis searching for some specific conceptual elements that we consider as key points for the users' understanding. These elements are the following: a) the incoming of light in the eye, b) the finite nature of the speed of light, c) the distance between the pucks and the user, d) the object discretization (a set of points as punctual sources of photons), e) the geometrical relativistic effects explained by the Lorentz transform. Following users reactions during the immersion, we detected which situations favour the emergence of the conceptual elements that allow a relevant interpretation of the situation. As a conclusion the immersion of users in the 3D environment where they are confronted to relativistic phenomena favours lines of reasoning that take into account the light travel time and the arrival of light into the users eyes. We believe that using our application to experience these effects without thinking will help to develop intuition on relativistic behaviours while trying to play billiard properly at relativistic velocities. It is expected to help students in their efforts to understand Einstein s theory from

79 FFP Udine November 2011 a practical point of view. This will be tested by the EVEILS group through a dedicated research work in formal evaluations on physics students. References: Biseci, E. & Michelini, M. (2008). Comparative teaching strategies in special relativity, GIREP. Conference, August 18-22, Nicosia, Cyprus. de Hosson, C., Kermen, I., Parizot, E. (2010). Exploring students understanding of reference frames and time in Galilean and special relativity. European Journal of Physics, 31 (12), Doat T., Parizot E., & Vezien J.-M. (2011). A carom billiard to understand special relativity. Virtual Reality Conference (VR), IEEE, Hewson, P.W. (1982). A case study of conceptual change in special relativity: the influence of prior knowledge in learning. International Journal of Science Education, 4 (1), Saltiel, E., & Malgrange, J.L. (1980). Spontaneous ways of reasoning in elementary kinematics. European Journal of Physics, 1, Savage, C. M., Searle, A., & McCalman L. (2007). Real Time Relativity: exploration learning of special relativity. American Journal of Physics, 75, Scherr, R., Schaffer, P., & Vokos, S. (2001). Student understanding of time in special relativity: simultaneity and references frames. American Journal of Physics, 69, Villani, A., & Pacca, J.L.A. (1987). Students' spontaneous ideas about the speed of light. International Journal of Science Education, 9 (1),

80 INVESTIGATING TEACHER PEDAGOGICAL CONTENT KNOWLEDGE OF SCIENTIFIC INQUIRY Claudio Fazio, Giovanni Tarantino, Rosa M. Sperandeo Mineo, Dipartimento di Fisica- Università di Palermo, Italia. ABSTRACT This paper analyses the scientific inquiry in the context of modelling physical systems and point out some related teaching/learning strategies. Two relevant points are also discussed: i)- the different representations of scientific inquiry held by a sample of secondary school physics teachers and their competencies in conducting investigations aimed at solving scientific problems; ii)- what kind of supports can be supplied to teachers in order to modify their misconceptions about scientific inquiry and develop adequate competences related to Pedagogical Content Knowledge of Scientific Inquiry. The topic will be analysed in the light of some preliminary results of a course for in-service physics teacher education held at University of Palermo in the framework of Establish, an FP7 EU Project. 1. Introduction In designing innovative pedagogical materials usually supports for teachers subject matter knowledge and pedagogical content knowledge for students ideas (e.g., misconceptions), but rarely for pedagogical content knowledge of scientific inquiry, are provided. By inquiry we refer to learning experiences that engage students in various integrated activities of identifying questions, collecting and interpreting evidence, formulating explanations, and communicating their findings, that are consistent with science standards and recent reports (Duschl, Schweingruber,& Shouse, 2007; National Research Council, 1996; Singer, Hilton, & Schweingruber, 2005). Many researchers have shown that Inquiry-Based (IB) approaches to learning are able to increase student motivation, interest, understanding and development (Collins, 1997; Singer et al., 2005). However, despite the consensus found in educational research, teachers may have different ideas about the meaning of inquiry-based instruction and it has been shown that misconceptions abound (National Research Council. 2000). These mistaken notions about inquiry can, perhaps, deter efforts to reform science education. In this paper, we analyse scientific inquiry in the context of modelling physical systems and point out some related teaching/learning strategies and how these are perceived by physics teachers. The topic will be analysed on the light of some preliminary results of a course for in-service physics teacher education held at University of Palermo in the framework of Establish (ESTABLISH, 2010), an FP7 Project aimed at extending the use of IBSE (Inquiry Based Science Education) in second level schools across Europe. 2. Theoretical Background Despite the consensus found in educational research about the efficacy of IB approaches, it has been pointed out that misconceptions about IB instruction abound and serve to deter efforts to reform science education (National Research Council. 2000). The more relevant misconception involves the idea that IB instruction is the application of the scientific method Many teachers learned as students that the process of science can be reduced to a series of five or six simple steps. It has been shown that the notion that scientific inquiry can be reduced to a simple step-bystep procedure is misleading and fails to acknowledge the creativity inherent in the scientific process. Research has connected this view with what science s nature is perceived from samples of pupils and teachers (Akerson, et al., 2000; Lederman, 1992). Some studies showing that science teachers persist in holding views about nature of science qualified as empirical or naïvely empirical, identify two peculiar aspects in their thinking, involving: - the role accorded to observation, which is seen as giving experimental data an absolute value; - the role played by theory in conducting experiments and in making observations, along with the value of scientific knowledge as a means of explaining and predicting (Van Driel, et al., 1998; Glasson and Bentley, 2000; Abd-El-Khalick, 2005). Other studies report that teachers carry positivistic views of their discipline: i. e. they teach only the knowledge aspects of science and emphasize vocabulary rather than balance knowledge claims with knowledge generation and evaluation, and present science as the method of understanding the world (Gess-Newsome, 1999). Additional classroom consequences may include a decreased emphasis on inquiry-oriented and problem solving teaching methods that positively impact pupils conceptions of science (Gess-Newsome, 1999; Lederman, 1992).

82 to find an explanation or to compare different kinds of explanations. All the discussions have been audio taped and the detailed analysis is in progress. Figure 1. The experimental situation, first only theoretically proposed, then practically analyzed by the teacher group. Some ice cubes are placed on plates of different material, mass, area and thickness, identified by letters A, B, C, D, E, F, G, H. 4. Results A detailed analysis of teacher incorrect predictions with the related explanations is in progress. We note here that all teachers correctly predicted that the ice cubes melt sooner when placed on aluminum plates, but only 3 teachers performed the correct prediction by appropriately ranking plates with different masses and thickness. By considering at a finer grain detail the predicted melting time sequence for the three aluminum plates (of different geometrical characteristics), as well as the teachers explanations about the parameters that have to be taken into account for the correct description and explanation of the phenomenon, some considerations can be drawn about the procedures they followed in formulating their hypotheses. The correct analysis of the proposed situation must take into account several parameters, i.e. the plates geometrical characteristics (surface area and thickness), thermal capacity, thermal conductivity and the temperature difference between the two plate faces. All these parameters have to be considered together to correctly explain the phenomenon. An approach based on the typical theoretical knowledge about thermology, resulting from classical university education can easily guide the teacher to search for an explanation to the phenomenon based on thermal conduction, i.e. on the idea of heath flux between two surfaces at different temperatures. From the Fourier law, we know that this flux is proportional to the surface area and inversely proportional to the plate thickness, so thinner plates should make ice cubes melt quicker, as they allow a bigger heath flow. Indeed, for a clear understanding of the phenomenon other factors involved in the analysis of thermal interaction between two bodies (the ice cube and the plate that exchange thermal energy until equilibrium is reached) must be taken into account. This second kind of approach actually produces opposite predictions with respect to the previous ones, i.e. a greater melting speed of ice cubes placed on the thicker (and so heavier) plate. Our data show that 6 teachers appear to describe the phenomenon only on the basis of the Fourier law by considering as more relevant factor the plate thickness and 5 teachers that seem to consider thermal capacity as relevant for the explanation of the phenomenon, but not plate thickness or thermal conductivity coefficient. It is worthy to note that the 4 teachers performing the correct ranking (2 graduated in Biology, 1 in Physics and 1 in Natural Science) analysed the experimental situation by trying to point out the different characteristics of the various bodies and evaluating the relevance of each of them and not by searching, in their knowledge, for laws to apply to the experimental situation. In comparing their predictions with experimental results, the majority of teachers that predicted the melting of the ice cubes on the basis of a partial acknowledgement of relevant variables expressed, in different ways, a sort of surprise with respect to the results. They did not try to search for an explanation of the differences by showing that their predictions were mainly driven by memory of subjects studied during past courses. Only a few made reference to the necessary comparison between what they remembered from textbooks and real-life experience. The great majority of teachers at last commented that the approach to the proposed situation posed them some difficulties, as it is one that is not part of their theoretical knowledge about thermology. As previously mentioned, the group discussion during the second phase of the W was devoted to deepen the teacher conception of the different procedures involving in a scientific investigation stimulated by the real situation analysed. A preliminary analysis of the audio taped discussion shows the relevant factors described in the following. During the discussion, all teachers mentioned (at least one time) the need to apply the Scientific Method (SM) in order to solve the different problems involved in the analysis of an experimental

84 resources necessary for a profitable development of their Pedagogical Content Knoeledge about SI. Moreover teachers need to gain a more epistemically congruent representation of how contemporary science is done by developing activities across different domains of inquiry and many types of investigations. References Abd-El-Khalick, F., & Lederman, N. G. (2000). The influence of history of science courses on students' views of nature of science. Journal of Research in Science Teaching, 37(10), Abd-El-Khalick, F. (2005), Developing deeper understandings of nature of science: the impact of a philosophy of science course on preservice science teachers views and instructional planning, International Journal of Science Education, 27(1), Abd-El-Khalick, F., Waters, M., & Le, A.-P. (2008). Representations of nature of science in high school Chemistry textbooks over the past four decades. Journal of Research in Science Teaching, 45(7), Akerson, V. L., Abd-El-Khalick, F., Lederman, N.G. (2000), Influence of a Reflective Explicit Activity-Based Approach on Elementary Teachers Conceptions of Nature of Science, Journal Of Research In Science Teaching, 37(4), Beyer, C. J., Delgado, C., Davis, E. A., & Krajcik, J. S. (2009). Investigating teacher learning supports in high school biology curricular programs to inform the design of educative curriculum materials. Journal of Research in Science Teaching, 46(9), Borko, H., & Putnam, R. T. (1996). Learning to Teach. In D. C. Berliner & R. C. Calfee (Eds.), Handbook of Educational Psychology, New York: Macmillan Clark, D. B., & Linn, M. C. (2003). Scaffolding knowledge integration through curricular depth. Journal of Learning Sciences, 12(4), Collins, A. (1997). National science education standards: Looking backward and forward. The Elementary School Journal, 97(4), Duschl, R. A., Heidi A. Schweingruber, H. A. & Shouse, A. W. (2007) (Eds.), Taking Science to School: Washington, D. C: National Academies Press ESTABLISH (2010), Gess-Newsome, J. & Lederman, N. G.,(1999) (Eds.), Examining Pedagogical Content Knowledge. Dordrecht, The Netherlands: Kluwer Academic Publishers.Linn, M.C, Hsi, S. (2000). Computers, Teachers, Peers. Mahwah, NJ Erlbaum Glasson, G.E. & Bentley, M.L. (2000), Epistemological undercurrents in scientists' reporting of research to teachers, Science Education, 84, Lederman, N.G. (1992) Students' and teachers' conceptions of the nature of science: A review of the research, Journal of Research in Science Teaching, 29, Lederman, N. G. (2004). Syntax of nature of science within inquiry and science instruction. In L. B. Flick & N. G. Lederman (Eds.), Scientific Inquiry and Nature of Science (pp ). Bordrecht: Kluwer Academic Publishers Magnusson, S., Krajcik, J., & Borko, H. (1999). Nature, sources, and development of pedagogical content knowledge for science teaching. In J. Gess-Newsome & N. G. Lederman, N. G.,(1999) (Eds.), Examining Pedagogical Content Knowledge. Dordrecht, The Netherlands: Kluwer Academic Publishers. Marx, R.W., Blumenfeld, P.C., Krajcik, J.S.,&Soloway, E. (1997). Enacting project-based science. The Elementary School Journal, 97(4), McComas, W.F., Almazroa, H. and Clough, M.P. (), The Nature of Science in Science Education: An Introduction, Science & Education, 7(6), National Research Council. (2000). Inquiry and the National Science Education Standards.Washington, DC: National Academy Press.Singer, S., Hilton, M. & Schweingruber, H. (2005). Needing a New Approach to Science Labs. The Science Teacher, 72(7), 10. Pinto, R. (2004). Introducing curriculum innovations in science: Identifying teachers transformations and the design of related teacher education. Science Education, 89, Roehrig, G.H.,& Luft, J.A. (2004). Constraints experienced by beginning secondary science teachers in implementing scientific inquiry lessons. International Journal of Science Education, 26(1), Shulman, L.S. (1986). Those who understand: Knowledge growth in teaching. Educational Researcher, 15(5), Singer, S.R., M.L. Hilton, and H.A. Schweingruber, eds.( 2005) America s lab report: Investigations in high school science. Washington, DC: The Nat. Acad. Press. Van Driel, J. H., Verloop, N., & De Vos, W. (1998). Developing science teachers' pedagogical content knowledge. Journal of Research in Science Teaching, 35(6), Windschitl, M. (2003). Inquiry projects in science teacher education: What can investigative experiences reveal about teacher thinking and eventual classroom practice? Science Education, 87(1),

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100 FFP Udine November 2011 TEACHING MODERN PHYSICS FOR FUTURE PHYSICS TEACHERS E. F. Nobre, A. O. Feitosa, M. V. P. Lopes, D. B. Freitas, R. G. M. Oliveira, M. C. Campos Filho, Federal University of Ceará, Campus do Pici, Fortaleza, Ceará, Brazil N. M. Barone, R. Monteiro, M. F. S. Souza, Virtual Institute-Federal University of Ceará, Campus do Pici, Fortaleza, Ceará, Brazil,l Abstract In this work we present a new way for teaching Modern Physics to students of course for teachers formation. One of the great difficulties faced by the teachers of the secondary schools of the State of Ceará, in Brazil, may have its origin in their formations during the undergraduate courses. We aim to offer a course in modern physics so that future physics teachers can transmit this knowledge to their future students in a clear, simple way and with good results, awakening in their future pupils the interest by the study of Sciences in general and in Physics, in particular. 1. Instructions Since year 2000, when Modern Physics was included in the program of examination for entrance of students in the Federal University of Ceará (UFC), the secondary schools began to include Modern Physics into their programs. The big problem was that this inclusion has been made simply Since year 2000, when Modern Physics was included in the program of examination for entrance of students in the Federal University of Ceará (UFC), the secondary schools began to include Modern Physics into their programs. The big problem was that this inclusion has been made simply as an appendix to the contents of the third year of high school, almost in the end of the school year. In addition, many teachers were not familiar with this subject and thus the inclusion of Modern Physics was compromised. Aiming to help solving this problem, we developed a discipline of Modern Physics to be applied to students of the Undergraduate Course in Physics from the Federal University of Ceará (UFC), sponsored by the Virtual UFC Institute. The course is specific for training teachers in pre-service and is applied to 5 cities in the State of Ceará. The Virtual Institute of Federal University of Ceará (Virtual-UFC), besides creating, implementing and maintaining the technological core, organizes the physical structure, logic and the educational progress of each course. All over the world, the electronic learning (e-learning), a model of teaching based on the online environment, leveraging the ability of the Internet for communication has been an excellent way to open the University to the student's home, allowing a high degree of flexibility in any academic program. There is no doubt that the multimedia technology is a powerful tool in the development of e-learning contents, which is inserting deep changes at higher education [BAGGALEY, Jon; KIRKUP, Gill]. The use of e-learning mediated by the Information and Communication Technologies (ICTs) in the educational process, has favored the inclusion of a contingent of people in higher education, contributing to increase places in higher education. This work is part of project developed at the Learning Environment of the Virtual Institute of the Federal University of Ceará, in Brazil, called SOLAR. The most part of courses which are being developed in this e-learning environment are focusing on the teachers formation. The production of teaching materials for e-learning plays a more significant relevance role than those ones we use for presence courses. In this process the steps of planning, construction, mediation and evaluation inherent to the traditional teaching, plus the endless possibilities of combining the diversity of information available and the benefits of interactivity, flexibility and dynamism favored by the ICTs require the implementation of this material in a more safely way. We must consider that the students are not face to face with their teachers, so they need a material absolutely clear and easy of understanding. By developing the didactic material that is used at the distance courses, the didactic transition of the Virtual UFC, uses as a

101 FFP Udine November 2011 methodologic reference the socio-interactionism (FIORENTINI, L). In this context, the education becomes a dynamic task that allows to the students, themselves, the active construction of their knowledge, according to their experiences in different situations where they liveby the Virtual UFC Institute. The course is specific for training teachers in pre-service and is applied to 5 cities in the State of Ceará. The Virtual Institute of Federal University of Ceará (Virtual-UFC), besides creating, implementing and maintaining the technological core, organizes the physical structure, logic and the educational progress of each course. All over the world, the electronic learning (e-learning), a model of teaching based on the online environment, leveraging the ability of the Internet for communication has been an excellent way to open the University to the student's home, allowing a high degree of flexibility in any academic program. There is no doubt that the multimedia technology is a powerful tool in the development of e-learning contents, which is inserting deep changes at higher education [BAGGALEY, Jon; KIRKUP, Gill]. The use of e-learning mediated by the Information and Communication Technologies (ICTs) in the educational process, has favored the inclusion of a contingent of people in higher education, contributing to increase places in higher education. This work is part of project developed at the Learning Environment of the Virtual Institute of the Federal University of Ceará, in Brazil, called SOLAR. The most part of courses which are being developed in this e-learning environment are focusing on the teachers formation. The production of teaching materials for e-learning plays a more significant relevance role than those ones we use for presence courses. In this process the steps of planning, construction, mediation and evaluation inherent to the traditional teaching, plus the endless possibilities of combining the diversity of information available and the benefits of interactivity, flexibility and dynamism favored by the ICTs require the implementation of this material in a more safely way. We must consider that the students are not face to face with their teachers, so they need a material absolutely clear and easy of understanding. By developing the didactic material that is used at the distance courses, the didactic transition of the Virtual UFC, uses as a methodologic reference the socio-interactionism (FIORENTINI, L). In this context, the education becomes a dynamic task that allows to the students, themselves, the active construction of their knowledge, according to their experiences in different situations where they live. 2 Methodology and discussions The material concerning of Modern Physics was developed by a team composed by a Professor, who was in charge of all the contents of Physics; Educators, experts at Pedagogy who are in charge of the Didactic Transition (DT) and technicians in ICT. This last team is composed by specialists in various areas of computing who use the multimedia technologies in order to transform the texts written by the Professor in e-materials. So, each class presents animations, retractable texts, vector animations, links to additional readings, chats, forums, and all kinds of resources currently used in digital technologies focusing on education. The use of these tools enables the integration of various resources in a single learning environment, and encourages the adoption and understanding of audiovisual language. Our main goal is develop e-lessons in order they are presented to the students in a very clear, simple and attractive way. Nevertheless our structure is developed based on the idea of the use of on line material, all materials of the virtual courses of the UFC, are available on line and in printed form for the students. The course was developed in six lessons covering the following topics: Special Relativity; Thermal Radiation and the Origin of Quantum Theory; Atomic Models: Thomson, Rutherford, Bohr; Wave- Particle Duality and the Principles of Quantum Theory and Nuclear Physics Topics. The teaching material was structured in order to bring a bit of History of Physics in all contents. In some subjects, we developed animated lessons, with audio. In all classes the subjects were presented with simulations, animations, music, challenges that were intended to excite students to seek solutions to challenging problems, suggestions of sites for complementary reading and research

102 FFP Udine November 2011 and much supporting material, discussion forums, chats and an special virtual space called portfolio, where the students presented the solutions of the exercises. In nuclear physics lessons we also developed a special tool called Dynamic Book, which are virtual books to complement the covered topics. The structure of lessons is showed below: Lesson 1: Special Relativity 8 Topics Topic 01: Galileo s Transformations Review Topic 02: The Michelson-Morley Experiment Topic 03: The Postulates of Relativity; Simultaneity Topic 04: Time dilatation Topic 05: Space Contraction Topic 06: Lorentz transformations Topic 07: Relativistic Dynamics - Relativity of mass Topic 08: Relativistic Dynamics: Energy Lesson 02: Thermal Radiation and the Origin of Quantum Theory 5 Topics Topic 01: Black Body Radiation - Principles Topic 02: Black Body Radiation - Theory Topic 03: Photoelectric Effect, The Quantum Theory of Radiation Topic 04: Compton Effect Topic 05: X-Rays Lesson 03: Atomic Models I 4 Topics Topic 01: Thomson Model Topic 02: Successes and Failures of the Thomson model Topic 03: Model of Rutherford. Experiment Rutherford Planetary Model Topic 04: Successes and Failures of the Rutherford model Lesson 04: Atomic Models II - Atom quantized Model 4 Topics Topic 01: Atomic Spectra, Spectrum Series, Bohr's Postulates Topic 02: Electron orbits, the Hydrogen Atom Topic 03: Energy Quantization Topic 04: Successes and Failures Lesson 05: Wave-Particle Duality - 4 Topics Topic 01: de Broglie Waves Topic 02: Diffraction Particle Topic 03: Principle of Uncertainty Topic 04: The Wave Function Lesson 06: Topics in Nuclear Physics - 4 Topics Topic 01: The Nucleus Topic 02: Nuclear Reactions Topic 03: Radioactivity Topic 04: Radiological Risks to Health The lessons were developed in order the students could use the new multimedia technologies with the use of computers and the internet, but besides these virtual recourses, the students had six meetings at presence moments with their Teachers/Tutors during the application of discipline. There was one Teachers/Tutors for each city. These teachers/tutors, some of them are specialists in Physics Teaching, some with Master's degree in Physics and some finishing his doctoral degree, accompanied the students during the time that the lessons are applied. This team of Teachers/Tutors was coordinated by the same Professor who produced and wrote all the material and by the Coordinator of tutors, a Professor whose function is to monitor the progress of the work of Teachers/Tutors. In addition to face meetings, the students were oriented at a distance by their teachers/tutors through the discussion forums, chats, and all the resources currently available by the digital technologies. The method of evaluation of the course in e-learning is different from what is done in traditional courses. In this course the students were evaluated personally and virtually. The students had several small conceptual evaluations at the end of each face meetings with their teachers/tutors.

103 FFP Udine November 2011 The goal of these evaluations was to provide students a better understanding of physical concepts. There was no math in these evaluations. Besides these small tests, the students did a big test which required physical concepts and also the mathematical tools. The virtual evaluations were distributed among the six discussion forums and the six lists of exercises solved by students and posted on the solar environment. The final average was formed with the results of all evaluations: in presence and virtual evaluations. For the student who did not achieve the minimum average, a kind of GPA (Grade Point Average) required for approval by the University, was offered a final exam: an written exam covering all topics studied. In addition to the material produced especially for this course, the students were guided to consult also some basic textbooks of Modern Physics (Acosta, Beiser, Eisberg-Resnick, Serway, Tipler). In addition to basic textbooks on the subject, the student also received a printed manual containing all lessons that were available on the website of the course. This greatly favored those students who still not have a computer in their homes. The results of 31 students who participated of this project were excellent. The most of them was approved with A concept and there was a minimal dropout. These results are showed in figure bellow. 31 STUDENTS A B dropout Figure 1. General results of all students 3 Concluding Remarks As a result of this way of teaching Modern Physics, we obtained one of the best results so far, since the beginning of the Physics Course in The students felt truly attracted by this subject. The abandonment was minimal, the frequency in the meetings was almost total and participation in activities was massive. The results showed approval ratings in the discipline between 80% to 100%. Next year, first half of 2012, we will apply the lessons of Modern Physics to all students, around 11 cities, involving about 100 students. We hope that the project will have again the same acceptance by the students and they are as successful as were the students of this first class. Another expected result is the involvement of students who attended the course acting as monitors in the next application of the project in Playing a role as monitors of the discipline, they will act under the guidance of teacher / tutor of the class, helping the new students, answering questions, forming study groups, which may bring greater benefits in learning. We also expect that helping to colleagues, may promote a consolidation in student learning that will act as a monitor. References Acosta, Virgilio; Cowan, Clyde L; Grahan, B. J. Curso de fisica moderna.. Mexico: Harla, p Baggaley, Jon; Hoon, M., Ng Lee (2005) Pandora's Box: Distance Learning Technologies in Asia, Learning, Media & Technology, v30, (1) Beiser, Arthur. Concepts of Modern Physics. 3rd ed. New York: McGraw-Hill, p. Eisberg, Robert, Resnick, Robert. Quantum Physics of Atoms, Molecules, Solidas, Nuclei and Particles, John Wiley & Sons Inc., p. Fiorentini, L. E Morais, R. (2000) Languages and interactivity in distance education, São Paulo, Brazil. P&D,

104 FFP Udine November 2011 Kirkup, Gill; Kirkwood, Adrian (2005) Information and communications technologies (ICT) in higher education teaching: a tale of gradualism rather than revolution. Learning, Media and Technology. Serway, A. Raymond, Moses J. Clement, Moyer A. Curt. Modern Physics. Saunders College Publishing, 2 a. ed p. Tipler, Paul Allen. Modern Physics. Rio de Janeiro: Guanabara Dois, p

105 MASS FROM CLASSICAL TO RELATIVISTIC CONTEXT A PROPOSAL OF CONCEPTUAL UNIFICATION EXPERIMENTED IN THE IDIFO3 SUMMER SCHOOL Emanuele Pugliese, Lorenzo Santi University of Udine Physics, Chemistry and Environment Department Division of Physics Abstract The IDIFO3 summer school proposed a basic concept construction in modern physics to young talented students. Inertial and gravitational mass concepts were examined in Newtonian paradigm, the former according to Mach as well. Temporal component of 4-momentum was interpreted as total relativistic energy, through series expansion at low velocities; we then built the relativistic conceptual extension of mass starting from rest energy. We integrated an analysis on how students consider mass with an inquiry on mass meanings produced by our path. The results allowed us to recognize 5 student profiles. 1. Why mass? «The concept of mass is one of the most fundamental notions in physics, comparable in importance only to the concepts of space and time» (Jammer, 2000). The historian of science Max Jammer puts mass on a level with the backstage of all physical phenomena: space-time. Mass plays such a founding role because massless macroscopic objects would be made of massless particles, without any form of interaction energy among them: we would have a Universe of free particles travelling at light speed 1. Mass is to be admitted to endow matter with something that characterizes it and that determines its motion in the absence of electromagnetic fields 2. This quantity has a manifold character. In Newtonian physics, it is essentially the metaphysical quantity of matter, deeply connected with inertia; the conception is explained in the opening paragraph of Principia: The quantity of matter is the measurement of the latter obtained from the product of density and volume. [ ] Air of double density, in a double space, is quadruple in quantity [ ] And the norma [Latin text] of all bodies, which be differently condensed for any causes, is identical [ ]. I will mean hereafter, everywhere, this quantity under the name of body or mass. Inertial mass acquires its gravitational meaning with the development of the universal gravitation law. In 1883, Mach criticized Newton s conception of mass, because it brought to a vicious circle, and proposed an operative definition through the measure of accelerations. He used a meaningful result derived from Newton s second and third laws, where m 1 can be put as unit of mass: m 2 = m 1 a a Eventually, in Special Relativity (SR) a variation of mass corresponds to a variation in the internal energy of a body 3 : the idea of mass becomes more and more subdivided. 2. Learning problems The concept of mass is very difficult to understand because of this multifaceted character and of the persistence of the ontological vision above as well. A further facet: in the theory of General Relativity, the momentum-energy density modifies the geometry of space-time, so matter exerts an active action on the last, differently from what happened with Newtonian space. In high-school textbooks, the following learning problems have been found: identification of weight with gravitational mass; belief that a beam balance measures weight; inertial mass defined operationally through F/a, where, as a literature example, F is the measurable force impressed by a compressed spring. On the subject matter ground, the conception of quantitas materiae has also generated misconceptions concerning the mass-energy relation: mass is converted into energy, in a generic sense (6 high-school textbooks); E=m c 2 represents conversion of mass into energy (1 high-school textbook); confusion/mistakes between energy conservation law and mass conservation law (3 textbooks) [Lehrman, 1982]. In matter of students learning difficulties, an 1 We take into account Special Relativity here. 2 Gamow considers indeed gravity as the force that rules Universe. 3 This quantity is usually called rest energy E 0 : the energy of a body measured in its rest reference frame (1)

106 experiment on 16 to 18-aged Spanish pupils was performed. They tend to prefer a teleologicalqualitative (pre-theoretical) view of mass to an operative-quantitative-formal (scientific) vision. For instance, most of them identified it with other quantities: volume or density on one side and weight on the other; in addition, quantitas materiae prevailed on inertia [Domenech, 1993]. 3. Relativistic mass debate In several treatises on relativity a quantity called "relativistic mass" appeared, whose expression is m r = m 0 u 1 c Using the latter, one can rewrite relativistic momentum in the form p = mr u Is it enough to justify the use of (2) as the expression for a proper physical quantity? Taylor and Wheeler underline that relativistic mass is the first component of relativistic quadrimomentum, while the invariant mass is the magnitude of the latter: a scalar. Actually, why should we interpret m in the following master equation as the inertial classical mass? 2 2. (3) (2) E p c = m 2 c 4. (4) In general, there are good reasons both to consider mass in continuity with Newtonian mechanics and to interpret it as a completely new quantity 4 in a new paradigm (Kuhn 1970): «the expression m0 is best suited for THE mass of a moving body», wrote R. C. Tolman (1934). From the 2 u 1 c 2 mathematical point of view, the two quantities are completely symmetric: the classical mass can be generalized to two quantities with different tensorial characters (Bickerstaff 1995) 5. One strong objection to relativistic mass is that m r = E 2 c varies with the reference frame: it should not define a physical property of a real particle or body. In spite of the last considerations, however, Richard Feynman (1969) claimed the importance of relativistic mass: Newton s Second Law [ ] was stated with the tacit assumption that m is a constant, but we now know that this is not true, and that the mass of a body increases with velocity [ ]. For those who want to learn just enough about it so they can solve problems, that is all there is to the theory of relativity it just changes Newton s laws by introducing a correction factor to the mass. 4. The activity We took a sample of 42 talented high-school students, aged 17 to 19, selected for attending a modern physics summer school. Our activity consisted in an interactive tutorial with proposals for individual reflections and group discussions. We gave each student worksheets containing open and closed questions; for the discussions, we asked for conceptual differences among the (classical) notions of mass examined first, then what classical meanings of mass underwent a change passing to relativity. We began the rationale of our path with the quotation in section 1, together with another one containing an operative definition of mass through weight, from Principia. An analysis of inertial ( 5) 4 The philosopher of science Feyerabend considered it a «relation, involving relative velocities, between an object and a coordinate system», instead of a «property of the object itself» (Feyerabend 1965). 5 Similarly, you can generalize the classical quantity time either to the proper time or to the zero component of (ct, x, y, z).

107 and gravitational mass concepts in Newtonian theory followed, as well as an overview of Mach s definition of the former 6. We then introduced proper time using light clock, built 4-displacement through analysis of the world lines, and came to quadrimomentum dividing by proper time, in analogy with classic momentum. We finally performed a Taylor series expansion of the temporal component in the Newtonian limit and defined this quantity as relativistic kinetic energy, apart from an additive constant. The identification 2 E 0 = mc, as well as its interpretation, were straightforward. We asked the students some questions after the path: 1) When does mass plays a role in your everyday life and which phenomena is it involved in? 2) "What theories of physics do study these phenomena? 3) What do you mean when you talk about the quantity of matter? 4) What connotations and definitions of mass do you know? 5) Does the inertial mass of a body change in function of its energy, apart from the kinetic energy? 6) "In many textbooks relativistic mass is mentioned. Explain what it is". 5. Data Histograms from the collected data follow. For the analysis, we used vertical and horizontal modalities. The former consists in a separate examination of every answer: we aimed at categorizing them and finding their distribution in the student sample. The latter was a search for each student s own way of looking at mass examining the correlations among answers to recognize the five profiles, or levels of «physical representation», pointed out by Doménech et al. They are ontological (pre-theoretical 7 ): mass is either a general property of matter or completely identified with bodies/matter/particles. In this case quantitas materiae becomes the best example; functional: the quantity is identified with a property, behaviour or trend of a physical system, for instance inertia or heaviness. An implicit theoretical framework is already present here; translational: the quantity is identified with another one whose definition is assumed as well stated, for instance mass as either density/ volume, or weight; relational: the quantity is defined through precise conceptual relationships inside a formal theory made by a number of mathematical laws 8. In our analysis this representation was reduced to a clear and well-defined outline of conceptual relations; operational: the quantity is a number that you obtain through a series of explicit and achievable operations; gravitational mass comes out of a measure with an equal arm balance. Moreover, answers to questions 4 to 6 allowed us to recognize several scientific meanings of mass: inertial, gravitational, Mach s empirical meanings, mass in SR (referred as «energy», «rest energy», «rest mass» by students), relativistic mass. 9, together with quantitas materie Analysis and Results For questions 1 to 3, we divided described phenomena in classes; then physical theories that students join to phenomena were grouped, as well as things described as quantity of matter. The categories are not mutually exclusive. We found out that in free fall mass is taken into account by a high percentage (17%) of students, showing they haven t understood the physical meaning of the Equivalence Principle. (9) 6 We also inserted a reference to mass as the result of interaction with all other masses in the Universe. 7 It does not refer to a theoretical framework. The definition is then concrete, in spite of appearance: it s implied by a concrete view of physical world. 8 Mass is given by F a in the inertial sense and by P g, or through universal gravitation law as well, in the gravitational sense. 9 We took these meanings from (Domenech 1993) and extended it to Special Relativity.

109 50% 45% 40% 35% 30% 25% 20% 15% 10% 5% 0% Quantitas materiae Inertial Gravitational Mach's definition Mass in S R (TOT) Energy Rest energy/internal energy "rest mass" circular definitions Figure 2: Meanings of the idea of mass (IV question) 25% 20% 15% 10% 5% 0% Energy (ontological + relational) Rest energy/internal energy Mass depending on v Mass at relativistic v Mass in Relativity (not- Newtonian) Mixed replies (min 2 categories) Unclassifiable replies Figure 3: Relativistic mass (VI question). The first category includes both a generic relation with energy (ontological level) and a formula connecting E, m, E 0 and m 0 in every possible combination (relational level) 6. Conclusions Our first research question was How and in which contexts did our students relate themselves to the term mass and make use of it? Some interesting elements came out. Phenomena as well as physical quantities recalled in familiar semantic areas are in large part mechanical ones, except for quantity of matter, "force fields, "gravitational field". The pre-theoretical conception quantitas materiae is rooted in most of students minds; however, students that answered the first group question never made use of it, as verified in the analysis of video recordings. A few students are aware of the importance of mass in electromagnetism and no one can contextualize it in familiar phenomena. Ubiquity is a character of mass expounded by 6 students, but it s not been rationalized by the half of them. In the end, mental representations of mass seem to be strongly affected by learning areas, so it is important to design integrated teaching (Fabri 2007). We noted in particular a local view of the mass in SR in

110 a context defined by speed and a grasping of the concept of mass in SR as limited to a "chapter" of physics. Our second research question was How did students interpret the extension of the concept mass to mass as rest energy in the relativistic context (under the influence of our path)? It came out that the young talents show very good ability to formalize, the relationship (9) being an important exception in this regard. Besides, terminology plays an important role in the proper understanding of mass in relativity and in framing its conceptual relations with total energy, rest energy, "relativistic mass. Indeed, wrong answers about the latter, reporting learning difficulties concerning relation (2), have been found in 26% of the students. Moreover, probably misconceptions found about mole were generated by the identity between the word for the physical quantity and for the unit. References Mach E (1883) La meccanica nel suo sviluppo storico-critico, trad. it. di Die Mechanik in ihrer Entwicklung historisch-kritisch dargestellt; Lehrman R L (1982) Confused physics: A tutorial critique, The Physics Teacher (20), ; Doménech A, Casasús E, Doménech M T (1993) The classical concept of mass: theoretical difficulties and students definitions, Int. J. Sci. Educ., 15 (2), ; Fabri E (2007) Insegnare relatività nel XXI secolo, selected contributions from A.I.F summer school; Okun L (2001) Photons, Clocks, Gravity and the Concept of Mass. Transparencies of the 18th Henry Primakoff lecture at the University of Pennsylvania, Philadelphia, USA, April 11, 2001 and of the Special lecture given at the 7th International Workshop on Topics in Astroparticle and Underground Physics, Laboratori Nazionali del Gran Sasso, Italy, September 9, 2001; Jammer M (2000) Concept of mass in contemporary physics and philosophy, Princeton University Press, Princeton, New Jersey; Kuhn T S (1970) The Structure of Scientific Revolutions, University of Chicago Press, Chicago, USA; Tolman R C (1934) Relativity, Thermodynamics, and Cosmology, Clarendon Press, Oxford, UK; Bickerstaff R P, Patsakos G (1995) Relativistic Generalization of Mass, European Journal of Physics (16), 63-68; Feyerabend P K (1965) Problem of Empiricism in Colodny R G, Beyond the Edge of Certainty, Prentice-Hall, Englewood Cliffs, New Jersey; Richard P. Feynman, Robert B. Leighton, Matthew Sands (1969) Feynman lectures on physics, vol.i, Addison Wesley, Reading, Massachussets.

111 STORIES IN PHYISCS EDUCATION Federico Corni, Department of Education and Humanities, University of Modena and Reggio Emilia, Italy Abstract. Narratives are at the basis of every effort of making sense of human experience. Stories are a particular type of narrative particularly suitable for children because they involve them affectively and cognitively. This contribution deals with the opportunity of using stories for physics education in primary school. It is introduced a disciplinary approach suitable to be shaped into stories, relying on cognitive linguistics results, as well as some elementary features of the stories are compared to the fundamental caracteristics of scientific inquiry. Hints are proposed for possible research programs and some examples of stories, coherent with this program, are presented and discussed. 1. Introduction It is out of doubts that children love stories. A fact presented to children in form of a story gains a great attraction and becomes very interesting to their eyes. The emotions that stories stimulate make the children fully involved, not only affectively, but also cognitively (Egan, 1986). Stories, however, are not only to be mentioned when speaking about children, since they represent, in a more general meaning, the way humans express their thought through language. As the stories frame the narrated facts into a certain structure and with a certain sequence, in an analogous way also concepts and thoughts are explicated within a certain structure and sequence. When we explain a scientific theory or a model, we organize our talk in a suitable way, with assumptions, logical connections, references, and support of experimental observations: everything follows a coherent sequence, necessary and sufficient to support the thesis. A mathematical theorem is expressed as a logical path we have to follow, starting from the hypotheses, to show the evidence of the thesis. From this point of view, stories and theories or models in physics are not two distinct species; rather they are the ends of a spectrum of how humans create meanings. From this qualitative consideration, we could start to fund the use of stories in physics education as form in which to shape the disciplinary contents and the way they are presented and taught to children. Not every type of story, however, is adequate to host science contents, as well as not every approach to science is suitable to be framed into a story form. On one side, the contents to be delivered have to be suitable to be shaped in narrative form; on the other side, the stories have to meet certain conditions to be suitable for the construction of meanings. In this contribution both aspects will be treated, that of the stories and that of the contents: in the next two sections will be introduced a disciplinary framework coherent with the basic forms of language as they are recognized and studied in cognitive linguistics, then will be evidenced features of stories that make them akin to the process of scientific inquiry; in the further next section will be proposed general hints for possible research programs and experimentations of the production and use of stories for physics education; finally, will be illustrated and discussed examples of stories developed according to this program. 2. The disciplinary framework for physics education in primary school Cognitive sciences and cognitive linguistics in particular (Lakoff and Johnson 1999; Johnson 1987) evidence recursive and pervasive patterns, called image schemata, we use to express our thought with language and to understand the world. An image schema is a recurrent pattern, shape and regularity in, or of, our actions, perceptions and conceptions (Johnson 1987). We extract these structures from bodily experiences: they are developed very early in the life and are already present in a rudimental form also in small children. They are simple yet have enough internal structure so that more elaborate mental structures can be built upon them. In other words, they make reasoning possible (Johnson, 1987). Image schemata are more general than every particular mental image we form; this means that any effort to represent them, even schematically, trivializes them. We have the concept of triangle, but whenever we draw a triangle, we reduce its generality. Moreover, image schemata are more basic than any proposition, explainable with a definite statement; rather they emerge with analog fashion from our thought.

112 These recurrent patterns are relatively few in number and we use them in a variety of situations. In synthesis, image schemata are real abstractions, gestalts: structures simpler than the sum of their parts. Tab.1 reports the most important image schemata evidenced by Johnson (1987), Croft and Cruse (2004), Evans and Green (2006). Tab.1 Main image schemata form specific literature Children, in this sense, are abstract thinkers, since they make sense of reality employing image schemata, the same used by adults, just less featured or differentiated. They and we speak about natural, emotional, and social phenomena using metaphorical projection of image schemata (Lakoff and Johnson, 1980). The abstract elementary concepts developed by the children minds should not be regarded as obstacles for scientific thought; on the contrary, these concepts share the same seeds of adults formal science and are to be regarded as resources to build upon. Fuchs (2007) has identified an important gestalt, named Force Dynamic Gestalt (FDG), particularly relevant to scientific understanding. It is structured with three main aspects that we use to understand the most diverse complex phenomena, from psychological ones - fear, happiness, pain - to social ones - justice, the market -, and that can be strictly related to basic and simple concepts useful to explain the behaviour of various natural phenomena, such as those involving fluids, electric charge, heat, motion, and chemicals. The FDG has the aspects of fluid substance, polarity of verticality, and force. The image schema of fluid substance has the character of an amount or quantity, and is related to the scientific concept of extensive quantity. The verticality image schema has the character of intensity or scale, and corresponds to the scientific concept of intensive quantity or generalized potential. Finally, the image schema of force, that summarizes various more simple schemata of direct manipulation (see Tab.1), is related to causation and the scientific transversal concept of energy (Fuchs, 2007). So, we think to and speak about the increase of temperature of a pot of water while it is heated on a heater in the same way we think to and speak about the increase of the level in a glass while we pour water, or in the same way we think to the rise of electrical potential while we charge a capacitance. Again, we refer to heat as a fluid substance that flows from hot to cold things in contact, in a similar way as we speak about water or air that flow from a high-pressure vessel connected to low-pressure one. Water, heat and electric charge, in these examples, are conceptualized as fluid substance-like quantities with a qualitative vertical scale of intensity. The physics contexts that are usually regarded as distinct fields of knowledge, especially in the school curricula, with this approach are seen in a unitary fashion, evidencing their analogical structure of contents. Every context has its characteristic extensive quantity that accumulates into containers; the current of such quantity flows through the cotainer boundaries directed by a negative difference of the conjugate generalized potential; the scientific concept of capacitance relates to the way in which the potential increases when the extensive quantity accumulates into a container; and the concept of resistance relates to the way in which the potential difference

113 increases to increase the current intensity. Tab.2 shows the analogical correspondence between the main experience contexts relevant for primary school physics education. Tab.2 Correspondence between the fundamental concepts of the contexts relevant for primary school Context Substance-like Potential Capacitance Current Resistance Fluids Volume Pressure Section Current Hydraulic impedance Heat Entropy Temperature Specific heat Thermal current Thermal resistance Electricity Electric charge Electric Electric Capacitance Electric current potential resistance Motion Momentum Velocity Mass Force At primary school level, physics education has the general aim to help children to identify, differentiate, and recognize and master the relationships among the aspects of the FDG. 3. Stories and scientific inquiry The connecton between image schemata and phenomena are the metaphors we produce, through imagination, expressing the affinity among the various experience contexts. When we want to express our thought to ourselves or to someone else we are forced to put in a narrative form (verbal and sequential) what is an imaginative pattern in our mind. At the basis of every effort of making sense of human experience there is the narrative thought, which searches for meanings and establishes relations (Bruner, 1986). Narratives are spontaneous and natural tools, very appropriate to the human being. Historically they appeared with the development of oral cultures, well after writing. In the child they appear very early, without external influence or instruction. Over time the child refines his ability of employing this tool and learns the mechanisms that regulate narratives in order to produce them more and more effectively. Many research efforts are devoted to the study of narratives, especially by humanists, but also by scientists. A particular kind of narrative is the story form. We know that children love stories: let s discuss if and how such a tool may be useful for physics education purposes. The most common and basic features of stories can be summarized as follows (Mandler, 1984; Egan, 1986, 1988). 1. Stories have a general structure called story grammar or story schema. If we listen to someone speaking, not necessarily we feel the speech as meaningful and interesting for us, and this even if he is treating a particular argument and does not beat around the bush. A narrative results meaningful for us when it is dressed in a form that attracts our attention, that is affectively attractring. Stories build upon, engage, and strengthen affective meaning. Basically, a story consists in a beginning that is set up by a problem created by a polarity, a middle part where the problem is elaborated, and an ending where the problem is solved. It is ultimately a technique for organizing events, facts, ideas, characters, and so on, whether real or imagined, into meaningful units that shape our affective responses. As the events of the story proceed, the story tells us how to feel about the whole picture. 2. A story tells about a limited world, with a created and given context. It is a simplified and delimited environment where events stand out and are clearly focused so as we can grasp the necessary information and search for their meaning. The story form provides a suitable environment to exercize imagination. Things become meaningful within contexts and limited spaces. 3. Stories are narratives that have an end. A story does not necessarily end when we are told that all lived happily ever after, but when we have grasped all the presented events and situations and we know how to feel about them. A narrative is a story because its ending balances the initial tension and completes all what was elaborated in the middle. Stories grant us the satisfaction of being sure how to feel about events and characters. To answer the question about the utility of stories for phyisics education we can draw a sort of parallel between the peculiarities of the story form listed above and those of scientific inquiry.

114 1. An investigation is an action that developes in time. It starts from a problem, it provides for the elaboration of information, and ends with a solution of the problem. Not all data are appropiate for a given inquiry, but only those ones that are coherent with the need of clarification and rational understanding the problem rises. An inquiry is the story of the information that involves us cognitively. Affectivity and cognition are two ways of grasping things. 2. Within an investigation, the interest is focused and the object of the study is limited. In order to converge to a solution, the investigation field must be defined and limited, the phenomenon object of inquiry have to be insulated form the rest of the world, and, in some cases, it can be reproduced and studied in the laboratory. A circumscribed world and the laboratory work favor the research. 3. An investigation does not end arbitrarily, but when all needed information is known and it is organized in a coherent and meaningful picture. We do not feel to have finished an inquiry until we have arrived to a satisfactory solution of the initial and of all open problems. Completeness and coherence are the final gaols of any inquiry. As a story is a search for affective meaning in a context, analogously, science is a search for rational meaning of a phenomenon. As, over the primary school years, children interests evolve from affective to rational values, the story form can fairly be regarded as the suitable environment to match both kinds of meaning. Stories could be used as a vehicle to guide the children in their growth between the affective needs of early childhood and the developing interest in rational and scientific understanding of the world of higher ages. Stories can be employed at all grades of primary school as transversal educational tools that develops together with children. 4. Hints for education research Joining the disciplinary approach to physics and the methodological value of stories discussed in the previous two sections, we may argue that well crafted stories can be tools useful to clarify and make apparent the aspects of the FDG and their relationships about an object of interest. This makes our thought clear to ourselves and to the listeners of our stories. Telling good stories about natural phenomena, i.e., using a correct natural language for it, is the first step toward a useful conceptualization of the processes of nature. A story is in general narrated in natural language, but if one seeks to clarify the aspects of the FGD, he needs and makes use of a language which becomes increasingly clear, accurate, unambiguous, and, finally, formal. For the purpose of science education, stories can be either told by the teacher, or told by the children, or both. The fauncions are different, of course, and fairly important. In the rest of this paper stories will be treated in their particular value for the teacher side. Just a little parenthesis about children storytelling. We have observed that children are competent narrators even in the early childhood: stories can be used by the teacher as instrument for the assessment of the children learnings. This is a valuable function of stories. In fact, a scientific education driven by the cognitive processes that occur in the children minds rather than oriented to the mere acquisition of contents, as outlined in the second section, cannot be assessed with the conventional instruments (tests, prolems, ) suitable to quantify notion knowledge and scientific competence. The children thought can be accessed and qualitatively evaluated through the analysis of their discourses, through the observation of the language naturally employed to explain their stories of any type (concerning fantastic facts, or explaining an experimental inquiry). In this sense, grids of language analysis, either from a syntax point of view, or from a semantic point of view, can be useful (Laurenti et al., 2011; Mariani et al., 2011). Back to our focus, stories to be told by the teacher in primary school have to be suitably designed, with developing structure, content, characters and language according to the children mind development and age. Research efforts should be addressed to the development of the story form from early childhood up to the last years of primary school. With increasing age of the children: 1. the story structure should change from simple and elementary to articulated and complex, hosting pupils activities (experiments, texts, drawings, games, ) and teacher intervention (discussions, explanations, ) 2. the content should concern increasningly abstract topics (water, food,, heat,, energy)

115 3. in the first years, the affective characters of the story can be fantastic characters, then, in the last years, some of them could remain as friends of children, but the effective central characters should become the natural phenomena and the physical quantities 4. language should become more and more precise and specific to the particular cases, as well as decontextualized referring to general conceptual organizers. This research program contributes to overcome the notion of a dichotomy between narrative and paradigmatic thought. In contrast to Bruner (1986), narrative and paradigmatic forms of understanding represent a polarity that opens up a spectrum between the poles of narrative and paradigmatic thought. At the same time, this program works in the direction of giving a unitary fashion to primary school instruction, and of strengthening the marriage between sciences and humanities. In the next section, will be synthetically presented the stories produced within the project Piccoli Scienziati (Little Scientists), developed at the University of Modena and Reggio Emilia in the last three years, proposed to a large number of teachers in training courses in various Italian Regions, and experimented by the teachers themselves in their classes (see Corni et al. 2011). 5. The Pico s stories Lots of examples of use of stories are present in science education literature, most of them with historical background (Jung, 1994; Kipnis, 2002; Tselfes and Paroussi, 2009; Adúriz-Bravo and Izquierdo-Aymerich, 2009; Klassen, 2010; Kubli, 2000; Lehane and Peete, 1977; Lebofsky and Lebofsky, 1996; Ellis, 2001; Sallis et al., 2008). The Pico s stories (Corni et al., 2011) - Pico and his friends at the Luna Park, Rupert and the dream of a swimming pool, and The Rupert efforts and the mountain trip (Fig.1) - are designed to be used at the various levels of primary school according to the above research program. Fig.1 Scenes of the three stories of the Piccoli Scienziati project They are animated stories in slideshow format that tell the adventures of some friends in various situations. They are corredated by two cases full of materials toys, simple models, devices of everiday use for experimental and laboratory activities. Each character of the stories has evident features in which a child, according to his temperament and disposition, can easily identify with. Pico is the archetypal child, lively and creative, who may become a scientist. He acts as mediator between the story world of characters and the real world of children. He is always present in the story scene and promptly intervenes in the problematic situations. He poses the right question at the right time and involves children in the solution inviting them to make hypotheses, draw pictures and schemes, perform experimental activities with the help of the materials of the cases. A second key character is Blackbird that flies high and looks at the situations from different points of view. Rupert, a frog, pretty, ingenuous and a bit unlucky, is the carefree child that, together with his bear friends Thomson and Aielmo, undergo to various vicissitudes and adventures. In the folowing, the features of the three stories will be synthetically presented with reference to the variables evidenced in the previous section concerning the education research. The structure concerns the complexity of the scene and the presence of children and teacher activities, the content relates to the aspects of the FDG, the characters are the effective entities over wich the children attention is focused, and the language concerns the linguistic competencies needed. 5.1 Pico and his friends at the Luna Park (1st 2nd grades)

116 The friends buy ice creams, exchange ice cream balls and reflect on the quantities and flavors. Then they buy drinks and reflect on their quantities and on their levels in various glasses. Structure. The scene is still and and centered on the friends. Content. Identification and differentiation of the aspects of quantity and intensity of substances. Characters. The friends who want to have fun at the Luna Park. The story is centered on their wishes. Language. Elementary and with common terms. 5.2 Rupert and the dream of a swimming pool (3 rd 4 th grades) Rupert wishes to place a swimming pool in his garden. It positions the pool at various heights and tries to fill it by connecting with pipes to a nearby aqueduct. Structure. The scene is structured and focused on essential elements (aqueduct, pool, pipe). Children are invited to do experiments with a model apparatus. Content. Identification and differentiation of difference of potential, current, resistance and their relationships in the context of water. Characters. The Rupert s problem to be solved. Language. Elementary with the introduction of terms of common language like to empty the pool, water flow, water level The Rupert efforts and the mountain trip, part I (4 th 5 th grades) Rupert struggles to push a wheelbarrow full of flower pots to emebllish the pool. He tries various strategies, using the wind and an inflated balloon. Structure. The scene is complex and offers various cues. Children are invited by Pico to study and explore (following some worksheets) a hairdryer and toys (a car and a balloon) to understand their functioning and to find the way to better employ them to obtain a result. Content. Identification, differentiation of differences of intensive quantities and currents of substances in interactions (wind generated by the hairdryer with the car; air blowon by an on board inflated balloon with the car); relationships between current and changes in potentials of the cause (wind) and of the effect (motion). Characters. The wind and the car in the experimental activities. Rupert and his wishes are on the background. Language. Precise and specific to express clearly the various conditions met The Rupert efforts and the mountain trip, part II (second part) Rupert and his friends go for a mountain trip and see and experience various natural phenomena (rivers flowing downhills, a melting glacier, a dam with an artificial lake, a landslide, operating wind mills, a lightning during a storm, food, photovoltaic panels, ). The story is an adventure to discover nature. Structure. The numerous scenes are rich and offer many cues. Children are invited by Pico to discuss about natural phenomena and their effects on one another. Content. Decontextualization and generalization of the relationships between differences of potential and currents in the interactions. Characters. Natural phenomena. Rupert and his frined are on the background. Language. Specific to every phenomenon, but more and more decontextualized referring to potencials and currents. 5.4 Results of experimentations with children The results of the use of the swimming pool story in fourth grade classes have been evaluated (Corni et al., 2010) with reference to the following research issues: 1. children involvement in problem solving, 2. transition from the description to the interpretation levels, 3. identification of the relevant variables, 4. generalization of meanings Children involvement in problem solving By analyzing the children s drawings evolution, context items are less and less present and sidelined in the background, while elements that relate to the problematic situation are highlighted

117 and centered. The drawings made by each child evidence a gradual ability to identify and focus on significant story objects Transition from the description to the interpretation levels The analises of children drawings and texts show (see Fig.2) a gradual shift from phenomenological description to formulation of hypotheses and interpretations. Fig.2 Percentages of descriptive and interpretative children drawings and texts Identification of the relevant variables During the evolution of the story, children begin to identify some concepts, mainly those of water level difference, current and resistance. The learning analysis of every child shows a generally increasing trend with some fluctuations Generalization of meanings Generalization is a high goal and it cannot be reached effectively with a single activity or series of activities limited in time. However, two children in particular have developed a remarkable level. They represent and explain by words that if the pipe is full of water, its shape does not influence the water flow, even if in some points the pipe is lifted above the water level of the aqueduct. This conclusion is not directly deducible from the story and the provided activities with the hydraulic model. 6. Conclusions Support for the use of stories in physics education has been outlined. Research in this direction must be conducted in order to define how to design stories, how to employ them in the didactical practice, and what results can be obtained. The project Piccoli scienziati has started a research program since three years ago and some encouraging results are now being obtained. References Adúriz-Bravo A and Izquierdo-Aymerich M (2009) A Research-Informed Instructional Unit to Teach the Nature of Science to Pre-Service Science Teachers, Science & Education 18, Bruner J (1986) Actual Minds, Possible Worlds. Cambridge, Harvard University Press Corni F, Mariani C, Laurenti E (2011) Innovazione nella didattica delle scienze nella scuola primaria: al crocevia fra discipline scientifiche e umanistiche, Artestampa, Modena Corni F, Giliberti E, Mariani C (2010) A story as innovative medium fro science education in primary school, GIREP 2010 proceedings. Croft W, Cruse D A (2004) Cognitive Linguistics, Cambridge University Press, Cambridge, UK Egan K (1988) in De Castell S C, Luke A, Luke C (Eds.) Language, authority, and criticism: readings on the school textbook. London, Falmer

118 Egan K (1986) Teaching as Story Telling An Alternative Approach to Teaching and Curriculum in the Elementary School. The University of Chicago Press Ellis B (2001) Cottonwood: How I learned the importance of storytelling in science education, Science and Children, 38 (4), Evans V and Green M (2006) Cognitive Linguistics. An Introduction. Edinburgh University Press, Edinburgh. Fuchs H U (2007) From Image Schemas to Dynamical Models in Fluids, Electricity, Heat, and Motion. ZHAW, Winterthur, Switzerland. Literature.html, accessed december 2011 Johnson M (1987) The Body in the Mind. University of Chicago Press, Chicago. Jung W (1994) Toward preparing students for change: A critical discussion of the contribution of the history of physics in physics teaching, Science & Education, 3(2), Kipnis N (2002) A history of science approach to the nature of science: learning science by rediscovering it. The Nature of Science in Science Education, 5(II) Klassen S (2010) The Relation of Story Structure to a Model of Conceptual Change in Science Learning, Sci & Educ 19, Kubli F (2000) Can the Theory of Narratives Help Science Teachers be Better Storytellers? Science & Education 10, , Lakoff G, Johnson M (1999) Philosophy in the Flesh, Basic Books, New York, NY Lakoff G, Johnson M (1980) Metaphors we live by, University of Chicago Press Laurenti E, Mariani C, Corni F (2011) A qualitative look on children s conversations about processes involving the concept of energy, GIREP 2011 Proceedings Lebofsky L A and Lebofsky N R (1996) Celestial storytelling. Science Scope, 20(3), Lehane S, Peete M (1977) The Amazing Adventures of Erik Stonefoot. Language Arts, 54(4), Mandler J M (1984) Stories, Scripts, and Scenes. Psychology Press, New York, NY Mariani C, Laurenti E, Corni F (2011) Hands-on, minds-on activities to construct the concept of energy in primary school: Experiments, games and group discussions, ICPE 2011 Proceedings Sallis D A (2008) Humorous cartoons for teaching science concepts to elementary students: Process and products. First Annual Graduate Student Research Symposium, University of Northern Iowa, Cedar Falls, IA. Tselfes V, Paroussi A (2009) Science and Theatre Education: a cross-disciplinary approach of scientific ideas addressed to student teachers of early childhood education, Science & Education, 18(9),

119 FFP Udine November 2011 GRAVITATIONAL MODEL OF THE THREE ELEMENTS THEORY Frederic Lassiaille, PolytechSophia, University of Nice Sophia-Antipolis Abstract The gravitational model of the three elements theory is an alternative theory to dark matter. It uses a modification of Newton's law in order to explain gravitational mysteries. The results of this model are explanations for the dark matter mysteries, and the Pioneer anomaly. The disparities of the measurements of G might also be explained. Meanwhile, this gravitational model is perfectly compatible with restricted relativity and general relativity, and is part of the three element theory, a unifying theory. 1. Introduction The three elements theory is a unifying theory. The gravitational model of the three elements theory is based on this theory. From construction, this theory is completely compatible with general and special relativity. This article compares the predictions of this model with the following topics: dark matter mysteries, Pioneer anomaly, Saturn flyby by Pioneer 11, Earth flyby anomalies, perihelion advance or precession of Mercury and Saturn, disparity of the gravitational constant measurements. 2. The gravitational model This gravitational model is based on the inner mechanisms of restricted and general relativity. As a first consequence, it is perfectly consistent with relativity. Noticeably, PPN parameters (Parameterized Post-Newtonian formalism) are exactly the same as for relativity. Each of the Lorentz transformation details, space-time deformation by energy and following geodesics principles have been used in order to construct this gravitational model. Moreover, the attempt is to explain Lorentz transformation with the help of space-time deformation by energy principle. During this attempt, some inconsistencies are found. The resolution of these errors leads to stating that matter is composed of indivisible particles, called luminous points. Those points are always travelling in space at the speed of light. They generates around them a radical space-time deformation. These deformations are propagated at the speed of light in every space directions, and are combined together with the help of a non-associative and non-linear operator, called the relativistic operator. This operator is only applied once, at any point of space and at any time. Therefore, the shape of space-time at any space-time point is determined by the propagated space-time deformations coming from the luminous points in the universe, along the relativistic cone centered on this point. Therefore, the local gravitation law depends strongly on the energy distribution among the universe. By construction, this mechanism retrieves Lorentz transformation details. But moreover, it allows us to recover Newton's law for long distances and for a constant and homogeneous distribution of energy in the universe. The key point is that this Newton's law is only retrieved for long distances, and for this special energy distribution. For short and intermediate distances, Newton's law is no longer retrieved. That's what is called the first modification of Newton's law. And for a non constant distribution of energy, the Newton's law is even more modified. This one is called the second modification of Newton's law. 3. Dark matter mysteries The theoretical curves are close to the measured ones. For example, Fig.1 shows the NGC 7541 theoretical speed profile.

120 FFP Udine November 2011 Figure 1: Theoretical speed profile for the NGC 7541 galaxy. X-coordinate is the distance from the galactic center, in kpc, and y-coordinate is the tangential velocity of the stars, in km/s. There are x-coordinate values corresponding each of them with more than one y-coordinate value. This is because this curve results from a simulation. This theoretical curve must be compared with an experimental one (Kyazumov 1989, Fig.3). There is an issue of sign for those theoretical speed profiles. The measured curves are retrieved, but only when using an opposite sign for the equation of the gravitational force. A plausible explanation of this error is an occultation mechanism during the propagation of the space-time deformations coming from the galaxy luminous points. This mechanism has been partially validated by calculations. Moreover, the theoretical speed values are retrieved with 17 % and 64 % of precision, respectively for NGC 3310 and NGC 1068 galaxies. Let's note that those values are interesting only for their order of magnitude, because of the opposite sign issue. Concerning the mystery of the velocity of a galaxy inside its group, the explanation is more direct. For this mystery, the study calculates a greater value for G, the gravitational constant. Indeed, as soon as the local space-time deformations rely strongly on the surrounding matter densities, the gravitation force is completely different between those two cases: inside a galaxy, and outside any galaxy. The model is even calculating G as a direct function of the surrounding distribution of luminous points in the universe and along the relativistic cone: 4 c G = 2 (1) 8e p x p p In Eq. (1), c stands for the speed of light, ande p stands for the energy of a luminous point. x p is the distance between the considered luminous point and the point in space where G is calculated. The sum is done for each luminous point along a fixed width small space solid angle centered on the point in space where we want to calculate G. 4. The Pioneer anomaly For the Pioneer anomaly (Turyshev 2010), the first modification of Newton's law is enough to retrieve a theoretical anomaly value equal to At = 7.25 x 10 m / s, in place of the measured value Am = 8.74 x 10 m / s. But the shape of the theoretical curve is not perfect. Fig.2 shows this theoretical curve.

121 FFP Udine November 2011 Figure 2: Theoretical curve of the Pioneer anomaly, using the first correction of Newton's law''. X-coordinate -10 is the distance from the sun, in AU, y-coordinate is the added acceleration toward the sun, in 10 m/s². This theoretical curve must be compared with an experimental one (Turyshev 2010, Fig.5.1). There is also a negative predicted anomaly for the probe's trajectory between the sun and Saturn (x-coordinate lesser than 10 AU). This work is in progress, and noticeably exact comparison with ephemerides must be done. In order to retrieve the perfect curve, the second modification of Newton's law must be used also, taking into account the Kuiper belt in the distribution of matter of the solar system, rather than the constant uniform distribution of matter (for the first modification of Newton's law ). The Kuiper belt is a belt of asteroids located beyond the location of Saturn, along the ecliptic plane. Now the result, on Fig.3, is very encouraging. Figure 3: Theoretical curve of the Pioneer anomaly, using the second correction of Newton's law''. X- coordinate is the distance from the sun, in AU, y-coordinate is the added acceleration toward the sun, in m/s² On this theoretical curve, the maximum value is exactlyam = 8.74 x 10 m / s, the measured value. But this theoretical curve has been obtained with fitted values for the Kuiper belt space-time deformation contributions. On the contrary, the curve of Fig.2 was calculated without any fitting. When applying this gravitational model to the case of Pioneer 11 trajectory, an added anomaly is found. It is an acceleration anomaly during the flyby of Saturn. The model is calculating an anomalous decrease before the Saturn encounter, and an anomalous increase after the encounter.

122 FFP Udine November 2011 Figure 4: Theoretical curve of the Pioneer 11 anomaly, in the vicinity of Saturn. The calculation is using the first and second modifications of Newton s law. X-coordinate is the distance from the sun, in AU, and y-coordinate is the anomalous acceleration toward the sun, in m/s². The represented anomaly is the sum of the Pioneer anomaly and the Saturn flyby anomaly. 5. Saturn perihelion and Earth flyby anomalies For the Saturn perihelion anomaly (Iorio 2009), as well as for earth flyby anomalies (Anderson et al 2008), the first modification of Newton's law is not enough for giving explanations. The second modification of Newton's law must be conducted, which represents difficult calculations and needs an exact knowledge of the probes trajectories, and the ephemerides. 6. Measurements of G The issue is well defined. For nearly three centuries, G has been measured, without getting roughly a better precision than 0.7 %. The reason of this poor precision is the fact that many measurements values are contradicting each others, taking into account their confident interval (Gillies 1997). The first modification of Newton s law is not able to explain this issue. Indeed, the order of magnitude of the theoretical error is far below the experimental one. But the second modification of Newton s law retrieves the same order of magnitude as the measured one. Let s compare first with sideral gravity experimental data (Gershteyn et al 2002). A prediction of the model of the three elements theory is that the G value is depending of the surrounding matter distribution. This is not a new idea, it has been suggested before (Gershteyn et al 2004). Moreover, it has been proven by experimental data (Gershteyn et al 2002), that the value of G depends of the orientation. This behavior is also a consequence of this theoretical idea. The amplitude of the variation of the G value is more than 0.054% as compared to its absolute value (Gershteyn et al 2002). The prediction of the gravitational model of the three elements theory for this value is between 1% and 0.013% depending of extra-galactic contributions. Therefore, the order of magnitude of this model prediction is compatible with experimental data. Now let s try another estimation, not taking in account the presence of the stars, but the presence of mountains around the apparatus during the measurement of G. The ratio below is the relative difference between two values of G, G (Michaelis et al 1995), and G (Bagley and Luther 1997). 1m 2m G1m -G 2 m 3 = 6.5 x 10 (2) G 1m Those two official measurements of G have been chosen in order to get the greatest possible ratio above.

123 FFP Udine November 2011 Let us assume that those two experiments have been done in completely different places. And the important difference between them is the distribution of matter in the surrounding neighborhood. Experiment {1}, for example, is done at the very top of a hill on the floor of a desert, and this floor is completely plane outside of the hill on which we are located. Experiment {2}, is done in the middle of a valley, which is surrounded by mountains. The interesting thing is that the measured value of G will be completely different between those two cases even if exactly the same experiment apparatus is used and the same measurement procedure is applied. Indeed, the presence of the surrounding mountains in the second measurement has an important effect on the final measured value: G1-G x 10 (3) G 1 This theoretical value is not far from the measured one, of equation (2). This proves that the order of magnitude of the measured difference can be explained by our correction of Newton s Law. As an intermediate conclusion, the gravitational model of this study might explain very precisely the great historical disparity between the measurements of G. It has been shown that this theoretical value of G might depend on the distribution of matter in the surrounding neighborhood (buildings, hills, mountains, and sea) of the place where the measurement of G is done. Moreover, this study might give, with the help of our database of today G measurements, a precise value for the G gravitational constant. This G value is the classical gravitational constant G, but valid only for an empty neighborhood, inside the galaxy. Thereafter, with this value of G, it will be possible to predict the exact value of G at any distances and in any cases inside the galaxy. Noticeably, whatever the distributions of the surrounding matter in the neighborhood, it would be possible to calculate the value of G. This value of G will be valid only for the local application of Newton s law. In this case, there is no doubt that the accuracy of the measurement of G, and the following calculations of G, will be much better than today, and with no longer disparities. 7. Conclusion The gravitational model of the three elements theory is compatible by construction with restricted and general relativity. The PPN parameters are exactly the same as for relativity. It seems to be compatible also with gravitation experimental measurements. This model might explain, after some calculations, the following anomalies: Earth flyby anomalies, perihelion precession of Saturn, disparity of the gravitational constant measurements. But this model is actually giving an explanation for the following mysteries: dark matter mysteries, Pioneer anomaly. As a conclusion, the gravitational model of the three elements theory seems to be validated. As such, this is a validation of the three elements theory itself. The next step will be to realize the two specific measurements of G in order to check the prediction of this model. 8. Bibliography Kyazumov G.A. (1989), Soviet Astronomy Letters, vol. 6, p. 220 Turyshev S. G. (2010), Living Reviews in Relativity, vol. 13, 4 Iorio L. (2009), The Astronomical Journal, Anderson J. D. et al (2008), Phys. Rev. Lett., Gillies G. T. (1997), Rep. Prog. Phys., Gershteyn M. L. et al (2002), arxiv Gravitation & Cosmology, vol. 8, issue 3 Gershteyn M. L. et al, Australian National University (Publisher Redaktsiya Izvestiya Vuzov Radiophysica, 2004) Michaelis W. et al (1995), Metrologia, Bagley C. H. and Luther G. G. (1997), Phys. Rev. Lett.,

125 FFP Udine November 2011 unlike the North South combined magnetic poles we encounter. Such monopoles have failed to show up even after seventy five years. Some of these problems were sought to be explained by what has been called inflationary cosmology whereby, early on, just after the Big Bang the explosion was super fast (Zee 1982, Linde 1982). What would happen in this case is, that different parts of the Universe, which could not be accessible by light, would now get connected. At the same time, the super fast expansion in the initial stages would smoothen out any distortion or curvature effects in space, leading to a flat Universe and in the process also eliminate the monopoles. Nevertheless, inflation theory has its problems. It does not seem to explain the cosmological constant observed since. Further, this theory seems to imply that the fluctuations it produces should continue to indefinite distances. Observation seems to imply the contrary. One other feature that has been studied in detail over the past few decades is that of structure formation in the Universe. To put it simply, why is the Universe not a uniform spread of matter and radiation? On the contrary it is very lumpy with planets, stars, galaxies and so on, with a lot of space separating these objects. This has been explained in terms of fluctuations in density, that is, accidentally more matter being present in a given region. Gravitation would then draw in even more matter and so on. These fluctuations would also cause the cosmic background radiation to be non uniform or anisotropic. Such anisotropies are in fact being observed. But this is not the end of the story. The galaxies seem to be arranged along two dimensional structures and filaments with huge separating voids. From 1997, the conventional wisdom of cosmology that had concretized from the mid sixties onwards, began to be challenged. It had been believed that the density of the Universe is near its critical value, separating eternal expansion and ultimate contraction, while the nuances of the dark matter theories were being fine tuned. But that year, the author proposed a contra view, which we will examine. 2. Cosmology To proceed, as there are 80 N ~ 10 such particles in the Universe, we get, consistently, Nm = M (1) where M is the mass of the Universe. It must be remembered that the energy of gravitational interaction between the particles is very much insignificant compared to electromagnetic considerations. In the following we will use N as the sole cosmological parameter. We next invoke the well known relation (Sidharth 2003, Nottale 1993, Hayakawa 1965) GM R (2) 2 c where M can be obtained from (1). We can arrive at (2) in different ways. For example, in a uniformly expanding Friedman Universe, we have 2 2 R& = 8 πgρr / 3 In the above if we substitute R& = c at R, the radius of the universe, we get (2). We now use the fact that given N particles, the (Gaussian)fluctuation in the particle number is of the order N (Hayakawa 1965, Huang 1975, Sidharth 1998a, Sidharth 1998b, Sidharth 1998c, 2 Sidharth 1999), while a typical time interval for the fluctuations is ~ h / mc, the Compton time, the fuzzy interval within which there is no meaningful physics as argued by Dirac and in greater detail by Wigner and Salecker. So particles are created and destroyed - but the ultimate result is that N particles are created just as this is the nett displacement in a random walk of unit step. So we have,

126 FFP Udine November 2011 dn N = (3) dt τ whence on integration we get, (remembering that we are almost in the continuum region that is, 23 τ ~ 10 sec 0), T = h N (4) 2 mc We can easily verify that the equation (4) is indeed satisfied where T is the age of the Universe. Next by differentiating (2) with respect to t we get dr HR dt (5) where H in (5) can be identified with the Hubble Constant, and using (2) is given by, 3 Gm c H = (6) h Equation (1), (2) and (4) show that in this formulation, the correct mass, radius, Hubble constant and age of the Universe can be deduced given N, the number of particles, as the sole cosmological or large scale parameter. We observe that at this stage we are not invoking any particular dynamics the expansion is due to the random creation of particles from the quantum vacuum background. Equation (6) can be written as 1 2 Hh 3 m (7) Gc Equation (7) has been empirically known as an "accidental" or "mysterious" relation. As observed by Weinberg (Weinberg 1972), this is unexplained: it relates a single cosmological parameter H to constants from microphysics. In our formulation, equation (7) is no longer a mysterious coincidence but rather a consequence of the theory. As (6) and (5) are not exact equations but rather, order of magnitude relations, it follows, on differentiating (5) that a small cosmological constant is allowed such that 2 0( H ) This is consistent with observation and shows that is very small this has been a puzzle, the so called cosmological constant problem because in conventional theory, it turns out to be huge (Weinberg 1979). But it poses no problem in this formulation. This is because of the characterization of the ZPF or quantum vacuum as independent and primary in our formulation this being the mysterious dark energy. Otherwise we would encounter the cosmological constant 120 problem of Weinberg: a that is some 10 orders of magnitude of observable values! To proceed we observe that because of the fluctuation of ~ N (due to the ZPF), there is an excess electrical potential energy of the electron, which in fact we identify as its inertial energy. That is (Sidharth 1998a, Hayakawa 1965), 2 2 N e / R mc. On using (2) in the above, we recover the well known Gravitation-Electromagnetism ratio viz., e / Gm ~ N 10 (8) or without using (2), we get, instead, the well known so called Weyl-Eddington formula, R = Nl (9) (It appears that (9)) was first noticed by H. Weyl (Singh 1961). Infact (9) is the spatial counterpart of (4). If we combine (9) and (2), we get, Gm 1 1 = T (10) 2 lc N 2

127 FFP Udine November 2011 where in (10), we have used (4). Following Dirac (cf.also (Melnikov 1994)) we treat G as the variable, rather than the quantities m, l, c and h which we will call micro physical constants because of their central role in atomic (and sub atomic) physics. Next if we use G from (10) in (6), we can see that c 1 H = (11) l N Thus apart from the fact that H has the same inverse time dependence on T asg, (11) shows that given the microphysical constants, and N, we can deduce the Hubble Constant also, as from (11) or (6). Using (1) and (2), we can now deduce that ρ m 1 3 l N (12) Next (9) and (4) give, R = ct (13) Equations (12) and (13) are consistent with observation. Finally, we observe that using M, G and H from the above, we get M = GH This relation is required in the Friedman model of the expanding Universe (and the Steady State model too). In fact if we use in this relation, the expression, H = c / R which follows from (11) and (9), then we recover (2). We will be repeatedly using these relations in the sequel. As we saw the above model predicts a dark energy driven ever expanding and accelerating Universe with a small cosmological constant while the density keeps decreasing. Moreover mysterious large number relations like (6), (12) or (9) which were considered to be miraculous accidents now follow from the underlying theory. This seemed to go against the accepted idea that the density of the Universe equalled the critical density required for closure and that aided by dark matter, the Universe was decelerating. However, as noted, from 1998 onwards, following the work of Perlmutter, Schmidt and Riess, these otherwise apparently heretic conclusions have been vindicated by observation. It may be mentioned that the observational evidence for an accelerating Universe was the American Association for Advancement of Science's Breakthrough of the Year, 1998 while the evidence for nearly seventy five percent of the Universe being Dark Energy, based on the Wilkinson Microwave Anisotropy Probe (WMAP) and the Sloan Sky Digital Survey was the Breakthrough of the Year, 2003 (Science 1998, Science 2003). The trio got the 2011 Nobel for Physics. 3. Discussion 1. We observe that in the above scheme if the Compton timeτ τ P, we recover the Prigogine Cosmology (Nicolis and Prigogine 1989, Tryon 1973). In this case there is a phase transition in the background ZPF or Quantum Vacuum or Dark Energy and Planck scale particles are produced. On the other hand if τ 0 (that is we return to point spacetime), we recover the Standard Big Bang picture. But it must be emphasized that in neither of these two special cases can we recover the various so called Large Number coincidences for example Equations like (4) or (6) or (8) or (9). 2. The above ideas lead to an important characterization of gravitation. This also explains why it has not been possible to unify gravitation with other interactions, despite nearly a century of effort. Gravitation is the only interaction that could not be satisfactorily unified with the other fundamental interactions. The starting point has been a diffusion equation 3 c

128 FFP Udine November x =< x >= ν t ν = h / m, ν lv (14) This way we could explain a process similar to the formation of Benard cells (Sidharth 2008, Nicolis and Prigogine 1989) -- there would be sudden formation of the ``cells" from the background dark energy, each at the Planck Scale, which is the smallest physical scale. These in turn would be the underpinning for spacetime. We could consider an array of N such Planckian cells (Sidharth 2010). This would be described by 2 r = N x (15) = (16) 2 B ka k x k T where kb is the Boltzmann constant, T the temperature, r the extent and k is the spring constant given by 2 k ω 0 = (17) m 1 k = a = a ω 0 m ω r r We now identify the particles or cells with Planck masses and set (18) x a = lp, the Planck length. It 2 may be immediately observed that use of (17) and (16) gives k T ~ m c, which ofcourse agrees with the temperature of a black hole of Planck mass. Indeed, Rosen (Rosen 1993) had shown that a Planck mass particle at the Planck scale can be considered to be a Universe in itself with a Schwarzchild radius equalling the Planck length. We also use the fact alluded to that a typical 40 elementary particle like the pion can be considered to be the result of n ~ 10 Planck masses. Using this in (15), we get r ~ l, the pion Compton wavelength as required. Whence the pion mass is given by m = m / n which of course is correct, with the choice of n. This can be described by l = nl, τ = nτ, (19) l 2 P P P = h P m τ The last equation is the analogue of the diffusion process seen, which is in fact the underpinning for particles, except that this time we have the same Brownian process operating from the Planck scale to the Compton scale (Cf. also (Sidharth 2002, Sidharth 2001)). We now use the well known result alluded to that the individual minimal oscillators are black holes or mini Universes as shown by Rosen (Rosen 1993). So using the Beckenstein temperature formula for these primordial black holes (Ruffini 1983), that is 3 c kt = h 8πGm we can show that Gm 2 ~ h c (20) 5 We can easily verify that (20) leads to the value m ~ 10 gms. In deducing (20) we have used the typical expressions for the frequency as the inverse of the time the Compton time in this case and similarly the expression for the Compton length. However it must be reiterated that no specific values for l or m were considered in the deduction of (20). P P B P

129 FFP Udine November 2011 We now make two interesting comments. Cercignani and co-workers have shown (Cercignani 1998, Cercignani 1972) that when the gravitational energy becomes of the order of the electromagnetic energy in the case of the Zero Point oscillators, that is 2 3 Gh ω ~ h ω (21) 5 c then this defines a threshold frequency ω above which the oscillations become chaotic. In other words, for meaningful physics we require that ω ω max. max Secondly as we can see from the parallel but unrelated theory of phonons (Huang 1975, Reif 1965), which are also bosonic oscillators, we deduce a maximal frequency given by 2 2 c ω max = (22) 2 l In (22) c is, in the particular case of phonons, the velocity of propagation, that is the velocity of sound, whereas in our case this velocity is that of light. Frequencies greater than ω in (22) are again meaningless. We can easily verify that using (21) in (22) gives back (20). In other words, gravitation shows up as the residual energy from the formation of the particles in the universe via Planck scales (Benard like) cells. 3. It has been mentioned that despite nearly 75 years of search, Dark Matter has not been found. More recently there is evidence against the existence of Dark Matter or its previous models. The latest LHC results for example seem to rule out SUSY. On the other hand our formulation obviates the need for Dark Matter. This follows from an equation like (10) which shows a gravitational constant decreasing with time. Starting from here it is possible to deduce not just the anomalous rotation curves of galaxies which was the starting point for Dark Matter; but also we could deduce all the known standard results of General Relativity like the precession of the perihelion of mercury, the bending of light, the progressive shortening of the time period of binary pulsars and so on (Cf.ref.(Sidharth 2008)). References Cercignani C., Galgani L. and Scotti A. (1972) Phys.Lett. 38A, pp.403. Cercignani C (1998) Found.Phys.Lett. Vol.11, No.2, pp Hayakawa S (1965) Suppl of PTP Commemmorative Issue pp Huang K. (1975) Statistical Mechanics, Wiley Eastern, New Delhi, pp.75ff. Linde A.D. (1982) Phys.Lett. 108B, 389. Melnikov V N. (1994) Int.J.of Th.Phys. \underline{33}, (7), pp Nicolis G. and Prigogine I. (1989) Exploring Complexity, W.H. Freeman, New York, p.10. Nottale L. (1993) Fractal Space-Time and Microphysics: Towards a Theory of Scale Relativity, World Scientific, Singapore, pp.312. Reif F (1965). Fundamentals of Statistical and Thermal Physics, McGraw-Hill Book Co., Singapore. Rosen N (1993) Int.J.Th.Phys., 32, (8), pp Ruffini R and Zang L.Z (1983) Basic Concepts in Relativistic Astrophysics, World Scientific, Singapore, p.111ff. Science, December Science, December 2003 Sidharth B G (1998a) Int.J. of Mod.Phys.A, 13, (15), pp.2599ff. Sidharth B G (1998b) International Journal of Theoretical Physics, Vol.37, No.4, pp Sidharth B G (1998c) Frontiers of Quantum Physics (1997) Lim, S.C., et al. (eds.), Springer Verlag, Singapore. Sidharth B G (1999) Proc. of the Eighth Marcell Grossmann Meeting on General Relativity (1997) Piran, T. (ed.) World Scientific, Singapore, pp Sidharth B G (2001) Chaotic Universe: From the Planck to the Hubble Scale, Nova Science, New York. Sidharth B G (2002) Found.Phys.Lett. 15, (6), pp Sidharth B G (2003) Chaos, Solitons and Fractals 16, (4), pp Sidharth B G (2007). Encounters:Abdus Salam in New Advances in Physics Vol.1, No.1, March 2007, pp max

130 FFP Udine November 2011 Sidharth B G (2008) The Thermodynamic Universe World Scientific, Singapore. Sidharth B G (2010) Int.J.Th.Ph Vol.49(10), pp Singh J (1961) Great Ideas and Theories of Modern Cosmology Dover, New York, pp.168ff. Tryon E P (1973) Is the Universe a Vacuum Fluctuation? in Nature Vol.246, December , pp Weinberg S (1972) Gravitation and Cosmology John Wiley & Sons, New York, p.61ff. Weinberg S (1979) Phys.Rev.Lett. 43, pp Zee A (1982) Unity of Forces in the Universe (Vol.II) World Scientific, Singapore, p.40ff.

131 FFP Udine November 2011 BEHAVIOUR AT ULTRA HIGH ENERGIES Sidharth Burra Gautam, International Institute for Applicable Mathematics and Information Sciences,Hyderabad (India) & Udine (Italy),B.M. Birla Science Centre, Adarsh Nagar, Hyderabad , (India) Abstract The Large Hadron Collider has already attained an unprecedented enery of 7 TeV. By 2013 it is expected to reach its peak energy of double this figure. We can hope that many surprises and discoveries are waiting to happen in the years to come. In this context we explore the behaviour of particles, particularly fermions at these Ultra High Energies. In particular two aspects will be touched upon : The Feshbach-Villars formulation for high energies and also considerations at the Planck length. Some new insights are explored thereby. 1. Introduction The LHC in Geneva is already operating at a total energy of 7 TeV and hopefully after a pause in 2012, it will attain its full capacity of 14 TeV in These are the highest energies achieved todate in any accelerator. It is against this backdrop that it is worthwhile to revisit very high energy collisions of Fermions. We will in fact examine their behaviour at such energies. To get further insight, let us consider the so called Feshbach-Villars formulation (Feshbach and Villars 1958) and analyze the problem from this point of view rather than that of conventional Field theory. In this case with an elementary transformation, the equations for the components ϕ and χ of the Dirac wave function can be written as 2 ih ( φ / t) = ( 1/ 2m)( h / i \ ea/ c) ( φ + χ ) 2 ( e φ + mc )χ ih 2 ( χ / t) = ( 1/ 2m)( h / i ea/ c) ( φ + χ ) 2 ( eφ mc )φ (1) What Feshbach and Villars did was give a particle interpretation to the Klein-Gordon and Dirac equations without invoking field theory or the Dirac sea. In this case φ represents the "low energy" solutions, that is the normal solution and χ represents the "high energy" solutions. It must be remembered that at our usual energies it is the wave function φ, the so called positive energy solution that dominates, χ being of the order of v 2 / c 2 of φ. On the other hand at very "high energies" χ the so called negative energy solution dominates. Feshbach and Villars identified these two solutions with particles and antiparticles respectively. We have φ0 i / h( p. Et ) ϕ = e, χ ( ) ( i/ h( p Et ) φ = φ0 p e 0 We consider separately the positive and negative values of E (coming from (2)), viz., 2 2 = ; E p ( cp) + ( mc ) E ±E p The solutions associated with these two values of E are 1 2 [ ] 2 = (3) (2) E + mc mc E 2 2 φ ( + ) p ( + ) p 0 = χ 2 1/ 2 0 = 2 1/ 2( mc E p ) 2( mc E ) 2 p

132 FFP Udine November 2011 For E = ± E. p mc E E + mc 2 2 φ ( ) p ( ) p 0 = χ 2 1/ 2 0 = 2 1/ 2( mc E p ) 2( mc E ) 2 p As is well known the positive solution E = E ) and the negative solution E = E ) represent ( p ( p solutions of opposite charge. We also mention the well known fact that a meaningful subluminal velocity operator can be obtained only from the wave packets formed by positive energy solutions. However the positive energy solutions alone do not form a complete set, unlike in the non relativistic theory. This also means that a point description in terms of the positive energy solutions alone is not possible for the K-G (or the Dirac) equation, that is for the position operator, r δ r ( X ) In fact the eigen states of this position operator include both positive and negative solutions. All this is well known (Cf.ref.(Feshback 1958)). This matter was investigated earlier by Newton and Wigner too (Newton and Wigner (1949) from a slightly different angle. Some years ago the author revisited this aspect from yet another point of view (Sidharth 2002) and showed that this is symptomatic of noncommutativity which is exhibited by 2 [ x, x ] = O( l ) Θ X 0 ı j ıj and is related to spin and extension. The noncommutative nature of spacetime has been a matter of renewed interest in recent years particularly in Quantum Gravity approaches. At very high energies, it has been argued that (Sidharth 2008a) there is a minimum fuzzy interval, symptomatic of a non commutative spacetime, so the usual energy momentum relation gets modified and becomes (Sidharth 2008b) E = p + m + αl p (4) the so called Snyder-Sidharth Hamiltonian (Sidharth 2008c, Glinka 2008, Raolina 2011). It has been argued that for fermions α > 0 while α can be < 0 for bosons. Using (4) it is possible to deduce the ultrarelativistic Dirac equation (Sidharth 2005a, Sahoo 2010) 2 5 ( D βlp γ ) ψ 0 + = (5) β = α. In (5) D is the usual Dirac operator while the extra term appears due to the new dispersion relation (4). As indicated above α is positive. It is known that (Sidharth 2010c), in this case equation (5) can be written in Hamiltonian form 0 2 γ p ψ = ( D + ıα lp γ ) ψ (6) 0 5 where D ı ı γ p. Further it is well known that the Hamiltonian is given by ı H = ıγ uur r r r p = ıγ p s( p) 5 5 (7) It can be seen from (7) that the Dirac particle acquires an additional mass. However what is very interesting is that the extra mass term is not invariant under parity owing to the presence of γ 5. Indeed as we know from the theory of Dirac matrices Pγ = Pγ (8) 5 5

133 FFP Udine November 2011 In the case of a massless Dirac particle, it was argued that this leads to the mass of the neutrino (Sidharth 2010b). Thus the mass m gets split into m + m and m m with two states, Ψ L and Ψ R. Remembering that a dominant φ and a dominant χ respectively represent particle and antiparticle in this Feshbach- Villars formulation and also remembering that under reflection, as is well known, φ φ, χ χ (9) we can see that this means that the particle and antiparticle have different masses, namely m + m and m m '. Indeed this conclusion was anticipated earlier (Sidharth 2010d). The difference would be minute but in principle can be observed. Already there have been reports of such mass asymmetry being observed in the MINOS Fermi Lab experiment with neutrinos and anti neutrinos 2 (Report 2010). What the MINOS team recorded was a difference in the m value for neutrinos and anti neutrinos by as much as forty percent. It is expected that more definitive results would be available by It has been pointed out that the fact that equations like (7) and the following applied to neutrinos which are massless suggests one (or more) neutrinos. This is brought out more clearly in the above. Remarkably there seems to be very recent confirmation of such an extra or sterile neutrino (Roe ).. The above discussion brings out ultra high energy effects in Fermionic behavior. Already equation (4) shows modifications to Lorentz symmetry, as has been discussed in detail in several places, for example (Cf.ref.(Sidharth 2008b, Sidharth 2008c) and references therein). This exposes the limits of strict special relativistic considerations. 2. Extra Relativistic Effects It has just been announced that the OPERA (Oscillation Project with Emulsion Tracking Apparatus) experiment, 1400 meters underground in the Gran Sasso National Laboratory in Italy has detected neutrinos travelling faster than the speed of light, which has been a well acknowledged speed barrier in physics. This limit is ,458 meters per second, whereas the experiment has detected a speed of 299,798,454 meters per second. In this experiment neutrinos from the CERN Laboratory 730 kilometers away in Geneva were observed. They arrived 60 nano seconds faster than expected, that is faster than the time allowed by the speed of light. The experiment has been measured to 6σ level of confidence, which makes it a certainty (Adam ) and has been repeated again. However it is such an astounding discovery that the OPERA scientists would like further confirmation from other parts of the world. In 2007 the MINOS experiment near Chicago did find hints of this superluminal effect. It must be reported that the author had predicted such deviations from Einstein's Theory of Relativity, starting from 2000 (Cf.eq.(4)). (4) shows that the energy at very high energies for fermions is greater than that given by the relativity theory so that effectively the speed of the particle is slightly greater than that of light. For example, if in the usual formula, we replace c by c + c, then, comparing with the above we would get: c = αl p [4m c + 2 p c] The difference is slight, but as can be seen is maximum for the lightest fermions, viz., neutrinos which are in any case already travelling with the velocity c. We could also argue that the extra term in the Snyder-Sidharth Hamiltonian can contribute partly to an oscillating mass of the neutrino oscillations and partly to a fluctuating super luminal velocity.

134 FFP Udine November Ultra High Energy Particles Let us look at all this differently. Following Weinberg (Weinberg 1972) let us suppose that in one 0 0 reference frame S an event at x 2 is observed to occur later than one at x 1, that is, x2 > x1 with usual notation. A second observer S moving with relative velocity v r will see the events separated by a time difference α α x x = Λ ( v)( x x ) 2 1 α 2 1 β where Λ α ( v) is the "boost" defined by or, r x x = γ ( x x ) + γ v ( x x ) and this will be negative if We now quote from Weinberg (Weinberg 1972): '0 ' v ( x x ) < ( x x ) (10) "Although the relativity of temporal order raises no problems for classical physics, it plays a profound role in quantum theories. The uncertainty principle tells us that when we specify that a particle is at position x1 at time t 1, we cannot also define its velocity precisely. In consequence there is a certain chance of a particle getting from x1 to x2 even if x1 x2 is spacelike, that is, 0 0 x1 x2 > x1 x2. To be more precise, the probability of a particle reaching x 2 if it starts at x1 is nonnegligible as long as ( x x ) ( x x ) h (11) m where h is Planck's constant (divided by 2π ) and m is the particle mass. (Such space-time intervals are very small even for elementary particle masses; for instance, if m is the mass of a 14 proton then / m w h = cm or in time units 6 10 sec. Recall that in our units 10 1sec = 3 10 cm). We are thus faced again with our paradox; if one observer sees a particle emitted at x 1, and absorbed at x 2, and if h / m ), then a second observer may see the particle absorbed at 2 ( x x ) ( x x ) is positive (but less than x at a time t 2 before the time t 1 it is emitted at x 1 ". To put it another way, the temporal order of causally connected events cannot be inverted in classical physics, but in Quantum Mechanics, the Heisenberg Uncertainty Principle leaves a loop hole. As can be seen from the above, the two observers S and S see two different events, viz., one sees, in this example the protons while the other sees neutrons. Moreover, this is a result stemming from (11), viz., < ( x x ) ( x x ) ( h ) (12) m The inequality (12) points to a reversal of time instants ( t1, t 2) as noted above. However, as can be seen from (12), this happens within the Compton wavelength. We now observe that in the above formulation for the wave function φ ψ =, χ where, as noted, φ and χ are, for the Dirac equation, each two spinors.φ (or more correctly 0 φ ) represents a particle while χ represents an antiparticle. So, for one observer we have

135 FFP Udine November 2011 and for another observer we can have χ 0 (13) φ 0 (14) that is the two observers would see respectively a particle and an antiparticle. This would be the same for a single observer, if for example the particle's velocity got a boost so that (14) rather than (13) would dominate after sometime. Interestingly, just after the Big Bang, due to the high energy, we would expect, first (14) that is antiparticles to dominate, then as the universe rapidly cools, particles and antiparticles would be in the same or similar number as in the Standard Model, and finally on further cooling (13) that is particles or matter would dominate. Finally we now make two brief observations, relevant to the above considerations. Latest results in proton-antiproton collisions at Fermi Lab have thrown up the Bs mesons which in turn have decayed exhibiting CP violations in excess of the predictions of the Standard Model, and moreover this seems to hint at a new rapidly decaying particle. Furthermore, in these high energy collisions particle to antiparticle and vice versa transformations have been detected. 4. Ultra High Energy Particles Some years ago (Sidharth 2006), we explored some intriguing aspects of gravitation at the micro and macro scales. We now propose to tie up a few remaining loose ends. At the same time, this will give us some insight into the nature of gravitation itself and why it has defied unification with other interactions for nearly a century. For this, our starting point is an array of n Planck scale particles. As discussed in detail elsewhere, such an array would in general be described by (Jack Ng and Van Dam (1994) l 2 = n x (15) ka k x = k T (16) B where kb is the Boltzmann constant, T the temperature, r the extent and k is the analogues of the spring constant given by 2 k ω 0 = (17) m ω 1 k = = a a 0 m ω r r (18) We now identify the particles with Planck masses and set x a = lp, the Planck length. It may be 2 immediately observed that use of (17) and (16) gives k T ~ m c, which ofcourse agrees with the temperature of a black hole of Planck mass. Indeed, Rosen (Rosen 1993) had shown that a Planck mass particle at the Planck scale can be considered to be a Universe in itself with a Schwarzchild radius equalling the Planck length. Whence the mass of the array is given by m = m / n (19) while we have, P B P

136 FFP Udine November 2011 In the above 5 33 P l = nl, τ = nτ, (20) l 2 P P = h m τ P P 42 m ~ 10 gms, l ~ 10 cm andτ P ~ 10 sec, the original Planck scale as defined by P Max Planck himself. We would like the above array to represent a typical elementary particle. Then we can characterize the number n precisely. For this we use in (19) and (20) 2`GmP lp = (21) 2 c which expresses the well known fact that the Planck length is the Schwarzchild radius of a Planck mass black hole, following Rosen. This gives 2 lc 40 n = ~ 10 (22) Gm where l and m in the above relations are the Compton wavelength and mass of a typical elementary particle and are respectively ~ 10 cms and 10 gms respectively. Before coming to an interpretation of these results we use the well known result alluded to that the individual minimal oscillators are black holes or mini Universes as shown by Rosen (Rosen 1993). So using the Beckenstein temperature formula for these primordial black holes (Ruffini and Zang 1983), that is 3 c kt = h 8πGm we can show that Gm 2 ~ h c (23) 5 We can easily verify that (23) leads to the value m = m ~ 10 gms. In deducing (23}) we have used the typical expressions for the frequency as the inverse of the time the Compton time in this case and similarly the expression for the Compton length. However it must be reiterated that no specific values for l or m were considered in the deduction of (23). We now make two interesting comments. Cercignani and co-workers have shown (Cercignani 1998, Cercignani 1972) that when the gravitational energy becomes of the order of the electromagnetic energy in the case of the Zero Point oscillators, that is 2 3 Gh ω ~ h ω (24) 5 c then this defines a threshold frequency ω above which the oscillations become chaotic. In other words, for meaningful physics we require that ω ω max. max where ω max is given by (24). Secondly as we can see from the parallel but unrelated theory of phonons (Huang 1975, Reif 1965), which are also bosonic oscillators, we deduce a maximal frequency given by 2 2 c ω max = (25) 2 l In (25) c is, in the particular case of phonons, the velocity of propagation, that is the velocity of sound, whereas in our case this velocity is that of light. Frequencies greater than ω in (25) are again meaningless. We can easily verify that using (24) in (25) gives back (23). As Large Number sense, (23) can also be written as, 2 2 Gm ~ e P P P max e, h c = in a

137 FFP Udine November 2011 That is, (23) expresses the known fact that at the Planck scale, electromagnetism equals gravitation in terms of strength. In other words, gravitation shows up as the residual energy from the formation of the particles in the universe via Planck scales particles. The scenario which emerges is the following. Analogous to Prigogine cosmology (Prigogine 1996, Tryon 1973), from the dark energy background, in a phase transition Planck scale particles are suddenly created. These then condense into the longer lived elementary particles by the above process of forming arrays. But the energy at the Planck scales manifests itself as gravitation, thereafter. We will further discuss this in the next section. 5. Discussion Equation (22) can also be written as Gm ~ 2 lc 80 where N ~ 10 is the Dirac Large Number, viz., the number of particles in the universe. There are two remarkable features of (22) or (26) to be noted. The first is that it was deduced as a consequence in the author's 1997 cosmological model (Sidharth 2005b). In this case, particles are created fluctuationally from the background dark energy. The model predicted a dark energy driven accelerating universe with a small cosmological constant. It may be recalled that at that time the prevailing paradigm was exactly opposite -- that of a dark matter constrained decelerating universe. As is now well known, shortly thereafter this new dark energy driven accelerating universe with a small cosmological constant was confirmed conclusively through the observations of distant supernovae. It may be mentioned that the model also deduced other inexplicable relations like the Weinberg formula that relates the microphysical constants with a large scale parameter like the Hubble Constant: 1 2 Hh 3 m (27) Gc While (27) has been loosely explained away as an accidental coincidence Weinberg (Weinberg 1972) himself emphasized that the mysterious relation is in fact unexplained. To quote him, "In contrast (this) relates a single cosmological parameter (the Hubble Constant) to the fundamental constants h, G, c and m and is so far unexplained." The other feature is that (26) like (27) expresses a single large scale parameter viz., the number of particles in the universe or the Hubble constant in terms of purely microphysical parameters. As we saw the scenario is similar to the Prigogine cosmology in which out of what Prigogine called the Quantum Vacuum, or what today we may call Dark Energy background, Planck scale or Planck mass are created in a phase transition, very similar to the formation of Benard cells (Nicolis and Prigogine 1989). The energy at the Planck scale, given by (24) then gets distributed in the universe -- amongst all the particles, as the Planck particles form these various elementary particles according to equations (15) to (20). This is brought out by the fact that equation (26) can also be written as the well known Eddington formula: Gm / e ~ (28) N which was believed to be another ad hoc coincidence unrelated to (27). Equation (28) shows how the gravitational force over the cosmos is weak compared to the electromagnetic force. In other words the initial "gravitational energy" on the formation of the Planck scale particles, that is (23) is distributed amongst the various particles of the universe (Sidharth 2005c). From this point of view while l, m, c etc. are indeed microphysical constants as Dirac characterized them, G is not. It is related to the Large Scale cosmos through the Dirac Number N of particles in the universe. This N (26)

138 FFP Udine November 2011 would also explain the Weinberg puzzle: In this case in equation (27), there are the large scale parameters namely G and H on right side of the equation. Once we recognize this, we can easily see that unlike what was thought previously, the Weinberg formula (27) is in fact the same as the Dirac formula (28). To see this, we use in (27) two well known relations from cosmology (Cf.eg.(Weinberg 1972)), viz., GM R ~ andm = Nm 2 c where R is the radius of the universe ~ 10 cm, M its mass ~ 10 gm and m is as before the mass of a typical elementary particle. Then (27) will reduce to (28). Thus, there is only one relation -- (27) or (28), and they express the fact that rather than being a microphysical parameter, G rather than representing a fundamental interaction is related to the large scale cosmos via either of these equations. It must be observed that this conclusion resembles that of Sakharov (Sakharov 1968), for whom Gravitation was a secondary force like elasticity. References Adam T. et al., arxiv 1109,4897. Cercignani C., Galgani L. and Scotti A. (1972) Phys.Lett. 38A, pp.403. Cercignani C (1998) Found.Phys.Lett. Vol.11, No.2, pp Freshbach H and Villars F (1958) Rev.Mod.Phys. Vol.30, No.1, January 1958, pp Glinka L.A. (2008) Apeiron 2, April 2008; arxiv hep-ph Huang K. (1975) Statistical Mechanics, Wiley Eastern, New Delhi, pp.75ff. Jack Ng Y. and Van Dam H. (1994) Mod.Phys.Lett.A. 9, (4), pp Newton T D and Wigner E.P (1949) Reviews of Modern Physics Vol.21, No.3, July 1949, pp Nicolis G. and Prigogine I. (1989) Exploring Complexity, W.H. Freeman, New York, p.10. Prigogine I (1996) The End of Certainty The Free Press, New York, pg.172ff. Raoelina A. and Rakotonirina C (2011) EJTP Vol.8 (25), May Reif F (1965) Fundamentals of Statistical and Thermal Physics McGraw-Hill Book Co., Singapore. Report (2010) : Physics World, IOP June 18, Roe B to appear in Phys.Rev.Lett. Rosen N (1993) Int.J.Th.Phys., 32, (8), pp Ruffini R and Zang L.Z. (1983) Basic Concepts in Relativistic Astrophysics World Scientific, Singapore, p.111ff. Sahoo S (2010) Ind.J.Pure and Appld.Phys. 48, October 2010, pp Sakharov A D (1968) Soviet Physics Doklady Vol.12, No.11, pp Schweber S S (1961) An Introduction to Relativistic Quantum Field Theory Harper and Row, New York. Sidharth B G (2002) Foundation of Physics Letters 15 (5), 2002, pp.501ff. Sidharth B G (2005a) Int.J.Mod.Phys.E 14 (6), 2005, 927ff. Sidharth B G (2005b) The Universe of Fluctuations Springer, Netherlands. Sidharth B G (2005c) Found.Phys.Lett. 18(4),2005. Sidharth B G (2006) Int.J.Mod.Phys.A 21(31), December 2006, pp Sidharth B G (2008a) Thermodynamic Universe World Scientific, Singapore and other references therein. Sidharth B G (2008b) Foundation of Physics 38 (8), 2008, pp Sidharth B G (2008c) Foundation of Physics 38 (1), 2008, pp89ff. Sidharth B G (2010b) Int.J.Mod.Phys.E 19 (1), 2010, pp Sidharth B G (2010c) Int.J.Mod.Phys.E 19 (11), 2010, pp.1-8. Sidharth B G (2010d) EJTP, Vol. 7, (24), July Tryon E P (1973) Is the Universe a Vacuum Fluctuation? in Nature Vol.246, December , pp Weinberg S (1972) Gravitation and Cosmology John Wiley & Sons, New York, p.61ff.

139 LEARNING KNOTS ON ELECTRICAL CONDUCTION IN METALS Giuseppe Fera, Physics Education Research Unit of University of Udine Abstract A number of studies related to the conceptual understanding of DC circuits highlighted several difficulties in learning about: the closing of the circuit (Osborne, 1983), differentiation of the concepts of electrical energy, voltage and current (Psillos, 1998), the determination of the equivalent circuit as a descriptor of multiple loop circuit and use of systemic reasoning instead of local reasoning (McDermott & Shaffer, 1992). Others researchers identified the following issues, which are relevant to learning: the phenomenological relationship between electrostatic and electrodynamics (Eylon & Ganiel, 1990; Stocklmayer & Treagust, 1996; Viennot, 2001; Chabay & Sherwood, 2007) and the link between the macroscopic and microscopic level of processes description (Licht, 1991; De Posada, 1997; Thacker, Ganiel & Boys, 1999; Wittmann, Steinberg & Redish, 2002). The dilemma of displaying the microscopic world is still under discussion (Mashhadi & Woolnough, 1999; Seifert & Fischler, 2003): students are confronted every day with colorful pictures of atoms and molecules which are spread both by the media and the textbooks, not in accordance with the actual proportions between the physical properties of the particles (size and speed) and can generate misunderstandings about the microscopic processes. The comprehensive discussion of these issues leads to design learning paths on those specific concepts which favor a parallel approach on macroscopic and microscopic levels of the phenomena. Introduction This study is founded on literature survey and presents some research results on teaching/learning of electric conduction in solids. Electricity is one of the most important topics in physic, both looking at the wide contexts of everyday life where it is applied, both for the relevant role of the involved concepts (charge, current, field, potential, ). These aspects make electricity a subject for teaching in every level of instruction. It is also one of the most investigated fields as regards the learning difficulties of students of any age. From the perspective of physics education it s important to recognize learning knots of the specific context. As concern the electric conduction in metals, there are several knots related to different aspects: 1) Electrical circuits, 2) Phenomenological relationship between electrostatic and electrodynamics, 3) Link between the macroscopic and microscopic level of processes description, 4) Use of simulations and analogies. 1. Electrical circuits The researches developed starting from 1980 (Fredette & Clement, 1981; Osborne, 1983; Shipstone, 1984) have analyzed how children working with batteries, wires and light bulbs to build simple circuits. A number of studies shows that the basic concepts related to the d.c. circuits are

140 not intuitive and that electrical phenomena are interpreted by young people through persistent and widespread non scientific schemes. These alternative conceptions coexist in parallel with the scientific view and are not integrated into it, even to the end of education path. The transition from the common sense to the scientific view, according to Psillos (1998), is favored by the measurement procedures that allow to identify and recognize the physical quantities voltage and current and their role in explaining the functioning of the circuit. As in the context of electrostatics, learning difficulties are compounded by the fact that the processes involved in the electrical phenomena are far from direct sensory perception. It is not easy to connect the observable effects to abstract concepts like "electric charge", "current" and "energy". Since the students know that the batteries "are consumed" and you pay for household electricity consumption, they believe that in the circuits "something is consumed" and, for many of them, the most reasonable thing to "consume" is the electricity itself, conceived as energy at times, sometimes as electric current. Some authors find that the concept of conservation of electric charge is intuitive for children and yet coexists with the view that the current is consumed. In addition, the concept of electric potential is already quite difficult to acquire in the context of electrostatics, for most students its relevance and applicability in the case of electrical circuits remains unclear. The main learning knots identified, as regards electrical circuits, are: a) need of closing the circuit for the current circulation b) topological knot: recognizing the equivalent circuit as describer of many loop circuit (McDermott & Shaffer, 1992) c) sequential knot: the current in a series circuit element is independent of its position d) construction of formal thinking: representing the circuit with a symbolic scheme e) role of the battery: it is not a source of constant current regardless of the connected circuit f) use of systemic reasoning in place of local reasoning: the change of a circuit element can generate changes in all other circuit elements g) interpretation and differentiation of the concepts of energy, voltage and electric current (Psillos, 1998) h) incorrect use of Ohm s law: the absence of current implies absence of voltage 2. Phenomenological relationship between electrostatic and electrodynamics Many authors (Eylon & Ganiel, 1990, Thacker, Ganiel & Boys, 1999; Tveita, 1997) tend to identify in the missing link between electrostatic and electrodynamics the main reason for incomplete students' conceptual knowledge about electrical circuits. These authors argue that students whose education has focused on the microscopic processes involved in the functioning of electrical circuits develop a better understanding not only of simple macroscopic phenomena, but also more complex phenomena such as transient. Specifically Chabay & Sherwood (2007) argue that provide students with an explanatory mechanism facilitates the understanding of physical phenomena. They, to teach the electrical

141 conduction, choose the electric field as conceptual referent and the surface charges as a mechanism. In this way, they also realize the connection between the macroscopic and microscopic description of the processes. Main learning knots of context are: a) transient processes occurring in electric circuits with capacitors b) electric charge concept and physical properties c) differentiation of the voltage propagation in the circuit by the current circulation d) voltage concept as engine of the charge transfer e) link between macroscopic concept of voltage and microscopic concept of electric potential f) needs of surface charges on the wire so that the electric field inside the wire is parallel to the wire longitudinal axis (Chabay & Sherwood, 2007) 3. Link between the macroscopic and microscopic level of processes description The link between electrostatics and electrodynamics for many authors is integrated with the connection between the macroscopic and microscopic description of the processes of electrical conduction. In contrast, other studies (Benseghir & Closset, 1996, Borges & Gilbert, 1999) show a cautious attitude on the educational value of the microscopic model of learning by identifying nodes as: a) overcoming the local model focused on the behaviour of single atom (as in chemistry) b) integration of different models: current as a flow of charge carriers, motion of electrons in metals (random motion in the absence of field, drift motion in the presence of field), description of states and processes in the quantum band model c) evidences for the existence of the electron and characterization of its physical properties both intrinsic and interaction with the ionic lattice of the material. 4. Use of simulations A model of conduction in metals based on the free electron gas (Stephens, McRobbie & Lucas, 1999) can be used to explain the observed relation between resistance and section of the wire (Ohm's second law): the resistance decreases by increasing the section not because the conduction electrons have more space available to move (the average number of collisions of an electron per unit time does not change with respect to the thin wire) but because a greater number of electrons through a section of the wire into time. Similar representations are proposed by the simulations of the interior of a wire with flowing current. According Sins & al. (2009), computer simulation tools can be an opportunity to promote the development of scientific thought in the students on issues of particular complexity, such as the relationship between macroscopic and microscopic properties of material. Simulations can show a phenomenon and/or enable one to make predictions about its trend. Phet Interactive Simulations or Easy Java Simulations provide a microscopic representation of electrical transport in solids. Showing current as moving electron spheres to directly counter the

142 spontaneous model that current is consumed, but the physical proportions among electrons and lattice ions are not respected. However such simulations can help to show how macroscopic properties emerge from a microscopic model at the level of material individual constituent and their interactions. The use of simulations also involves learning knots among which we highlight the following: a) physical proportions among electrons and lattice ions are not respected b) free electrons model of metals in Supercomet simulation: the average electric field into a metal is zero, then opposite sign charges don t interact by Coulomb law 5. Analogies In teaching practice the use of hydraulic analogies is common in teaching of electrical conduction in wires. Under this analogy the wires are likened to pipes and current to a fluid flowing in them, the battery to a pump, etc. These analogies are founded, from the mathematical perspective, on the fact that the transport equations defined by the laws of Fourier, Ohm, Poseuille have the same form to describe the flow of heat, electricity and a viscous fluid, respectively. Chiu & Lin (2005) argue that using analogies not only promoted profound understanding of complex scientific concepts (such as electricity), but it also helped students overcome their misconceptions of these concepts. However, we can identify knots in the context: a) water is a liquid substance and electric current is not b) the current repartition between two parallel bulbs connected to the same battery c) the current repartition between two different bulbs in series 6. Implications for physics education Common reasoning must be taken into account in teaching: they should be considered not only as creating an obstacle to learning physics, but also as resources at the learner s disposal. The comprehensive discussion of these issues leads to design learning paths on those specific concepts which favor a parallel approach on macroscopic and microscopic levels of the phenomena. The relative learning process should be planned as an aware and systematic transition between different levels of description and interpretation of phenomena. References Osborne, R. J., Towards modifying children's ideas about electric current. Research in Science and Technological Education, 1, 1, (1983) Psillos, D., Teaching Introductory Electricity. In Andrée Tiberghien, E. Leonard Jossem, Jorge Barojas (eds), Connecting Research in Physics Education with Teacher Education, 1998 McDermott, L. C. & Shaffer, P. S., Research as a guide for curriculum development: An example from introductory electricity. Part I: Investigation of student understanding. Part II: Design of instructional strategies. American Journal Of Physics 60, 11, (1992) Eylon, B. S. & Ganiel, U., Macro-micro relationship: the missing link between electrostatics and electrodynamics in students' reasoning. International Journal of Science Education, 12, 1, (1990)

143 Stocklmayer S. M. & Treagust D. F., Images of electricity: How do novices and experts model electric current? International Journal of Science Education, 18, 2, (1996) Viennot L., Reasoning in Physics. The Part of Common Sense, Kluwer, 2001 Chabay, R. W., & Sherwood, B. A., Matter and interactions II: Electric and magnetic interactions (Wiley, New York, 2007) Licht, P., Teaching electrical energy, voltage and current: an alternative approach. Physics Education, 25, (1991) De Posada, J. M., Conceptions of High School Students Concerning the Internal Structure of Metals and Their Electric Conduction: Structure and Evolution, Science Education, 81(4) (1997) Thacker, B.A., Ganiel U. & Boys D., Macroscopic phenomena and microscopic processes: Student understanding of transients in direct current electric circuits. American Journal Of Physics 67, 7, S25-S31 (1999) Wittmann M. C., Steinberg R. N., Redish E. F., Investigating student understanding of quantum physics: Spontaneous models of conductivity, Am. J. Phys. 70(3), (2002) Mashhadi A. & Woolnough B. Insights into students' understanding of quantum physics: visualizing quantum entities, Eur. J. Phys. 20, (1999) Seifert, S. & Fischler H. A multidimensional approach for analyzing and constructing teaching and learning processes about particle models, online (2003) Fredette, N. H., Clement, J. J. Students Misconceptions of an Electric Circuit: What Do They Mean? Journal of College Science Teaching, 10, 5, (1981) Shipstone, D. M., A study of children s understanding of electricity in simple DC circuits. European Journal of Science Education, 6, 2, (1984) Tveita J., Constructivistic teaching methods helping students to develop particle models in science (1997) < visited May 2011 Benseghir, A. & Closset J. L., The electrostatics-electrokinetics transition: historical and educational difficulties. International Journal of Science Education, 18, 2, (1996) Borges, A. T. & Gilbert, J. K., Mental models of electricity, International Journal of Science Education, 21, 1, (1999) Stephens, S.-A., McRobbie, C., Lucas, K., Model-based reasoning in a year 10 classroom, Research in Science Education, 29(2), (1999) Sins P. H. M., Savelsbergh E. R., van Joolingen W. R. & van Hout-Wolters B. H. A. M. The Relation between Students Epistemological Understanding of Computer Models and their Cognitive Processing on a Modelling Task, International Journal of Science Education, 31:9, (2009) Chiu, M. H., & Lin, J. W. Promoting Fourth Graders Conceptual Change of Their Understanding of Electric Current via Multiple Analogies. Journal of Research in Science Teaching, 42, 4, (2005)

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152 FFP Udine November 2011 QUANTUM PHYSICS IN TEACHER EDUCATION Gesche Pospiech, Matthias Schöne, TU Dresden, Faculty of Science Abstract Teacher education forms the decisive link between physics education research and the realization of research results in school. The actions of teachers are influenced by their own experience from school, from study at university and the need to follow the school curriculum. Discrepancies between these factors have to be overcome. In quantum physics, the gap between teaching traditions at university, recent experimental results and the possibilities of teaching at school is especially big. Therefore it seems desirable to study this gap in detail in order to diminish it within given resources. 1. Introduction Quantum physics is the basic theory of physics. During the last decades there has been made significant progress in understanding its fundamental features as opposed to classical physics. Especially experiments concerning uncertainty, entanglement and quantum information allow for deeper insight and have found their way into popular science (e.g. Zeilinger 2003). In this way the fascination of youth towards quantum physics is raised. However, in general these new developments do not yet find their way into the school curricula and also not into teacher education. On the contrary the teaching at school is strongly influenced by the historical developments, such as the atomic model, the representation of e.g. an electron by the so-called wave-function or the statistical interpretation of the uncertainty relation of position and momentum as central parts. Whereas it is undoubted that quantum physics as a basic theory has to be taught at school it is not equally clear which aspects to choose as the most relevant content. Therefore the implementation of a modern course in quantum physics be at school or at university requires big efforts in several aspects: first it needs the definition of an agreed upon "canon" for school and university alike and secondly identifying appropriate learning pathways. The decision of teachers regarding the focus of their lessons relies on several factors: their own experience as students in school, their own convictions about good physics teaching and its related goals, what they have learned at university and last but not least the school curriculum with the expected teaching. In addition, teaching quantum physics is strongly connected with the personal views on the role of physics and its statements it can make as a science and where its borders are. Teacher education has to cover mastering the basics of quantum physics - including some mathematical formalism - and on the other hand being able to explain it to students. In university lectures on quantum physics these both combined goals mostly are not reached in such a way that the teachers transfer their knowledge into the classroom. So the adequate fitting of content knowledge and pedagogical content knowledge remains an open problem. The goal of the present study is to go a first step towards its solution in describing the relation between the conceptions of the lecturers at university and the students conception of the relevant quantum physics terms and the interrelations. 2. Theoretical Framework Teacher education lies in the triangle between the professional knowledge teachers are expected to acquire at university, their own experiences from the physics lessons they attended themselves at school and the demands of school curricula, lesson plans and school authorities, (Etkina 2010). In order to devise their teaching teachers have to combine their content knowledge, the pedagogical knowledge and the pedagogical content knowledge. This is seen through the glasses of personal convictions, teaching habitudes and cornered by the school curriculum Teaching of Quantum Physics Quantum physics proves to be a special area in physics teaching because of its abstract formalism, the differences to classical physics, the nearly complete absence of real experiments suitable for school and the strong traditions in teaching partly due to the long unclear history of

155 FFP Udine November 2011 interview results concept maps of the categories were built with the program Atlas.ti. The main categories were defined as best linked with other categories or often mentioned in the interviews. Based on this analysis we created a main quantitative questionnaire containing questions on the main contents of the lectures, the demanded content for an additional didactical course and suitable topics for a school course in quantum physics, a rating of imparted PCK and CK knowledge after a standard course and a self rating of the quantum physical understanding. 4. Results 4.1. Results of analysis of lectures The main concepts of the standard courses in quantum physics from lecturers' point of view are shown in Fig. 1 as a concept map. Lecturers put a great focus on mathematical techniques. One possible reason may be that lecturers often rate students' mathematical knowledge as insufficient and that it has to be enlarged. So the ability for doing formal calculations and mathematical analysis of quantum physics are two of the main goals of the lecturers. Other important goals are the knowledge of basic concepts and example applications. But no lecturer mentioned methodical or didactical concepts as important. Lecturers often feel there is too few time for these additional questions and/or they are not trained in this respect. Figure 1: Main concepts according to lecturers. The most important topics are marked grey Results of pilot study with teacher students As a first result of the pilot study the five different interviews with the students were analyzed and different categories for the content of theoretical physics courses, the demands for an additional

156 FFP Udine November 2011 didactical course and possible topics for teaching quantum physics in school were created. The main content of the lectures from the students point of view are shown in Fig.2 as a concept map. Figure 2: Main contents of lectures seen by students. The most important topics are marked grey. The identified categories were used as different choices in the quantitative main questionnaire, so a rank order of these topics could be constructed. In contrast to the lecturers' view students see more details like the Compton or photo effect than main principles. They separate more between physical model and mathematical formalism. Also students connect general quantities to concrete examples, e.g. Spin is related to hydrogen atom, but indistinguishability not at all. Also the main demands of the students for an additional course that improves the didactical education were categorized and are shown in Fig. 3, the main demands also marked grey. Figure 3: Students demands for an additional didactical course As a second part the answers in the test for understanding were analyzed. Three of five students formally knew the problems of the measuring process in quantum physics, but none respectively

157 FFP Udine November 2011 only one could apply this knowledge to two questions about energy measurement (see Robertson/Kohnle 2010, question 1.1 and 1.2). In contradiction 90% of the comparable experimental group of Robertson/Kohnle could give the right answer to question no. 1.1 and 35% to question no Concerning the explanation of quantum properties the uncertainty could be described by four of five students, but none could describe the difference between the uncertainty in a classical Galton Board experiment (for further details see Baily/Finkelstein 2009, essay question no. 2) and quantum uncertainty. This is almost in agreement to the result of Baily/Finkelstein with 15% right answers. Only two of five students in our survey could describe the indeterminism and the entanglement correctly. None of the students knew the significance of the EPR Experiment Results of Questionnaire In the main study 20 master students participated, mainly from the fifth year at universities in Dresden and Leipzig. The content view of the master students in the quantitative survey is similar to the qualitative one. Main subjects like the Dirac formalism, the Schrödinger equation and calculus of potential wells are mentioned often, but historical experiments seem not so important for this group. In addition they put more attention to angular momentum/spin and mathematical formalism, e.g. the description of the time evaluation, so they are closer to the lecturers view. The main demands of the students are also a stronger explanation of the physical model in interpretation questions but mainly PCK questions in practical teaching methods (18 of 20 students), concepts for a teaching unit (17 of 20 students), application examples (14 of 20 students) and general educational viewpoints (16 of 20 students). This is in accordance to the rating of students own learned PCK in quantum physics lectures. 70 % of the students think the given knowledge is not sufficient for teaching quantum physics well in school. In contradiction students feel well prepared in their content knowledge. 16 of the 20 interviewed students think the main concepts of quantum physics became clear. But not all subdomains were equally rated, only 11 of 20 of the students feel well prepared in the topic of entanglement, 12 of 20 in uncertainty and only 8 of 20 in the understanding of the measurement process. In accordance to the lecturers' view only 12 of 20 students feel sufficiently prepared in the mathematical formalism. The analysis of suitable topics for school courses in quantum physics show that the uncertainty principle, application of laser and properties of quantum objects are recommended by all students. Also the photo effect (19 of 20 students) and interference experiments (17 of 20) were seen as suitable topics. Many topics from the university lectures like angular momentum, Schrödinger equation, Bose Einstein Condensation, probability distribution and equations of radiation, which are difficult, were rejected by the students for school education. Visionary application topics like quantum computer or cryptography are rated ambiguous. This shows that students often reconstruct their own school experience, especially in historical experiments, but are open for new subjects like applications if they are well constructed. The analysis of the central points of the saxonian school curriculum (2011) shows similar topics with photon as a quantum object, the photo effect, interference of photon, electron and neutron and Heisenberg's uncertainty principle. Only the measurement process was not mentioned by the students, but is a main topic in the curriculum. It is remarkable that lecturers are focused on mathematical formalism, which cannot be used in school, because only basics of vector algebra are known by the pupils. 5. Future work In order to compare content structures of university and school lectures it is planned to analyze two different concept maps. One is built up from lecture scripts and exercises of university lectures, the

158 FFP Udine November 2011 other from school books. Hendrik Härtig has shown in his dissertation (2010), that school books are with high significance a good indicator for the average school lesson. To validate both concept maps experts will analyze and verify them. To improve the pedagogical content education of teacher students we are planning an additional course. Main goals of this course is the discussion of basic ideas of quantum physics, fundamental experiments and possible problems in the teaching learning process. But also teaching concepts in quantum physics and practical developments by students will be included. The course will be evaluated with a pre- and posttest with qualitative instruments and a questionnaire. 6. Conclusion This study shows that lecturers and teacher students have different views and demands for a reasonable course in quantum physics. Lecturers see basic mathematical techniques and quantum physics formalism as main goals. Teacher students want to understand the basic principles, interpretational problems and like to discuss PCK relevant questions, like practical teaching methods, teaching concepts and applications. It was shown that teache students feel well prepared in the content knowledge but poorly in pedagogical content knowledge with the consequence that they fell not prepared to teach quantum physics in school. We will analyze the effects of an additional course especially for teacher students in this respect. References Akarsu B (2011) Instructional Designs in Quantum Physics: a Critical Review of Research, Asian Journal of Applied Sciences 4(2), Fanaro M, Otero M R & Arlego M (2009) Teaching the foundations of quantum Mechanics in Secondary School: a Proposed Conceptual Structure, Investigações em Ensino de Ciências 14(1), Eylon B, Cohen E, Bagno E (2010) The interplay of physics and mathematics in a graduate quantum mechanics course for physics teachers, talk given in GIREP-conference 2010 in the Symposium "Addressing the role of mathematics in physics education. Baily C & Finkelstein N (2009) Development of quantum perspectives in modern physics, Physical Review Special Topics - Physics Education Research 5, Baily C & Finkelstein N (2010) Refined characterization of student perspectives on quantum physics, Physical Review Special Topics - Physics Education Research 6, Baumert J, Kunter M, Blum W, Brunner M, Voss T, Jordan A et al. (2010) Teachers' mathematical knowledge, cognitive activation in the classroom, and student progress, American Educational Research Journal, 47(1), Borowski A, Kirschner S, Liedtke S & Fischer H E (2011) Vergleich des Fachwissens von Studierenden, Referendaren und Lehrenden in der Physik, Physik und Didaktik in Schule und Hochschule 10(1), 1-9 Deslaurier L & Wieman C (2011). Learning and retention of quantum concepts with di erent teaching methods. Physical Review Special Topics - Physics Education Research, 7, Darling-Hammond L (2000) Teacher Quality and Student Achievement: A Review of State Policy Evidence, Education Policy Analysis Archives 8(1), 1-44 Demtröder et al (2006) Thesen für ein modernes Lehramtsstudium im Fach Physik, DPG Etkina E (2010) Pedagogical content knowledge and preparation of high school physics teachers, Physical Review Special Topics Physics Education Research 6, Kalkanis G, Hadzidaki P & Stavrou D (2003) An Instructional Model for a Radical Conceptual Change Towards Quantum Mechanics Concepts, Science & Education 87, Levrini O, Fantini P, Pecori B (2008) The problem is not understanding the theory, but accepting it : a study on students difficulties in coping with Quantum Physics, in GIREP-EPEC Conference, Frontiers of Physics Education Selected contributions, RIJEKA Zlatni rez, Mannila K, Koponen I T & Niskanen J A (2002) Building a picture of students conceptions of wave- and particle-like properties of quantum entities, European Journal of Physics 23, Mayring P (2010) Qualitative Inhaltsanalyse: Grundlagen und Techniken, Beltz Michelini M, Ragazzon R, Santi L & Stefanel A (2000) Proposal for quantum physics in secondary school, Physics Education 35(6),

159 FFP Udine November 2011 Müller R, Wiesner H, (1998) Vorstellungen von Lehramtsstudenten zur Interpretation der Quantenmechanik H. Behrendt (Hrsg): Zur Didaktik der Physik und Chemie, Alsbach, S Pospiech G (1999) Teaching the EPR--Paradox at High School?, Physics Education 34, Riese J (2010) Empirische Erkenntnisse zur Wirksamkeit der universitären Lehrerbildung - Indizien für notwendige Veränderungen der fachlichen Ausbildung von Physiklehrkräften, Physik und Didaktik in Schule und Hochschule, 9, Robertson E, Kohnle A (2010) Testing the Development of Student Conceptual Understanding of Quantum Mechanics, GIREPEPEC PHEC 2009 Conference (2010), Santi L, Michelini M, Stefanel A & Meneghin G (2011) A resource environment for preservice teacher education to introduce quantum physics in secondary school, in 'MPTL conferrence 2010'

160 NEW STRENGTH TO PLANCK S LENGTH CHOICE Giuseppe Fazio, Mauro Giaconi, Davide Quatrini, Electronic Engineering Department, Faculty of Engineering, University of Rome Tor Vergata, via del Politecnico Rome Abstract The calculation of the total mass of a spacetime in which an observer-independent scale of length exist is possible through the use of Heisenberg Uncertainity Principle. In the following paragraphs we show that such a spacetime has a total mass equal to the mass of our Universe observed by the WMAP Nasa spacecraft if the cited observer-independent scale of length is determined by the Planck s length lp. This result gives new strength to fundamental theories that make use of lp as a length with a special role, like the various string theories. 1. Introduction The theoretical description of a general class of spacetimes in which observer-independent scales of both velocity and length exist can be found in recent-years published papers (Amelino-Camelia 2001). Besides, from another point of view, it is possible to affirm that a stringy nature (i.e. a Universe in which a string theory can be successfully applied) needs only two constants: a maximum velocity for satisfying the relativistic invariance principle and a minimum length for quantization (Veneziano 1986). For these reasons the two cited values are of great importance in the search for a theory that binds together the small and the large scales of the Universe, i.e. a theory that could lead to sinergies between General Relativity and Quantum Mechanics. The debate related to the value of the maximum velocity needed for the relativistic invariance was solved long time ago with the selection of c, i.e. the speed of light in vacuum. This value is universally accepted because of its role in General Relativity. Regarding the minimum length the debate is still open, but the most appreciated candidate is Planck's length lp, the only length that can be obtained combining General Relativity constants (c and G) together with the Quantum Mechanics constant ħ: lp = (ħg/c 3 ) ~ m (1) The aim of the present work is giving an evidence based on real experimental data to support the choice of Planck's length as the correct minimum length in our Universe. We give a simple way for calculating the mass of the Universe through the use of Planck's length and then we compare the obtained result with the data provided by the Nasa WMAP spacecraft. The selection of this process was obvious, for us, because we believe that the properties of such a unifying constant like the Planck s length can emerge only applying the rules and laws of Quantum Mechanics to large systems mainly ruled by General Relativity (like our Universe at large scale is). 2. Universe Mass' calculation through Heisenberg Uncertainity Principle The calculation of the Universe mass is possible using the Heisenberg Uncertainity Principle (HUP). It is sufficient obtaining an expression for the mass from the best-case HUP for the lenghtmomentum couple (i.e. selecting the = operator instead of the ) and then maximizing the obtained expression: ΔxΔp = ħ/2 (2) mδxδv = ħ/2 (3) m = (ħ/2)/δxδv (4) The maximization of this expression requires the selection of minimum values for Δx and Δv. According to (Amelino-Camelia 2001) it is possible to select the Planck's length for the minimum value of Δx, and then proceed from there for the selection of the minimum value of Δv. Using the simplest definition for velocity, i.e. v = s/t, we can minimizing it through the selection of the biggest amount of time and the smallest length. For this last one our selection is coherent with the above considerations, so we choose lp. For the biggest amount of time we can use the age of the

161 Universe, or its good approximation given by the 1/H value, where H is the Hubble constant. In this way we obtain the smallest observable velocity at any given time in our spacetime, because calculated through the ratio between the smallest selected length (lp) and the largest possible observation period (the age of the Universe). Our research group is at work for finding deeper implications on the existence of the cited minimum value of velocity, however, such as possible impacts on the quantization of fundamental quantities. With the chosen values the mass obtained is: m = (ħ/2)/(lp*(lp*h)) (5) m = (c 3 )/(2GH) ~ Kg (6) This is the largest calculable mass in a spacetime in which HUP is valid and the Planck's length is the minimum length. 3. Universe Mass' calculation through 2008 WMAP Nasa Spacecraft Data 2008 WMAP NASA spacecraft data (NASA/WMAP Team 2008) confirmed that the the density of the Universe (ρ) is equal to its critical density calculated by the Friedmann equations (Friedmann 1922), i.e. 3(H 2 )/8πG. This fact confirmed also the Euclidean geometrical structure for our Universe, leading to an estimation of Universe mass based on the assumption that its volume could be calculated as the volume of a sphere, i.e. V=(4/3)π(r 3 ). In other words: m = ρv (7) m =((H 2 )(r 3 ))/(2G) (8) The selection of the value for the radius r can be done considering the space travelled by a ray of light for the entire duration of the Universe (Ua), i.e. r = c*ua. Using the above mentioned approximation for which Ua = 1/H we obtain the following value for the mass of the Universe: m = (c 3 )/(2GH) ~ Kg (9) Please note that this value, and the expressions used for its calculation, are widely accepted by the scientific community, and are also used by Nasa itself in their official publications (NASA Glenn Research Centre). 4. Conclusions A spacetime in which Relativity applies, in which observer-independent scales of both velocity and length exist and are respectively determined by c and lp, and in which HUP is valid has a a total mass equal to: m = (c 3 )/(2GH) ~ Kg (10) In our Universe Relativity applies, an observer-independent scale of velocity exist and is determined by c, HUP is valid and the total mass is equal to: m = (c 3 )/(2GH) ~ Kg (11) For these reasons it is very probable that in our Universe an observer-independent scale of length exists and it is determined by Planck's length lp. References Amelino-Camelia, G. (2001): Testable scenario for relativity with minimum length. Physics Letters B 510, Veneziano, G. (1986): A Stringy Nature Needs Just Two Constants. Europhys. Lett NASA / WMAP team (2008), WMAP Press Release: WMAP reveals neutrinos, end of dark ages, first second of universe. Friedmann, A. (1922): Über die Krümmung des Raumes. Z. Phys , NASA Glenn Research Centre: On the expansion of the Universe. 12/Numbers/Math/documents/ON_the_EXPANSION_of_the_UNIVERSE.PDF

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174 How Physics Education Research contributes to designing teaching sequences Jenaro Guisasola. Applied Physics Department. University of the Basque Country. Spain Abstract Over the last decades, a growing number of physicists have taken up the challenge of implementing the same research discipline in physics learning and teaching as applied to traditional research in the field. This commitment is widely known as Physics Education Research (PER). In this paper, I will single out some of the directions taken by present and emerging research that I deem promising. I will also discuss the impact of research on the educational practice of physics teaching at university level. This paper presents evidence from different studies to demonstrate the potential positive impact of research into teaching and learning physics on students understanding of physics. Finally, I will show some practical challenges and propose some steps that could be taken to ensure PER growth and productivity. 1. Introduction In recent decades a growing number of physicists have taken up the challenge of applying an approach to physics education research, just as rigorous as traditional physics research, concerning problems relating to learning and teaching physics. This commitment is widely known as Physics Education Research (PER). PER concentrates on understanding and improving how physics is learnt by studying the contents of the physics curriculum and what teachers and students do when teaching and learning in schools. The research field relating to teaching science, and in particular Physics Education, has been well-established for some decades. It attempts to integrate knowledge from different fields of research; such as physics, the psychology of learning, the epistemology of science or the pedagogies of the teaching-learning process (Abell and Lederman 2007, McDermott 1997) in a non-mechanical way. The objective of improving student learning lies at the heart of all education research. As Hurd de Hart (1991) says there is a little reason to do research, unless there is a pay-off in the classroom. PER has developed around intensifying the belief that this goal is possible. Its objective is usually formulated as reforming or changing the teaching of physics causing PER to go beyond identifying student learning difficulties found in traditional teaching. Research has developed didactic materials and strategies, which have been repeatedly submitted to examination, assessment and redesign. A number of different studies have had a positive impact on physics teaching and learning. One example is research into student's difficulties in terms of learning physics concepts which resulted in designing new instruments to assess students' knowledge and the effectiveness of teaching. Halloun and Hestanes (1985) used the results obtained from an investigation into university students' ideas, within the field of Mechanics, to design the Force Concept Inventory formative assessment test. Since its design, there has been an increase in the number of physics programmes and textbooks, which pay greater attention to conceptual differences. Recently, in the field of electromagnetism, Maloney and al. (2001) have developed formative assessment instruments with similar aims. However, difficulties in carrying out research into teaching science, aimed at improving the practice of teaching, must not be underestimated. Lijsen and Klaassen (2004) argue that designing learning sequences requires a complex process of applying the general principles of didactics to specific teaching contexts for teaching the subjects on the curriculum. They point out that this task is not linear but rather a cyclical process with the aim of generating knowledge about teaching and learning, implementing improved teaching methods appropriately in the classroom. Designing teaching sequences is not a mechanical process involving transferring pedagogical principles and research results to teaching specific science subjects. On the contrary, teaching sequence design is a creative process, which considers not only research but also the classroom culture and the circumstances of both teachers and pupils. 1

175 In this paper I will discuss the impact of research covering educational practice of teaching physics on university courses. I will look at some practical challenges and propose some steps that could be taken to ensure PER growth and productivity. 2. Implications of P.E.R. to designing teaching sequences Designing research-based teaching sequences takes into account two kinds of research recommendations: a) results of empirical studies on students ideas and reasoning; b) technical contributions connected to the nature of science and how it is learnt and taught. Both contributions are connected, as the principles deriving from the latter influence how empirical studies are analysed on the former. Analysing students' ideas involve not only conceptual aspects but also epistemological and ontological aspects. On this point, it is necessary to analyse the historical development of the topic to be taught, the difficulties that the scientific community had to overcome and the arguments used to construct new concepts and explanatory models. Working from this epistemological analysis on the scientific content of school curriculum, it is possible to define the teaching-learning aims in a well-founded way. In other words, it is possible to justify choosing these aims based on epistemological evidence of the discipline and not idiosyncratically or based on the educational programme's tradition. The research recommends sequencing the main stages that teacher must work through when designing the teaching programme. In our research group we use the so-called learning indicators to specify what students should learn on the topic in accordance with the school curriculum (Guisasola et al 2010). The learning indicators include ontological aspects (values and attitudes) that must be consciously taken into account. Research into teaching sciences shows that the emotional and value-related aspects cannot be considered without making a close connection to cognitive processes when students are working on their activities in science classes (Zembylas, 2005). In this respect, designing activities that relate aspects of science, technique, society and the environment to each other means supporting a presentation of socially contextualised science that encourages students' interest in the scientific topic being taught. Including activities related to Science-Technology-Society-Environment generates interest among students on the study topics, encouraging them to get involved in the solving task (Simpson and Oliver 1990). The design and development of the research-based teaching sequences imply that creative integration must be carried out taking into account the teaching difficulties, learning indicators and ontological aspects. This creative integration leads to a close analysis of the differences between the learning difficulties and the teaching targets set in the curriculum. This analysis should build bridges between activity design and research work. In this respect we can say that we are using evidence from the empirical research when designing the sequence. It is necessary to highlight that the curriculum standards provide information on "what should be taught" generally. On the contrary, teaching aims for the panel that we have defined using the research evidence specify even further what students should learn and justify why they should do it. Although many of the activities from the innovating educational sequences are common to the questions and exercises from the text books used in regular teaching, it is necessary to highlight that they are used differently. Maybe the most significant differences revolve around the time dedicated to student difficulties, plus activities that aim to interest the students on the topic and justify introducing new models and concepts. The sequence activities are designed with the aim of providing students with opportunities to understand and apply the same model repeatedly. On the other hand, the activities also tackle epistemological aims by getting students to appreciate the power of a scientific model capable of explaining a great number of experimental cases. As a conclusion, all the above can be organised into table 1. 2

176 Table 1. Use of research evidence to design teaching sequences Students ideas and reasoning Difficulties in learning Epistemological analysis of the contents of the school curriculum Learning indicators Teaching goals Interests, attitudes, values and standards S-T-S-E Aspects Set out specific problems and aims in a sequence Interactive learning environment Teaching strategies 3. Teaching sequence of electromagnetic induction (EMI) Over recent years, within the Physics Education Research Group of the University of the Basque Country (PERG-UBC), I have carried out different research projects, which have developed teaching sequences for classroom implementation and subsequent assessment as one of their main objectives. See, for example, Guisasola et al. 2010a, 2010b, 2009, 2008 and Furio et al In this paper, I will describe the processes involved in designing teaching sequences following the instruments within chart 1, with specific reference to teaching Electromagnetic Induction (EMI) in an Introductory Physics Course at university level. Regarding students' ideas, previous works (Guisasola et al. 2011) have identified their conceptual and epistemological difficulties in understanding the theory of EMI. On this point, it is necessary to analyse the historical development of the topic to be taught, the difficulties that the scientific community had to overcome and the arguments used to construct new concepts and explanatory models. Working from this epistemological analysis of the scientific content of the school curriculum, it is possible to define the learning indicators and justify their choice based on epistemological evidence from the discipline and not idiosyncratically or based on the tradition of the education programme. We will present the learning indicators drawn up for teaching EMI given in Table 2 below: Table 2. Description of learning objectives and difficulties in EMI topic Learning indicators i1) Be able to weigh up how useful it is to solve the proposed problem i2) Be familiar with the experimental phenomena of electromagnetic induction i.2.1) Find out about electromagnetic induction phenomena in spirals and solenoids crossed by variable magnetic fields. i.2.2) Find out about electromagnetic induction in circuits that are moving within a stationary magnetic field. i.2.3) Find out about electromagnetic induction phenomena caused by a combination of the aforementioned effects. i.2.4) Be able to tell the difference between the phenomenon of producing induced electromotive force and generating induced current intensity. Students difficulties - EMI presentation is not significant to involve students in the study. - Lack of familiarity with the scientific lab methodology (compiling data, handling apparatus, analysing results) - Difficulties to distinguish between the empirical level (use of multi-meters and interpretation of measurements) and the interpretative level that uses concepts such as variable magnetic and electric fields over time, Lorentz Force, magnetic flow and electromotive force. - Not used to working in a group. 3

177 i3) Be able to analyse electromagnetic induction phenomena qualitatively and later quantitatively. i.3.1) Using macroscopic modelling correctly i.3.2) Using microscopic modelling correctly - Students find it hard to: a) Interpret simple induction phenomena properly. b) Attribute EMI to the presence of a variable magnetic field or to the variation of magnetic flux c) Apply Faraday s Law correctly, interpreting motional electromotive force phenomena - Students tend to use a field model more frequently than a force model when explaining EMI phenomena. i4) Repeatedly use scientific work strategies to find the solution to the set problems. - Few students show that they know how to explain the phenomena from a microscopic and macroscopic point of view - Lack of familiarity with the scientific methodology in problem solving. Difficulties to: o Carry out qualitative analysis o Make a hypothesis o Draw up alternative strategies o Analyse results o Handle apparatus - Not used to working in a group. Analysing the differences between the learning difficulties and the teaching aims set provides us with bridges between sequence design and the research work. In this respect we can say that we are using evidence from the empirical research when designing the sequence. Table 3 below shows the sequence for EMI topic. Table 3. Map of the didactic sequence for the study on EMI Problem sequence What is the interest and/or use of the EMI study? When does an EMI phenomenon occur and when does it not? How can the EMI be quantified? Science Procedure to be learnt by the students - Science is interested in natural phenomena and their social implications. The origin of the interest in a phenomenon, situation or fact can vary depending on: the science topic, technological problems, local or global problems, etc. - Make empirical observations and take down information on the phenomena that occur and make predictions on what might happen. - Science is able to measure the phenomena that are seen and give quantitative answers from the phenomena seen. Explanations for the students to understand - Acquiring a preliminary conception of the study that is going to be carried out and promoting interest on the EMI study. - Macroscopic and qualitative study of EMI. EMI occurs when there is a variable B over time and/or when a conductor moves in a magnetic field. There is no induction if the conductor is in a stationary magnetic field. Macroscopic perspective of EMI. - Faraday's Law - Induced emf - Induced emf induced I - Energy conservation law (Lenz Law) 4

180 Continuing with the design of materials and strategies, as well as their assessment in extensive samples of schools and at different educational levels seems crucial to me, as in research we base ourselves on the fact that if the science teaching (physics) were as it should be, it would not be necessary to spending time getting a better understanding of how, when or why students learn. References Abell S K and Lederman N G (2007) Handbook of Research on Science Education, Lawrence Erlbaum, Mahwah, NJ Furió C, Guisasola J, Almudi J M and Ceberio M (2003) Learning the electric field concept as oriented research, Science Education 87(5), Guisasola, J., Almudi, J M and Zuza, K. (2011) University students understanding of electromagnetic induction, International Journal of Science Education, DOI / accesed 2011 november Guisasola, J, Almudí, J M and Zuza K (2010a) The design and evaluation of an instructional sequence on Ampere's law, American Journal of Physics 78(11), Guisasola, J, Zubimendi, J L and Zuza K (2010b) How much have students learned? Research-based teaching on electrical capacitance, Physicsl Review Special Topics: Physics Education Research 6(2), Guisasola J, Furió C and Ceberio, M (2009) Science Education based on developing guided research, in Science Education in Focus, Thomase M V ed, Nova Science Publisher, Guisasola J, Almudi J M, Zuza K and Ceberio M (2008) The Gauss and Ampere laws: different laws but similar difficulties for students learning, European Journal of Physics 29, Halloun I and Hestane D (1985) Common sense conceptions about motion, American Journal of Physics 53, Hurd de Hart P (1991) Issues in linking research to science teaching, Science Education 75, Leach J and Scott P (2002) Designing and evaluating science teaching sequences: an approach drawing upon the concept of learning demand a social constructivism perspective on learning, Studies in Science Education 38, Lijnse P and Klaassen K (2004) Didactical structures as an outcome of research on teaching-learning sequences? International Journal of Science Education, 26(5), Maloney D P, O kuma T L, Hieggelke C J and Van Heuvelen A (2001) Surveying students conceptual knowledge of electricity and magnetism. American Journal of Physics Suppl, 69(7), McDermott L C (1997) Bringing the gap between teaching and learning: The role of research, in The changing role of physics departments in modern universities, American Institute of Physics, New York L.C. McDermott, P.R.L. Heron, P.S. Shaffer, and M.R. Stetzer (2006) Improving the preparation of K-12 teachers through physics education research, American Journal Physics 74 (9) Mellado V (2003) Cambio didáctico del profesorado de ciencias experimentales y filosofía de la ciencia, Enseñanza de las Ciencias 21(3), Michelini M. (2003) New approach in physics education for primary school teachers: experimenting innovative approach in Udine University, Inquiries into European Higher Education in Physics, H.Ferdinande et al. eds. EUPEN, 7, Pintó R (2005) Introducing curriculum innovations in science: identifying teachers transformations and the design of related teacher education, Science Education 89(1), 1-12 Simpson R D and Oliver J E (1990). A Summary of Major Influences on Attitude Toward and Achievement in Science Among Adolescent Students. Science Education, 74(1), 1-18 Zembylas, M. (2005). Three Perspectives on linking the Cognitive and the Emotional in Science Learning: Conceptual Change, Socio-Constructivism and Poststructuralism. Studies in Science Education, 41,

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189 FFP Udine November 2011 DIRECTION OF TIME FROM THE VIOLATION OF TIME REVERSAL INVARIANCE Joan A. Vaccaro, Centre for Quantum Dynamics, Griffith University Abstract We show that the violation of time reversal invariance (T) observed in meson decay may have a profound effect on the direction of time. Our starting point is a universe without a presumed direction of time. Time evolution is modeled by taking an equal superposition of steps in both directions of time. This gives rise to multiple paths through time. The presence of T violating processes is shown to give rise to destructive interference between all paths except for the two that comprise continuously forwards and continuously backwards steps. We show that this leads to a new kind of irreversibility that is quite unlike that found in thermodynamics. 1. Introduction The direction of time is typically studied in terms of various arrows of time (Price 1996). Each arrow is associated with some time-asymmetric phenomenon and is oriented to point to the future. Perhaps the best known is the thermodynamic arrow which points in the direction of an increase in the entropy of an isolated system (Eddington 1928). Essentially, an increase in entropy is the effect that results from a fixed direction of time and an incomplete knowledge of a system. The thermodynamic arrow requires the universe to be initially in a low-entropy non-equilibrium state and evolve towards equilibrium. This implies an asymmetry between the past and future boundary conditions. The time asymmetry of other arrows of time can also be attributed to asymmetric temporal boundary conditions. For example, the cosmological arrow points in the direction of the expansion of the universe and so relies on a small universe in the past and a large universe in the future. All such arrows are based on time-symmetric dynamical laws. The one notable exception is the matter-antimatter arrow of time (also called the weak arrow of time) which partially explains the dominance of matter over antimatter in the universe. It arises from the observed violation of charge conjugation and parity inversion invariance (CP) in meson decay due to the weak interaction (Christenson 1964). Under the CPT theorem, CP violation is equivalent to violation of time reversal invariance (T). This means that meson decay is described by a time-asymmetric dynamical law and so the matter-antimatter arrow originates from the time asymmetry of a dynamical law rather than boundary conditions. Of all the arrows of time, it is the only one to have the potential of being the cause of the direction of time, rather than being simply an effect. Indeed T violation in meson decay has recently been shown to have potentially large scale physical effects in a model of the universe in which the direction of time is not predefined (Vaccaro 2011). Here we re-examine the implications of the model for the direction of time. In Section 2 we briefly review the model that was introduced in (Vaccaro 2011). Next in section 3 we show how a new kind of irreversibility emerges from the model when sufficient T violation processes are present. In section 4 we explore the repercussions of baryogenesis and end with a discussion of the implications for the direction of time in section Effect of T violation in a universe without a presumed temporal direction We want to explore how T violation can give rise to a preferred direction of time. For this we need to model the universe in such a way that it has no predefined direction of time if T violation processes are absent. We have two choices: the universe does not evolve at all, in which case there is nothing more to be said, or both directions of time have an equal footing. To incorporate the second option, rather than take time steps in a particular direction, time evolution must somehow be in both directions of time. In other words we must allow for the possibility of multiple paths through time. Feynman s sum over paths (Feynman 1948) provides the mathematical framework for dealing with such situations. We let the evolution in one direction of time, which we shall call forwards and associate with the positive time direction, be described by F U ( F ) 0, (1)

190 FFP Udine November 2011 and in the other direction of time, which we shall call backwards and associate with the negative time direction, by B U ( B ) 0, (2) where represents a time interval and UF ( ) exp( ihf ) 1 1 U ( ) TU ( ) T exp( ith T ) exp( ih ). B 1 F Here, H F and HB THFT are the Hamiltonians for the universe with respect to the forward and backward directions of time, respectively, T is Wigner s time reversal operator (Wigner 1959) and we use units where 1. The Hamiltonian violates time reversal invariance if H H. B The expressions U ( F ) 0 and U ( B ) 0 represent the probability amplitudes for the universe to evolve from the state 0 to the state via two different paths, each of which corresponds to a different direction of time. We have no reason to favor one path over the other so, according to Feynman s sum over paths method, we take the total probability amplitude to evolve from 0 to as the sum UF ( ) 0 U B( ) 0 UF ( ) U B( ) 0. As this result is true for all states we can write the unbiased evolution of the universe as F F B ( ) [ U ( ) U ( ) ]. (3) F B 0 We call the process described by eq. (3) symmetric time evolution. This analysis can be repeated for an additional step of symmetric time evolution. After N such steps the state of the universe is given by ( ) [ ( ) ( N N UF UB ) ]. 0 (4) To simplify the description we have ignored the normalization of the state. Eq. (4) is represented as a binary tree in fig. 1(a). (a) 0 (b) 0 U B U F ( N ) U ( ) 0 U N ( ) 0 N B F Figure 1: (a) and (b) are representations of eqs. (4) and (5), respectively, as binary trees. Leftward arrows ( ) represent the application of U B and rightward arrows ( ) represent the application of U F. Expanding the N-fold product on the right side of eq. (4) gives a series of 2 N terms each of which represents a different path through time. For example the terms N N 1 U F ( ) 0 and UF ( ) U B( ) 0 represent paths of N steps containing zero and one backward step, respectively. The series can be separated into subseries whose terms involve the same number of steps in the forward and backward directions, but with steps in different orders. For example, the small circle in fig. 1(a) represents a subseries characterized by two forward steps and one backward step: U ( ) U ( ) U ( ) U ( ) U ( ) U ( ) U ( ) U ( ) U ( ). F F B 0 F B F 0 B F F 0 Note that the operators U ( F ) and U ( B ) do not commute if the Hamiltonian is not T invariant. In that case the terms of the subseries have been shown to interfere destructively (Vaccaro 2011).

191 FFP Udine November 2011 In order to model the interference effect we use the conventional phenomenological model of neutral kaon decay (Lee 1973, Yao 2006) as the prototypical T violation process. We quantify the number of kaons as f 10 where f is the fraction of the estimated 10 total number of particles in 44 the universe and set the size of the time step to the Planck time, i.e. 10 s. With these 1/ 2 13 conditions, a relatively lengthy calculation (Vaccaro 2011) reveals that for N f 10 s eq. (4) becomes N N ( N ) [ U ( ) U ( )]. (5) F B 0 The right side comprises only two parts, one representing consistent evolution in the forward direction, U N F ( ) 0, and the other representing consistent evolution in the backward direction, N U B ( ) 0, as illustrated in fig. 1(b). Destructive interference has eliminated all other paths. This should be contrasted with the situation in eq. (4) for which the direction of the evolution can change from step to step as illustrated in fig. 1(a). 3. New kind of irreversibility We now turn to new results. The derivation of eq. (4) relies on the construction of paths through time and it is this feature we wish to explore in more detail here. If the direction of time is known to be the forwards direction, say, then we can determine whether a universe that is in state A will be found in a different state B after a time interval t from the probability 2 BU () F t A. (6) A nonzero probability value means that there is a temporal path from A to B. Expression (6) is essentially the probability of transitioning from A to B after a time delay of t. In the case where the direction of time is not predetermined, we have seen in the previous section that the construction of temporal paths of N steps each of duration is given by [ U ( ) U ( )] N. F With this operator we can check whether a universe that is in state A can evolve to state B in N time steps of unspecified direction by seeing if the (un-normalized) probability N P ( B A) B [ U ( ) U ( )] A N F B B 2 N N A [ U ( ) U ( ) ] B B [ U ( ) U ( )] A (7) F B is nonzero. The converse situation where the universe can evolve from B to A is given by checking for a nonzero value of F B N P ( A B) A [ U ( ) U ( )] B N F B N N B [ U ( ) U ( )] A A [ U ( ) U ( ) ] B. F B 2 F B (8) For a non T-violating universe HB H F and so U F ( ) exp( ihf ) UB( ) and U B( ) exp( ihb ) U ( F ), and thus PN( B A) PN( A B ). In other words, if a non T-violating universe can evolve from A to B it can also evolve from B to A, as one would expect if both directions of time are allowed. However, the situation is very different for a T violating universe, i.e. one for which for HB H F. In this case the right sides of eqs. (7) and (8) are not equal in general. Indeed, in the extreme case, one may be zero while the other is not. To see this let B be given by where M is an integer, for which we find B [ U ( ) U ( )] M A (9) F B

192 FFP Udine November 2011 P ( B A) B [ U ( ) U ( )] N F B N M N A [ U ( ) U ( )] [ U ( ) U ( ) ] A. F B A F 2 B 2 In the special case where N M P ( B A ) B B M 2 which is the square of the norm of B. Normalizing the probability results in a value of unity. This implies that the evolution from A to B is guaranteed as would be expected from eq. (9). However for the converse case, N P ( A B) A [ U ( ) U ( )] B N F B N M A [ U ( ) U ( )] A. F B 2 2 (10) If the value of M is sufficiently large that destructive interference eliminates all paths but two, as illustrated by eq. (5), then the state B can be written as and it follows that eq. (10) becomes [ M M B U ( ) U ( )] A F B 2 2 F B F B ( ) N M ( ) N M ( ) N M ( ) N M P A B A U U A A U A A U ( ) A. (11) N Moreover, let the value of M be such that AB 0 and so P0 ( A B ) 0. It would be reasonable to assume that the observable universe is not cyclical and never returns to the same state in a fixed direction of time. For our model this means that N M N M AUF ( ) A 0 and AUB ( ) A 0, and so from eq. (11) P ( A B ) 0 N for any positive value of N. We have succeeded in showing, therefore, that the universe does not evolve from B to A despite the fact that it is guaranteed to evolve from A to B. This is a new kind of irreversibility. But to fully appreciate what it means we first need to consider the relationship between the two terms on the right side of eq. (5). 4. Superposition of matter and antimatter If the state 0 is T invariant, i.e. if 0 T 0, the two components of the right side of Eq. (5) are related by the time reversal operation U ( N ) TU ( N ). B 0 F 0 Multiplying both sides by CP, where C and P are the charge conjugation and parity inversion operators, and invoking the CPT theorem shows that U ( F N ) 0 and U ( B N ) 0 are related by charge-parity conjugation. It is interesting to consider the case where the evolution U ( F N ) 0 symbolizes baryogenesis. In this case, if U ( F N ) 0 is dominated by matter then U ( B N ) 0 is dominated by antimatter. Every particle in U ( F N ) 0 has a corresponding partner antiparticle in U ( B N ) 0 and vise versa, and so a planet composed of matter in U ( F N ) 0 is correspondingly composed of antimatter in U ( B N ) 0. The two versions of the planet evolve independently of each other according to eq. (5) and so particle-antiparticle annihilation between the two versions of the planet would not occur. Moreover the evolution of the matter-dominated universe in U ( F N ) 0 is equivalent, by the CPT theorem, to the evolution of the antimatter-dominated universe in U ( B N ) 0. This symmetry between U ( F N ) 0 and U ( B N ) 0 implies that

193 FFP Udine November 2011 evolution in the forward direction is equivalent to the evolution in the backward direction. The directions forward and backward are redundant in this sense. The two arrows in fig. 1(b) can therefore be considered to be one as illustrated in fig past A B future U U F B ( N ) 0 matter ( N ) antimatter 0 Figure 2. A reinterpretation of eq. (5) treating the two directions, forward and backward, as equivalent. The states A and B are related by the relationship given in eq. (9) 5. Discussion We began with a model of the universe that has no presumed direction of time. Time evolution is described by a superposition of all possible paths that zigzag forwards and backwards through time as illustrated in fig. 1(a). We found that by incorporating sufficient T-violating processes only two paths representing always forwards and always backwards evolution survive destructive interference, as illustrated in fig. 1(b). We also showed that the two directions of time are essentially equivalent, as illustrated in fig. 2. Moreover we found that the universe can evolve from one state A to another B, as illustrated in fig. 2, but not vice versa. This endows the universe with a temporal direction pointing from a past to a future. We have therefore described a physical mechanism which, if it exists in nature, could account for the directedness of time that we observe in our universe. This mechanism has the potential of being the cause of the direction of time. It is quite different to the asymmetry of the thermodynamic arrow which is an effect of a fixed direction of time and an incomplete knowledge of a system. References 0 Christenson J H, Cronin J W, Fitch V L and Turlay R (1964) Evidence for the 2 decay of the K 2 meson, Phys. Rev. Lett. 13, 138 Eddington A S (1928) The nature of the physical world, Macmillan, New York, USA Feynman R P (1948) Space-time approach to non-relativistic quantum mechanics, Rev. Mod. Phys. 20, Lee T D and Wolfenstein L (1965) Analysis of CP-noninvariant interactions and the K1, K 2 system, Phys. Rev. 138, B1490-B1496 Price H (1996) Time s arrow and Archimedes point, Oxford University Press, New York, USA Vaccaro J A (2011) T violation and the unidirectionality of time, Found. Phys. 41, Wigner E P (1959) Group theory and its application to the quantum mechanics of atomic spectra, Academic Press, New York, USA Yao W -M et al. (2006) Review of particle physics, J. Phys. G: Nucl. Part. Phys. 33,

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207 Relativistic Classical and Quantum Mechanics: Clock Synchronization and Bound States, Center of Mass and Particle Worldlines, Localization Problems and Entanglement Luca Lusanna, Sezione INFN di Firenze, Via Sansone 1, Sesto Fiorentino (FI), Italy Abstract: Parametrized Minkowski theories allow to describe isolated systems in global non-inertial frames in Minkowski space-time (defined as M/oller-admissible 3+1 splittings with clock synchronization and radar 4-coordinates), with the transitions among frames described as gauge transformations. The restriction to the intrinsic inertial rest frame allows to formulate a new relativistic classical mechanics for N-particle systems compatible with relativistic bound states. There is a complete control on the relativistic collective variables (Newton-Wigner center of mass, Fokker-Pryce center of inertia, M/oller center of energy) and on the realization of the Poincare algebra (with the explicit form of the interaction-dependent Lorentz boosts). The particle world-lines are found to correspond to the ones of predictive mechanics and localization problems are clarified. The model can be consistently quantized avoiding the instantaneous spreading of the center-of-mass wave packets (Hegerfeldt theorem), because the non-local non-covariant center of mass is a nonmeasurable quantity. The basic difference with non-relativistic quantum mechanics is that the composite N-particle system cannot be represented as the tensor product of single particle Hilbert spaces, but only as the tensor product of the center-of-mass Hilbert space with the one of relative motions. This spatial non-separability (due to the Lorentz signature of space-time) makes relativistic entanglement much more involved than the non-relativistic one. Some final remarks on the emergence of classicality from quantum theory are done. I. INTRODUCTION Predictability in classical and quantum physics is possible only if the relevant partial differential equations have a well posed Cauchy problem and the existence and unicity theorem for their solutions applies. A pre-requisite is the existence of a well defined 3-space (i.e. a clock synchronization convention) supporting the Cauchy data. In Galilei space-time there is no problem: time and Euclidean 3-space are absolute. Instead there is no intrinsic notion of 3-space, simultaneity, 1-way velocity of light (two distant clock are involved) in special relativity (SR): in the absolute Minkowski space-time, only the conformal structure (the light-cone) is intrinsically given as the locus of incoming and outgoing radiation. The light postulate says that the 2-way (only one clock is involved) velocity of light c is isotropic and constant. Its codified value replaces the rods (i.e. the standard of length) in modern metrology, where an atomic clock gives the standard of time and a conventional reference frame centered on a given observer is chosen as a standard of space-time (GPS is an example of such a standard). The standard way out from the problem of 3-space is to choose the Euclidean 3-space of an inertial frame centered on an inertial observer and then use the kinematical Poincaré

208 group to connect different inertial frames. This is done by means of Einstein convention for the synchronization of clocks: the inertial observer A sends a ray of light at x o i towards the (in general accelerated) observer B; the ray is reflected towards A at a point P of B worldline and then reabsorbed by A at x o f ; by convention P is synchronous with the mid-point between emission and absorption on A s world-line, i.e. x o P = xo i (xo f xo i ) = 1 2 (xo i + x o f ). This convention selects the Euclidean instantaneous 3-spaces x o = ct = const. of the inertial frames centered on A. Only in this case the one-way velocity of light between A and B coincides with the two-way one, c. However, if the observer A is accelerated, the convention breaks down, because if only the world-line of the accelerated observer A (the 1+3 point of view) is given, then the only way for defining instantaneous 3-spaces is to identify them with the Euclidean tangent planes orthogonal to the 4-velocity of the observer (the local rest frames). But these planes (they are tangent spaces not 3-spaces!) will intersect each other at a distance from A s world-line of the order of the acceleration lengths of A, so that all the either linearly or rotationally accelerated frames, centered on accelerated observers, based either on Fermi coordinates or on rotating ones, will develop coordinate singularities. Therefore their approximated notion of instantaneous 3-spaces cannot be used for a wellposed Cauchy problem for Maxwell equations. Therefore in Refs.(Alba and Lusanna, 2007, 2010) a general theory of non-inertial frames in Minkowski space-time was developed. As shown in Refs. (Lusanna, 1997; Alba and Lusanna 2010) this formulation allowed to develop a Lagrangian description (parametrized Minkowski theories) of isolated systems (particles, strings, fluids, fields) in which the transition from a non-inertial frame to another one is formalized as a gauge transformation. Therefore the inertial effects only modify the appearances of phenomena but not the physics. Then the restriction to inertial frames allowed to define the Poincaré generators starting from the energy-momentum tensor. The intrinsic inertial rest frame of every isolated system allowed the definition of the rest-frame instant form of dynamics where it is possible to study the problem of the separation of the collective relativistic variable of an isolated system from the relative ones living in the so called Wigner 3-spaces in accord with the theory of relativistic bound states (Alba, Crater and Lusanna, 2010, 2001; Crater and Lusanna, 2001). The ordinary world-lines of the particles can then be reconstructed (Alba, Crater and Lusanna, 2007). The rest-frame instant form allows to give a formulation of relativistic quantum mechanics (RQM) (Alba, Crater and Lusanna,2011) which takes into account of the problems of relativistic causality and relativistic localization implied by the Lorentz signature of Minkowski space-time. Also a definition of relativistic entanglement can be given: the Lorentz signature implies features of non-locality and spatial non-separability absent in the non-relativistic formulation of entanglement. Elsewhere we extend the definition of non-inertial frames to general relativity (GR) in asymptotically Minkowskian space-times. In GR there is no absolute notion since also spacetime becomes dynamical (with the metric structure satisfying Einstein s equations): however in this class of space-times it is possible to make a Hamiltonian formulation and to separate the inertial degrees of freedom of the gravitational field from the tidal ones (the gravitational waves of the linearized theory).

209 II. NON-INERTIAL FRAMES IN MINKOWSKI SPACE-TIME AND RADAR 4- COORDINATES A metrology-oriented description of non-inertial frames in SR can be done with the 3+1 point of view and the use of observer-dependent Lorentz scalar radar 4-coordinates. Let us give the world-line x µ (τ) of an arbitrary time-like observer carrying a standard atomic clock: τ is an arbitrary monotonically increasing function of the proper time of this clock. Then we give an admissible 3+1 splitting of Minkowski space-time, namely a nice foliation with space-like instantaneous 3-spaces Σ τ : it is the mathematical idealization of a protocol for clock synchronization. All the clocks in Σ τ sign the same time of the atomic clock of the observer: it is the non-factual idealization required by the Cauchy problem generalizing the existing protocols for building coordinate system inside the future light-cone of a time-like observer. On each 3-space Σ τ we choose curvilinear 3-coordinates σ r having the observer as origin. These are the radar 4-coordinates σ A = (τ; σ r ). If x µ σ A (x) is the coordinate transformation from the Cartesian 4-coordinates x µ of a reference inertial observer to radar coordinates, its inverse σ A x µ = z µ (τ, σ r ) defines the embedding functions z µ (τ, σ r ) describing the 3-spaces Σ τ as embedded 3-manifold into Minkowski space-time. The induced 4-metric on Σ τ is the following functional of the embedding 4 g AB (τ, σ r ) = [z µ A η µν z ν B ](τ, σr ), where z µ A = zµ / σ A and 4 η µν = ɛ (+ ) is the flat metric (ɛ = ±1 according to either the particle physics ɛ = 1 or the general relativity ɛ = 1 convention). While the 4-vectors z µ r (τ, σ u ) are tangent to Σ τ, so that the unit normal l µ (τ, σ u ) is proportional to ɛ µ αβγ [z α 1 z β 2 z γ 3 ](τ, σ u ), we have z µ τ (τ, σ r ) = [N l µ + N r z µ r ](τ, σ r ) with N(τ, σ r ) = ɛ [z µ τ l µ ](τ, σ r ) and N r (τ, σ r ) = ɛ g τr (τ, σ r ) being the lapse and shift functions. The foliation is nice and admissible if it satisfies the conditions: 1) N(τ, σ r ) > 0 in every point of Σ τ (the 3-spaces never intersect, avoiding the coordinate singularity of Fermi coordinates); 2) ɛ 4 g ττ (τ, σ r ) > 0, so to avoid the coordinate singularity of the rotating disk, and with the positive-definite 3-metric 3 g rs (τ, σ u ) = ɛ 4 g rs (τ, σ u ) having three positive eigenvalues (these are the Møller conditions); 3) all the 3-spaces Σ τ must tend to the same space-like hyper-plane at spatial infinity (so that there are always asymptotic inertial observers to be identified with the fixed stars). These conditions imply that global rigid rotations are forbidden in relativistic theories. In Ref.(Alba and Lusanna, 2010) there is the expression of the admissible embedding corresponding to a 3+1 splitting of Minkowski space-time with parallel space-like hyper-planes (not equally spaced due to a linear acceleration) carrying differentially rotating 3-coordinates without the coordinate singularity of the rotating disk. It is the first consistent global noninertial frame of this type. Each admissible 3+1 splitting of space-time allows to define two associated congruences of time-like observers: a) the Eulerian observers with the unit normal to Σ τ as 4-velocity; b) the non surface forming observers with 4-velocity proportional to z µ τ (τ, σ r ). Therefore starting from the four independent embedding functions z µ (τ, σ r ) we obtain the ten components 4 g AB (τ, σ u ) of the 4-metric: they play the role of the inertial potentials generating the relativistic apparent forces in the non-inertial frame (the usual Newtonian inertial potentials can be recovered by doing the non-relativistic limit). The extrinsic curvature tensor 3 K rs (τ, σ u ) = [ 1 2 N (N r s + N s r τ 3 g rs )](τ, σ u ), describing the shape of the

210 instantaneous 3-spaces as embedded 3-sub-manifolds of Minkowski space-time, is a secondary inertial potential functional of the inertial potentials 4 g AB. III. CLASSICAL RELATIVISTIC ISOLATED SYSTEMS In these global non-inertial frames of Minkowski space-time it is possible to describe isolated systems (particles, strings, fields, fluids) admitting a Lagrangian formulation by means of parametrized Minkowski theories. The matter variables are replaced with new ones knowing the clock synchronization convention defining the 3-spaces Σ τ. For instance a Klein- Gordon field φ(x) will be replaced with φ(τ, σ r ) = φ(z(τ, σ r )); the same for every other field. Instead for a relativistic particle with world-line x µ (τ) we must make a choice of its energy sign: then the positive- (or negative-) energy particle will be described by 3-coordinates η r (τ) defined by the intersection of its world-line with Σ τ : x µ (τ) = z µ (τ, η r (τ)). Differently from all the previous approaches to relativistic mechanics, the dynamical configuration variables are the 3-coordinates η r (τ) and not the world-lines x µ (τ) (to rebuild them in an arbitrary frame we need the embedding defining that frame!). Then the matter Lagrangian is coupled to an external gravitational field and the external 4-metric is replaced with the 4-metric g AB (τ, σ r ) of an admissible 3+1 splitting of Minkowski space-time. With this procedure we get a Lagrangian depending on the given matter and on the embedding z µ (τ, σ r ), which is invariant under frame-preserving diffeomorphisms. As a consequence, there are four first-class constraints (an analogue of the super-hamiltonian and super-momentum constraints of canonical gravity) implying that the embeddings z µ (τ, σ r ) are gauge variables, so that all the admissible non-inertial or inertial frames are gauge equivalent, namely physics does not depend on the clock synchronization convention and on the choice of the 3-coordinates σ r. Even if the gauge group is formed by the frame-preserving diffeomorphisms, the matter energy-momentum tensor allows the determination of the ten conserved Poincare generators P µ and J µν (assumed finite) of every configuration of the system (in non-inertial frames they are asymptotic generators at spatial infinity like the ADM ones in GR). If we restrict ourselves to inertial frames, we can define the inertial rest-frame instant form of dynamics for isolated systems by choosing the 3+1 splitting corresponding to the intrinsic inertial rest frame of the isolated system centered on an inertial observer: the instantaneous 3-spaces, named Wigner 3-spaces due to the fact that the 3-vectors inside them are Wigner spin-1 3-vectors, are orthogonal to the conserved 4-momentum P µ of the configuration. The embedding corresponding to the inertial rest frame is z µ (τ, σ) = Y µ (τ) + ɛ µ r ( h) σ r, where Y µ (τ) is the Fokker-Pryce center-of-inertia 4-vector, h = P / ɛ P 2 and ɛ µ A=ν( h) is the standard Wigner boost for time-like orbits sending P µ = ɛ P 2 ( 1 + h 2 ; h) to (1; 0). In Ref.(Alba and Lusanna, 2010) there is also the definition of the admissible non-inertial rest frames, where P µ is orthogonal to the asymptotic space-like hyper-planes to which the instantaneous 3-spaces tend at spatial infinity. This non-inertial family of 3+1 splittings is the only one admitted by the asymptotically Minkowskian space-times without supertranslations in GR. Finally in Ref.(Alba, 2006) there is the definition of parametrized Galilei theories, non relativistic limit of the parametrized Minkowski theories. Also the inertial and non-inertial frames in Galilei space-time are gauge equivalent in this formulation.

211 IV. RELATIVISTIC CENTER OF MASS, PARTICLE WORLD-LINES AND BOUND STATES The framework of the inertial rest frame allowed the solution of the following old open problems: A) The classification (Alba, Lusanna and Pauri, 2002) of the relativistic collective variables (the canonical non-covariant Newton-Wigner center of mass (or center of spin), the non-canonical covariant Fokker-Pryce center of inertia and the non-canonical non-covariant Møller center of energy), replacing the Newtonian center of mass (and all tending to it in the non-relativistic limit), that can be built only in terms of the Poincare generators: they are non measurable quantities due to the non-local character of such generators (they know the whole 3-space Σ τ ). There is a Møller world-tube around the Fokker-Pryce 4- vector containing all the possible pseudo-world-lines of the other two, whose Møller radius ρ = S / ɛ P 2 ( S is the rest angular momentum) is determined by the Poincaré Casimirs of the isolated system. This non-covariance world-tube is a non-local effect of Lorentz signature of the space-time absent in Euclidean spaces. The world-lines x µ i (τ) of the particles are derived (interaction-dependent) quantities and in general they do not satisfy vanishing Poisson brackets (Alba, Crater and Lusanna, 2007): already at the classical level a non-commutative structure emerges due to the Lorentz signature! B) The description of every isolated system as a decoupled (non-measurable) canonical non-covariant (Newton-Wigner) external center of mass (described by frozen Jacobi data) carrying a pole-dipole structure: the invariant mass and the rest spin of the system expressed in terms of suitable Wigner-covariant relative variables of the given isolated system inside the Wigner 3-spaces (after the elimination of the internal center of mass with well defined rest-frame conditions). The invariant mass is the effective Hamiltonian inside these 3-spaces. C) The formulation of classical relativistic atomic physics (the electro-magnetic field in the radiation gauge plus charged scalar particles with Grassmann-valued electric charges to make a ultraviolet and infrared regularization of the self-energies and with mutual Coulomb potentials) and the identification of the Darwin potential at the classical level by means of a canonical transformation transforming the original system in N charged particles interacting with Coulomb plus Darwin potentials and a free radiation field (absence of Haag s theorem at least at the classical level). Therefore the Coulomb plus Darwin potential is the description as a Cauchy problem of the interaction described by the one-photon exchange Feynman diagram of QED (all the radiative corrections and photon brehmstrahlung are deleted by the Grassmann regularization). In Ref.(Lusanna, 2011) there are the references for all the interacting systems for which the explicit form of the interaction-dependent Lorentz boosts is known. V. RELATIVISTIC QUANTUM MECHANICS In Ref.(Alba, Crater and Lusanna, 2011) there is a new formulation of relativistic quantum mechanics in inertial frames englobing all the known results about relativistic bound states due to the use of clock synchronization for the definition of the instantaneous 3-spaces, which implies the absence of relative times in their description.

212 In Galilei space-time non-relativistic quantum mechanics, where all the main results about entanglement are formulated, describes a composite system with two (or more) subsystems with a Hilbert space which is the tensor product of the Hilbert spaces of the subsystems: H = H 1 H 2. This type of spatial separability is named the zeroth postulate of quantum mechanics. However, when the two subsystems are mutually interacting, one makes a unitary transformation to the tensor product of the Hilbert space H com describing the decoupled Newtonian center of mass of the two subsystems and of the Hilbert space H rel of relative variables: H = H 1 H 2 = H com H rel. This allows to use the method of separation of variables to split the Schroedinger equation in two equations: one for the free motion of the center of mass and another, containing the interactions, for the relative variables (this equation describes both the bound and scattering states). A final unitary transformation of the Hamilton-Jacobi type allows to replace H com with H com,hj, the Hilbert space in which the decoupled center of mass is frozen and described by non-evolving Jacobi data. Therefore we have H = H 1 H 2 = H com H rel = H com,hj H rel. While at the non-relativistic level these three descriptions are unitary equivalent, this is no more true in RQM. The non-local and non-covariant properties of the decoupled relativistic center of mass, described by the frozen Jacobi data z and h = P / ɛ P 2, imply that the only consistent relativistic quantization is based on the Hilbert space H = H com,hj H rel (in the non-relativistic limit it goes into the corresponding Galilean Hilbert space). We have H H 1 H 2, because, already in the non-interacting case, in the tensor product of two quantum Klein-Gordon fields, φ 1 (x 1 ) and φ 2 (x 2 ), most of the states correspond to configurations in Minkowski space-time in which one particle may be present in the absolute future of the other particle. This is due to the fact that the two times x o 1 and x o 2 are totally uncorrelated, or in other words there is no notion of instantaneous 3-space (clock synchronization convention). Also the scalar products in the two formulations are completely different. We have also H H com H rel, because if instead of z = Mc x NW (0) we use the evolving (non-local and non-covariant) Newton-Wigner position operator x NW (τ), then we get a violation of relativistic causality because the center-of-mass wave packets spread instantaneously as shown by the Hegerfeldt theorem. Therefore the only consistent Hilbert space is H = H com,hj H rel. The main complication is the definition of H rel, because we must take into account the three pairs of (interaction-dependent) second-class constraints eliminating the internal 3-center of mass inside the Wigner 3-spaces. When we are not able to make the elimination at the classical level and formulate the dynamics only in terms of Wigner-covariant relative variables, we have to quantize the particle Wigner-covariant 3-variables ηi r, κ ir and then to define the physical Hilbert space by adding the quantum version of the constraints a la Gupta-Bleuler. Relativistic quantum mechanics in rotating non-inertial frames by using a multi-temporal quantization scheme is defined in Ref.(Alba and Lusanna, 2006). In it the inertial gauge variables are not quantized but remain c-numbers; the known results in atomic and nuclear physics are reproduced.

213 VI. RELATIVISTIC ENTANGLEMENT AND LOCALIZATION PROBLEMS Since we have that the Hilbert space H = H com,hj H rel is not unitarily equivalent to the one H 1 H 2..., where H i are the Hilbert spaces of the individual particles, at the relativistic level the zeroth postulate of non-relativistic quantum mechanics does not hold (the necessity to use H rel implies a type of weak form of relationism different from the formulations connected to the Mach principle). Since the Hilbert space of composite systems is not the tensor product of the Hilbert spaces of the sub-systems, we need a formulation of relativistic entanglement taking into account this spatial non-separability and the nonlocality of the center of mass, both coming from the Poincare group, i.e. from the Lorentz signature of space-time. Since the center of mass is decoupled, its non-covariance is irrelevant. However its nonlocality is a source of open problems: do we have to quantize it (in the preferred momentum basis implied by the quantum Poincaré algebra) and if yes is it meaningful to consider centerof-mass wave packets or we must add some super-selection rule? The wave function of the non-local center of mass is a kind of wave function of the 3-universe: who will observe it? As a consequence in SR there are open problems on which type of relativistic localization is possible. There are strong indications that the Newton-Wigner position operator cannot be self-adjoint but only symmetric with the implication of a bad localization of relativistic particles. The existing problems with relativistic position operators (like the Newton-Wigner one), deriving from the Lorentz signature of Minkowski space-time which forbids the existence of a unique collective variable with all the properties of Newton center of mass, point towards a non-measurability of absolute positions (but not of the relative variables needed to describe the spectra of bound states). This type of un-sharpness should be induced also in non-relativistic quantum mechanics: in atomic physics the crucial electro-magnetic effects are of order 1/c. Experiments in atomic and molecular physics are beginning to explore these localization problems, in frameworks dominated by a Newtonian classical intuition and taking into account the experimental quantum limits for atom localization. Therefore SR introduces a kinematical non-locality and a kinematical spatial nonseparability, which reduce the relevance of quantum non-locality in the study of the foundational problems of quantum mechanics. Relativistic entanglement will have to be reformulated in terms of relative variables also at the non-relativistic level. Therefore the control of Poincare kinematics will force to reformulate the experiments connected with Bell inequalities and teleportation in terms of the relative variables of isolated systems containing: a) the observers with their measuring apparatus (Alice and Bob as macroscopic quasi-classical objects); b) the particles of the protocol (but now the ray of light, the photons carrying the polarization, move along null geodesics); c) the environment (macroscopic either quantum or quasi-classical object). VII. OPEN PROBLEMS The main open problem in SR is the quantization of fields in non-inertial frames due to the no-go theorem of Ref.(Torre and Varadarajan, 1999) showing the existence of obstructions to the unitary evolution of a massive quantum Klein-Gordon field between two space-like surfaces of Minkowski space-time. Its solution, i.e. the identification of all the 3+1 splittings allowing unitary evolution, will be a prerequisite to any attempt to quantize canonical gravity

214 taking into account the equivalence principle (global inertial frames do not exist!). Moreover entanglement in non-inertial frames without Rindler observers is still to be formulated. Alba.D (2006), Quantum Mechanics in Noninertial Frames with a Multitemporal Quantization Scheme: II. Nonrelativistic Particles, Int.J.Mod.Phys. A21, 3917 (hep-th/ ). Alba D., Crater H.W. and Lusanna L. (2001), The Semiclassical Relativistic Darwin Potential for Spinning Particles in the Rest Frame Instant Form: Two-Body Bound States with Spin 1/2 Constituents, Int.J.Mod.Phys. A16, 3365 (arxiv hep-th/ ). Alba D., Crater H.W. and Lusanna L. (2007), Hamiltonian Relativistic Two-Body Problem: Center of Mass and Orbit Reconstruction, J.Phys. A40, 9585 (arxiv gr-qc/ ). Alba D, Crater H.W. and Lusanna L. (2010), Towards Relativistic Atom Physics. I. The Rest- Frame Instant Form of Dynamics and a Canonical Transformation for a system of Charged Particles plus the Electro-Magnetic Field, Canad.J.Phys. 88, 379 (arxiv ) and II. Collective and Relative Relativistic Variables for a System of Charged Particles plus the Electro- Magnetic Field, Canad.J.Phys. 88, 425 (arxiv ). Alba D., Crater H.W. and Lusanna L. (2011), Relativistic Quantum Mechanics and Relativistic Entanglement in the Rest-Frame Instant Form of Dynamics, J.Math.Phys. 52, (arxiv ). Alba D. and Lusanna L. (1998), The Lienard-Wiechert Potential of Charged Scalar Particles and their Relation to Scalar Electrodynamics in the Rest-Frame Instant Form, Int.J.Mod.Phys. A13, 2791 (arxiv hep-th/ ). Alba D. and Lusanna L. (2006), Quantum Mechanics in Noninertial Frames with a Multitemporal Quantization Scheme: I. Relativistic Particles, Int.J.Mod.Phys. A21, 2781 ( arxiv hep-th/ ). Alba D. and Lusanna L. (2007), Generalized Radar 4-Coordinates and Equal-Time Cauchy Surfaces for Arbitrary Accelerated Observers, Int.J.Mod.Phys. D16, 1149 (arxiv gr-qc/ ). Alba D. and Lusanna L. (2010), Charged Particles and the Electro-Magnetic Field in Non- Inertial Frames: I. Admissible 3+1 Splittings of Minkowski Spacetime and the Non-Inertial Rest Frames, Int.J.Geom.Methods in Physics 7, 33 (arxiv ) and II. Applications: Rotating Frames, Sagnac Effect, Faraday Rotation, Wrap-up Effect, Int.J.Geom.Methods in Physics, 7, 185 (arxiv ). Alba D., Lusanna L. and Pauri M. (2002), Centers of Mass and Rotational Kinematics for the Relativistic N-Body Problem in the Rest-Frame Instant Form, J.Math.Phys. 43, (arxiv hep-th/ ). Crater H.W. and Lusanna L. (2001), The Rest-Frame Darwin Potential from the Lienard- Wiechert Solution in the Radiation Gauge, Ann.Phys. (N.Y.) 289, 87. Lusanna L. (1997), The N- and 1-Time Classical Descriptions of N-Body Relativistic Kinematics and the Electromagnetic Interaction, Int.J.Mod.Phys. A12, 645. Lusanna L. (2011), Canonical Gravity and Relativistic Metrology: from Clock Synchronization to Dark Matter as a Relativistic Inertial Effect (arxiv ). Torre C.G. and Varadarajan M. (1999), Functional Evolution of Free Quantum Fields, Class. Quantum Grav. 16, 2651.

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235 TWO-DIMENSIONAL ANHARMONIC CRYSTAL LATTICES: SOLITONS, SOLECTRONS, AND ELECTRIC CONDUCTION Manuel G. Velarde, Werner Ebeling, Alexander P. Chetverikov, Instituto Pluridisciplinar, Universidad Complutense de Madrid (Spain) Abstract Reported here are salient features of soliton-mediated electron transport in anharmonic crystal lattices. After recalling how an electron-soliton bound state (solectron) can be formed we comment on consequences like electron surfing on a sound wave and ballistic transport, possible percolation in 2d lattices, and a novel form of electron pairing with strongly correlated electrons both in real space and momentum space. 1. Introduction Electrons, holes, or their dressed forms as quasiparticles, in the approach introduced by Landau (Landau 1933, Kaganov 1979), play a key role in transferring charge, energy, information or signals in technological and biological systems (Slinker 2011). Engineers have invented ingenious methods for, e.g., long range electron transfer (ET) such that an electron and its carrier, forming a quasiparticle, go together all along the path hence with space and time synchrony. Fig. 1 illustrates the simplest geometry between a donor (D) and an acceptor (A). Velocities reported are in the range of sound velocity which in bio-systems or in GaAs layers are about Angstrom/picosecond (Km/s). Such values are indeed much lower than the velocity of light propagation in the medium. Thus at first sight, leaving aside a deeper discussion concerning specific purposes (Pomeau 2007, Chetverikov 2012a), controlling electrons seems to be more feasible with sound (or even supersonic) waves than with photons. Electron surfing on an appropriate highly monochromatic, quite strong albeit linear/harmonic wave has recently being observed (Hermelin 2011, McNeil 2011). Earlier the present authors have proposed the solectron concept as a new quasiparticle (Velarde 2005, 2006, 2008a, 2008b, 2010a, 2010b, Chetverikov 2006a, 2009, 2010, 2011a, 2011b, Cantu Ros 2011, Hennig 2006, Ebeling 2009) encompassing lattice anharmonicity (hence invoking nonlinear elasticity beyond Hooke s law) and (Holstein-Fröhlich) electron-lattice interactions thus generalizing the polaron concept and quasiparticle introduced by Landau and Pekar (Landau 1933, Pekar 1954, Devreese 1972). Anharmonic, generally supersonic waves are naturally robust due to, e.g., a balance between nonlinearity and dispersion (or dissipation). In the following Sections we succinctly describe some of our findings and predictions for one-dimensional (1d) crystal lattices for which exact analytical and numerical results exist (Section 2) and, subsequently, for two-dimensional (2d) lattices for which only numerical results are available (Sections 3, 4 and 5). Comments about theory and experiments are provided in Section 6 of this text. Figure 1: Electron transfer from a donor (D) to an acceptor (A) along a 1d crystal lattice. The springs mimic either harmonic interactions or otherwise. In this text they are assumed to correspond to (anharmonic) Morse potentials. The figure also illustrates the electron-soliton bound state (solectron) formation. Depending on the material ways other than photoexcitation at the donor site could lead to the same consequences. 2. Soliton assisted electron transfer in 1d lattices

236 Although the basic phenomenological theory exists (Marcus 1985) yet long range ET (beyond 20Å) in biomolecules is an outstanding problem (Gray 2003, 2005). Recent experiments by Barton and collaborators (Slinker 2011) with synthetic DNA show an apparent ballistic transport over 34 nm for which no theory exists. Let us see how we can address this question building upon our solectron concept (Fig. 1). We consider the 1d-crystal lattice with anharmonic forces described by the following Hamiltonian H lattice 2 Mvn = + D n n n 2 ( 1 exp[ B( x x )] ) 1 2 σ, (1) where x n, v n, M, D, B and σ denote, respectively, space lattice coordinates/sites, lattice particle/unit velocities, unit masses (all taken equal), the potential depth or dissociation energy of the Morse potential (akin to the 12-6 Lennard-Jones potential), lattice stiffness constant and interparticle equilibrium distance or initial lattice spacing. For our purpose here we introduce q = B x nσ. Around the minimum of the suitably rescaled relative lattice displacements, ( ) 2 potential well we can define ( ) 1 2 n n ω 0 = 2DB M as the linear (harmonic) vibration frequency. For 13 1 biomolecules like azurin ω 0 = 10 s, M 100 amu. Then for D = 0. 1 ev ( D = 1. 0 ev ) we can set B = 2. 3Å 1 ( B = Å 1 ) (Velarde 2010b). If an excess electron is added to the lattice we can take it in the tight binding approximation (TBA) and hence * electron = En CnCn n n * * ( q )( C C C C ) H V, (2) n, n 1 k n n 1 + with n denoting the lattice site where the electron is placed (in probability density, 2 wn = Cn, wn = 1). We want to emphasize the significance of hopping in the transport process n relative to effects due to onsite energy shifts and hence we assume E n = E0 for all n save those referring to D and A. The quantities V n, n 1 belong to the transfer matrix or overlapping integrals. They depend on actual relative lattice displacements, and we can set (Slater 1974) [ ( q q )] n, n 1 = V0 exp n n 1 n n 1 V α, (3) where V 0 and α account for the electron-lattice coupling strength. Accordingly, τ = V0 hω0 provides the ratio of the two dynamical time scales (electronic over mechanical/sound). From (1)-(3) follow the equations of motion in suitable dimensionless form: d 2 qn 2 dt + 2αV ( q ) [ ] ( ) ( ) n qn+ 1 qn qn+ 1 qn 1 qn = 1 e e 1 e [ ] ( q ) n 1 qn e * α ( qn qn+ 1) * α ( qn 1 q ) [ Re( C C ) e Re( C C ) e ] n dc dt n+ 1 n n n 1 α ( qn qn+ 1) α ( qn qn ) iτ [ C e + C e ]. (5) n 1 = n+ 1 n 1 It is worth recalling that if rather than the Morse potential (1) we use a similar potential introduced by Toda the lattice dynamic problem defined by Eqs. (4) in the absence of the added electron ( α = 0) is exactly solvable (Toda 1989, Chetverikov 2006b). Thus we know analytical expressions for lattice motions and, moreover, for the thermodynamics/statistical mechanics (including specific heats, dynamic structure factor, etc.) of such 1d many-body problem. For the Morse potential (1) it has been numerically shown that no significant differences exist for lattice motions and other physical quantities (Dancz 1977, Rolfe 1979). Temperature can be incorporated in the dynamics by adding to Eqs. (4) Langevin sources by using an appropriate heat bath (delta-correlated Gaussian + (4)

237 white noise) and using Einstein s relation between noise strength and temperature. To avoid redundancy we illustrate this point in Sect. 3. The implementation of the scheme shown in Fig. 1 is one prediction with velocities in the sonic and supersonic range. Fig. 2 illustrates the possibility using Eqs. (4)-(5) of extracting an electron placed in a potential well in the 1d Morse lattice by a generally supersonic soliton. For the geometry of Fig. 1 we can use it to estimate the ballistic process time lapse to go from the donor to the acceptor. For the computation with a lattice of N = 100 units the well is assumed Gaussian of depth E (in units of h ω0 ) with E < O localized at site 50. The soliton initially spans a few lattice sites (two or three) excited at site 40. If the well depth is shallow enough the extraction is ensured up to 100% whereas if the well is too deep no extraction occurs. Needless to say extraction is possible with probability varying from zero to unity as the well depth is decreased. Time lapse from D to A is obtained by simply dividing length over soliton speed. Illustration is provided in Fig. 3 where l (see Fig. 1) accounts for the distance travelled (in principle from D to an appropriately placed acceptor A). Comparison is provided between the ballistic case and other possibilities like diffusion-like transport with thermally (hence randomly) excited solitons (Chetverikov 2010) and tunneling transport (Chetverikov 2012b)... Figure 2: Extraction of an electron from a potential well (a donor) and ballistic transport to an acceptor 2 observed using the electron probability density C n. Left panel: shallow well E = 10, extraction 100%. Right panel: deep well E = 18, no extraction. Parameter values: α = 1.75, V 0 = 0.35 and τ = 10. Figure 3: Logarithm of reciprocal time lapse (in seconds) which an electron bound to a soliton needs to travel a distance l (in Angstrom) for the geometry of Fig. 1. The upper dotted (blue) curve corresponds to a sound velocity of 17 Angstrom/ps, illustrating a ballistic transport. The second dotted (green) curve from above shows the reciprocal time needed if the electron hops stochastically between thermally excited solitons. The bottom solid line embraces data illustrating a tunneling process. The dots are reciprocal times measured for natural bio-molecules (Gray 2003, 2005). The transfer times found for synthetic DNA are much shorter (Slinker 2011) bearing similarity to our model findings upper dotted (blue) line for solectron transfer.

238 3. Two-dimensional crystal lattices Recently, two groups of experimentalists have observed how an electron can surf on a suitably strong albeit linear, highly monochromatic sound wave (in GaAs layers at 300 mk). Sound demands lattice compressions and hence is accompanied by electric/polarization fields which for piezoelectric crystals integrate to macroscopic level. Our theoretical solectron approach targets sound-wave electron surfing due to nonlinear soliton excitations in 2d-anharmonic crystal lattice layers, with velocities ranging from supersonic to sub-sonic (Chetverikov 2006b, Hennig 2006, Makarov 2006). We do not pretend here to explain the GaAs experimental results. We simply wish to point out that appropriate sound waves in suitable nonlinear crystalline materials, could provide long range ET in 2d, with sonic or supersonic velocities for temperatures much higher than that so far achieved in experiments. In 2d the Morse potential needs to be truncated to avoid overcounting lattice sites. Then using complex coordinates Z = x + iy, where x and y are Cartesian coordinates, the equations of motion replacing Eqs. (4) are d 2 Z n dz = F 2 nk nk nk v nx ny k dt dt F Z dv dr = with nk ( nk ) ( ) r Znk z n ( Z ) z + γ + 2D ( ξ + iξ ) =, nk ( n k ) n k, (6) = Z Z Z Z. In Eqs. (6) we have incorporated thermal effects. The quantities γ (friction coefficient), D ν (diffusion coefficient) and the ξ s (noise generators) characterize the Gaussian noise. Dv = k BTγ M is Einstein s relation with k B, Boltzmann constant. To illustrate lattice motions we consider each lattice unit as a sphere representing the core electron Gaussian distribution at the corresponding site: ρ ( Z, t) = 2 2 exp Z Z ( ) 2 j t λ with λ Z Z ( t ) < 1.5 i a parameter. Thus overlapping of two such Gaussians permit to detect the expected mechanical compression of two lattice units as Fig. 4 illustrates (Chetverikov 2011c, 2011d). The evolution of the electron follows Eq. (5) for the 2d lattice geometry. Figure 4: Cumulative sequence of snapshots using ρ ( Z,t) to track a soliton running along the x-axis of a triangular lattice using Eqs. (4), (6). Parameter values: Bσ = 4, T = Two-dimensional crystal lattices. Pauli s master equation approach Continuing with the 2d case, we now consider an alternative approach to using the Schrödinger equation (5). We shall consider how transport is achieved following Pauli s master equation Z. The energy levels are taken in V nn approach (Ebeling 2009). Eq. (2) is now considered with ( ) n,n'

239 2 4 the polarization approximation 2 2 E n = E0 U eh Z n, n' + h, where U e is the electric n potential strength and h defines the range of the electric field polarization interaction. Rather than relying on the Schrödinger description of the TBA we follow Pauli s master equation approach with transition probabilities W 2 ( t ) exp [ 2α Z ] E( n, ; β ) n, n = 0 n, n' n h, (7) dw dt [ W w W w ] n = nn n n' n n ', (8) where E ( n, n ; β ) = 1 if E n < En and E( n, n ; β ) = exp[ β ( E n En' )] if E n En (7)-(8) are solved with Eqs. (6) to obtain the electron probability density ( t) >, β = 1 T. Eqs. k B w n neglecting the feedback of the electron on the lattice dynamics. Figs. 5 and 6 illustrate electron and solectron evolution along a 2d lattice. Fig. 5 refers to electron taken alone while Fig. 6 illustrates how, after switching-on the electron-lattice interaction, the soliton from Fig. 4 is able to trap the electron from Fig. 5 and after forming the solectron transports charge along the lattice (see also Velarde 2008a). Figure 5: An electron alone placed at a given lattice site (left panel). The quantity w here accounts for the 2 probability density (otherwise C n in Fig. 3). As time progresses the electron spreads over the slightly T = D following Pauli s equation from the initial condition (left panel) to a subsequent time instant (right panel). heated lattice ( ) Figure 6: Solectron formation and eventual evolution when the electron-phonon (here electron-soliton) interaction is switched-on ( α 0 in the corresponding to Eqs. (4) for the 2d case). We see that the electron is trapped by a soliton (like that of Fig. 4) thus forming the solectron which transfers the electron probability density without spreading at variance with the result illustrated in Fig. 5 (right panel).

240 5. Percolation and other features in 2d-lattices Solitons can be excited in a crystal lattice by several actions. One is to add finite momentum to a group of nearby lattice units, another is by heating the crystal all-together. Then one expects quite many excitations including phonons and solitons randomly appearing along the 2d lattice and having finite life times thus leaving finite-length traces. Fig. 7 illustrates thermal excitations leading n e x, y; t due to higher than equilibrium electric/polarization to spots of instant electron density ( ) field maxima. Here in the simplest Boltzmann approximation 0 nel ( Z x, y ; t) = { exp [ U ( Z, t) k BT ] nel }, (9) 0 0 With n el the normalizing factor, nel = exp[ U ( Z, t) k BT ]dz. Only at temperatures high enough one expects a distribution of local spots permitting in kind of zig-zag the occurrence of an infinite path thus percolating from side to side of the 2d lattice (Chetverikov 2009, 2011a). Indeed by increasing temperature one increases the significance as well as the density of soliton excitations/traces. If percolation does occur by adding an excess electron and playing with an external field we have a novel way of one-sided electric conduction mediated by the solitons. We have just explored this possibility but have not yet been able to draw conclusions about the scaling laws of the process. On the other hand since percolation is expected as a second-order phase transition it seems worth investigating the possible connection with the pseudo-gap transition observed in such superconducting materials as cuprates. Figure 7: Towards percolation. Instantaneous space distribution of electron probability density n ( x, y) associated to lattice solitons (sound) in a triangular Morse lattice (N = 100) at, respectively, low (T = 0.02 D) (left panel) and high (T = 0.4 D) (right panel) temperatures. The latter exhibits an almost percolating path. Parameter values: Bσ = 3, U e = 0.4D, h = 0.7σ. 6. Concluding remarks We have illustrated how lattice solitons arising from finite amplitude compression-expansion longitudinal motions bring sound and also create electric polarization fields (Landau 1933). The latter are able to trap charges and provide long-range ET in a wide range of temperatures (up to e.g. 300K for bio-molecules). Such sound waves could exhibit subsonic, sonic or supersonic velocity, whose actual value depends on the strength of the electron-phonon/soliton interaction. Noteworthy is that such interaction and subsequent electric transport, in the most general case, embraces a genuine polaron effect (Pekar 1954, Devreese 1972) and also a genuine soliton/solectron effect (Cantu Ros 2011). For piezoelectric materials like GaAs that sound waves can transport electrons there is now experimental evidence (Hermelin 2011, McNeil 2011). This was achieved by means of strong albeit linear/infinitesimal, highly monochromatic waves appropriately creating the electric/polarization field that due to the specificity of the crystal symmetry and other features integrate to macroscopic level. These experiments done at 300 mk due to quantum limitations imposed to the set-up provide hope for similar long-range ET at high temperatures. Indeed the limitations are only due to the electron entry and exit/detector gates. The solectron theory predicts such a possibility in appropriate non-linearly elastic crystal materials

241 capable of sustaining lattice solitons. Recent experiments using synthetic DNA (Slinker 2011) show a kind of ballistic ET over 34 nm which as Fig. 3 illustrates bears similarity with a prediction of our solectron theory (Velarde 2008a). In 2d crystal lattices the solectron theory predicts the possibility of percolation as a way of long range charge transport when the material is heated up to the range of robustness/stability of lattice solitons, as Fig. 7 illustrates. Work remains to be carried out to assess the corresponding percolation scaling laws. Finally, we have recently shown that the solectron theory offers a new way of electron pairing by having two electrons strongly correlated (both in real space and in momentum space with due account of Pauli s exclusion principle and Coulomb repulsion using Hubbard s local approximation) due to their trapping by lattice solitons (Velarde 2006, 2008c, 2011a, 2011b, 2011c, Hennig 2008, Brizhik 2012). This feature shows the quite significant role played by the lattice dynamics well beyond the role played in the formation of Cooper pairs (in momentum space) underlying the BCS theory (Cooper 2011) or in the bipolaron theory (Alexandrov 2007) and much in the spirit of Fröhlich approach to the problem unfortunately using a harmonic lattice Hamiltonian at a time before (lattice) solitons were known (Fröhlich 1950, 1952, 1954a, 1954b, Zabusky 1965, 2005, Toda 1989; see also Nayanov 1986). Incidentally, Einstein (1922) was the first who used the concept of molecular conduction chains trying to understand superconduction. Thus it is reasonable to expect that a soliton-mediated Bose-Einstein condensation could take place in appropriate 2d anharmonic crystal lattices well above absolute zero. This is yet to be shown. Acknowledgments The authors are grateful to A. S. Alexandrov, E. Brändäs, L. Brizhik, L. Cruzeiro, F. de Moura, J. Feder, D. Hennig, R. Lima, R. Miranda, R. McNeil, D. Newns, G. Röpke and G. Vinogradov for enlightening discussions. This research has been sponsored by the Spanish Ministerio de Ciencia e Innovación, under Grants EXPLORA FIS and MAT References Alexandrov A S (Ed.) (2007) Polarons in Advanced Materials, Springer, Dordrecht Brizhik L, Chetverikov A P, Ebeling W, Röpke G and Velarde M G (2012) Electron pairing and Coulomb repulsion in one-dimensional anharmonic lattices (in preparation) Cantu Ros O G, Cruzeiro L, Velarde M G and Ebeling W (2011) On the possibility of electric transport mediated by long living intrinsic localized solectron modes, Eur. Phys. J. B 80, Chetverikov A P, Ebeling W and Velarde, M G (2006a) Nonlinear excitations and electric transport in dissipative Morse-Toda lattices, Eur. Phys. J. B 51, Chetverikov A P, Ebeling W and Velarde, M G (2006b) Disipative solitons and complex currents in active lattices, Int. J. Bifurcation Chaos 16, Chetverikov A P, Ebeling W and Velarde, M G (2009) Local electron distributions and diffusion in anharmonic lattices mediated by thermally excited solitons, Eur. Phys. J. B 70, Chetverikov A P, Ebeling W and Velarde, M G (2010) Thermal solitons and solectrons in nonlinear conducting chains, Int. J. Quantum Chem. 110, Chetverikov A P, Ebeling W and Velarde, M G (2011a) Soliton-like excitations and solectrons in two dimensional nonlinear lattices, Eur. Phys. J. B 80, Chetverikov A P, Ebeling W, Röpke G and Velarde, M G (2011b) Hopping transport and stochastic dynamics of electrons in plasma layers, Contrib. Plasma Phys. 51, Chetverikov A P, Ebeling W and Velarde, M G (2011c) Localized nonlinear, soliton-like waves in twodimensional anharmonic lattices, Wave Motion 48, Chetverikov A P, Ebeling W and Velarde, M G (2011d) Propeties of nano-scale soliton-like excitations in twodimensional lattice layers, Physica D 240, Chetverikov A P, Ebeling W and Velarde, M G (2012a) Controlling electron transfer at the nano-scale by soliton-like excitations in two-dimensional nonlinear lattices (in preparation) Chetverikov A P, Cruzeiro L, Ebeling W and Velarde, M G (2012b) Studies on electron transfer and tunneling in inhomogeneous nonlinear lattices (in preparation) Cooper L N and Feldman D (2011) BCS: 50 years, World Scientific, London Dancz J and Rice S A (1977) Large amplitude vibrational motion in a one dimensional chain: Coherent state representation, J. Chem. Phys. 67, Devreese J T L (Ed.) (1972) Polarons in Ionic Crystals and Polar Semiconductors, North Holland, Amsterdam

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243 FFP Udine November 2011 THEORIES AS CRUCIAL ASPECTS IN QUANTUM PHYSICS EDUCATION Marco Giliberti, Physics Department University of Milan Abstract A lot of difficulties aroused in interpreting quantum physics from its very beginnings up today are still at the core of most educational presentations. In fact, usually people try to grasp theoretical concepts by referring to a blend of ideas taken from scientific, pre-scientific and common sense schemes. In this paper a pre-condition for every educational approach will be proposed: to rigorously keep to quantum mathematical formalism in order to understand the meaning and the "reality" of quantum physics. It will be argued that the addition to quantum theories of most extraneous concept or common sense scheme comes from an ambiguous idea of nature, scope and aims of science itself. 1. Introduction Many difficulties faced by most educational reconstruction of quantum theories have epistemological reasons, mainly embedded in the so called paradoxes of the quantum world. Even the educational paths that do not follow a historical (more often a pseudo-historical) approach, often lack a crucial point: that the meaning and the "reality" of quantum physics, just as the meaning of every physics topics, must be read from the theory, i. e.: the meaning and the concept of force should to be stressed from newtonian mechanics and the "reality" of the electric field should, in physics, be understood by Maxwell equations. In this paper the lack of consciousness about the proper and exclusive nature of physical theories is related to thess well known difficulties and it is proposed that, in order to avoid many misinterpretations, which especially come by connecting quantum theories with classical physics concepts in wrong ways and by trying to reduce it to common sense schemes, a pre-condition is crucial: in quantum physics education please rigorously keep to quantum mathematical formalism (that is that of Quantum Mechanics or that Quantum Field Theory) and its interpretation.. 2. Nature of difficulties: theories First of all our language describes an image. [...] It's clear that, if we want to understand the meaning of what we say, we must explore the image. But the image seems to save us this effort; it already hints at a determined use. So it mocks at us (Wittgestein 1967). The undetermined mess" to which we give the name "reality" is subjected to continuous changes because the status of the supposed entities that should form this "reality" is very flexible. Important and very known examples of these changes are the idea of absolute time, that of a luminiferous ether as a medium for the electromagnetic waves and the very idea and structure of atoms (Bellone, 2006). In fact, it is not possible to separate the object of knowledge from the instrument of knowledge: they must be considered as a whole. And this aspect is one of the main teaching of quantum physics from its very beginning. Theories are but mental constructions that help us find and define reality and utilize its resources. Physics inquiry begins with schemas and concepts that need not to be explained by theories; they come from common notions and language. Then proceeds introducing other concepts by means of what we can call "pretheories", that is already known physics theories stated as granted. Pretheories are unavoidable features and, for instance they are at the basis of our understanding and designing of measuring devices. Finally, our inquiry arrives at physics theories, that are determined by the basic concepts introduced before (common schemas plus pre-theories) plus formalized well defined new disciplinary concepts (Ludwig 2008). Actually we can imagine facts like icebergs, submerged under the surface of the sea of immediate experience that is perceived through common schemas. The submerged part of these facts can only be hypothesized, or, in a sense, imagined. A coherent and formalized imagination of this under-the-sea level of reality gives rise to physics theories. We can even think of these imagined realities as fairy tales; in this sense a scientific explanation is a story about how some entities, that are imagined but considered as real, would, by their very hypothesized nature, have worked together to generate the phenomenon to be explained (Ogborn 2010). A very simple and clear example of what is stated above about the construction of useful entities in theories as fairy tales

245 FFP Udine November 2011 In my opinion the framework of quantum field theory is best suited than quantum mechanics for quantum physics education in general and, more specifically at high school. One of the reason is that only in quantum electrodynamics the concept of photon can be well defined. At the University of Milano we have been working on this subject since 1995 and many encouraging results have come (Giliberti 1996; Giliberti 1997a; Giliberti 1997b; Giliberti 1997c; Giliberti 1998; Bergomi 2001; Giliberti 2002a; Giliberti 2002b; Bartesaghi 2004;Giliberti 2004a; Giliberti 2004b,; Giliberti 2007; Giliberti 2008). Anyway in general, whatever reference theory is chosen, in physics education one has to try to avoid misunderstanding of ideas and words that are used by the theory, but come from preceding conceptions (pre-theories, or even common sense) rooted in the biased idea that physics reality can be identified even out of a formal theory. A clear example of this can be seen in the idea of particle in quantum physics. When we speak of particles we should clearly state what we mean, what the theory allows us to think of, and what it does not. We must be careful that with that word we do not implement the idea that particles indicate physics entities in the sense of an ingenuous realism, and that these (ingenuous) entities coincide with the quanta of the theory. As it has been previously pointed out, in scientific construction, reality comes out of a set of coherent interpretations of the formalism of the reference theory. In this sense the common word "particle" is nothing but a useful metaphor of what is meant by the theory. In quantum mechanics particles of the same "kind" are identical, not only because they have the same charge, the same mass, the same spin,... but also because they are indistinguishable even through their position. They are identical because they have the same physical properties. We could think of a system with two electrons of different energy. In this situation we can say that one electron has a certain energy while another one has another energy, but it could be impossible to answer to the question "Which electron has which energy?". In more formal terms, the wave function obtained by the exchange of this two particle would yield the same previsions for the measurements of every observable. It is thus clear that the intuitive semantic content of the word "particle" given to the quantum mechanics quanta is, in general not adequate. From an educational point of view I believe it could be much clearer if we spoke of quanta as linked to the excitations of the normal mode, as it is done in quantum field theory, in this way it would be evident that they are identical and indistinguishable... and have little to do with the "usual" particles. It is the event of revelation of a quantum in a device that drives us to use the word "particle", giving a metaphorical sense to a word coming from classical physics (and from common language). If one strictly follows the guide of a theory most of the paradoxes (in quantum physics more or less all coming from wave/particle dualism) become not so central, with a great help for teaching. 5. A more deep difficulty I'm not claiming that keeping in close touch with a quantum theory all difficulties run away. In fact an objective problem remains and it comes from the theory itself. In formulating quantum mechanics (and even quantum theory of fields) the world must be split in two parts; in this way we have a dichotomy: a microscopic quantum physics description for the system we are studying and a macroscopic classical description for measuring devices; experimental context and results must be described in classical terms. Quantum mechanics cannot even be formulated without this distinction. The problem is that the theory gives no indication on how "to cut the world". Quite obviously to get information from a microscopic experiment we must produce an amplification process that leads to a macroscopic change; but, as even the apparatus are in principle describable in terms of quantum mechanics, this leads us to an aporia. Is there a macroscopicity parameter? Not in the theory. Nor it seems in experiments (see for instance experiments of diffraction of macromolecules that show their wavelike proprieties (Arndt 1999) or the quantum macroscopic proprieties of a superconductor). So some deep difficulties are still rooted in quantum mechanics itself: there's really no need of making educational path that instead of presenting difficulties where they are, generate confusion mixing aspects, ideas and words coming from a too ingenuous vision of reality.

246 FFP Udine November Conclusions In conclusion this paper is a call to realism. I would like to stress that, as we already (some time unconsciously) do for classical physics, when dealing with quantum physics education we keep closely to a specific formulation of the theory. We can choose among one of the many formulation of quantum mechanics or one of the formulation of quantum filed theory (we in Milan suggest the latter, for his more easily grasped epistemology, linked to specific space time field instead of the configuration space wave function of quantum mechanics).the path to follow and the results obtained in experimentations are but a secondary problem in this perspective: they come after. Historical or conceptual presentations and educational reconstructions of the topic cannot in our opinion skip this point, as instead many times they do keeping an eye closer to the path than to the goal. References Arndt M, Olaf Nairz O, Voss-Andreae J, Keller C,Gerbrand van der Zouw G, Zeilinger A (1999), Waveparticle duality of C60, Nature 401, , 14. Bartesaghi P, Camera F, Cazzaniga L, Costigliolo M, Giliberti M, Labanca I (2004), Alpha Natural Radioactivity in the Air: an Experiment for High-School Students, Proceedings GIREP Conference Teaching and Learning Physics in new Contexts, E. Mechlovà (editor), University of Ostrava (Ostrava), ISBN Bellone E (2006), L'origine delle teorie, Codice, 118. ISBN Bergomi N, Giliberti M (2001), Modelling Rutherford Scattering: A Hypertext for High-School Teachers Nettraining, International Conference Physics Teacher Education Beyond Selected Contributions, International ICPE-GIREP Conference, Barcellona 28 Agosto-1 Settembre 2000 R. Pinto & S. Surinach (eds), Elsevier Editions, Parigi, ISBN Cavallini G, Giliberti M (2008), La lezione della fisica quantistica, Epistemologia (XXXI), Giliberti M (1997a), Teaching about Heisenberg s Relations ; Proceedings of the G.I.R.E.P. I.C.P.E. International Conference: New Ways of Teaching Physics Ljubljana agosto 1996;, Board of Education of Slovenia, Lubiana (1997), ISBN X Giliberti M (1997c): Popularisation or Teaching? How much Math in Physics Courses? ; Proceedings of the G.I.R.E.P. I.C.P.E. International Conference: New Ways of Teaching Physics Ljubljana August 1996; Board of Education of Slovenia, Lubliana (1997), pag ISBN X. Giliberti M (1998), Il ruolo della matematica nell insegnamento della Fisica Moderna, La Fisica nella Scuola, XXXI, 1 Supplemento (1998), Giliberti M (2002), A Modern Teaching for Modern Physics in Pre-Service Teachers Training First International GIREP Seminar Developing Formal Thinking in Physics; Selected Contributions M. Michelini M, Cobal M; Udine, Forum, Editrice Universitaria Udinese, ISBN Giliberti M (2002b): Alcuni elementi di didattica della fisica moderna nel corso di Teorie Quantistiche della S.I.L.S.I.S.-MI; La Fisica Nella Scuola, XXXV, 2 Supplemento (2002), Giliberti M (2007), Elementi per una didattica della fisica quantistica, Milano, CUSL, Materiali didattici dell'indirizzo Fisico Informatico e Matematico SILSIS-MI, 2. ISBN Giliberti M (2008), Campi e particelle: introduzione all insegnamento della fisica quantistica, in Approcci e proposte per l insegnamento apprendimento della fisica a livello preuniversitario, a cura di P. Guidoni e O. Levrini, FORUM Editrice Universitaria Udinese, Giliberti M, Barbieri S (2004a), Teorie Quantistiche, CUSL, Milano. ISBN Giliberti M, L. Cazzaniga, G. M. Prosperi, G. Rinaudo, L. Borello, A. Cuppari, G. Rovero, M. Michelini, L. Santi, A. Sciarratta, S. Sonego, R. Ragazzon, A.Stefanel, G. Ghirardi (2002a): Proposte per l insegnamento della Meccanica Quantistica in SeCiF ; Atti del Congresso Didamatica 2002 Informatica per la Didattica; Liguori Editore, Napoli, luglio 2002, ISBN Giliberti M, Lanz L, Cazzaniga L (2004b), Teaching Quantum Physics to student Teachers of S.I.L.S.I.S.- MI, Second International GIREP Seminar Quality Development in Teacher Education and Training, Selected Contributions M. Michelini (editor), FORUM Editrice Universitaria Udinese, Udine, ISBN Giliberti M, Marioni C (1996), Revisiting the H Atom and Electron Configurations in Atoms: a Case Study at High-School Level, Proceedings of the G.I.R.E.P.-I.C.P.E. International Conference: Teaching the Science of Condensed Matter and new Materials (Udine 24-30, August 1995); Forum Editrice Universitaria Udinese, Udine; ISBN Giliberti M, Marioni C, (1997b), Introduzione di alcuni elementi della fisica dei quanti nella scuola secondaria superiore, La fisica nella scuola, XXX (3) Supplemento, Q7, Lorenz K (1973) L'altra faccia dello specchio, Adelphi, Torino, 25-26, translated by the author.

247 FFP Udine November 2011 Ludwig G. Thurler G (2008), new foundation of physical theories, Springer-Verlag, Heidelberg, 4. Ogborn J (2010), Science and Commonsense, in Connecting Research in Physics Education with Teacher Education,Vicentini M. and Sassi E. eds, New Dehli, Gautam Rachmandani, 7. Stenholm S (2002), Are there Measurements?", in Bertelmann R. A. and Zeilingfer A. (eds) Quantum Unspeakables: from Bell to quantum information, Springer, Berlin Heidelberg,188. Styer D et al (2002), Nine formulation of quantum mechanics, A. J. Phys 70 (3), Weinberg S (1995), The quantum theory of fields, I, Cambridge university press, Cambridge, 2. Wittgenstein L (1967), Ricerche filosofiche, Einaudi, Torino, 244, translated by the author.

250 FFP Udine November 2011 lead [emphasis added] (Anderson 1932, 238) according to a process similar to that formerly suggested by Millikan to explain the alleged proton tracks. In 1934, Anderson was still moving within Millikan s theoretical framework when he wrote that the simplest interpretation of the nature of the interaction of cosmic rays with the nuclei of atoms, lies in the assumption that when a cosmic-ray photon impinges upon a heavy nucleus, electrons of both sign are ejected from the nucleus [ ]. [The photographs] point strongly to the existence of nuclear reactions of a type in which the nucleus plays a more active role than merely that of the catalyst [emphasis added] (Anderson et al 1934). Figure 1. Anderson s cloud chamber photograph of a positron traversing the 6 mm lead plate upwards, discussed in September 1932 in Science (Anderson 1932b) and submitted in February 1933 to Physical Review (Anderson 1933) Figure 2. Anderson s cloud chamber photographs, submitted in June 1932 to Physical Review (Anderson 1932a), showing a particle of uncertain sign of charge that suffers a deflection of 0.5 degrees in traversing the 6 mm lead plate. 4. Blackett and Occhialini synthesis By their discovery of plenty of electron-positron pairs within the new phenomenon of cosmic-ray showers (Figure 3), made possible by their efficient triggered cloud chamber (Leone 2011), Blackett & Occhialini s constructed a forceful case (Kuhn 1962) for the existence of positron. Furthermore, they grasped that Anderson s positive electron and Dirac s anti-electron were the same particle, a view supported by the fact that Dirac s theory predicted the successful detection of a positron by the cloud chamber method. As reported by Blackett and Occhialini, Dirac s calculation of probability of electron/positron annihilation process led to a positron mean life in water close to 3.6 x s. Thus, the Cavendish researchers concluded, in [Dirac s theory] favour is the fact that it predicts a time of life for the positive electron that is long enough for it to be

251 FFP Udine November 2011 observed in the cloud chamber but short enough to explain why it had not been discovered by other methods (Blackett and Occhialini 1933, 716). Notwithstanding their support to Dirac s theory, Blackett and Occhialini s original paper provides reasons to believe that, in some respects, they departed from Dirac s pair production mechanism. In one instance, they suggest indeed that electron and positron may be born in pairs during the disintegration of light nuclei [emphasis added] since the showers had been observed to arise in air, glass, aluminum and copper (Blackett and Occhialini 1933, 713). Within another context, namely F. Joliot and I. Curie s April 1932 observation of electrons moving towards a poloniumberyllium neutron source (Leone and Robotti 2010), they concluded that Joliot and Curie s electrons were actually positrons arising from the action of neutrons (as opposed to γ rays). Both instances reveal a production mechanism dissimilar of the one originally put forward by Dirac. Figure 3. Photographs of electron-positron showers captured by Blackett and Occhialini via their triggered cloud chamber technique (Blackett and Occhialini 1933). 5. Concluding remarks As pointed out by Kuhn, it might be safely concluded that Dirac s theory preceded positron discovery (to Rutherford s regret); that Anderson s experiment was done in complete ignorance of Dirac s theory; and that Blackett and Occhialini made use of Dirac s theory to corroborate the positron existence. However, as we have evidenced above, on the one hand a wrong theory (Millikan s one) guided Anderson s discovery of the positron and interpretation of the tracks. And on the other hand, a correct theory (Dirac s one) was not fully exploited by Blackett and Occhialini in interpreting the positron production mechanism. This case study raises therefore a number of relevant issues about NoS. Firstly, that the theory vs. experiment relationship is not always a two-poles one (often in actual science, as in the actual discovery of positron, more theories and more experiments are involved). Secondly, that sometimes wrong theories led to correct discoveries. And, finally, that original historical sources tell histories in some respects at odds with textbooks histories or with later recollections by the protagonists themselves. References Anderson C D 1932a Energies of cosmic-ray particles, Physical Review (41), Anderson C D 1932b The apparent existence of easily detectable positives, Science (76), 238 Anderson C D 1933 The positive electron, Physical Review (43), Anderson C D 1936 The production and properties of positrons, Nobel Lectures, Physics , Elsevier Publishing Company, Amsterdam 1965 Anderson C D 1961 Early work on the positron and muon, American Journal of Physics (29), Anderson C D, Millikan R A, Neddermeyer S and Pickering W 1934 The mechanism of cosmic-ray counter action, Physical Review (45), Blackett P M S and Occhialini G 1932 Photography of penetrating corpuscular radiation, Nature (130), 363

252 FFP Udine November 2011 Blackett P M S and Occhialini G 1933 Some photographs of the tracks of penetrating radiation, Proceedings of the Royal Society of London (A139), De Maria M and Russo A 1985 The discovery of the positron, Rivista di Storia della Scienza (2), Dirac P A M 1931 Quantised singularities in the electromagnetic field, Proceedings of the Royal Society of London (A133), 61 Duit R, Gropengießer H and Kattmann U 2005 Towards science education research that is relevant for improving practice: The model of educational reconstruction. In: Fischer H E (ed.) Developing standards in research on science education (1-9), Taylor & Francis, London Galili I 2011 Promotion of Cultural Content Knowledge through the use of the history and the philosophy of science, Science & Education (20) Galili I and Hazan A 2000 The influence of a historically oriented course on students content knowledge in optics evaluated by means of facets-schemes analysis, American Journal of Physics 68(7), S3-S15 Galili I and Hazan A 2001 Experts views on using history and philosophy of science in the practice of physics instruction, Science & Education 10(4), Halloun I A and Hestenes D 1985 Common sense concepts about motion, American Journal of Physics (53), Hanson N R 1961 Discovering the positron, British Journal for the Philosophy of Science (12), , Hanson N R 1963 The concept of the positron. A philosophical analysis, Cambridge University Press, New York Kipnis N 1998 A history of science approach to the nature of science: Learning science by rediscovering it. In: McComas W F (ed) The nature of science in science education ( ), Kluwer, Dordrecht, The Netherlands Kragh H 1990 Dirac: a scientific biography, Cambridge University Press, 107 Kuhn T S 1962 Historical structure of scientific discovery, Science (136), Institut International de Physique Solvay 1934 Structure et Propriétés des Noyaux Atomiques, Gauthier- Villars, Paris, Leone M 2011 Particles that take photographs of themselves: The emergence of the triggered cloud chamber technique in early 1930s cosmic-ray physics, American Journal of Physics (79), Leone M and Robotti N 2008 P.M.S. Blackett, G. Occhialini and the invention of the counter-controlled cloud chamber ( ), European Journal of Physics (29), Leone M and Robotti N 2010 Frédéric Joliot, Irène Curie and the early history of the positron ( ), European Journal of Physics (31), Matthews M 2000 Time for science education: How teaching the history and philosophy of pendulum motion can contribute to science literacy, Plenum Press, New York McCloskey M 1983 Intuitive physics, Scientific American 248(4), McComas W F 2008 Seeking historical examples to illustrate key aspects of the nature of science, Science & Education 17(2-3), Millikan R A and Anderson C D 1932 Cosmic-ray energies and their bearing on the photon and neutron hypotheses, Physical Review (40), Monk M and Osborne J 1997 Placing the history and philosophy of science on the curriculum: A model for the development of pedagogy, Science & Education 81(4), National Research Council 2011 A Framework for K-12 Science Education: Practices, Crosscutting Concepts, and Core Ideas. Committee on a Conceptual Framework for New K-12 Science Education Standards. Board on Science Education, Division of Behavioral and Social Sciences and Education, The National Academies Press, Washington, DC Roqué X 1997 The manufacture of positron, Studies in History and Philosophy of Modern Physics (28),

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258 Dark Energy from Curvature and Ordinary Matter Fitting Ehlers-Pirani-Schild: Foundational Hypotesis M. De Laurentis 1,2, Lorenzo Fatibene 3,4, Mauro Francaviglia 3,4 1 Dipartimento di Scienze Fisiche, Università di Napoli Federico II, Compl. Univ. di Monte S. Angelo, Edificio G, Via Cinthia, I Napoli, Italy 2 INFN Sez. di Napoli, Compl. Univ. di Monte S. Angelo, Edificio G, Via Cinthia, Naples, Italy 3 Dipartimento di Matematica, Universit di Torino, Turin, Italy 4 INFN Sez. di Torino, Via Carlo Alberto 10, Turin, Italy (Dated: May 22, 2012) We discuss in a critical way the physical foundations of geometric structure of relativistic theories of gravity by the so-called Ehlers-Pirani-Schild formalism. This approach provides a natural interpretation of the observables showing how relate them to General Relativity and to a large class of Extended Theories of Gravity. In particular we show that, in such a formalism, geodesic and causal structures of space-time can be safely disentangled allowing a correct analysis in view of observations and experiment. As specific case, we take into account the case of f(r) gravity. PACS numbers: Keywords: g I. INTRODUCTION Einstein General Relativity (GR) is a self-consistent theory that dynamically describes Space, Time and Matter under the same standard. The result is a deep and beautiful scheme that, starting from some first principles, is capable of explaining a huge number of gravitational phenomena, ranging from laboratory up to cosmological scales. Its predictions are well tested at Solar System scales and give rise to a comprehensive cosmological model that agrees with the Standard Model of particles, with the recession of galaxies, with the cosmic nucleosynthesis and so on. Despite these good results, the recent advent of the so-called Precision Cosmology tests from astrophysics (rotation curves of galaxies) and possible some tests coming from the Solar System outskirts (e.g. the Pioneer anomaly) entail that the self-consistent scheme of GR seems to disagree with an increasingly high number of observational data, as e.g. those coming from IA-type Supernovae, used as standard candles, large scale structure ranging from galaxies up to superclusters. Furthermore, being not renormalizable, GR seems to fail to be quantized in any classical way (see [4]). In other words, it seems, from ultraviolet up to infrared scales, that GR is not and cannot be the definitive theory of Gravitation heven if it successfully addresses a wide range of phenomena. Many attempts have been therefore made both to recover the validity of GR at all scales, on one hand, and to produce theories that suitably generalize Einsteins one, on the other hand. In order to interpret a large number of recent observational data inside the paradigm of GR, the introduction of DarkMatter (DM) and Dark Energy (DE) seemed to be necessary: the price of preserving the simplicity of the Hilbert Lagrangian has been, however, the introduction of rather odd-behaving physical entities which, up to now, have not been revealed by any experiment at fundamental scales. In other words, we are observing the large scale effects of missing matter (DM) and the accelerating behaviour of the Hubble flow (DE) but no final evidence of these ingredients exists, if we want to deal with them as standard quantum particles or fields. However, from an observational point of view, considering GR + cosmological constant+dm gives an extremely good snapshot of the currently observed Universe. The problem is that dynamics of previous epochs cannot be reconstructed and addressed in a self-consistent way starting from the present status of observations. Furthermore, it seems that the type of DM to be considered strictly depends on the size of selfgravitating structures (e.g. the dynamical behavior of DM in small galaxies, in giant galaxies and in galaxy clusters is completely different). So, besides the issue to find out DM and DE at fundamental scales, it seems hard to find out a general dynamics involving such components working at all cosmic epochs and at any astrophysical size. With these considerations in mind, one can wonder if extending gravity sector could be a more economic and useful approach which does not involve too much exotic ingredients but retains all the good results achieved by GR (for a review, see e.g. [5]). In this paper we address some of the recent issues concerning the geometrical structure of physically reasonable gravitational theories, starting from the fundamental work of Elehers-Pirani-Shild [6 8] about the geometric and physical foundations of relativistic theories of gravitation and revisiting them, á la Palatini, in view of applications to the new challenges discussed above [9, 10].The outline of the paper is as follows. In Section II we introduce the EPS framework. Section III is devoted to the EPS formalism in GR while Section IV is a critical discussion of such an approach. In Section V, we discuss EPS from the point of view of Extended Theories of Gravity (ETG). In particular, the straightforward extension of GR, f(r)-gravity, is taken into account. Section VI and VII is devoted to discussion and conclusions. A new paradigm for gravitational

259 2 theories is proposed assuming the EPS paradigm. II. EHLERS-PIRANI-SCHILD-THEORY We first summarize Ehlers-Pirani-Schild (EPS) analysis of the mathematical structures that lie at the basis of all reasonable relativistic gravitational theories [6 8]. In early 70s EPS started from a set of well motivated physical property of light rays matter in a relativistic framework to derive the geometrical structure of spacetime from potentially observable objects. This is particularly suitable to discuss which geometric structure is observable and which are conventionals. In this way it provides stronger physical motivation and understanding not only of space-time geometry as such, but also in comparison with more general geometries (as candidates for mathematically modeling physical space-time ). EPS specifically highlighted the potential role of spacetime models based on Weyl geometry. Supplying this new axiomatic characterization of the otherwise mathematically familiar space-time geometry structure, EPS also brings relevant new insight even from a strictly mathematical (geometrical) standpoint. Einstein s GR uses ses advanced mathematical ideas. Things like 4-dimensional curved spacetime are not easy to grasp. Even if one masters the math behind it, the essential physical meaning and content is not obvious. EPS is one of a series of attempts to clarify the physics behind the math. Unfortunately and unavoidably, getting there requires even more abstract math. At first sight, this seems self-defeating; however, some of these mathematical ideas are chosen so as to be closer to operational physical interpretation, representing more elementary physical observation, measurement and construction. In the upshot, EPS ends up with Lorentzian metric (L4), rather then of accepting it as starting point: the idea is to rebuild L4 from scratch, using only bricks with intuitively clear physical meaning to the extent possible, and at the cost of some extra math. For example as far as metric structure L4 is concerned EPS clearly showed that what is physically well defined is a conformal structure (the class of all a conformally equivalent Lorentzian metrics) such a representative g (i.e. a specific Lorentzian metric) can be singled out only by convention of an observer. As is typical in axiomatic reconstructions like EPS, one exploits the benefit of hindsight, as the intended result (in this case: L4 spacetime of General Relativity) is already known. So this in no way detracts from Einstein s original feat, on the contrary. The scope of EPS is limited to the kinematics of space-time itself; the problem of any possible axiomatic derivation or reconstruction of Einstein field equations ( thet is dynamics) governing matter and gravity within such a space-time model, is left open. The approach shows how quantitative measures of time, angle and distance, and a procedure of parallel displacement... can be obtained constructively from geomtry free assumptions about light-rays and freely falling particles; pseudo-riemannian (or Weylian) geometry is recognized even more clearly than before as the appropriate language for a generalized kinematics which allows for the unavoidable and ever-present distortions called gravitational fields. (Ehlers) A. Outline of Ehelers-Pirani-Shild construction With the above considerations in mind, let us outline the main points of EPS conceptual construction. The construction of EPS space-time proceeds in steps as sketched below, each one enriching the axiomatic content of the underlying set of events. Roughly, the underlying idea is the following. From differential geometry, one knows that the geodesics determine their affine connection (assuming torsion to be zero, for instance) and hence a corresponding metric. Now, in contrast to the metric itself, these geodesics do possess an immediate physical interpretation (as light ray worldlines for null geodesics or particle world lines for timelike ones). So in very general terms, one tries to reconstruct the sought-after metric from known geodesics that fulfill certain qualitative criteria (postulates), which are themselves physically meaningful and plausible. Particles and light rays in event space. EPS adopts a set M of events (to become the spacetime manifold) as its backdrop. On this, a set of particles p and a set of light rays l are assumed given. Each particle and each light ray are identified with their world line of events. Smooth radar coordinates for events As subsets of the space of events, particle and light ray world lines are taken to be smooth one dimensional manifolds. A permissible local coordinate represents time as measured by a (possibly irregular) local clock. Light ray messages between particles p and q smoothly relate their private time parameters, the timing of echoes received back by p also relate smoothly to that of the message flashes it sent out to q to begin with. Using radar soundings in this way, pairs of observer particles set out to map surrounding events by assigning 2 time values each, or a total of 4 coordinates each. Postulating that this process may cover the entire event set, the events form a smooth 4-dimensional space (manifold). Light propagation ensures local validity of pointwise causality At each point of space-time (event), the propagation of light determines an infinitesimal null cone, amounting to a conformal structure C of Lorentzian signature. This assertion is stated operationally, demanding that one may (topologically) distinguish between C-time-like, space-like and null vectors, directions and curves at an event. Null curves

260 3 lying on a null hypersurface are singled out as null geodesics. Free falling particles encode influence of gravity on particle motion Among the timelike curves, the free-falling particles form a preferred family of wordlines. Imposing a generalized law of inertia provides a projective structure, with freefall world lines as its (C-time-like) geodesics. Fre fall implicitly define a projective structure P. They in turns determine, by a canonical gauge fixing, a preferred connection space-time. Light and particle motion agree Then one can define two compatible conformal and projective structures on space-time. The choice of representatives is a conventional gauge fixings. The conventional nature of metrics and connections is important to be noticed in view of which quantities are to be considered physically sound. In particular, one can choose canonically a standard representative of projective structure imposing (Γ) µ g αβ = 2V µ g αβ (1) for some covector V. Then there is a canonical connection Γ α βµ which, of course, depends on extra degrees of freedom depending on A. The triple (M, C, Γ) is called a Weyl-geometry. It is called metric if there exists a representative g C such that Γ = {g} coincides with the Christoffel symbols of the metric g. In this case, the metric g describes light rays and particles free fall, as it is assumed in standard GR. However, in general one needs two different (still compatible) structures to describe light rays and matter free fall. Let us stress once again that there is no reason at this stage to assume that the Weyl-geometry obtained on spacetime is metric. A Weyl space possesses a unique affine structure A: A geodesics are P and A parallel displacement preserves C nullity. In a Weyl space, one may construct a proper time arc length (up to linear transformation) along non-null curves by purely geometrical means (i.e. using light rays reflected from particles only, so without any need for atomic clocks). In technical terms, one employs affine parallel displacement, and congruence in the tangent space, as defined by C. This geodesic clock is known as the Marke-Wheeler clocks [11]. B. Hypotesis In summary EPS analysis is based on a number of assumptions: It physically distinguishes the Principle of Equivalence from the Principle of Causality and investigates the need of measuring and describing SpaceTime structure through light rays. The need of measuring in SpaceTime and using light rays requires that SpaceTime carries a (Lorentzian) metric while the Principle of Equivalence and interaction with matter ( Free Fall under gravitational pull) requires that SpaceTime carries also a (Linear or Affine) Connection. The Connection, an object that can be reduced to be zero at each single point, is the potential of the gravitational field. The Metric determines causality and photon propagation. According to EPS analysis, in order for a Gravitational Theory being physically reasonable, compatibility conditions should exist between the Metric and the Connection. The Connection defines a family of autoparallel lines (also called improperly geodesics). They establish the free fall of pointlike (in principle massive) test particles. The Metric defines light cones and a family of geodesics. Null geodesics of the Metric are paths of light rays (photons). The family of autoparallel lines of the Connection determine an equivalence class of Projectively Equivalent Connections. Along them free fall is the same, only proper time changes. The light cones of the Metric define an equivalence class of Conformally Equivalent Metrics. Along them units and measuring devices change point by point, but light rays and photon trajectories are the same. The required compatibility condition amounts to pretend that the two families of autoparallel lines of the (projective equivalence class of) Connections and the family of null geodesics of the conformal equivalence class of Metrics are in a precise relation: each null geodesic of the Metric has to be one of the autoparallel lines of the Connection At this point of their fresh analysis all Foundational Axioms have been satisfied. In order to recover General Relativity as the Unique Relativistic Theory of Gravitation Elhers, Pirani and Schild make some further axiomatic hypotheses: Speed of time does not depend on path A final physical assumption (expressed mathematically as an axiom) ensures the existence of a Lorentzian metric, which determines both light cones and free fall. Equally spaced clock ticks along one particle world line are transported to a nearby particle by Einstein simultaneity. Imposing that this must generate (approximately) equidistant ticks also for the second particle and applying the equation of geodesic deviation for the curvature tensor given by A implies (through the vanishing of the Weyl track curvature ) the existence of a single Lorentzian metric compatible to both C with A. This finally reduces Weyl space to L4. Requiring in this way that time runs equally fast along all paths amounts to denying the existence of a second clock effect. Indeed, in (Lorentzian) GR, only the time interval between 2 events is path dependent (i.e. the first clock effect ); not the speed of time. Metricity Axiom : a single Metric is chosen in the

261 4 conformal class and the Connection is chosen while be the Levi-Civita Connection of this Metric. With this above hypothesis the compatibility conditions are met. Notice, however, that this just amounts to say that the gravitational theory is of purely metric nature. To recover GR as the unique Relativistic Theory of Gravitation one has in fact to make a further assumption. III. ELEHERS-PIRANI-SCHILD AND GENERAL RELATIVITY In order to recover General Relativity as the Unique Relativistic Theory of Gravitation one has in fact to make the following further axiomatic hypotheses: Lagrangian Axiom : the Lagrangian that governs gravitational field equations (in absence of Matter) is the Scalar Curvature. The Metricity Axiom has in fact no real physical grounds. According to EPS (and to physical needs) a Metric has to exist to define rods and clocks, but there is no need to pretend from the very beginning that it defines also the gravitational potential, i.e. the Connection. Assuming that the Metricity Axiom holds is just a matter of taste and in a sense it corresponds to have a great mathematical simplification. From the viewpoint of Lagrangian Mechanics it is a purely kinematical restriction imposed a priori on Dynamics. Physically spacing, it is much better not to impose a priori purely kinematical restriction on Dynamics. Physics requires that possible restrictions should be obtained from dynamics rather than imposed a priori as a constrains. This point was perfectly clear to Albert Einstein when, in 1923, he tried to establish a more general setting for Gravity (and Electromagnetism) by assuming a priori that both a Metric and a Connection must be chosen, from the beginning, as dynamical variables. So-called Palatini formalism was born. Also the Lagrangian Axiom had in fact no real physical grounds. Once it is clear which are the variables that have to enter dynamics, the choice of a Lagrangian for them is again a matter of taste or it should be at least determined on the basis of Phenomenology, in order to fit observational data. When Hilbert, in 1916, in the purely metric framework (the only one that was available before 1919 and Levi-Civitas work on Linear Connections) assumed the Lagrangian to be the Scalar Curvature of the Metric this was, in a sense, an obliged choice. Dictated by simplicity. The choice of the Hilbert-Einstein Lagrangian R(g), made in 1916, was not only the simplest one. It satisfied the will to obtain second-order field equations suitably generalizing Newtons law, fitting all astronomical predictions, satisfying conservation of matter and being compatible with Maxwell. G αβ = R αβ 1 2 R g αβ = kt αβ, (2) where G αβ is the Einstein tensor, a combination of curvature invariants derived from Bianchi s identities, T αβ is the stress-energy momentum tensor and κ = 16πG is the gravitational coupling constant. In the new framework introduced by assuming both metric and connection among the variables, Einstein decided to take into account again, in 1923, the Lagrangian to be the scalar curvature (of metric and connection), again for the sake of simplicity. At that time there were very few observations fine enough to be used as tests and all of them agree with purely metric predictions. Thus the first test for an extended theory was to reproduce standard GR in purely metric formalism. When Einstein, in 1923, in the new framework he introduced by assuming both a Metric and a Connection among the variables he decided to assume again the Lagrangian to be the Scalar Curvature of the Metric and the Connection, again for the sake of simplicity. In this new framework and with the Linear Lagrangian R(g, Γ) he proved that no realy new Physics comes on stage. Field equations impose in fact, a posteriori, hat the Connection is nothing but the Levi-Civita Connection of the Metric, so that GR is eventually recovered. Einstein did not investigate, however, what happens when Matter is coupled to the Linear Lagrangian R(g, Γ). In this case just a few slight changes are necessary if Matter couples with the Metric g but great difficulties arise if Matter couples with the Connection Γ (as it should). Around the sixties a number of mathematical papers were written about possible generalizations of Einsteins Theory by reverting to Non-Linear Lagrangians, more complicated than R(g). These Higher Order Theories remained just as a mathematical game for long time. Renewed interest towards Non-Linear Lagrangians more complicated than R(g) (Higher Order Theories) was lately determined by new phenomenology, such as: Inflation, Acceleration in the Expansion, Dark Matter, Quantum Gravity, Low Energy Limit of String Models [12, 13]. IV. ELEHERS-PIRANI-SCHILD REVISED FORMALISM The analysis of EPS concerning the mathematical and physical foundations of relativistic theories of gravitation and the compatibility between (conformal classes of) metrics and (projective classes of) connections is worth of being revisited. EPS have shown that the family of gravitational theories that satisfy all of their Axioms (with the exception of the Metricity Axiom and the Lagrangian Axiom ) includes many (but not all) of the currently investigated frameworks for (relativistic) gravitation. First of all, it suggest that the correct and most general framework for dealing with gravity is the Palatini formalism, since it is based on the physical and mathematical distinction between the Principle of Equivalence and the Principle of Causality that for obvious reasons are mathematically and physically distinct. They imply

262 5 the necessity of introducing a priori distinct and separate structures to full fill them, even if compatibility is required a posteriori on the mathematical and physical structures they induces on space-time. Within this formalism, the most general class of theories that care be considered without renauncing to the physical requirements point out by EPS analysis, is the family of so-called Further Exteneded Theories of Gravity that has been explicitly introduced in [14]. This class includes all gravitational theories in which the gravitational Lagrangian depend on g and the (Ricci) curvature of the connection, the matter Lagrangian interacts allows in principle interaction of matter with both g and Γ and, a posteriori or a priori, field equations imply EPS compatibility. Of course one is free to work in more general frameworks for gravitation, but in such a case one has to remind that at list one EPS requirements will fail. FIG. 1. the eps theories Our choice wil be more restrictive and will be therefore based on three assumptions: 1. assume Palatini - EPS framework and accept the view that in Palatini formalism the gravitational field is encoded in to the dynamical connection (i.e. free fall) while the dynamical metric has more to do with measures, rods, clocks and causality. We accept moreover that dynamics, and in particular the interaction with matter, will determine a posteriori the relations between the metric and the connection, so to satisfy EPS compatibility requirements. 2. assume that the Lagrangian is a possibly non-linear function of curvature of g and Γ; 3. assume that the Lagrangian is simple - the simplest choice is of course f(r(g, Γ) but it is not the only simple case. V. THE EHELERS-PIRANI-SCHILD APPROACH FOR f(r)-gravity With this choice one can show that if no Matter is present, purely gravitational equations entail that the Connection Γ entering dynamics is still the Levi-Civita Connection of the Metric g while a (quantized) Cosmological Constant enters the game and somehow determines the asymptotic freedom for Gravity. One should remark that if Matter is present and couples only to the Metric g things change if and only if the trace t of the Stress Tensor is different from zero. In particular, thence, nothing changes when Electromagnetism couples, so that the light cone structure and photons are not affected when passing to Palatini framework. It is however known that - both in purely metric formalism (higher order gravity) and in the Palatini approach (first order gravity) - coupling with matter generates relativistic effects that are not present in vacuum. This is particularly evident when one relies on non-linear Lagrangians of the typef(r). In f(r) gravity in the Palatini approach in presence of Matter coupled only with g field equations still imply that the Connection is metric, but now it is the Levi-Civita Connection of a new Metric h, conformally related with the Metric g given in the Lagrangian. Being this the core point of our discussion, we want to derive in details the field equations of f(r) gravity in Palatini formalism and then perform the EPS analysis in this framework. Let us first consider on M a metric field g, a torsionless connection Γ and a generic tensor density A of rank 1 and weight 1. The covariant derivative of A µ is then defined as Γ µ A ν = d µ A ν Γ λ νµa λ + Γ λ λµa ν, (3) Accordingly, we have Γ ( ) (µ A ν) = d (µ A ν) Γ ɛ νµ δ(ν ɛ Γλ µ)λ A ɛ = where we set u ɛ µν := Γ ɛ µν δ ɛ (µ Γλ ν)λ. = d (µ A ν) u ɛ µνa ɛ, (4) Let us consider the following Lagrangian (density) L = 1 κ gf(r) + gg µν Γ µ A ν, (5) where g = det(g µν ), R = g µν R µν (Γ) is the scalar curvature of (g, Γ), and f(r) is a generic (analytic) function. By variation of this Lagrangian and usual covariant in-

263 6 tegration by parts one obtains g δl = (f (R)R (αβ) 12 ) κ f(r)g αβ κt αβ δg αβ + g gg αβ A λ δu λ αβ + κ gαβ f (R) Γ λ δu λ αβ + +gg µν Γ µ δa ν = g = (f (R)R (αβ) 12 ) κ f(r)g αβ κt αβ δg αβ + 1 ( Γ λ ( gg αβ f (R) ) ) + κgg αβ A λ δu λ αβ + κ Γ µ (gg µν ) δa ν + ( ) + Γ g λ κ gαβ f (R)δu λ αβ + gg λν δa ν, (6) where we used the well-known identity δr (αβ) = Γ λ δu λ αβ and we set for the energy-momentum tensor T αβ := g ( g αβ g µν Γ µ A ν Γ (α A β) ). Field equations are f (R)R (αβ) 1 2 f(r)g αβ = κt αβ, Γ ( λ gg αβ f (R) ) = α λ gg αβ f (R), Γ µ (gg µν ) = 0, where we set α λ := κ g f (R) A λ. Notice that the third equation (that is the matter field equation) is not enough to fix the connection due to the contraction. Notice also that these are more general than field equations of standard f(r) theories due to the rhs of the second equation (that is originated by the coupling between the matter field A and the connection Γ). Nevertheless one can analyze these field equations along the same lines used in f(r) theories. Let us thence define a metric h µν = f (R)g µν and rewrite the second equation as (7) Γ ( ) λ hh αβ = α λ hh αβ. (8) According to the analysis of EPS-compatibility done in [14] this fixes the connection as Γ α βµ := {h} α βµ κ ( ) 2f h αɛ h βµ 2δ(β α (R) δɛ µ) a ɛ, (9) where for notational convenience we introduced the 1- form a ɛ := ga ɛ. For later convenience let us notice that we have Kβµ α Γ α βµ {h} α βµ = κ 2f (R) ( h αɛ h βµ 2δ α (β δɛ µ) ) (10) a ɛ. Now we can define the tensor Hβµ α := Γα βµ {g}α βµ and obtain Hβµ α = Kβµ α 1 [ ] g αλ g βµ 2δ(β λ 2 δα µ) δ λ ln f (R) = = 1 [ [ 2f g αɛ g βµ 2δ(β ɛ (R) δα µ) [κa ɛ + δ ɛ f (R)], (11) By substituting into the third field equation we obtain g ( µ (gg µν ) + g H µ λµ gλν + Hλµg ν µλ 2Hλµg λ µν) = 0, Hλµh ν µλ Hλµh λ µν = 0, 1 [( 2f h νɛ h λµ 2δ(λ ɛ (R) δν µ) ( + h λɛ h λµ 2δ(λ ɛ δλ µ) ) h µɛ ) h µν] (κa ɛ + δ ɛ f (R)) = 0, 3 f (R) hνɛ (κa ɛ + δ ɛ f (R)) = 0, a ɛ = 1 κ δ ɛf (R), (12) where g µ is now the covariant derivative with respect to the metric g. Hence the matter field A ɛ = ga ɛ = g κ δ ɛf (R) has no dynamics and it is completely determined in terms of the other fields. We can also express the connection as a function of g alone (or, equivalently, of h alone) ( h αɛ h βµ 2δ(β α δɛ µ) Γ α βµ := {h} α βµ ) δ ɛ ln f (R) {g} (13) α βµ This behaviour, which has been introduced by the matter coupling, is quite peculiar; the model resembles in the action an f(r) theory but in solution space the connection is directly determined by the original metric rather than by the conformal metric h as in f(r) theories. Still the metric g obeys modified Einstein equations. In fact, we have the first field equation which is now depending on g alone, since the matter and the connection have been determined as functions of g. The master equation is obtained as usual by tracing (using g αβ ) f (R)R 2f(R) = κt f(r) = 1 2 (f (R)R κt ), (14) where we set T := T αβ g αβ the trace of the stress-energy tensor. By substituting back into the first field equation, being T = 3 κ f (R), we obtain f (R) [ R αβ 1 4 Rg αβ = (α β) f (R) f (R)g αβ, ] 3 4 f (R)g αβ = R αβ 1 2 Rg αβ = 1 [ f (α β) f (R) 14 ] (R) ( f (R) + f R) g αβ,(15)

264 7 where now the curvature and covariant derivatives refer to g. These are exactly the field equations obtained in the corresponding purely-metric f(r) theory [5]. Hence we have that, regardless of the function f(r), when there is no matter field other than the field A all these models behave exactly as metric f(r) theories. The conformal factor is f (R) and R can be calculated in terms of T, i.e. φ(t ) = f (R(T )). Field equations then imply that Einstein equations hold for the new metric ĝ (corresponding to the above h), with a suitably modified stress-energy tensor that takes into account extra effects due to the conformal factor. The previous Einstein equations are recovered with ˆR αβ 1 2 ˆR ĝ αβ = κ ˆT αβ (16) ˆT αβ = 1 ] [T f αβ + T (grav) αβ, (17) where the first term on the rhs is due to a standard matter term- Clearly Eq.(17) means that the extra degrees of freedom coming from f(r) gravity can be managed as a further contribution to the stress-energy tensor and the above observational shortcomings, related to GR (e.g. DM and DE), can be, in principle, solved in a geometrical way. Which are the physical implications from the EPS formalism point of view? 1. being g and ĝ conformally related photon propagation does not change; 2. Einstein equations hold for the new metric ĝ with extra stress-energy tensor directly generated by ordinary matter T ; 3. rods and clocks change pointwise, by a factor depending on T. In summary, EPS formalism works also for ETG and further information can be always enclosed in a suitable definition of stress-energy tensor. VI. A NEW PARADIGM FOR GRAVITY The coupling of a non-linear gravitational Lagrangian f(r) with matter Lagrangians, depending on the metric g in an arbitrary way or even on the connection Γ in a peculiar way (dictated by EPS compatibility), generate a set of modified Einstein equations in which the following effects are easily recognizable: A new metric ĝ - conformally related to the original metric g - arises. The conformal factor is a computable function of curvature and, through functional inversion, of the trace of the stress tensor that corresponds to the ordinary matter distribution (including possible DM and DE effects). In the Palatini approach, the new metric generates the connection Γ as its Levi-Civita connection, so that it describes the free fall of ordinary matter. This new metric induces, in fact, a change of rulers and clocks that affects measurements and conservation laws, while the original g is directly related to light propagation. Due to conformal equivalence, light propagates on the same null geodesics of both g and ĝ, although clock rates are different in presence of matter. The net effect of non-linearity and of (non trivial) interaction with matter resides in a change of the stress tensor that couples to the Einstein tensor of ĝ; a change that induces additions to the previously existing one (directly generated from the matter Lagrangian as discussed above). This new stress-energy tensor defines conservation laws that are fully covariant with respect to the Einstein frame of ĝ. Furthermore, it contains an additional term, that can be interpreted under the form of a space-time varying cosmological constant Λ(x) - in turn determined by distribution of ordinary (and Dark) Matter - so that the residual amount could be interpreted as a net curvature effect (DE) due to the change of rules and clocks induced by EPS compatibility [5]. In other words, the observational effects of such a dynamics are the clustering of astrophysical structures (DM) and the revealed cosmic speed up (DE). VII. CONCLUSIONS AND REMARKS To conclude, we can say that very likely Einstein today, after the new phenomenological evidences would much probably come back onto his own steps and accept, as he always did, that models are not eternal and should be dictated by phenomenology rather than by preestablished rules and prejudices. Why should we insist on pre-judicial rules that impose metricity a priori (and metricity with respect to a given metric!) and insist on the choice of the simplest Hilbert-Lagrangian, when cosmology, quantum Issues and strings suggest instead to us to strictly follow the beautiful analysis of EPS, and work at least a priori, in the extended framework of Palatini-EPS formalism and in a much larger class of Lagrangians? Moreover, let us remark that working in the extended setting suggested by the Palatini-EPS framework requires to reconsider all the machinery and settings of the observational paradigms and protocols have to be carefully analyzed to disentangle purely metrical effects from effects that measure the interaction with free-fall (and therefore with the connection) that in purely metric formalism GR are necessarily mixed up and entangled by the a priori requirement that free-fall is also driven by the metric.

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271 ENRICO FERMI AND ETTORE MAJORANA: SO STRONG, SO DIFFERENT Francesco Guerra, (Department of Physics, Sapienza University of Rome and INFN, Rome Section) Nadia Robotti, (Department of Physics, University of Genova, Italy) Abstract By exploiting primary sources we will analyze some of the aspects of the very complex relationship between Enrico Fermi and Ettore Majorana, from 1927 (first contacts of Majorana with the Institute of Physics of Rome, and with Fermi) until 1938 (disappearance of Majorana). The relationship between Fermi and Majorana can not be interpreted in the simple scheme Teacher-Student. Majorana, indeed, played an important role in the development of research in Rome in the field of the statistical model for the atom and in nuclear physics. Our current research concerns the development of Nuclear Physics in Italy in the Thirties of XXth Century, and is based exclusively on primary sources (archive documents, scientific literature printed on the journals of the time, and so on). In this framework, we will try to outline some aspects of the complex topic concerning the relationship between Enrico Fermi and Ettore Majorana. Of course, this paper will touch only some of the most important issues. For convenience, our exposure will be connected to a periodization of Majorana scientific activity, which we used in previous works. One of our important results is the reassessment of the role played by Majorana for the decisive orientation of the research in Rome (statistical model of the atom, nuclear physics). This role is obscured in discussions which place emphasis on the (alleged) "genius" of Majorana, usually associated with his (alleged) "lack of common sense". Perhaps we could summarize, in a very concise and expressive manner, the nature of the relationship between Majorana and Fermi, and in general the Physical Institute of Rome, presenting these two important documents stored in the Archive Heisenberg, at the Max Planck Institute of Munich. On November 9, 1933, the Sweden Academy of Sciences announced that the Nobel Prize for Physics for the year 1932 was awarded to Werner Heisenberg. Immediately, all the exponents of world culture send their congratulations, in various forms. Heisenberg stored in a folder of his personal archive all received messages. Among them, a telegram by the Deutsche Reichspost, from Rome and dated 11-XI-33 (Fig.1) written in a very cold and formal style, in German: MOST CORDIAL CONGRATULATIONS CORBINO FERMI RASETTI SEGRE AMALDI WICK We note the order of signing according to the close rank of academic seniority at the Institute of Physics in Rome (Orso Mario Corbino, Enrico Fermi, Franco Rasetti, Emilio Segre, Edoardo Amaldi, Gian Carlo Wick). Fig.1. Roma Congratulations (Heisenberg Archive, Max Planck Institut, Munchen)

272 Ettore Majorana (who is in Rome) is not included in the list, not even at the very last place. But Majorana, who had left Leipzig in early August, sends his "Gratulationen", according to his style. He sends a small personal business card, dated Rome, (Fig. 2) with the title "Dr." canceled by hand, written in a poignant Italian. It is a very intense letter. Dear Professor, Let me (if you have not forgotten me!) allow to express my greetings on the occasion of the new formal recognition of your prodigious work. With deep admiration Yours Ettore Majorana. Fig.2 Majorana Congratulations (Heisenberg Archive, Max Planck Institut, Munchen) 1.The formation years until the doctoral degree (1929) The scientific research activities of Ettore Majorana develop immediately at the highest international standard, while it is still a university student at the School of Engineering, in His first research concern the applications, the improvement, and the extension of the statistical model for atoms, introduced by Enrico Fermi at the end of 1927, only few months before! (Fermi 1927). This model is now known as the Thomas-Fermi model. As it is well known, Enrico Fermi was called as full Professor in Theoretical Physics in Rome in 1927, through the effort of the Director of the Institute Orso Mario Corbino to develop advanced modern physics in Rome. The main Fermi achievements concern: the so called Fermi-Dirac statistics, the statistical model for the atom, the theory of weak interactions, the discovery of the neutron-induced radioactivity, the effect of the slowing down of neutrons, (after the 1938 Nobel Prize and the emigration to the U.S.A.) the atomic pile, the Manhattan project, the elementary particle physics, the computers, and so on). The first involvement of Majorana on the statistical model subject is a paper in collaboration with his friend Giovanni Gentile jr., published on the Proceedings of the Accademia dei Lincei, presented on July 24th, 1928 by Orso Mario Corbino. (Gentile, Majorana 1928). They calculate the splitting of the spectroscopic energy levels due to the hypothesis of the spinning electrons as recently developed by Dirac. It is a well received paper, developed completely in the frame of Fermi aproach. Then Majorana continues his research alone, with full autonomy and effectiveness. He proposes an improvement of the model (he changes the expression of the effective potential acting on the optical electron) and includes also positive ions (it is the first treatment made). Some of the results are communicated to the XXII General Meeting of the Italian Society of Physics (Rome, December 1928).

273 At that time Majorana was still a student, and had recently officially moved from Engineering to Physics. The communication was presented by Majorana in front of an audience of famous Physicists and Mathematicians, as O.M. Corbino, T. Levi Civita, V. Volterra, G. Polvani, Q. Majorana, A. Carrelli, E. Fermi, (he was not a timid person!) This communication is regularly published on Il Nuovo Cimento, the Journal of the Society (Majorana 1929), but it has been never mentioned in any scientific paper, nor by any of the many historians who wrote on the life and activity of Majorana. Historical analysis is in a paper by F.G. and N.R. (Guerra, Robotti 2008). The acknowledgements are very interesting: The A. thanks Professor Fermi, for the advice and suggestions around new applications of this statistical method that has thrown much light on the atomic physics, and whose fertility, appears far from exhausted, still waiting to be ventured into investigation of fields of larger scope and more full of promise. " The results contained in Majorana notebooks (at Domus Galiaeana in Pisa) and in the communication show that Majorana reached a fully developed scientific personality, completely independent from Fermi. However, Majorana does not publish the results announced in the communication, nor the other results contained in his notebooks. An enlarged paper on the subject would have given a complete representation of the statistical model, his applications and extensions. As a matter of fact, Majorana does not work anymore on these subjects. Fermi convinces himself of Majorana improvement only in late 1933 and puts it at the basis of the monumental conclusive paper of the Rome School, co-authored by Fermi and Amaldi in 1934, without any reference to Majorana (Fermi, Amaldi 1934). It is amazing to note that all formulas of the general Fermi-Amaldi paper (1934) coincide with the corresponding formulas of Majorana communication (1929). Majorana earns his doctoral degree in Physics (July 6th, 1929) with a Thesis on Nuclear Physics (with the title "On the mechanics of radioactive nuclei"). Fermi is the supervisor. Majorana Thesis is the first work on Nuclear Physics in Rome, and also in Italy. Majorana gives a rigorous justification to Gamow model of alpha decay based on the quantum tunnel effect. Anyway, even if original and internationally competitive results are achieved, they are not published. 2. From the doctoral degree to the private professorship ( ) After graduating a short period of silence: he does not deal at the moment with Nuclear Physics, but maturates new lines of research (Atomic Physics, Molecular Physics and Elementary Particle Physics) in complete autonomy from Fermi ( which deals with Quantum Electrodynamics and Hyperfine Structure of Atomic Spectra). Then follows a very intense activity, oriented toward the Il Nuovo Cimento. Between the end of 1930 and January 1931 there are ready two papers (Majorana 1931 A, B) on the quantum explanation of the chemical bond (formation of Helium molecular ion and Hydrogen molecule). The first is presented by Corbino at Accademia dei Lincei, December 7, 1930: "I warmly thank Professor Enrico Fermi, who gave me precious advice and aid. Majorana becomes a pioneer in Theoretical Chemistry (this discipline will develop in Italy only in the 50s of XX Century). In 1931, other two papers will be published, this time in Spectroscopy. Majorana provides a theoretical interpretation of two new lines of Helium, recently discovered (Majorana 1931 C) and some triplets of calcium (Majorana 1931 D). These two works are appreciated as elegant examples of applications of group theory. In reality they have a direct physical interest: for the first time the role of the phenomenon of self-ionization of atoms is recognized in Spectroscopy. Follows a very important work (published in 1932): "Atoms oriented in magnetic field" (Majorana 1932 A). In this paper, he proposes an optimal arrangement of the magnetic field to show the sudden flip in the spatial quantization of the spin of atoms, and other related effects. This arrangement is immediately adopted with great success in the laboratory of Otto Stern in Hamburg (where there is temporarily Emilio Segrè). Majorana gives the Announcement (never quoted in Literature), in the Journal Ricerca Scientifica (Majorana 1932 B), so as to publicize immediately his results and have priority of discovery! Then the fundamental work: "Theory of relativistic particles with arbitrary intrinsic momentum (Majorana 1932 C). He formulates a relativistic generalization of the Schrödinger equation that completely eliminates the existence of negative-energy solutions (provided instead by the Dirac equation) and valid for particles with arbitrary spin (Dirac equation instead is valid only for s = 1/2). It is expanded and systematized by Wigner in 1939, which recognizes the pioneering role of

276 On the relationship with Fermi, we have a reprint of the article with dedication (courtesy of Prof. Giacomo Morpurgo): "To His Excellency Enrico Fermi, with very best regards. Ettore Majorana. Recently we found a reprint with dedication to Gian Carlo Wick: "A Gian Carlo Wick with very best regards. Ettore Majorana. Note the nuance in the dedications: Fermi (His Excellency), expected President of the Evaluating Committee, Wick, a candidate in pole position. The competition for a full professorship in Theoretical Physics at the University of Palermo (deadline for applications: June 15th, 1937) is a particularly significant example of the academic practice of the time (rich documentation at the Central State Archives in Rome). Ettore diligently submits his application (it is the first occasion for a chance to be officially recognized). Attached to the application are: "Scientific activity" and "List of Publications". Obviously in the list it is missing the communication to Congress in 1928 and the two Announcements on the Ricerca Scientifica (not suitable for a Competition). On July 7, the Committee was appointed directly by the Minister (in a totalitarian regime). Of course Fermi is the President of the Committee, so composed: Fermi Enrico; Persico Enrico; Lazzarino Orazio; Polvani Giovanni; Carrelli Antonio. In the first meeting (October 25, 1937, Record 1) the Committee decides to propose to the Minister to appoint directly Majorana as full professor in some University of the Kingdom, outside the procedures of the competition. The scientific Report on Majorana is of course very laudatory, albeit with some important omissions. The Minister accepts the proposal immediately, and on November 2, 1937, Majorana is appointed Full Professor in Theoretical Physics at the Royal University of Naples. Having removed Majorana from the competition, the work of Committee resumes immediately beginning again with the record N 1! (not number 2! ). In November 1937 the Committee, according to the rules, selects the winning triplet in the following order: Wick Giancarlo; Racah Giulio; Gentile Giovanni. 6. The disappearence (1938) Majorana in Naples attends regularly to his course. The contents can be reconstructed from his notes in Pisa and from the testimony of his students, in particular Prof. Gilda Senatore. From the notes at the Domus, the intent of Majorana is clearly to differentiate his course from that of Fermi, in particular by developing in full detail, some of the important topics, that Fermi had just mentioned (as for example the relativistic corrections to the quantum Bohr atom). He disappeared in circumstances not yet clarified in late March After his disappearance, a presumed intervention by Fermi on Mussolini, to intensify the police researches, is not confirmed in the Central Archive in Rome (secretariat of the Duce). In fact, the alleged letter of Fermi to Mussolini (27 July 1938), preserved at the Domus (Fig.3), shows puzzling aspects. It contains a central body written with unknown handwriting, in a very rhetoric style, certainly completely unrelated to Fermi, as even Emilio Segrè recognized in a letter addressed to Amaldi, who had sent him a copy of the text of the letter. On the other hand, the first and last lines of the letter are written in the handwriting of Ettore s brother, Salvatore. Significant is the attempt to imitate the signature of Fermi. Notice the incredible addressing of Fermi toward Mussolini as Duce, while the correct addressing would have been Excellency. This letter is a further step in the complex research theme "Majorana". As a matter of fact, a careful comparison of the handwriting of the central body of the letter with the existing documents, at the Domus Galilaeana in Pisa and the Archive of the Department of Physics in Rome, leads to a surprising conclusion. The letter was in reality written by Giovanni Gentile jr, a close friend of Majorana strictly associated with his Family. The style of the letter is in complete agreement with this discovery. A full analysis of this disconcerting episode is contained in a forthcoming paper.

277 Fig. 3 Presumed Fermi letter to Mussolini (Domus Galilaeana, Pisa) 7. Conclusions In the confirmatory report (1930) for Fermi Professor of Theoretical Physics, the Commission (Chairman Orso Mario Corbino) recognized, among other things, the merits of the "construction of a school of young people vigorously trained in the study of the most advanced problems of modern physics". Ettore Majorana was certainly one of them. However, the complex issue of relations between Majorana and Fermi can not be simplistically interpreted in terms of Teacher-Student. Majorana since the early activity shows a wide autonomy and independence. His ability to "follow the scientific movement" allows him to have a significant influence on the development of research in Rome, mainly on the themes of the statistical model of the atom, first, and the physics of the nucleus, later. The international prestige achieved by the development of nuclear models directed him towards an effective role as leader of the research. The situation of apparent separation from active research, created after his return from Leipzig in August 1933, has prevented the occurrence of this event. Italy lost in 1938, almost simultaneously but in different circumstances, the two leading figures in the field of research in nuclear physics, Majorana and Fermi. References Fermi E (1927) Un metodo statistico per la determinazione di alcune proprietà dell atomo, Rendiconti dell Accademia dei Lincei (6), Fermi E, Amaldi E (1934) Le orbite s degli elementi, Memorie dell Accademia d Italia (6), Gentile G, Majorana E (1928) Sullo sdoppiamento dei termini Roentgen e ottici a causa dell elettrone rotante e sull intensità delle righe del Cesio, Rendiconti dell Accademia dei Lincei (8), Guerra F, Leone M, Robotti N (2006) Enrico Fermi s discovery of neutron-induced radioactivity: neutrons and neutron sources, Physics in Perspective (8), Guerra F, Robotti N (2008 A) Ettore Majorana s forgotten publication on the Thomas-Fermi model, Physics in Perspective (10), Guerra F, Robotti N (2008 B) Ettore Majorana. Aspects of his Scientific and Academic Activity, Edizioni della Normale, 243 pp

278 Guerra F, Robotti N (2009) Enrico Fermi's discovery of neutron-induced artificial radioactivity: the influence of his theory of beta decay, Physics in Perspective (2009)., Physics in Perspective (11), Majorana E (1929) Ricerca di un espressione generale delle correzioni di Rydberg, valevole per atomi neutri o ionizzati positivamente, Nuovo Cimento (6), XIV-XVI Majorana E (1931 A) Sulla formazione dello ione molecolare di elio, Nuovo Cimento (8), Majorana E (1931,B) Reazione pseudopolare fra atomi di idrogeno, Rendiconti dell Accademia dei Lincei (13), Majorana E (1931 C) I presunti termini anomali dell elio, Nuovo Cimento (8), Majorana E (1931 D) Teoria dei tripletti P incompleti, Nuovo Cimento (8), Majorana E (1932 A) Atomi orientati in campo magnetico variabile, Nuovo Cimento (9), Majorana E (1932 B) Atomi orientati in campo magnetico variabile, La Ricerca Scientifica (5), 329 Majorana E (1932 C) Teoria relativistica di particelle con momento intrinseco arbitrario, Nuovo Cimento (9), Majorana E (1933 A), Über die Kerntheorie, Zeitschrift für Physik, (82), Majorana E (1933 B) Nuove ricerche sulla teoria dei nuclei, La Ricerca Scientifica (4) 522. Majorana E (1937), Teoria simmetrica dell elettrone e del positrone, Nuovo Cimento (5),

279 ENERGY EXCHANGE BY THERMAL RADIATION: HINTS AND SUGGESTIONS FOR AN INQUIRY BASED LAB APPROACH Onofrio Rosario Battaglia, Claudio Fazio, Nicola Pizzolato, Dipartimento di Fisica- Università di Palermo ABSTRACT: In this paper we present some laboratory activities developed in the framework of an inquirybased, context-to-content teaching/learning approach to the study of energy exchange by thermal radiation. These activities have been developed in the context of Establish, a FP7 European Project aimed at promoting and developing Inquiry Based Science Education in European Secondary Schools. By starting from real life, meaningful situations, students are engaged in designing and carrying out laboratory activities by collecting, processing and analysing data. Particular attention is paid in building data interpretation by taking into account the effects of parameters like the environmental temperature, that, if not correctly considered, can lead to the application of wrong approximations and to an incomplete understanding of the physics underlying the analyzed situations. 1. Introduction Several researches in science education support the view that teaching science as inquiry enables students to obtain an experience that is similar to that of scientists, making their learning more meaningful and improving their scientific understanding (Minstrell and van Zee, 2000; Windschitl and Thompson 2006). By inquiry we refer to learning experiences that engage students in various integrated activities of identifying questions, collecting and interpreting evidence, formulating explanations, and communicating their findings, that are consistent with science standards and recent reports (Duschl et al, 2007; Singer et al, 2005). In the majority of the developed Inquiry Based (IB) approaches, a relevant role is played by laboratory activities where students are directly engaged in finding answers to their questions, developed in appropriate contexts, as well as in developing investigative competences. On the contrary, in traditional content-to-context based physics education, the focus of laboratory activities developed by students is mostly dedicated on verifying information previously transferred by the teacher. It is well acknowledged that science education has to make students aware that science is not just a body of knowledge that reflects current understanding of the world; it is also a set of practices used to establish, extend, and refine that knowledge. Both elements, knowledge and practice, are essential (NAS, 2011). In all IB approaches to science teaching, students are themselves engaged in the practices and do not merely learn about them second-hand. Students cannot comprehend scientific practices, nor fully appreciate the nature of scientific knowledge itself, without directly experiencing those practices for themselves. However, many problems exists in integrating contents and practices. In this view, we are developing a teaching-learning sequence regarding the investigation of energy exchange by thermal radiation, conduction and convection, within the context of ESTABLISH ( a FP7 European Project aimed at promoting and developing IBSE in European secondary schools. Our teaching-learning sequence is organized by firstly engaging students by means of scientifically oriented questions about real life situations, such as thermal insulation of houses and the use of energy saving materials. Students are invited to design and build by themselves a scale model of an energy-efficient house, through the understanding of relevant concepts regarding the energy flow in thermal systems. Then they are directly involved into modelling and practical activities and are stimulated to carry out their own experiments. As a part of this inquiry-oriented teaching/learning sequence, here we present a laboratory activity for secondary school level students, or even first-level university ones, aimed at investigating the physics of energy transfer by thermal radiation. A scientific investigation on the energy exchange between a powered resistor and its surrounding environment during the heating and cooling processes is proposed. Students are stimulated to plan and carry out their own laboratory activity by collecting, processing and analysing data, in order to learn new concepts or laws and obtain more meaningful conceptual understanding of the physics underlying the process of energy exchange by thermal radiation. In particular, we report as problems and experiments have been developed in a workshop for engineering students attending the second year of their university curriculum. In the following section, we present the theoretical framework that grounded the organization of the workshop. In Sections 3 and 4 we describe an example of inquiry-lab learning path where

280 engineering students are directly involved in experimental activities and in the related investigations aimed at giving theoretical meaning to their experimental results. Finally, we discuss the pedagogical experiment by drawing conclusions about the way to improve its efficacy. 2. Theoretical Background The term inquiry, has been interpreted over time in many different ways throughout the science education community. For this reason, many new documents better specify what is meant by inquiry in science and the range of cognitive, social, and physical practices that it requires. The new Framework for K-12 Science Education (NAS, 2011) describes a set of practices for K-12 students derived from those that scientists and engineers actually engage in as part of their work. It is recognized that students cannot reach the level of competence of professional scientists and engineers, as well as that to give students opportunities to immerse themselves in these practices and to explore why they are central to science and engineering is critical to appreciate the skill of the expert and the nature of his or her enterprise. Eight practices are considered to be essential elements of the K-12 science and engineering curriculum: 1. Asking questions (for science) and defining problems (for engineering) 2. Developing and using models 3. Planning and carrying out investigations 4. Analyzing and interpreting data 5. Using mathematics, information and computer technology, and computational thinking 6. Constructing explanations (for science) and designing solutions (for engineering) 7. Engaging in argument from evidence 8. Obtaining, evaluating, and communicating information. These may be, a fortiori, considered relevant for undergraduate students. However, typical pedagogical methods used in introductory physics courses mainly engage students in lectures and the related laboratory experiments can be grouped into two categories: A theoretical relationship between two or more variables is already known (or at least suspected) and an experiment is needed to verify or quantify this relationship. A theoretical relationship between two or more variables is not available but rather sought through an experiment. Such kind of activities obscure the inquiry characteristics of scientific investigation and very often students perform experiments by slavishly following the recipes supplied by instructor. Many papers have pointed out the fairly general process that one can follow to design an experiment under any circumstances (Winncy, et al., 2005; Hatzikraniotis et als., 2010). This process can serve as a tool for teaching students experimental design as well as to introduce in it many of the practices previously described (NAS, 2011). Relevant elements of such a process have been considered the following (Winncy, et al., 2005): 1. Define the goals and objectives of the experiment. 2. Research any relevant theory (or idea ) about what to expect from the experiment. 3. Select the dependent and independent variable(s) to be measured. 4. Select appropriate methods for measuring these variables. 5. Choose appropriate equipment and instrumentation. 6. Select the proper range of the independent variable(s). 7. Determine an appropriate number of data points needed for each type of measurement. An easy solution, of course, is to intend all the information in steps 1 through 7 as a prescriptions by providing a cookbook lab. However, this types of approach should not supply an environment suitable to develop competences in experimental design as well as to engage students in inquiring. Our approach is aimed at defining and characterising practices in parallel at applying them at the solution of well posed problems. We think that this approach is relevant for making students aware of the involved practical/thinking process in a way that make them able to transfer these in different contexts.

281 Here, we describe a 4-hour Worksop (W) where students were put in a real problematic situation that made them face with a real problem, whose solution required the activation of the typical operative processes (theoretical and experimental) of scientific investigation. 3. Method The W involved 60 engineering students that have completed their two courses of Introductory Physics mainly consisting in lectures and activities devoted to the solution of numerical problems. They participated to the W on a voluntary basis. The lab activities have been introduced through a phase of topic exploration, mainly based on students prior knowledge, that has been followed by a discussion about the relevant physical quantities which should be taken into account in a contest of a laboratory activity of measurements. From their previous studies, students should know that the energy exchange between an emitting body and its surrounding is caused by the processes of conduction, convection and radiation. By discussing between each other, students should gather that, in the absence of a medium, as in a vacuum system, the conduction and convection cannot take place. The inquiry lab activity on thermal radiation proceeded as following: 1. A scientific investigation on the energy exchange between a powered resistor and its surrounding environment during the heating and cooling processes was proposed. 2. Students were stimulated to design experimental activities aimed at testing previously formulated hypotheses. 3. Then, they were introduced to the laboratory equipment and invited to adapt their plans about conducting the experiments with the available measurement facility for the study of energy exchange by thermal radiation. 4. Finally, they carried out their own experiments by collecting, processing and analysing data, by focusing on new concepts and laws and constructing models supplying a meaningful conceptual understanding of the physics underlying the process of energy exchange by thermal radiation. 4. The lab activities At the end of the first two steps previously described, all students agreed that in order to eliminate effects due to conduction and convection, the cooling experiments have to be conducted in a vacuum environment. Moreover it is necessary to measure the variation of the resistor temperature as well as that of the environment (for example a glass tube surrounding the resistor). Figure 1 shows the proposed equipment. Figure 1. The vacuum tube containing the resistor and details of the ceramic-type resistor and the thermocouple. The system is composed by a vacuum pump which is connected to the vacuum tube containing the resistor; the pump can lower the pressure inside the tube up to 0.1 mbar. The temperature of the resistor is measured by means of a thermocouple which is placed inside the vacuum tube and whose tip is in contact with the resistor surface. The thermocouple is externally connected to an amplifier module and interfaced to the lab computer. The resistor is a ceramic-type with the following physical characteristics: 11 W, 150 Ω. The temperature of the external surface of the

282 vacuum tube is measured by using a surface temperature sensor connected to a Vernier LabQuest computer interface. A power management system drives the electric current into the resistor. The first experiment was carried out by the measuring the temperature of the resistor during the heating and cooling process. Figure 2 shows two typical results: the first one involving a modest variation of the environment temperature, the second one showing a relevant heating of the environment temperature. Figure 2. Heating and cooling curves of resistor and glass vacuum tube: on the left the temperature of the tube is almost constant; on the right the glass tube temperature increases and then decreases as the resistor temperature increases/decreases. Students decided to investigate the cooling process following the saturation regime, just after the power supply has been switched off. They discussed about the collected data and tried to compare their experimental findings with the theory of thermal emission of radiation expressed by the Stefan law. dt 4 4 C = εσ S ( T Tb ) dt where T is the temperature of the emitting object, C the heat capacity, ε the emissivity, σ the Stefan constant, S the object surface and T b the temperature of the surrounding ambient. Both the heat capacity and the emissivity are assumed to be equal to the constantlearning in education, Ha values C=1.5 J/K and ε = 0.9, respectively. Students realised that, in order to perform such comparison, it was necessary to integrate the Stefan law or to approximate it ( as reported in their textbooks) to an exponential law: t b b0 inf inf ( ) T ( t) = T T e + T (exponential decreasing) b where T b0 is the maximum value of ambient temperature, T inf is the equilibrium temperature and b is the time constant. They tried to change these parameters in order to match the theory with the experimental data, and found that in the case of Fig. 3a a satisfactory agreement is not obtained for the exponential approximation, but the agreement is acceptable in the case of numerical integration of Stefan law. For the case of Fig. 3b students tried to identify the mean temperature of the glass tube as environment temperature during the cooling, but in both case the agreement is not satisfactory. They deduced that the problem needed further investigations. By analysing their thinking-aloud reasoning we verified that the experiment made students aware that the interpretation of experimental finding needs to put in action a lot of approximations, in particular the need to understand the importance of the boundary conditions. A group of students analysing the cooling of the glass tube and hypothesizing that in this case the effect of conduction/convection can be the most relevant factor tried to fit data by imposing also an exponential decreasing of the glass tube temperature by obtaining the curve 3) (see Fig. 3b) that fits well experimental data.

284 Winncy Y. Du, Burford J. Furman, Nikos J. Mourtos (2005) On the ability to design engineering experiments.8th UICEE Annual Conference on Engineering Education, Kingston, Jamaica, 7-11 February 2005

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290 FFP Udine November 2011 EXTENDED GRAVITY FROM NONCOMMUTATIVITY Paolo Aschieri, Dipartimento di Scienze e Innovazione Tecnologica and INFN Gruppo collegato di Alessandria, Università del Piemonte Orientale, Viale T. Michel 11, Alessandria, Italy. Abstract We present a first order theory of gravity (vierbein formulation) on noncommutative spacetime. The first order formalism allows to couple the theory to fermions. This NC action is then reinterpreted (using the Seiberg-Witten map) as a gravity theory on commutative spacetime that contains terms with higher derivatives and higher powers of the curvature and depend on the noncommutativity parameter θ. When the noncommutativity is switched off we recover the usual gravity action coupled to fermions. The first nontrival corrections to the usual gravity action coupled to fermions are explicitly calculated. 1. Introduction In the passage from classical mechanics to quantum mechanics classical observables become noncommutative. Similarly we expect that in the passage from classical gravity to quantum gravity, gravity observables, i.e. spacetime itself, with its coordinates and metric structure, will become noncommutative. Thus by formulating Einstein gravity on noncommutative spacetime we may learn some aspects of quantum gravity. Planck scale noncommutativity is further supported by Gedanken experiments that aim at probing spacetime structure at very small distances. They show that due to gravitational backreaction one cannot test spacetime at those distances. For example, in relativistic quantum mechanics the position of a particle can be detected with a precision at most of the order of its Compton wave length λ C = /mc. Probing spacetime at infinitesimal distances implies an extremely heavy particle that in turn curves spacetime itself. When λ C is of the order of the Planck length, the spacetime curvature radius due to the particle has the same order of magnitude and the attempt to measure spacetime structure beyond Planck scale fails. This Gedanken experiment supports finite reductionism. It shows that the description of spacetime as a continuum of points (a smooth manifold) is an assumption no more justified at Planck scale. It is then natural to relax this assumption and conceive a noncommutative spacetime, where uncertainty relations and discretization naturally arise. In this way the dynamical feature of spacetime that prevents from testing sub-plankian scales is explained by incorporating it at a deeper kinematic level. A similar mechanism happens for example in the passage from Galilean to special relativity. Contraction of distances and time dilatation can be explained in Galilean relativity: they are a consequence of the interaction between ether and the body in motion. In special relativity they become a kinematic feature. The noncommutative gravity theory we present following (Aschieri 2009 a, Aschieri 2011) is an effective theory that may capture some aspects of a quantum gravity theory. Furthermore we reinterpret spacetime noncommutativity as extra interaction terms on commutative spacetime, in this way the theory is equivalent to a higher derivative and curvature extension of Einstein general relativity. We have argued that spacetime noncommutativity should be relevant at The results presented in these proceedings are based on joint work with Leonardo Castellani

291 Planck scale, however the physical phenomena it induces can also appear at larger scales. For example, due to inflation, noncommutativity of spacetime at inflation scale (that may be as low as Planck scale) can affect cosmological perturbations and possibly the cosmic microwave background spectrum; see for example (Lizzi 2002). We cannot exclude that this noncommutative extension of gravity can be relevant for advancing in our understanding of nowadays open questions in cosmology. In this contribution, after a short overview of possible noncommutative approaches, we outline the Drinfeld twist approach and review the geometric formulation of theories on noncommutative spacetime (Aschieri 2009 b ). This allows to construct actions invariant under diffeomorphisms. In Section 4 we first present usual gravity coupled to fermions in an index free formalism suited for its generalization to the noncommutative case. Then we discuss gauge theories on noncommutative space and in particular local Lorentz symmetry (SO(3, 1)-gauge symmetry), indeed we need a vierbein formulation of noncommutative gravity in order to couple gravity to spinor fields. The noncommutative Lagrangian coupled to spinor fields is then presented. In Section 5 we reinterpret this NC gravity as an extended gravity theory on commutative spacetime. This is done via the Seiberg-Witten map from noncommutative to commutative gauge fields. The resulting gravity theory then depends on the usual gravitational degrees of freedom plus the noncommutative degrees of freedom, these latter are encoded in a set of mutually commuting vector fields {X I }. The leading correction terms to the usual action are explicitly calculated in Section 6. They couple spinor fields and their covariant derivatives to derivatives of the curvature tensor and of the vierbein. It is interesting to consider a kinetic term for these vector fields, so that the noncommutative structure of spacetime, as well as its metric structure depend on the matter content of spacetime itself. A model of dynamical noncommutativity is presented in (Aschieri 2012 a ). Noncommutative vierbein gravity can also be coupled to scalar fields (Aschieri 2012 a ) and to gauge fields (Aschieri 2012 b ). 2. NC geometry approaches Before entering the details of the theory, we briefly frame it in the context of noncommutative geometry approaches. The easiest way to describe a noncommutative spacetime is via the noncommutative algebra of its coordinates, i.e., we give a set of generators and relations. For example [x µ, x ν ] = iθ µν canonical (2.1) [x µ, x ν ] = i f µν σx σ Lie algebra (2.2) x µ x ν qx ν x µ = 0 quantum (hyper)plane (2.3) where θ µν (a real antisymmetric matrix), f µν σ (real structure constants), q (a complex number, e.g. a phase) are the respective noncommutativity parameters. Quantum groups and quantum spaces (Feddeev 1090, Manin 1988) are usually described in this way. In this case we do not have a space (i.e. a set of points), rather we have a noncommutative algebra generated by the coordinates {x µ } and their relations; when the noncommutativity parameters (θ µν, f µν σ, q) are turned off this algebra becomes commutative and is the algebra of functions on a usual space. Of course we can also impose further constraints, for example periodicity of the coordinates describing the canonical noncommutative spacetime (2.1) (that typical of phase-space quantum mechanics) leads to a noncommutative torus rather than to a noncommutative (hyper)plane. Similarly, constraining the coordinates of the quantum (hyper)plane relations (2.3) we obtain a quantum (hyper)sphere. This algebraic description should then be complemented by a topological approach. One that

292 for example leads to the notions of continuous functions. This is achieved completing the algebra generated by the noncommutative coordinates to a C -algebra. Typically C -algebras arise as algebras of operators on Hilbert space. Connes noncommutative geometry (Connes 1994) starts from these notions and enriches the C -algebra structure and its representation on Hilbert space so to generalize to the noncommutative case also the notions of smooth functions and metric structure. Another approach is the -product one. Here we retain the usual space of functions from commutative space to complex numbers, but we deform the pointwise product operation in a -product one. A -product sends two functions ( f, g) in a third one ( f g). It is a differential operator on both its arguments (hence it is frequently called a bi-differential operator). It has the associative property f (g h) = ( f g) h. The most known example is the Gronewold- Moyal-Weyl star product on R 2n, Notice that if we set then ( f h)(x) = e i 2 θµν x µ y ν f (x)h(y) x=y. (2.4) F 1 = e i 2 θµν x µ y ν ( f h)(x) = µ F 1 ( f h)(x) x µ where µ is the usual product of functions µ( f g) = f h. The element F = e 2 i θµν y ν is an example of a Drinfeld twist. It is defined by the exponential series in powers of the noncommutativity parameters θ µν, F = e i 2 θµν x µ y ν = 1 1 i 2 θµν µ ν 1 8 θµ 1ν 1 θ µ 2ν 2 µ1 µ2 ν1 ν It is easy to see that x µ x ν x ν x µ noncommutative algebra (2.1). = iθ muν thus also in this approach we recover the In this paper noncommutative spacetime will be spacetime equipped with a -product. We will not discuss when the exponential series f g = f g i 2 θµν µ ( f ) ν (g) +... defining the function f g is actually convergent. We will therefore work in the well established context of formal deformation quantization (Kontsevich 2003). In the latter part of the paper we will consider a series expansion of the noncommutative gravity action in powers of the noncommutativity parameters θ, we will present the first order in θ (a second order study appears in (Aschieri 2011)), therefore the convergence aspect won t be relevant. The method of constructing -products using twists is not the most general method, however it is quite powerful, and the class of -products obtained is quite wide. For example choosing the appropriate twist we can obtain the noncommutative relations (2.1), (2.2) and also (depending on the structure constant explicit expression) some of the Lie algebra type (2.3). 3. Twists and -Noncommutative Manifolds Let M be a smooth manifold, a twist is an invertible element F UΞ UΞ where UΞ is the universal enveloping algebra of vector fields, (i.e. it is the algebra generated by vector fields on M and where the element XY YX is identified with the vector field [X, Y]). The element F must satisfy some further conditions that we do not write here, but that are satisfied if we consider abelian twists, i.e., twists of the form F = e i 2 θij X I X J = 1 1 i 2 θij X I X J 1 8 θi 1J 1 θ I 2J 2 X I1 X I2 X J1 X J2 +...

293 were the vector fields X I (I = 1,...s with s not necessarily equal to m = dim M) are mutually commuting [X I, X J ] = 0 (hence the name abelian twist). It is convenient to introduce the following notation F 1 = i 2 θij X I X J 1 8 θi 1J 1 θ I 2J 2 X I1 X I2 X J1 X J = f α f α (3.5) where a sum over the multi-index α is understood. Let A be the algebra of smooth functions on the manifold M. Then, given a twist F, we deform A in a noncommutative algebra A by defining the new product of functions f h = f α ( f ) f α (h), we see that this formula is a generalization of the Gronewold-Moyal-Weyl star product on R 2n defined in (2.4). Since the vector fields X I are mutually commuting then this -product is associative. Note that only the algebra structure of A is changed to A while, as vector spaces, A and A are the same. We similarly consider the algebra of exterior forms Ω with the wedge product, and deform it in the noncommutative exterior algebra Ω that is characterized by the graded noncommutative exterior product given by τ τ = f α (τ) f α (τ ), where τ and τ are arbitrary exterior forms. Notice that the action of the twist on τ and τ is via the Lie derivative: each vector field X I1, X I2, X J1, X J2... in (3.5) acts on forms via the Lie derivative. It is not difficult to show that the usual exterior derivative is compatible with the new -product, d(τ τ ) = d(τ) τ + ( 1) deg(τ) τ dτ (3.6) this is so because the Lie derivative commutes with the exterior derivative. We also have compatibility with the usual undeformed integral (graded cyclicity property): τ τ = ( 1) deg(τ)deg(τ ) τ τ (3.7) (the equality holds up to boundary terms). Finally we also have compatibility with the undeformed complex conjugation: (τ τ ) = ( 1) deg(τ)deg(τ ) τ τ. (3.8) Note. We remark that all these properties are due to the special nature of the -product we consider. As shown in (Kontsevich 2003) -products are in 1-1 correspondence with Poisson structures {, } on the manifold M. The Poisson structure the twist F induces is { f, g} = iθ IJ X I ( f )X J (g). However the twist F encodes more information than the Poisson bracket {, }. The key point is that the twist is associated with the Lie algebra Ξ (and morally with the diffeomorphisms group of M). Given a twist we can deform the Lie algebra Ξ (the diffeomorphisms group of M) and then, using the Lie derivative action (the action of the diffeomorphisms group) we induce noncommutative deformations of the algebra of functions on M, of the exterior algebra and more generally of the differential and Riemannian geometry structures on M (Aschieri 2006), leading to noncommutative Einstein equations for the metric tensor (Aschieri 2006).

294 4. Noncommutative vierbein gravity coupled to fermions 4.1. Classical action The usual action of first-order gravity coupled to spin 1 fields reads: 2 S = ε abcd R ab V c V d i ψγ a V b V c V d Dψ i(d ψ)γ a V b V c V d ψ (4.9) with R ab = dω ab ω a c ω cb, and the Dirac conjugate defined as usual: ψ = ψ γ 0. This action can be recast in an index-free form (Chamseddine 2004, Aschieri 2009), convenient for generalization to the non-commutative case: S = Tr (ir V Vγ 5 ) + ψv V Vγ 5 Dψ + D ψ V V Vγ 5 ψ (4.10) where with R = dω Ω Ω, Dψ = dψ Ωψ, D ψ = Dψ = d ψ + ψω (4.11) Ω 1 4 ωab γ ab, V V a γ a, R 1 4 Rab γ ab taking value in Dirac gamma matrices. Use of the gamma matrix identities γ abc = iε abcd γ d γ 5, Tr(γ ab γ c γ d γ 5 ) = 4iε abcd in computing the trace leads back to the usual action (4.9). The action (4.10) is invariant under local diffeomorphisms (because it is the integral of a 4-form on a 4-manifold) and under local Lorentz rotations. In the index-free form they read δ ε V = [V, ε], δ ε Ω = dε [Ω, ε], δ ε ψ = εψ, δ ε ψ = ψε (4.12) with ε = 1 4 εab γ ab The local Lorentz invariance of the index free action follows from δ ε R = [R, ε] and δ ε Dψ = εdψ, the cyclicity of the trace Tr and the fact that the gauge parameter ε commutes with γ Noncommutative gauge theory and Lorentz group Consider an infinitesimal gauge transformation λ = λ A T A, where the generators T A belong to some representation (the fundamental, the adjoint, etc.) of a Lie group G. Since two consecutive gauge transformations are a gauge transformation, the commutator of two infinitesimal ones [λ, λ ], closes in the Lie algebra of G. This in general is no more the case in noncommutative gauge theories. In the noncommutative case the commutator of two infinitesimal gauge transformations is [λ, λ ] λ λ λ λ = 1 2 {λa, λ B } [T A, T B ] [λa, λ B ] {T A, T B }. We see that also the anticommutator {T A, T B } appears. This is fine if our gauge group is for example U(N) or GL(N) in the fundamental or in the adjoint, since in this case {T A, T B } is again in the Lie algebra, however for more general Lie algebras (including all simple Lie algebras) we have to enlarge the Lie algebra to include also anticommutators besides commutators, i.e. we have to consider all possible products T A T B... T C of generators. Our specific case is the Lorentz group in the spinor representation given by the Dirac gamma matrices γ ab. The algebra generated by these gamma matrices is that of all even 4 4 gamma matrices. The noncommutative gauge parameter will therefore have components ε = 1 4 εab γ ab + iε 1 + εγ 5.

295 The extra gauge parameters ε, ε can be chosen to be real (like ε ab ). Indeed the reality of ε ab, ε, ε is equivalent to the hermiticity condition γ 0 εγ 0 = ε (4.13) and if the gauge parameters ε, ε satisfy this condition then also [ε, ε ] is easily seen to satisfy this hermiticity condition. We have centrally extended the Lorentz group to SO(3, 1) SO(3, 1) U(1) R +, or more precisely, (since our manifold M has a spin structure and we have a gauge theory of the spin group SL(2, C)) SL(2, C) GL(2, C). The Lie algebra generator i 1 is the anti-hermitian generator corresponding to the U(1) extension, while γ 5 is the hermitian generator corresponding to the noncompact R + extension. Since under noncommutative gauge transformations we have δ ε Ω = dε Ω ε + ε Ω (4.14) also the spin connection and the curvature will be valued in the GL(2, C) Lie algebra representation given by all the even gamma matrices, Ω = 1 4 ωab γ ab + iω 1 + ωγ 5, R = 1 4 Rab γ ab + ir 1 + rγ 5. (4.15) Similarly the gauge transformation of the vierbein, δ ε V = V ε + ε V, (4.16) closes in the vector space of odd gamma matrices (i.e. the vector space linearly generated by γ a, γ a γ 5 ) and not in the subspace of just the γ a matrices. Hence the noncommutative vierbein are valued in the odd gamma matrices V = V a γ a + Ṽ a γ a γ 5. (4.17) 4.3. Noncommutative Gravity action and its symmetries We have all the ingredients in order to generalize to the noncommutative case the gravity action coupled to spinors of Section 3: an abelian twist giving the star products of functions and forms on the spacetime manifold M (and compatible with usual integration on M); an extension to GL(2, C) of the Lorentz gauge group, so that infinitesimal noncommutative gauge transformations close in this extended Lie algebra. The action reads S = Tr (ir V Vγ 5 ) + ψ V V V γ 5 Dψ + D ψ V V V γ 5 ψ (4.18) with R = dω Ω Ω, Dψ = dψ Ω ψ, D ψ = d ψ + ψ Ω. (4.19) Gauge invariance The invariance of the noncommutative action (4.18) under the -variations is demonstrated in exactly the same way as for the commutative case: noting that besides (4.14) and (4.16) we have δ ε ψ = ε ψ, δ ε ψ = ψ ε, δ ε Dψ = ε Dψ, δ ε D ψ = D ψ ε, δ ε R = R ε+ε R, (4.20)

296 and using that ε commutes with γ 5, and the cyclicity of the trace together with the graded cyclicity of the integral with respect to the -product. Diffeomorphisms invariance The -action (4.18) is invariant under usual diffeomorphisms, being the integral of a 4-form. Under these diffeomorphisms the vector fields X I transform covariantly. We also mention that since the vector fields X I appear only in the -product, the action is furthermore invariant under -diffeomorphisms as defined in (Aschieri 2006), see discussion in (Aschieri 2009 b ), Section Under these deformed diffeomorphisms the vector fields X I do not transform. Reality of the action Hermiticity conditions can be imposed on the fields V and Ω as done with the gauge parameter ε in (4.13) : γ 0 Vγ 0 = V, γ 0 Ωγ 0 = Ω. (4.21) These hermiticity conditions are consistent with the gauge variations and can be used to check that the action (4.18) is real by comparing it to its complex conjugate (obtained by taking the Hermitian conjugate of the 4-form inside the trace in the integral). As previously observed for the component gauge parameters ε ab, ε, ε, the hermiticity conditions (4.21) imply that the component fields V a, Ṽ a, ω ab, ω, and ω are real fields. Charge conjugation invariance Noncommutative charge conjugation is the following transformation (extended linearly and multiplicatively 1 ): ψ ψ C C( ψ) T = γ 0 Cψ, V V C C V T C, Ω Ω C CΩ T C, θ C θ = θ, (4.22) and consequently θ C θ = θ. Then the action (4.18) is invariant under charge conjugation. For example S C bosonic = i Tr(R C θ V C θ V C γ 5 ) T = i Tr(R T θ V T θ V T Cγ 5 C 1 ) T = i Tr ( (V T θ V T γ T 5 )T R ) = i Tr ( (V T γ T 5 )T V R ) = i Tr(γ 5 V V R) = i Tr(R γ 5 V V) = i Tr(R V Vγ 5 ) = S bosonic (4.23) Similarly the fermionic part of the action satisfies S C f ermionic = S f ermionic. Charge conjugation conditions In the classical limit θ 0 the -product becomes the usual pointwise product. The noncommutative gauge symmetry becomes a usual gauge symmetry with gauge group GL(2, C) and the noncommutative vierbein in the classical limit leads to two independent vierbeins: V a and Ṽ a transforming both only under the SL(2, C) subgroup of GL(2, C). As observed in (Chamseddine 2004), this is problematic because we obtain two massless gravitons, and only one local Lorentz symmetry, that is not enough in order to kill the unphysical degrees of freedom. Either we concoct a mechanism such that the second graviton becomes massive or we further constraint the noncommutative theory so that in the classical limit the extra vierbein vanishes. 1 Multiplicativity and the transformation C θ = θ is equivalent to antimultiplicativity with respect to the -product: ( f g) C = g C f C (for f and g scalar fields that are not matrix valued).

297 The vanishing of the Ṽ a components in the classical limit is achieved by imposing charge conjugation conditions on the fields (Aschieri 2009 a ): CV θ (x)c = V θ (x) T, CΩ θ (x)c = Ω θ (x) T, Cε θ (x)c = ε θ (x) T (4.24) These conditions involve the θ-dependence of the fields. This latter is due to the -product θ-dependence (recall that the -product is defined as an expansion in power series of the noncommutativity parameter θ). Since noncommutative gauge transformations involve the - product, the gauge transformed fields will be θ-dependent and hence field configurations are in general θ-dependent. Conditions (4.24) are consistent with the -gauge transformations. For example the field CV θ (x) T C can be shown to transform in the same way as V θ (x) (Aschieri 2009 a ). For the component fields and gauge parameters the charge conjugation conditions imply that the components V a, ω ab are even in θ, while the components Ṽ a, ω, ω are odd: Vθ a = Va θ, ωab θ = ω ab θ (4.25) Ṽθ a = Ṽa θ, ω θ = ω θ, ω θ = ω θ. (4.26) Similarly for the gauge parameters: ε ab θ = ε ab θ, ε θ = ε θ, ε θ = ε θ. In particular, since the components Ṽ a are odd in θ we achieve their vanishing in the classical limit. We can also conclude that the bosonic action is even in θ. Indeed (4.24) implies V C = V θ, Ω C = Ω θ, R C = R θ. (4.27) Hence the bosonic action S bosonic (θ) is mapped into S bosonic ( θ) under charge conjugation. Also for the fermionic action, S f ermionic (θ), we have S f ermionic (θ) C = S f ermionic ( θ) if the fermions are Majorana, i.e. if they satisfy ψ C = ψ θ. From S bosonic (θ) = S bosonic ( θ) we conclude that all noncommutative corrections to the classical action of pure gravity are even in θ; this is also the case if we couple noncommutative gravity to Majorana fermions. 5. Seiberg-Witten map (SW map) In the previous section we have formulated a noncommutative gravity theory that in the classical limit θ 0 reduces to usual vierbein gravity. In the full noncommutative regime it has however a doubling of the vierbein fields. We can insist on a noncommutative gravity theory that has the same degrees of freedom of the classical one. This is doable if we use the Seiberg-Witten map to express the noncommutative fields in terms of the commutative ones. In this way the gauge group is the Lorentz group SL(2, C) and not the centrally extended one GL(2, C), indeed also the noncommutative gauge parameters ε ab, ε, ε are expressed in term of the commutative ones ε ab. Because of the SW map the noncommutative fields can therefore be expanded in terms of the commutative ones, and hence the noncommutative gravity action can be expanded, order by order in powers of θ, in terms of the usual commutative gravity and spinor field as well as of the noncommutativity vector fields X I. As we will see, we thus obtain a commutative gravity action that at zeroth order in θ is usual gravity (coupled to spinors) and at higher orders in θ contains higher derivative terms describing gravity and spinor fields coupled to the noncommutativity vector fields X I. The Seiberg-Witten map (SW map) relates the noncommutative gauge fields Ω to the ordinary Ω 0, and the noncommutative gauge parameters ε to the ordinary ε 0 and Ω 0 so as to satisfy: Ω(Ω 0 ) + δ ε Ω(Ω 0 ) = Ω(Ω 0 + δ ε 0Ω 0 ) (5.28)

298 with the noncommutative and ordinary gauge variations given by δ ε Ω = dε Ω ε + ε Ω, δ ε 0Ω 0 = dε 0 Ω 0 ε 0 + ε 0 Ω 0. (5.29) Equation (5.28) can be solved order by order in θ (Ulker 2008), yielding Ω and ε as power series in θ: Ω(Ω 0, θ) = Ω 0 + Ω 1 (Ω 0 ) + Ω 2 (Ω 0 ) + + Ω n (Ω 0 ) + (5.30) ε(ε 0, Ω 0, θ) = ε 0 + ε 1 (ε 0, A 0 ) + ε 2 (ε 0, Ω 0 ) + + ε n (ε 0, Ω 0 ) + (5.31) where Ω n (Ω 0 ) and ɛ n (ε 0, Ω 0 ) are of order n in θ. Note that ε depends on the ordinary ε 0 and also on Ω 0. The Seiberg-Witten condition (5.28) states that the dependence of the noncommutative gauge field on the ordinary one is fixed by requiring that ordinary gauge variations of Ω 0 inside Ω(Ω 0 ) produce the noncommutative gauge variation of Ω. This implies that once we expand, order by order in θ, the noncommutative action in terms of the commutative fields, the resulting action will be gauge invariant under ordinary local Lorentz transformations because the noncommutative action is invariant under the local noncommutative Lorentz transformations of Section 4.3. Following ref. (Aschieri 2011), up to first order in θ the solution reads: Ω = Ω 0 + i 4 θij {Ω 0 I, l JΩ 0 + R 0 J } (5.32) ε = ε 0 + i 4 θij {Ω 0 I, l Jε 0 } (5.33) R = R 0 + i ( 4 θij {Ω 0 I, (l J + L J )R 0 } [R 0 I, R0 J ]) (5.34) ψ = ψ 0 + i 4 θij Ω 0 I (l J + L J )ψ 0 (5.35) where Ω 0 A, R0 A are defined as the contraction along the tangent vector X I of the exterior forms Ω 0, R 0, i.e. Ω 0 A i IΩ 0, R 0 I i I R 0, (i I being the contraction along X I ). We have also introduced the Lie derivative l I along the vector field X I, and the covariant Lie derivative L I along the vector field X I. L I acts on R 0 and ψ 0 as L I R 0 = l I R 0 Ω 0 B R0 + R 0 Ω 0 I and L Iψ 0 = l I ψ 0 Ω 0 I ψ0. In fact the covariant Lie derivative L I has the Cartan form: L I = i I D + Di I where D is the covariant derivative. We refer to (Aschieri 2011) for higher order in θ expressions. All these formulae are not SO(1, 3)-gauge covariant, due to the presence of the naked connection Ω 0 and the non-covariant Lie derivative l I = i I d + di I. However, when inserted in the NC action the resulting action is gauge invariant order by order in θ. Indeed usual gauge variations induce the -gauge variations under which the NC action is invariant. Therefore the NC action, re-expressed in terms of ordinary fields via the SW map, is invariant under usual gauge transformations. Since these do not involve θ, the expanded action is invariant under ordinary gauge variations order by order in θ. Moreover the action, once re-expressed in terms of ordinary fields remains geometric, and hence invariant under diffeomorphisms. This is the case because the noncommutative action and the SW map are geometric, we indeed see that only coordinate independent operations like the contraction i I and the Lie derivatives l I and L I appear in the SW map. From (5.32) and (5.35) we also deduce Dψ = Dψ 0 + i 4 θij( Ω 0 I (l J + L J )Dψ 0 2R I L J ψ ). (5.36)

299 6. Action at first order in θ The expression of the gravity action, up to second order in θ, in terms of the commutative fields and of the first order fields (5.32)-(5.36) has been given in (Aschieri 2011). The action is gauge invariant even if the expression in (Aschieri 2011) is not explicitly gauge invariant. We here present the explicit gauge invariant expression for the action up to first order in θ. We replace the noncommutative fields appearing in the action with their expansions (5.32)-(5.36) in commutative fields and integrate by parts in order to obtain an explicit SO(3, 1) gauge invariant action. We thus obtain the following gravity action coupled to spinors S = Tr (irvvγ 5 ) + ψv 3 γ 5 Dψ + D ψv 3 γ 5 ψ (6.37) + i 4 θij( ψ{v 3, R IJ }γ 5 Dψ + D ψ{v 3, R IJ }γ 5 ψ ) + i 2 θij( 2L I ψr J V 3 γ 5 ψ 2 ψv 3 R I γ 5 L J ψ L I ψv 3 γ 5 L J Dψ L I D ψ V 3 γ 5 L J ψ + ψ({l I VL J V, V} + L I V VL J V)γ 5 Dψ + D ψ({l I VL J V, V} + L I V VL J V)γ 5 ψ ) + O(θ 2 ) where with obvious abuse of notation we have omitted the apex 0 denoting commutative fields, we also have omitted writing the wedge product, and V 3 = V V V. 7. Conclusions We have constructed an extended Einstein gravity action that: i) is explicitly invariant under local Lorentz transformations because expressed solely in terms of the gauge covariant operators L I, i I, D and fields R, V, ψ; ii) is diffeomorphic invariant; iii) has the same fields of classical gravity plus the noncommutative structure. This latter is given by the vector fields {X I } that choosing an appropriate kinetic term can become dynamical, the idea being that both spacetime curvature and noncommutativity should depend on matter distribution. This extended action has been obtained from considering gravity on noncommutative spacetime, that as argued in the introduction is a very natural assumption at high energies, like those close to the inflationary epoch. Acknowledgments We acknowledge the fruitful and pleasant atmosphere and the perfect organization enjoyed during the Udine Symposium. A. Gamma matrices in D = 4 We summarize in this Appendix our gamma matrix conventions in D = 4. η ab = (1, 1, 1, 1), {γ a, γ b } = 2η ab, [γ a, γ b ] = 2γ ab, (A.38) γ 5 iγ 0 γ 1 γ 2 γ 3, γ 5 γ 5 = 1, ε 0123 = ε 0123 = 1, (A.39) γ a = γ 0 γ a γ 0, γ 5 = γ 5 (A.40) γ T a = Cγ a C 1, γ T 5 = Cγ 5C 1, C 2 = 1, C = C T = C (A.41) References Aschieri P and Castellani L (2009 a ) Noncommutative D=4 gravity coupled to fermions, JHEP [arxiv: [hep-th]] Aschieri P and Castellani L (2011) Noncommutative gravity coupled to fermions: second order expansion via Seiberg-Witten map, arxiv: [hep-th] Lizzi F, Mangano G, Miele G and Peloso M (2002) Cosmological perturbations and short distance physics from noncommutative geometry, JHEP [hep-th/ ]

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306 New aspects of collective phenomena at nanoscales in quantum plasmas P. K. Shukla and B. Eliasson International Centre for Advanced Studies in Physical Sciences & Institut für Theoretische Physik, Fakultät für Physik und Astronomie, Ruhr Universität Bochum, D Bochum, Germany Abstract. We present two novel collective effects is quantum plasmas. First, we discuss a novel attractive force between ions that are shielded by the degenerate electrons in quantum plasmas. Here we show that the electric potential around an isolated ion has a hard core negative part that resembles the Lennard-Jones (LJ)-type potential. Second, we present a theory for stimulated scattering instabilities of electromagnetic waves off quantum plasma modes. Our studies are based on the quantum hydrodynamical description of degenerate electrons that are greatly influenced by electromagnetic and quantum forces. The relevance of our investigation to bringing ions closer for fusion in high-energy solid density plasmas at atomic dimensions, and for producing coherent short wavelength radiation in the x-ray regime at nanoscales are discussed. Keywords: Strongly coupled quantum plasmas, collective processes, novel attractive force, nonlinear effects PACS: Fp,53.35,Mw INTRODUCTION Recently, there has been a surge in the investigation of collective processes [1 3] in quantum plasmas. The latter are ubiquitous in a variety of physical environments, including the cores of Jupiter [4] and white dwarf stars [5 9], magnetars [10], warm dense matters [11]), compressed plasmas produced by intense laser beams [12 14], and processing devices for modern high-technology (e.g. semiconductors [15], thin films and nano-metallic structures [16]). In quantum plasmas, the electrons are degenerate and obey the Fermi-Dirac distribution function, while non-degenerate strongly correlated ions are coupled with the electrons via the electromagnetic fields. Quantum mechanical effects become important when the inter-electron distance is of the order of the thermal de Broglie wavelength. Here the overlapping of electron wave functions occurs due to Heisenberg s uncertainty and Pauli s exclusion principles [17]. The quantum statistical pressure law for relativistically degenerate electrons was obtained by Chandrasekhar in his classic paper [18]. He found for the electron pressure [18, 19] P e = πm4 ec 5 [ ] 3h 3 R(2R 2 3) 1 + R 2 + 3ln(R R 2 ), (1) where m e is the rest mass of the electrons, c the speed of light in vacuum, h the Planck constant, R = (n e /n c ) 1/3, n e the electron number density, and n c cm 3. The electron pressure P e can be represented in polytropic form P e = P 0 (n/n 0 ) γ, where we have P 0 = π 2/3 h 2 n 5/3 0 /5m e (3 hcn 4/3 0 /4) with γ = 5/3 (4/3) for non-relativistic

307 (ultra-relativistic) degenerate electrons, and h = h/2π is the reduced Planck s constant. The change in the equation of state (from 5/3 to 4/3) due to ultra-relativistic degeneracy of the electrons is responsible for the Chandrasekhar critical mass [18] M c ( hc/g) 3/2 m 2 n 1.4M, where M is the solar mass, of a white dwarf star, due to the balance between the gradient of ultra-relativistic electron pressure (3 hc/4)n 4/3 e and the gravitational force GM/R 2 s)m e m n, where G is the gravitational constant, M and R s are the mass and radius of a star, and m n mass of the nuclei (n e m n = M/R 3 c). Besides the quantum statistical electron pressure in dense plasmas, there is also a negative electron pressure P C due to Coulomb interactions [20], viz. P C = 8π3 m 4 ec 5 R 4 α 0 Z 2/3 ( ) i 4 1/3 h 3 10π 2, (2) 9π where α 0 = 1/137 is the fine structure constant and Z i the atomic number of ions. Furthermore, there are novel quantum forces associated with i) the quantum electron recoil/electron tunneling effect [21 26] caused by overlapping of electron wave functions due to the Heisenberg uncertainty and Pauli exclusion principles, and ii) electronexchange and electron correlations [27] caused by electron spin effect [28]. These quantum forces (the quantum pressure gradients, the quantum Bohm force due to the quantum electron recoil effect [21, 22], and the gradient of the the potential associated with electron-exchange and electron-correlations effects [27]), and electromagnetic forces control the dynamics of degenerate electrons in dense quantum plasmas. It turns out that the quantum forces produce new features to both electron and ion plasma oscillations [1-3] in an unmagnetized quantum plasma. In this paper, we present two novel aspects of collective interactions in quantum plasmas. First, we report on the discovery of novel attractive force [29] at atomic dimensions/nanoscales that can bring ions closer. Novel attractive force arises due to the interplay between the quantum Bohmian force [21, 22], as well as forces due to the quantum statistical pressure [18] of non-relativistic degenerate electrons and electronexchange and electron correlations effects [27]. Specifically, we shall use here the generalized quantum hydrodynamical (G-QHD) equations [1 3, 29] for non-relativistic degenerate electron fluids, the generalized viscoelastic (GV) ion momentum equations [1] including ion correlations and ion fluid viscosity effects, supplemented by Poisson s equation. Thus, our G-QHD-GV model captures the essential physics of electrostatic collective modes in quantum plasmas. The manuscript is organized in the following fashion. In Sec. II, we present the governing equations for electrostatic perturbations and derive the electron and ion susceptibilities in an unmagnetized quantum plasma. The linear response theory for quasistationary electrostatic perturbations involving immobile ions reveals that the electric potential around an isolated ion in a quantum plasma has a new profile [29], which resembles the Lennard-Jones-type potential. Thus, our newly found electric potential embodies a short-range negative hard core electric potential. The latter would be responsible for ion clustering and oscillations of ion lattices under the new attractive force we have recently discovered [29] in a strongly coupled quantum plasma. Section III contains an investigation of stimulated scattering instabilities of coherent high-frequency electromagnetic (EM) waves off quantum electron and ion plasma modes. Here we dis-

308 cuss stimulated Raman, stimulated Brillouin, and modulational instabilities of a constant amplitude EM pump wave. Explicit expressions for the growth rates are presented. The relevance of our investigation to the field of high-energy density quantum plasma physics is pointed out. THE GOVERNING EQUATIONS Let us consider an unmagnetized quantum plasma consisting of non-relativistic degenerate electrons and mildly coupled ions. In our quantum plasma, the electron and ion coupling parameters are Γ e = e 2 /a e k B T F and Γ i = Z 2 i e2 /a i k B T i, respectively, where a e a i = (3/4πn 0 ) 1/3 is the average interparticle distance, k B the Boltzmann constant, T F = ( h 2 /2m e k B )(3π 2 n 0 ) 2/3 the Fermi electron temperature, and T i the ion temperature. It turns out that Γ i /Γ e = Z 2 i T F/T i 1, since in quantum plasmas we usually have T F > T i. The dynamics of degenerate electron fluids is governed by the generalized quantum hydrodynamic (G-QHD) equations composed of the continuity equation the momentum equation du e m e n e dt and Poisson s equation n e t + (n e u e ) = 0, (3) = n e e ϕ P + n e V xc + n e V B, (4) 2 ϕ = 4πe(n e n i ) 4πQδ(r), (5) Here d/dt = ( / t) + u e, n e (n i ) is the electron (ion) number density, Q the test charge, u e the electron fluid velocity, and ϕ the electrostatic potential. Furthermore, we have denoted the quantum statistical pressure P = (n 0 m e v 2 /5)(n/n 0 ) 5/3, where v = h(3π 2 ) 1/3 /m e r 0 is the electron Fermi speed and r 0 = n 1/3 0 represents the mean inter-particle distance. The sum of the electron exchange and electron correlations potential is [27] V xc = 0.985(e 2 /ε)n 1/3 [1 + (0.034/a B n 1/3 )ln( a B n 1/3 )], where a B = h 2 /m e e 2 is the Bohr atomic radius. The quantum Bohm potential reads [1, 3, 21, 22] V B = ( h 2 /2m e )(1/ n e ) 2 n e. Equation (4) is valid if the plasmonic energy density hω pe is smaller than or comparable with the Fermi electron kinetic energy k B T F, where ω pe = (4πn 0 e 2 /m e ) 1/2 is the electron plasma frequency, and the electron-ion collision relaxation time is greater than the electron plasma period. The ion number density perturbation n i1 is obtained from the ion continuity equation n i t + (n iu i ) = 0, (6) where the ion fluid velocity u i is determined from the generalized viscoelastic ion momentum equation

309 ( )( ) D Du ( i 1 + τ m m i n i Dt Dt + P i + Z i en i ϕ η 2 u i ξ + η ) ( u i ) = 0, (7) 3 where D/Dt = ( / t) + u i, τ m is the viscoelastic relaxation time for ion correlations, P i = γ i µ i k B T i n i the ion thermal pressure involving strong ion coupling effects, m i the ion mass, γ i the adiabatic index, µ i [= (1/k B T i )( P i / n i ) kb T i ] 1 +U i (Γ i )/3 + (Γ i /9) U i (Γ i )/ Γ i the isothermal compressibility factor for non-degenerate ion fluids, the function U i (Γ i ) is the measure of the excess internal energy of the system, which is related with the correlation energy E c by U i (Γ i ) = E c /n i k B T i [ Γ i ( r 2 i /a2 i ), where r i is the ion core radius which depends on the degree of ion stripping [30]]. For one-component plasma model, we [31, 32] usually adopt U i (Γ i ) 0.9Γ i for Γ i 1 Furthermore, the coefficients of the shear and bulk ion fluid viscosities are denoted by η and ξ, respectively. We note that Eq. (7) is similar to that used by Frenkel [33] and Ichimaru [31, 32] in the context of ordinary fluids and one-component strongly coupled plasmas, respectively. Furthermore, Kaw and Sen [34] adopted a generalized viscoelastic dust momentum equation for studying the properties of dust acoustic waves [35] in a multi-component dusty plasma with highly- charged dust grains. Letting n j = n 0 + n j1, where n j1 n 0 and j equals e for electrons and i for ions, we linearize Eqs. (3) and (4) as well as (6) and (7) to obtain the electron and ion number density perturbations n j1. The latter can be substituted into the Fourier transformed version of Eq. (5), obtaining, in the linear approximation, the electric potential [29] ϕ(r) = Q exp(ik r) 2π 2 k 2 D(ω,k) d3 k, (8) where the dielectric constant D(ω,k) for an unmagnetized quantum plasma is D(ω,k) = 1 + χ e (ω,k) + χ i (ω,k). (9) Here the electron and ion susceptibilities are, respectively, and ω 2 pe χ e (ω,k) = ω 2 k 2 U 2 h 2, (10) k 4 /4m 2 e χ i (ω,k) = ω 2 pi ω 2 k 2 V 2 ti + iωη k 2 /(1 iωτ m ), (11) where U = (v 2 /3 + v 2 ex) 1/2, v ex = (0.328e 2 /m e r 0 ) 1/2 [ /( a B n 1/3 0 )] 1/2, ω pi = (4πn 0 Z 2 i e2 /m i ) 1/2, V ti = (γ i µ i k B T i /m i ) 1/2 the ion thermal speed, and η = (ξ + 4η/3)/m i n 0. The frequency and wave vector is denoted by ω and k, respectively.

310 Linear Quantum Plasma Waves Two cases are of interest. First, for high-frequency electron plasma waves with ω ω pi, we have from 1 + χ e (ω,k) = 0, the frequency of the electron plasma oscillations (EPOs) ω(k) = (ω 2pe + k 2 U 2 + h2 k 4 ) 1/2 Ω L (k). (12) Second for ω k(u 2 + h 2 k 2 /4m 2 e) 1/2 ku q, we have χ e (k) = ω 2 pe/k 2 U 2 q, so that 1 + χ e + χ i = 0 gives 4m 2 e ω 2 ω v + iω (1 iωτ m ) k2 Cs 2 h2 k 4 = 0, (13) 4m e m i for the ion plasma oscillations (IPOs). Here we have denoted ω v = η k 2 and C s = (m e U 2 /m i +Vti 2)1/2. In the hydrodynamic limit, viz. ωτ m 1, we have viscous damping of the quantum ion plasma mode. The real and imaginary parts of the frequencies (ω = ω r + iω i ) are, respectively, and ω r (k) = [k 2 C 2s + h2 k 4 ] 1/2 ωi 2 Ω s (k), (14) 4m e m i ω i (k) = ω v 2. (15) Furthermore, in the kinetic regime characterized by ωτ m 1, we have from (13) ω(k) = ( ωv + k 2 Cs 2 + h2 k 4 ) 1/2 Ω I (k). (16) τ m 4m e m i Generally, τ m = τ 0 Y g (k), where τ 0 = 1/ω v (1 µ i +4U i (Γ i )/15) and Y g (k) = exp( k/k g ) for a Gaussian distribution, and Y l (k) = (1+k 2 /k 2 L ) 1 for a Lorentzian distribution. Here k g and k l are the scale factors [32]. Potential Distribution Around an Isolated Ion Let us now present novel attractive force [29] in quantum plasmas, by using 1/D that is associated with quasi-stationary (ω 0)) propagating electrostatic perturbations in a quantum plasma with immobile ions. Accordingly, we have [29] 1 D = (k2 /ks 2 ) + αk 4 /ks (k 2 /ks 2 ) + αk 4 /ks 4, (17)

311 where k s = ω pe /U is the inverse modified (due to the potential of the electron-exchange and electron-correlations) Thomas-Fermi screening length, and α = h 2 ωpe/4m 2 2 eu 2. We note that α depends only on r 0 /a B. Inserting Eq. (17) into Eq. (8), we can express the latter as [29] ϕ(r) = Q [ (1 + b) 4π 2 k 2 + k+ 2 + (1 b) k 2 + k 2 where b = 1/ 1 4α, and k 2 ± = k 2 s [1 1 4α]/2α. The integral in Eq. (18) can be evaluated using the general formula exp(ik r) k 2 + k 2 ± ] exp(ik r)d 3 k, (18) d 3 k = 2π 2 exp( k ±r), (19) r where the branches of k ± must be chosen with positive real parts so that the boundary condition ϕ 0 at r is fulfilled. From Eqs. (18) and (19) we then have k 2 s ϕ(r) = Q r [cos(k ir) ibsin(k i r)]exp( k r r). (20) Several comments are in order. First, for α > 1/4, the solution of k± 2 = ( ) i 4α 1 /2α yields k± = (k s / 4α)[( 4α + 1) 1/2 i( 4α 1) 1/2 ] k r ik i, and the corresponding potential reads [29] ϕ(r) = Q r [ cos(ki r) + b sin(k i r) ] exp( k r r), (21) where b = 1/ 4α 1. Second, for α 1/4, we have k + = k = 2k s and ϕ(r) = Q ( 1 + k sr )exp( 2k s r). (22) r 2 Third, for α < 1/4, the expression 1 4α is real, and we obtain k ± = k s (1 1 4α) 1/2 / 2α. The resultant electric potential is ϕ(r) = Q 2r[ (1 + b)exp( k+ r) + (1 b)exp( k r) ]. (23) We note that in the absence of the quantum recoil effect, viz. α 0, we recover from Eq. (23) the modified Thomas-Fermi screened Coulomb potential ϕ(r) = (Q/r)exp( k s r). Finally, in the limit h 2 k 2 m 2 eu 2, the quantum recoil force dominates over the quantum statistical pressure and the forces associated with electron-exchange and electron correlations effects. Here we have the exponential cosine-screened Coulomb potential [36] where k q = (2m e ω pe / h) 1/2 is the quantum wave number. ϕ(r) = Q r cos(k qr)exp( k q r), (24)

312 α a /r B 0 FIGURE 1. The variation of α against a B /r 0. The critical value α = 1/4 is indicated with a dotted line. 2 x 10 3 φ/k s Q k r s FIGURE 2. The electric potential ϕ as a function of r for α = (dashed curve), α = 1/4 (solid curve) and α = 0 (dotted curve). The value is the maximum possible value of α in our model, obtained for a B /r After P. K. Shukla and B. Eliasson, Phys. Rev. Lett. 108, (2012). Figure 1 displays the variation of α against a B /r 0, with a maximum α at a B /r , corresponding to a number density n cm 3 (with a B = cm), a few times below solid densities. The value of α is above the critical value 1/4 only for a limited range of the plasma densities. In Fig. 2, we depict the profiles of the potential [given by Eqs. (21), (22) and (23) for α > 1/4, α = 1/4 and α < 1/4, respectively] for different values of α. We clearly see the new short-range attractive electric potential that resembles the LJ-type potential for 0.25 < α < 0.627, while for α 0.25, the attractive potential is absent. Figure 3(a) exhibits the distance r = d from the test ion charge where dϕ/dr = 0 and the electric potential has its minimum, and Figs. 3(b) and 3(c) show the values of ϕ and d 2 ϕ/dr 2 at r = d. The value of (d 2 ϕ/dr 2 ) determines the frequency and dispersion properties of the ion plasma oscillations, as shown below. The interaction potential energy between two dressed ions with charges Q i and Q j at the positions r i and r j can be represented as U i, j (R i j ) = Q j ϕ i (R i j ), where ϕ i is the potential around particle i, and R i j = r i r j. For α > 1/4, it reads, using Eq. (21), U i, j (R i j ) = Q iq j R i j exp( k r R i j ) [ cos(k i R i j ) + b sin(k i R i j ) ]. (25)

313 a) k s d b) φ(d)/k s Q c) φ (d)/k s 3 Q α x α x α FIGURE 3. a) The distance r = d from the test ion charge where dϕ/dr = 0 and the electric potential has its minimum, and b) the values of the potential ϕ and c) its second derivative d 2 ϕ/dr 2 at r = d. After P. K. Shukla and B. Eliasson, Phys. Rev. Lett. 108, (2012). On account of the interaction potential energy, ions would suffer vertical oscillations around their equilibrium position. The vertical vibrations of ions in a crystallized ion string in quantum plasmas are governed by m i d 2 δz j (t) dt 2 U i j (r i,r j ) =, (26) i j z j where δz j (t)[= z j (t) z j0 ] is the vertical displacement of the jth ion from its equilibrium position z j0. Assuming that δz j (t) is proportional to exp[ i(ωt jka)], where ω and k are the frequency and wave number of the ion lattice oscillations, respectively, and that Q i = Q j = Q, Eq. (26) for the nearest-neighbor ion interactions gives ω 2 = 4Q2 m i d 3 Sexp( k rd)sin 2 ( kd 2 ), (27) where S = [2(1 + k r d) + (kr 2 ki 2)d2 ](cosξ + b sinξ ) + 2k i d(1 + k r d)(sinξ b cosξ ). ξ = k i d, and d is the separation between two consecutive ions. We note that Eq. (27) can also be obtained from the formula [37] ω 2 = 4 m i [ d 2 W(r) dr 2 ] r=d sin 2 ( kd 2 where the inter-ion potential energy for α > 1/4 is represented as W(r) = (Q 2 /r)exp( k r r)[cos(k i r) + b sin(k i r)]. ), (28) Hence, the lattice wave frequency is proportional to [d 2 ϕ(r)/dr 2 ] 1/2 r=d [cf. Fig. 3(c)].

314 STIMULATED SCATTERING OF ELECTROMAGNETIC WAVES Large amplitude high-frequency (HF) electromagnetic (EM) waves are used for heating inertial confined fusion plasmas [38, 39], as well as for diagnostic purposes [11] in solid density plasmas that are created by intense laser and charged particle beams. The HF EM pulses also appear as localized bursts of x-rays and γ-rays from compact astrophysical objects. Furthermore, the generation of coherent HF EM waves is of great importance in the context of free-electron lasers (FELs) involving EM wigglers [40, 41, 42]. Therefore, studies of nonlinear phenomena (e.g parametric instabilities [43] and HF EM wave localizations [44]) associated with a large amplitude HF EM waves in dense quantum plasmas are of practical interest. In the following, we present an investigation of stimulated scattering instabilities of coherent circular polarized electromagnetic (CPEM) waves carrying orbital angular momentum (OAM) in an unmagnetized dense quantum plasma. The nonlinear interactions between the CPEM waves and electrostatic plasma oscillations in quantum plasmas is governed by the EM wave equation [42, 45] ( 2 ) t 2 c2 2 + ωp 2 A + ωpena 2 = 0, (29) which is derived from the Maxwell equation and the electron equation of motion with the EM fields E = c 1 A/ t and B = A, using the Coulomb gauge A = 0. Here A is the vector potential, N = n e1 /n 0 1, and n e1 the electron number density perturbation associated with low-frequency electrostatic plasma oscillations (EPOs) that are reinforced by the ponderomotive force of the CPEM waves. In the absence of nonlinear couplings between the latter and the EPOs, the paraxial EM wave solution, A(r,z)exp( iω e t + ik e z), where ω e = (kec ωpe) 2 1/2 and k e are the frequency and the wave number of the CPEM wave, respectively, of Eq. (28) is [46, 47] Here we have denoted A(r,z) = AF p,l (r,z)exp(ilφ). (30) F p,l (r,z) = 1 2 π [ (l + p)! p! ] 1/2 X l L l p (X)exp( X/2), (31) where X = r 2 /w 2 (z), w(z) the beam waist, and the associated Laguerre polynomial L l p (X) are defined by the Rodriguez formula, L l p (X) = (X l p!) 1 exp(x) d p [X (l+p) exp( X)]/dX p, p and l are the radial and angular mode numbers of the EM orbital angular momentum states, respectively, φ the azimuthal angle, r = (x 2 + y 2 ) 1/2 the radial of the cylindrical coordinates ((r,φ,z), so that 2 = 2 + z 2, where 2 = (1/r)( / r)(r / r) + (1/r2 ) 2 / φ 2. The Laguerre-Gauss (LG) solutions (30) describe CPEM waves with a finite OAM. The dynamics of the low-frequency (in comparison with the CPEM wave frequency ω e ) plasma oscillations involving degenerate electrons and non-degenerate ion fluids is governed by the G-QHD equations composed of the linearized electron continuity equation

315 n e1 + n 0 u e = 0, (32) t the electron momentum equation u e t and Poisson s equation e (ϕ ϕ p ) +U 2 n e1 + h2 m e 4m 2 2 n e1 = 0, (33) e 2 ϕ = 4πe(n e1 n i1 ), (34) together with the linearized Eqs. (6) and (7). Here ϕ p = e A 2 /m e c 2 is the ponderomotive potential of the CPEM waves. The EM ponderomotive force [38, 39] e ϕ p comes from the averaging (over the CPEM wave period) of the advection and nonlinear Lorentz forces involving the electron quiver velocity and the laser wave magnetic field. Let us now derive the governing equations for the electron and ion plasma oscillations in the presence of the ponderomotive force of the CPEM waves. First, we consider the driven electron plasma oscillations on the time scale of the electron plasma period, so that the ions do not have time to respond and the ion density perturbation is zero. We combine Eqs. (32) and (33) and substitute the resultant equation into (34) with n i1 = 0 to obtain the electron plasma wave equation [45] ( 2 ) t 2 + ω2 pe U h2 4m 2 4 N = e e2 2m 2 ec 2 2 A 2. (35) Second, we consider driven ion oscillations by supposing that 2 N/ t 2 U 2 2 N + ( h 2 /4m 2 e) 4 N. Here, one can neglect the right-hand side in Eq. (33), and use the resultant equation to eliminate the electric field ϕ from Eq. (7) to obtain, after linearization of the resultant equation under the quasi-neutral approximation n e1 = n i1, the driven ion oscillation equation [45] η m i n 0 N t ( ξ + η 3 m i n 0 ( 1 + τ m t ) 2 N t )( 2 ) t 2 C2 s 2 + h2 4 N 4m e m i ( ) Z 2 = 1 + τ i e 2 m t 2m 0 m i c 2 2 A 2, (36) where we have used the linearized Eq. (6) to eliminate u i from the ion momentum equation (7). Equations (29), (35) and (36) are the desired equations [45] for studying nonlinear effects (viz. parametric instabilities [38] and localization of light pulses [39] associated with L-G CPEM beams in quantum plasmas at nanoscales.

316 Nonlinear Dispersion Relations In the following, we present an investigation of stimulated Raman, stimulated Brillouin, and modulational instabilities [48] of L-G CPEM waves. Accordingly, we decompose the vector potential as A = A 0+ exp( iω 0 t + ik 0 r) + A 0 exp(iω 0 t ik 0 r) + A ± exp( iω ± t + ik ± r), (37) +, where the subscripts 0 and ± denote the CPEM pump and CPEM sidebands, respectively, and ω ± = Ω ± ω 0 and k ± = K ± k 0 are the frequency and wave vectors of the CPEM sidebands that are created by the beating of the pump (ω 0,k 0 ) and electrostatic oscillations (Ω,K). Inserting (37) into (29), (35), and (36), and supposing that N is proportional to exp( iωt +ik r), we Fourier decompose the resultant equations to obtain the nonlinear dispersion relations [44] and S R = ω2 pee 2 K 2 2m 2 ec 2 A 0 F p,l 2, (38) +, D ± S B = ω2 pez 2 i e2 K 2 2m e m i c 2 A 0 F p,l 2. (39) +, D ± Here we have denoted S R = Ω 2 ω 2 pe K 2 U 2 h 2 K 4 /4m 2 e and S B = Ω 2 K 2 C 2 s ( h 2 K 4 /4m e m i ) + iωη K 2 /m i n 0 (1 iωτ m ), and D ± = ω 2 ± k 2 ±c 2 ω 2 pe ±2ω 0 (Ω K V g δ), where V g = c 2 k 0 /ω 0 is the group velocity of the CPEM pump, ω 0 = (ω 2 pe + k 2 0 c2 ) 1/2 the pump wave frequency, and δ = K 2 c 2 /2ω 0 the small frequency shift arising from the nonlinear interaction between the CPEM pump and the electrostatic plasma oscillations in our quantum plasma. Growth Rates We now present a summary of formulas for the growth rates of stimulated Raman (SR) and stimulated Brillouin (SB) scattering instabilities, as well as of modulational instabilities of a constant amplitude pump that is scattered off a quantum electron plasma wave, a quantum ion mode, and a spectrum of non-resonant electron and ion density perturbations. For three-wave decay interactions, one assumes that D = 0 and D + 0. Thus, one ignores D + from Eqs. (38) and (39). Letting Ω = K V g δ + iγ R,B, and Ω = Ω L + iγ R and Ω = Ω s (Ω I ) + iγ B in the resultant equations, we obtain the growth rates for Raman and Brillouin backscattering ( K = 2k 0 ) instabilities, respectively,

317 γ R = ω pek 0 e A 0 F p,l 2ωe Ω R m e c, (40) and γ B = ω pek 0 Z i e A 0 F p,l 2ω0 Ω B m e m i c, (41) where Ω R = Ω L (K = 2k 0 ), Ω B = Ω s,i (K = 2k 0 ), and 2k 0 V g δ Ω R,Ω B. Since the growth rates of SR and SB scattering instabilities, given by Eqs. (40) and (41), respectively, are proportional to Ω R and Ω B, one notices that quantum and strong ion correlation effects significantly affect the e-folding time of the instabilities. Furthermore, the growth rates, which are proportional to F p,l, are minimum at the center of the vortex pump wave with OAM. Next, for the modulational instabilities, we have D ± 0 and S R,B 0. Here, we have to retain both upper and lower CPEM sidebands on an equal footing in (38) and (39), and write them as and S R [ (Ω K Vg ) 2 δ 2] = δω2 pee 2 K 2 A 0 F p,l 2 2ω 0 m 2 ec 2, (42) S B [ (Ω K Vg ) 2 δ 2] = δω2 pez 2 i e2 K 2 A 0 F p,l 2 2ω 0 m e m i c 2. (43) Equations (42) and (43) can be analyzed numerically to obtain the growth rates of the modulational instabilities. However, some analytical results follow for K V g = 0, in which case we have from (42) and (43), respectively, Ω 2 = 1 2 (Ω2 L + δ 2 ) ± 1 2 [ (Ω 2 L δ 2 ) 2 + 2δω2 pee 2 K 2 A 0 F p,l 2 ω 0 m 2 ec 2 ] 1/2, (44) and Ω 2 = 1 2 (Ω2 s,i + δ 2 ) ± 1 2 [ (Ω 2 s,i δ 2 ) 2 + 2δω2 pez 2 i e2 K 2 A 0 F p,l 2 ω 0 m e m i c 2 ] 1/2, (45) which exhibit oscillatory modulational instabilities. SUMMARY AND CONCLUSIONS In this paper, we have discussed two new aspects of collective interactions in an unmagnetized quantum plasma. First, we have reported the existence of novel attractive force that can bring ions closer. The appearance of the novel attractive force is attributed to

318 tunneling of degenerate electrons through the Bohmian potential on account of overlapping electron wave functions at atomic dimensions. There are several consequences of our newly found short-range attractive force at quantum scales. For example, due to the trapping of ions in the negative part of the exponential oscillating-screened Coulomb potential, there will arise ordered ion structures/ion clustering depending on the electron density concentration, which in fact controls the Wigner-Seitz radius r 0. The formation of ion clusters/ion atoms will emerge as new features in a dense quantum plasma. Furthermore, we have shown that both the electron and ion plasma oscillations in quantum plasmas can be excited by a large amplitude CPEM waves due to stimulated Raman and Brillouin scattering instabilities. We also have the possibility of the modulational instabilities of the CPEM waves, via which non-resonant electron density perturbations are created. Hence, there are enhanced electrostatic fluctuations at nanoscales in dense quantum plasmas. In conclusion, we stress that the results of the present investigation are useful for understanding the novel phenomena of ion clustering and the excitation of density fluctuations by the CPEM waves at nanoscales in high-energy density plasmas that are of interest for inertial confinement fusion schemes. Specifically, stimulated scattering instabilities of a coherent high-frequency short-wavelength radiation can offer a possible mechanism for the quantum free-electron lasers in the x-ray and gamma-ray regimes [42[, in addition to serving a diagnostic tool for determining the plasma parameters (viz, the plasma number density), once the spectra of enhanced density fluctuations are experimentally measured by using x-ray spectroscopic techniques. Finally, the present investigation, which has revealed the new physics of collective interactions between an ensemble of electrons at nanoscales, will open a new window for research in one of the modern areas of physics dealing with strongly correlated degenerate electrons and nondegenerate mildly coupled ions in dense quantum plasmas that share knowledge with cooperative phenomena (e.g. the formation of ion lattices) in condensed matter physics and in astrophysics. Acknowledgments This work was supported by the Deutsche Forschungsgemeinschaft through the project SH21/3-2 of the Research Unit The authors thank Lennart Stenflo and Massoud Akbari- Moghanjoughi for useful discussions. REFERENCES 1. P. K. Shukla and B. Eliasson, Phys. Usp. 53, 51 (2010); Rev. Mod. Phys. 83, 885 (2011). 2. S. V. Vladimirov and Yu. O. Tyshetskiy, Phys. Usp. 54, 1243 (2011). 3. F. Haas, Quantum Plasmas: An Hydrodynamical Approach (Springer, New York, 2011). 4. H. M. van Horn, Science 252, 384 (1991); V. E. Fortov, Phys. Usp. 52, 615 (2009). 5. E. Schatzman, White Dwarfs (Interscience Publishers, New York, 1958). 6. J. Liebert, Ann. Rev. Astron. Astrophys. 18, 363 (1980). 7. S. L. Shapiro and S. L. Teukolsky, Black Holes, White Dwarfs and Neutron Stars: The Physics of Compact Objects (John Wiley & Sons, New York, 1983). 8. A. M. Abrahams and S. L. Shapiro, Astrophys. J. 374, 652 (1991).

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332 didactical strategies have a greater potential to improve scientific knowledge in children. After the treatment, children showed not only the ability to learn new astronomical knowledge in their memory, but also to retrieve it easily by using scientific language and avoiding naïve explanation. These results support our expectation, indicating how social learning strategies and use of tools can improve children understanding of scientific events and, in particular, of astronomical concepts. It means that children s learning is more productive when teaching involves both social educational and interaction strategies among pairs with the support of tools that increase the scientific learning. Clearly, the level of acquisition of the new knowledge reflected the treatment carried out during the investigator activities. Children verbal descriptions and/or explanations represent important aspects to evaluate their scientific conceptual learning. For this latter aspect, examining children s open-ended statements, we found significant similarities in both theoretical and practical educational content delivered in the classroom. In addition, reading the children s descriptions it is possible to find also a narrative thinking that show the cognitive association between scientific concepts and quality of the learning (Boutler 2000). The current results indicate also that the integration of social learning methodologies and use of multimedia contents create a didactical environment, in which children are stimulated to extend their own scientific mental skills. This learning setting encouraged children in an active exploration of ideas affording their opportunity to learn Astronomy concepts in a new and dynamic way. However, the role of the schoolteacher is crucial. Schoolteachers should be able to experiment new didactical approaches to reinforce children interest, designing classroom laboratory to teach astronomical phenomena. The social learning approach gives to schoolteachers the possibility to fascinate children, and capture their attention and stimulate the imagination. Overall, this study demonstrated the successful use of the theory of social learning and tools for engendering playful astronomical learning. The didactical settings were designed to include theoretical and practical activities that mediated different astronomical concepts and phenomena. This enabled students to actively think on scientific concepts, thus providing their new way to explore scientific concepts and then to improve the learning outcomes. This combined use of social learning approach and tools means that the students themselves become a central part of the educational activity rather than just watching something in an inactive way. References Boulter C J (2000) Language, models and modeling in the primary science classroom, In J K Gilbert and C J Boulter (Eds) Developing models in science education (pp ), Kluwer Academic, Dordrecht Brewer W F, Chinn C A and Samarapungavan A (2000) Explanation in scientists and children, In F C Keil and R A Wilson (Eds) Explanation and cognition (pp ), Cambridge, The MIT Press Ehrlén K (2008) Children s Understanding of Globes as a Model of the Earth: A problem of contextualizing, International Journal of Science Education, 30(2), Iiskala T, Vauras M, Lehtinen, E and Salonen P (2010) Socially shared metacognition of dyads of pupils in collaborative mathematical problem-solving processes, Learning and Instruction, 21(3), Kallery M (2010) Astronomical Concepts and Events Awareness for Young Children, International Journal of Science Education, doi: / Kallery M and Psillos D (2004) Anthropomorphism and animism in early years science: Why teachers use them, how they conceptualize them and what are their views on their use, Research in Science Education, 34(3), Miller B W and Brewer W F (2010) RESEARCH REPORT Misconceptions of Astronomical Distances, International Journal of Science Education, 32(12), Nunnaly J (1978) Psychometric theory, McGraw-Hill, New York Plummer J D (2009) Early elementary students' development of astronomy concepts in the planetarium, Journal of Research in Science Teaching, 46(2), Skopeliti I and Vosniadou S (2007) Reasoning with External Representations in Elementary Astronomy, In S Vosniadou, D Kayser and A Protopapas (Eds) Proceedings of EuroCogSci07, the European Cognitive Science Conference (pp ), Delphi, Greece Stathopoulou C and Vosniadou S (2007) Exploring the Relationship between Physics Related Epistemological Beliefs and Physics Understanding, Journal of Educational Psychology, 32(3), Vosniadou S, Skopeliti I and Ikospentaki K (2004) Modes of knowing and ways of reasoning in elementary astronomy, Cognitive Development, 19(2), Vygotskij L S (1962) Thought and language, Chicago, MIT Press

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337 On INFN 2010 physics popularization school - Video report Santo REITO (Istituto Nazionale Fisica Nucleare) Abstract In this short video we report on the four days INFN school Comunicazione e Divulgazione della Fisica 2010 held in Perugia (Italy) on November 9/12. This meeting was conceived in the frame of a bigger training programme for INFN people interested in the popularization of physics. It was a full immersion school with traditional lectures, laboratories, working group and some sessions called A tavola con lo scienziato (i.e. dinners in the course of students could discuss with active scientist expressly invited). In the video are included summaries of the lessons and interviews of the teachers. In particular we paid some special attention on the A tavola con lo scienziato event. During the lunches the INFN school students met some teachers and student of the secondary school and ordinary people. This video is not just a chronicle of an event, it collects ideas, experiences and expectations of the people who shared this happening throughout four days. We were particularly interested in the opinions of the students and the teachers interviewed during A tavola con lo scienziato event because one of our goals was the understanding of the learning science problem directed to decrease the distance gap between science and youth in the era of learning in the wild and citizen-science projects. The science plays a crucial role in the modern society. How do people learn science? Nature in an editorial, titled Learning in the wild told us that much of what people know adout science is learned informally. The seemingly endless debate about how to improve US science education seems to make the tacit assumption that learning happens only in the classroom. As a result, the arguments tend to focus on issues such as curricula specifying, say, what information pre-college students should be expected to learn at each grade level and, as in US President Barack Obama's recent proposals to reform the No Child Left Behind policy, on the best way to hold schools to rigorous standards of student achievement. However, researchers who study learning are increasingly questioning this assumption. Their evidence strongly suggests that most of what the general public knows about science is picked up outside school, through things such as television programmes, websites, magazine articles, visits to zoos and museums and even through hobbies such as gardening and bird watching. This process of 'informal science education' is patchy, ad hoc and at the mercy of individual whim, all of which makes it much more difficult to measure than formal instruction. But it is also pervasive, cumulative and often much more effective at getting people excited about science and an individual's realization that he or she can work things out unaided promotes a profoundly motivating sense of empowerment. Nowadays science popularization plays a crucial role. The science popularization is an attempt to reduce the distance standing between science specialists and the public. Science popularization is interpretation of scientific information (science) intended for a general audience, rather than for other experts or students. The primary objective of the popularization of science is to increase public understanding of science. Since a few years ago, INFN has undertaken a strong program for training its Scientists in Science Outreach and supports many Outreach initiatives. In 2009 started a four days INFN school Comunicazione e Divulgazione della Fisica Goal of the Course was to provide participants with tools and insights instrumental to fulfil basic needs of Communication and Outreach about Science in general and Physics in particular. 4 days - full immersion 20 hours of plenary session lectures: Communication Psychology Targets of Outreach Writing Lab

339 FFP Udine November 2011 MAGNETIC FIELD AS PSEUDOVECTOR ENTITY IN PHYSICS EDUCATION Carlo Cecchini 2, Marisa Michelini 1, Alessandra Mossenta 1, Lorenzo Santi 1, Alberto Stefanel 1 Stefano Vercellati 1 Physics 1 and Mathematics 2 Department, University of Udine Abstract The analysis of the nature of the magnetic field offers the ideal framework in which students could address the mutual integration between mathematics and physical aspects facing experimentally the analysis of the phenomenology. Took look how students face experiments in which the phenomenology founds the theory and the mathematics offers its formalism as the best language to describe the explored phenomena, an activity concerning the pseudovectorial nature of the magnetic field was performed looking at the way in which the students reasoning evolve. 1. Introduction As in several physics situations, the knowledge of the symmetries of the physical systems analyzes the situations in a more effective and simple way, but the individuation of the symmetries could not be done without passing through the knowledge of the laws of transformation of the single entities that take part in the systems (Foot & Volkas, 1995; Kozlov & Valerij, 1995; Mohapatra & Senjanovi c, 1981; Redlich, 1984). For instance, symmetries are related with laws of conservation through the use of the Neother s first theorem (Noether et al 1918). Unfortunately the role of symmetries in high school physics education is often underestimated. The ways in which entities are transformed by symmetry operations are usually not explicitly adressed, the usual goals of the high school physics courses are only aimed to the definition of the structures of the entities, and this approach creates an intellectual gap between students studies in high school and university courses (Kolecki, 2002). The main example of this intellectual gap is related to the distinction of axial and polar vectors. In high school this distinction is usually neglected and this learning knot lies submersed until students had to face it in the university courses. This research is aimed to study how high school students reasoning evolves in the construction of formal thinking when they face an explicit situation in which the pseudovectorial nature of the magnetic field is highlighted. 2. Context and Sample Research work was done in the context of the Summer School of Modern Physics held in Udine during the summer of the The students participating to this school were the best 42 students coming from the last two years of the Italian high school (students are years old; school grades 12 th - 13 th ). The opening course of this intensive school was the course of electromagnetism. It was a 16 hours long course distributed on 3 days. 3. Instruments and Methods During this course of electromagnetism, students, divided in 7 groups of 6 components each one (4 groups of 12 th grade students and 3 of 13 th grade), follow an inquired based learning path that was constructed on the experimental exploration of the formal properties of the magnetic field with the aim to construct a formal representation of it. In particular the works on the exploration of the pseudovectorial properties of magnetic field was proposed in this learning path as an interactive lecture demonstration using an inquired based tutorial constructed with the Prevision-Experiment- Comparison (PEC) strategy (R. Martongelli, 2000; Michelini 2004). The situation proposed was the one represented in Figure 1. Taking into account the given circuit, students had to draw its mirrored image representing the needles of the compass, the needle of the

340 FFP Udine November 2011 reflected compass and analyze the picture (the compass is represented as the circle placed above the coils). Figure 1: Situation proposed to the students in the first step The second step was the realization of the experiment and the analysis of the original and the mirrored situations. Then, in the third part was asked to the students to rise up some considerations starting from the experimental observations. 4. Data Data were collected using personal inquired based worksheets. In the first phase, the provisional phase, 36% of the students represent the mirrored image of the needle as simple reflection, 28% represents the needle of the compass with opposite verse, 5% represent only the mirrored apparatus and 36% did not replay to this task (Graph 1). At the second phase, 19% of the students highlighted that the verse of the mirrored image of the needle is the same without justify it, while 14% highlights that the verse is different justifying their answers saying that the verse is different due to the right hand rule (2%), because the verse of rotation of the coils change (2%), because there is a change in the direction of the magnetic field (Graph 2). In phase three, 64% of the student highlight that the magnetic field is not a vector reporting only the representative matrix wrote by the teacher on the blackboard 31%, highlighting the rule in the recognition of the nature by the different rule of reflection 29%, pseudo magnetic vector is like the angular momentum or quantities that are defined through a vector product, 14% because the scalar product is anti-commutative 5%, or give other argumentation 4%, 5% did not motivate and the remaining 36% did not reply (Graph 3). Graph 1: Representation of the mirrored image of the needle done by the students Graph 2: Observation done by the students of the experimental results

341 FFP Udine November 2011 Graph 3: Consideration done by the students on the nature of vector of the magnetic field 5. Data analysis Neglecting the students who did not replay and the ones that represented only the mirrored apparatus, students spontaneous approaches to the analysis of the mirrored situation are almost distributed equally on two different way of analysis: the first one is the representation of the needle of the compass as a simple reflection of it and the other is the drawing of the compass needle starting from physical consideration of the mirrored apparatus. No significant differences were highlighted between the 13 th and 12 th grade students. In phase two, the analysis of the experimental situation, could be noticed how decrease the number of students that thought that the compass needle (i.e. the magnetic field) is reflected in the standard way, but not to a negligible percentage highlighting two main approaches. The first approach followed, by that the students of 13 th grade (that had already faced the magnetic field description during the previous school year), is to look at the experiment as a confirmation of their ideas and not as an investigation opened to new findings, do not allowing them to notice keys elements that are in opposite with their thesis. The second approach, followed mainly by the 12 th grade students is more open to the acceptance of eventually discording results respect to their prevision and so they can noticed and highlighted these discrepancies. After the discussion, in the third phase, no one of the students recognize the magnetic field as a standard vector, but the motivations that they gave are distributed on a wide spectrum that is reported in graph 3. Interesting to be noticed is that, even almost one third of the students justified this aspect reporting only the formal structure wrote on the blackboard by the teacher, the remaining part highlights experimental evidences to support this inference. In addition the data also highlight how mainly the 13 th grade students propose comparisons and analogies with other quantities that they had already faced during their previous study (as for instance the angular momentum).

342 FFP Udine November Conclusions and further remarks This experimentation, done on the introduction to high school students of the nature of pseudovector of the magnetic field, highlight how through a simple experiment student could face this formal characteristic, recognize it and do comparison between this characteristic and analogous previous physic entity that they already know. In addition, looking data concerning question two, were highlighted how the standard way of teaching used in the high school represent an obstacle to the acceptation of this new property by the student in they further studies. References Foot R, Volkas RR, Neutrino physics and the mirror world: How exact parity symmetry explains the solar neutrino deficit, the atmospheric neutrino anomaly, and the LSND experiment, Phys. Rev. D 52, (1995). Kolecki JC, An Introduction to Tensors for Students of Physics and Engineering, Tech. report, NASA Glenn Research Center, Cleveland, September (2002). Kozlov, Valerij V, Symmetries, topology and resonances in Hamiltonian mechanics, (Book) Berlin, Germany: Springer-Verlag, Inc, Martongelli R, Michelini M, Santi L, Stefanel A, Educational Proposals using New Technologies and Telematic Net for Physics, in Physics Teacher Education Beyond, 2000 Michelini, M, R Ragazzon, L. Santi, A. Stefanel (2004), Implementing a formative module on quantum physics for pre-service teacher training, in M. Michelini (ed.), Quality Development in the Teacher Education and Training, Girep Book of Selected Papers, Udine, Forum, pp Mohapatra R N, Senjanovi c G, Neutrino masses and mixings in gauge models with spontaneous parity violation, Phys. Rev. D 23 (1981) 165. Noether E, Invariante Variationsprobleme. Nachr. D. König. Gesellsch. D. Wiss. Zu Göttingen, Math-phys. Klasse 1918: , (1918). Redlich A N, Parity violation and gauge non-invariance of the effective gauge field action in three dimensions, Phys. Rev. D (1984).

344 FFP Udine November 2011 their activity. Two important principles make the system usable as a practical tool for identification of gifted pupils interested in a discipline. 1. Suggested relatively autonomous blocks of activities (topics, enquiry etc.) should be designed and offered in several different levels (different assignments) to be meaningful and at least partially completed by pupils of different (pre) knowledge, interests and ability levels. 2. The second principle: The available support and usage of the Talnet environment (including communications with experts) is limited only by pupil s purposive activity. Pupils, nominated mostly by their teachers or experienced friends, reveal important characteristics of gifted and talented pupil s profiles (cf.: Betts, 1988) in such an online environment and in a variety of structured activities. Keeping them in their school among classmates and simultaneously in their gifted pupil s (virtual) community we support them in accomplishment (filling up with people) their two different Zones of Proximal Development (Mcleod, 2010) in each of them they can play different roles. That is the important factor of their potential for communication of physics to their class or school mates. 2. Types of activities in Talnet Activities differ by subject (by subject are organized in series) and by individual design that should fit with the guide s (instructor the subject expert) intentions and styles. The types of activities (Table 1) differ namely by organizational forms, durations, outputs an expected assets to kids. Table 1: Types of activities in Talnet Activity Nature Main objective Structured, individual and team T-excursions (combined) Education Opportunity to demonstrate interest T-courses (online) T-pro-seminars (online) T-projects (combined) Online support (e.g. courses for Math. Olympiad participants ) International activities (combined) Education Research Education and training Study Research Education Research Communication & Research Knowledge deepening and to learn to study Application of ability/interest to learn to research Acquire general skills Deeper study of selected problem Authentic research To offer more active forms of preparation for existing activities To offer more active forms of preparation for existing activities To experience international interaction the need of language skills Loosely structured TalnetSpace, Café Talnet, T-journal (pending) Communication Assistance Production Games Own suggestions, information exchange within the community To apply the acquired knowledge in helping others Presentation of the results and the community To apply and develop the intellect

345 FFP Udine November 2011 Expedition Project To apply the range of knowledge, skills, and volatile qualities Workshop Experience To complement the online activities T-courses mostly offer online individual studies of several topics, based on communication between a student and an expert when working on assignments. The year-long study of such a course is divided into two online study period and a period to complete individual or team research like seminary work. When completed it is online discussed and defended among all pupils of all courses of Talnet. The authors are invited to present it at one (of three) face-to-face meetings of the courses faced to a jury. Talnet T-courses in physics (and mathematics support) realized annually follows: For ages 13-15: Mathematics 0 -Functions and Transformation For ages 14-15: Astro & Modeling I - Astronomy in Black & White for Now, and Learning Modeling on Computers Mathematics I - Mathematic Algorithms and Their Geometric Representations Materials and Crystals I - Experimenting and Measuring For ages 15-16: Astro & Modeling II Astronomy in Color and Our Models are Usable Mathematics II - Geometry in Standstill and Motion Materials and Crystals II Exploring Structures For ages 15-18: Doing Physics in a Team For ages 16-17: Mathematics III Combinatorics Games Meteo & Modern Physics World in Terms of Meteorology, Solar Energy Uses For ages 17-18: Astro & Modeling IV The Space in Ultra & Getting Better with Our Models Programmable Machines - Simulation and Control SW Mathematica for the Inquisitive - Application I Selected Chapters of the Theory of Relativity - Special and General The T-pro-seminars train more general types of skills, e.g. how to make presentations, to process measurement data, to use certain SW, etc. The online support of long term face-to-face activities, such as a series of lectures concerning the Mathematics Olympiad, allows the participants to practice, deepen and consult their knowledge gained during the traditional lectures. The most demanding activities are the multinational international activities (T-international). Another extreme, in terms of demands on knowledge, creativity, and observation, experimenting and research skills, are the team s enquiry activities (e.g. Doing Physics in a Team) Team s enquiry activities - Doing Physics in a Team This activity uses annually published problems of International Young Physicists' Tournament to be explored, investigated and explained in a virtual team consisting of pupils from different schools and regions. It is a very difficult task for facilitators to manage up to 30 pupils in such ambitious activities. The Tournament competition is developed for a five member team in a school. But lack of capacity of physics teachers and lack of pupils motivation (typically only one or two pupils of the school are interested and ready to spend so much time by in physics enquiry, limits the possibility of Young Physicists' Tournament in the school practice. The Talnet course Doing Physics in a Team get together individual pupils who have no partners in the school and on the other side small teams (typically a couple o triple) of interested pupils from the same school get needed support and training useful for participation in the Young Physicists' Tournament regardless if they are gifted or not. The course environment includes T-pro-seminars, sources, (modeling and other) tools carefully prepared to be used by students when needed in studying the problems.

346 Jan Krejci FFP Udine November 2011 Read+Write Tomas Rysavy Petr Vilasek Ondrej Winter Martin Zolnercik Lubomir Legemza 20 Jan Cesal Jakub Dolezal 0 Ales Neoral Adam Soukup Figure 1: Frequency of pupil s activities in the Team s Enquiry Course (by day and pupil) Figure 2: Special tools and meta layers were developed for massive communications

347 FFP Udine November 2011 The above figures 1 and 2 illustrate character of collaborative enquiry and mutual learning in the virtual environment. A closer analysis of communications shows that communications are horizontal (among pupils), more regular, more frequent and topic related. The difficulty and complexity of enquiry problems make pupils to meet more frequently than in other courses. This type of courses aims to create some opportunities to solve well known epistemological problems of school science (teaching) in almost all countries: the absence of development of science creativity by reducing science to the only moment of logical and experimental defense. This way the alumni (also future researchers, scientists) are not ready for discovering. In our seminars for school teachers they are (in their words) fascinated when see pupils - participants of the course - working and can interact with them. The teachers mention strong positive feelings and a will to look at their teaching and pupils in a new perspective. 3. Conclusions The recent experience and results, despite the limits given by computer mediated communication, shows that there are students who are willing and able to go deep in physics, does not matter whether we call them gifted or not (interestingly some seems to be independent learners). They have a great potential to communicate their live and authentic experience, gained in doing physics with a real physicist or in physics enquiry with mates sensitively guided by a physicist, with their classmates in an acceptable way and the language of the group knowledge and age level. In Talnet we offer study or developmental trajectories that include team s enquiry activities and simultaneously we open gates to real physics (as complement to school physics) not only to interested potentially gifted pupils (nominated by teachers) pupils but also their mates. For higher effect of gifted pupils influence in the classroom we offer teachers special opportunities (seminars with online support) to get experience in nomination of gifted pupils and get inspirations for involving pupils participating in Talnet to popularize physics in their classrooms or schools. There is still another critical point: limited possibilities and motivations of physicists to participate in such a complicated and time consuming intergeneration communications. References Renzulli, J. S. (1978). "What Makes Giftedness? Re-examining a Definition". Phi Delta Kappa, 60: pp FO - Physics Olympiad: SOC - Secondary School Expert Activity: Talnet: Betts, G., Neihart M. (1988): Profiles of the gifted and talented. Gifted Child Quarterly. NAGC. Mcleod, S. A. (2010). Simply Psychology; Zone of Proximal Development Talnet:

348 FFP Udine November 2011 SUSY RESULTS AT THE LHC WITH THE ATLAS DETECTOR Simone Brazzale Università di Udine e INFN Gruppo Collegato di Udine On behalf of the ATLAS Collaboration Abstract The data collected during 2011 with the ATLAS detector has been used to perform searches for signals due to R-parity conserving supersymmetry. The results of different analyses targeting jets, isolated leptons and missing transverse energy in the final state, a promising venue for the discovery of supersymmetry, are presented in some details in the following. 1. Introduction Supersymmetry (SUSY) is one of the most popular extensions of the Standard Model (SM). SUSY relates each elementary SM particle of one spin to another particle named superpartner, from which it differs by half a unit of spin (fig. 1). Figure 1: SM particles and their supersymmetric partners in the SUSY scenario In case of R-parity 1 conservation, charginos and neutralinos 2 can be produced directly in pairs or in the decay chains of squarks and gluinos. Moreover, the Lightest Supersymmetric Particle (LSP) is stable and therefore escapes any detector, while charginos decay into LSPs can yield high-p T leptons. A common signature for SUSY searches at high energy colliders is thus a high Transverse Missing Energy (E T miss) due to the LSPs, multiple high energetic jets from quark hadronization and eventually additional leptons coming from χ ± 1 decay to the LSP (fig. 2). The ATLAS Collaboration has searched these final states during the first year of proton-proton collisions at a centre-of-mass energy of 7 TeV, which have been delivered by the LHC in Figure 2: Example of proton-proton interactions where supersymmetric partners are produced in pairs: the final state includes E T miss, some jets and eventually one or more leptons. 1 R=(-1) 3(B-L)+2s, where B,L and S are respectively the barionic number, the leptonic number and the spin. 2 Charginos (χ ± i) and neutralinos (χ 0 j) are mass eigenstates of the superpartners of the bosons.

349 FFP Udine November 2011 During this year, the ATLAS detector has worked remarkably well, recording around 5 fb -1 of collisions with an overall data taking efficiency of 94%. The proton run ended in October, when the LHC switched to heavy ions collisions lepton final state When squarks and gluinos decay directly to quarks and LSPs (fig. 2), leptons are not produced in the decay chain and therefore no electrons nor muons appear in the final state. This case is favoured when the squark and gluino masses are not heavy enough to decay into heavier neutralinos or charginos. With 1 fb -1 of ATLAS data, the number of expected SM events has been compared with the number of observed events in the 0-lepton final state, and no excess has been found over the expectation. The SM background processes, such as W/Z+jets, top pair production and QCD multijets, have been estimated by defining different Control Regions (CR) and by extracting the amount in the Signal Regions (SR) by means of transfer factors 4. With no excess, it has been possible to set limits on the SUSY particle masses and subsequently to interprete the results within different theoretical models. Gluino and squark masses below 1000 GeV have been excluded at the 95% Confidence Level (C.L.) within the main models. The most important uncertainties taken into account in this analysis are Jet Energy Scale (JES), Jet Energy Resolution (JER), pileup effect, luminosity, Monte Carlo (MC) statistics and the uncertainty on cross sections lepton final state In case a squark decays via chargino and a quark (fig. 2), then the chargino may produce a high-p T lepton in the final state. Scenarios with large E T miss, jets and one electron or muon have been studied by ATLAS, but no excess has been found with respect to the SM predictions. The main background processes for this final state are W+jets, top pair production with a semileptonic decay and QCD processes with a jet misidentified as a lepton. To measure them, the number of expected background events in the defined SR has been extracted with data-driven techniques from different CRs. Good agreement was also observed between predicted and observed events in every CR. Systematic uncertainties for this analysis are mainly due to theoretical uncertainties such as the prediction of cross section, scale factors and PDFs (20-30%), MC statistics (15%), JES and JER (1-10%), pileup (1-10%) and luminosity (3.7%). Final results led to exclude masses of gluinos and squarks up to 875 GeV at the 95% C.L. (fig. 3). Figure 3: Observed and expected CL s limits at 95% C.L., as well as the ±1 sigma variation on the expected limits, in the combined 1-lepton channel for an integrated luminosity of about 1 fb All the analysis presented in this paper apply to 1fb - 1 of proton- proton collisions. 4 A Control (Signal) Region is defined with loose (tight) cuts to enhance the presence of background (signal).

350 FFP Udine November leptons final state The final state with two high-p T, well-isolated leptons is a very promising venue to discover SUSY at the LHC. Two leptons can emerge either from the chargino decay into lepton-neutrino-neutralino in both legs or from a single chargino decay into two leptons plus a lighter chargino (fig. 2). The SRs in this analysis have been optimized with studies on specific models and include final states with opposite sign leptons (OS), same sign (SS) and a flavour subtraction of events with same flavour and different flavour leptons. The major source of background for the OS SR comes from top pair production with a di-leptonic decay. SS leptons events have smaller SM background, but they suffer from low statistics. The estimate of all these processes has been handled with datadriven or semi data-driven techniques. In particular, the QCD misidentified leptons have been measured with a data-driven technique called Matrix Method. The total systematic uncertainty on the number of expected events for this channel ranges from 25% to 70%, depending on the cuts in the SR. As for the other channels, no excess over SM background has been observed and limits on the masses and cross sections have been set. Charginos masses up to 200 GeV are excluded at 95% C.L. 5. Conclusions A wide range of SUSY signatures has been investigated by ATLAS with 1 fb -1 of proton-proton collisions data. No hints for supersymmetry was observed in all channels and limits on the supersymmetric particles masses were set depending on the theoretical model. At 95% C.L. squarks and gluinos masses up to 1000 GeV are excluded in most principal models. For the year 2012, the ATLAS Collaboration is moving to study all the remaining range of SUSY signatures as well as to improve the existing analyses using the full 2011 statistics. The parameter space where SUSY particles are hiding is being continuously reduced. References The ATLAS Coll. (2011) Search for squarks and gluinos using final states with jets and missing transverse momentum with the ATLAS detector in sqrt(s) = 7 TeV proton-proton collisions, ArXiV: The ATLAS Coll. (2011) Search for supersymmetry in pp collisions at sqrt(s) = 7 TeV in final states with missing transverse momentum, b-jets and no leptons with the ATLAS detector, ATLAS-CONF The ATLAS Coll. (2011) Search for supersymmetry in final states with jets, missing transverse momentum and one isolated lepton in sqrt(s) = 7 TeV pp collisions using 1 fb-1 of ATLAS data, ArXiV: The ATLAS Coll. (2011) Searches for supersymmetry with the ATLAS detector using final states with two leptons and missing transverse momentum in sqrt(s) = 7 TeV proton-proton collisions, ArXiV: The ATLAS Experiment, accessed 2011 October

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363 On Fluid Maxwell Equations Tsutomu Kambe, (Former Professor, Physics), University of Tokyo, Abstract Fluid mechanics is a field theory of Newtonian mechanics of Galilean symmetry, concerned with fluid flows represented by the velocity field such as v(x, t) in space-time. A fluid is a medium of continuous mass. Its mechanics is formulated by extending discrete system of point masses. Associated with two symmetries (translation and space-rotation), there are two gauge fields: E (v )v and H v, which do not exist in the system of discrete masses. One can show that those are analogous to the electric field and magnetic field in the electromagnetism, and fluid Maxwell equations can be formulated for E and H. Sound waves within the fluid is analogous to the electromagnetic waves in the sense that phase speeds of both waves are independent of wave lengths, i.e. non-dispersive. 1 Introduction Fluid mechanics is a field theory of Galilean symmetry. Two symmetries are known as subgroups of the Galilean group: translation (space and time) and space-rotation. From this point of view, a gauge-theoretic study has been developed by Kambe (2008) for flows of an ideal compressible fluid with respect to both global and local invariances of the flow fields in the space-time (x, t), where x = (x 1, x 2, x 3 ) is the three-dimensional space coordinates. Suppose that we have a velocity field v(t, x) depending on x and time t. Then one can define the convective derivative D t (i.e. the Lagrange derivative in the fluid mechanics) by D t t + v, where t = / t, = ( / x i ). (1) It is remarkable that D t is gauge-invariant, namely D t is invariant with respect to local gauge transformations (Kambe 2008). This implies that the gauge theory can be applied to the fluid mechanics, since the covariant derivative is an essential building block of the gauge theory (Kambe 2010a). The fluid velocity v and acceleration A are defined by v = D t x, and A = D t v = t v + (v )v, respectively. The second term on the right of A can be transformed by the following vector identity: (v )v = v ( v) + ( 1 2 v 2 ). (2) It is verified by Kambe (2008, 2010a) that the vorticity defined by ω = v = curl v (3) is a gauge field with respect to the rotation symmetry, and the acceleration A is given by the following gauge-invariant expression: D t v = t v + (v )v = t v + ω v + ( 1 2 v 2 ). In a dynamical system of n discrete point masses with their positions denoted by X k (t) (k = 1,, n), the equations of motion are described by the form, (d/dt) 2 X k (t) = F k (X 1,, X n ), where F k is the force acting on the k-th particle. For a fluid of continuous mass distribution, the discrete positions X k (t) are replaced by a continuous field representation X(a, t), where a is the Lagrange parameters a = (a 1, a 2, a 3 ), or labels identifying fluid particles. Associated with the field representaton, the time derivative d/dt should be replaced by D t, and the acceleration is given by t v + (v )v. Thus, we have a connection term (v )v, which is the second gauge field. Clearly, the two gauge fields (v )v and v do not exist in discrete systems of point masses. They are the fields defined only in continuous differentiable fields such as the fluid flow. In fact, it is remarkable that they are analogous to the electric field and magnetic field in the electromagnetism, and a system of fluid Maxwell equations can be formulated for E (v )v and H v (Kambe 2010b). 1

364 It is shown in the next section 2 that there exists a similarity between the wave equations of electromagnetism and fluid mechanics. This suggests existence of correspondence between the variables of electromagnetism and fluid-mechanics. The section 3 describes that the correspondence permits formulation of a system of fluid Maxwell equations. In the section 4 a wave equation is derived from the system for a vector field. However in the fluid case, the vector form reduces to a scalar wave equation for sound waves, since all the terms of the equation are expressed by gradients of scalar fields. In the last section 5, another analogy between fluid mechanics and electromagnetism is presented for the equations of motion of a test particle in flow field and in electromagnetic field. 2 Equations of Fluid mechanics, compared with Electromagnetism 2.1 Equations of Fluid mechanics Equation of motion of an ideal fluid is given by the Euler s equation of motion: This equation is supplemented by the followings: t v + (v )v = 1 p. (4) ρ t ρ + v ρ + ρ v = 0, (5) t s + v s = 0, (6) t ω + (ω v) = 0, (7) where ρ is the fluid density, s the entropy per unit mass, p the pressure. The equation (5) is the continuity equation, and (7) is the vorticity equation, while (6) is the entropy equation, stating that each fluid particle keeps its initial entropy (i.e. adiabatic). If initial entropy field is uniform with a constant value s 0, the fluid keeps the isentropic state s = s 0 at any later time and everywhere. In this case, we have (1/ρ) p = h by the thermodynamics where h is the enthalpy per unit mass. 1 In isentropic flows, an enthalpy variation h and a density variation ρ are related by h = 1 ρ p = a2 ρ ρ, where p = a2 ρ, a 2 = ( p/ ρ) s. (8) The notation ( p/ ρ) s denotes partial differentiation with s fixed, and a = ( p/ ρ) s is the sound speed, as becomes clear below. From the above, we have t ρ = (ρ/a 2 ) t h and ρ = (ρ/a 2 ) h. Therefore, the equation (5) is transformed to (ρ/a 2 ) ( t h + v h + a 2 v ) = 0. Thus, the fluid equations (4) (7) reduce to the following three equations: Using (2), the equation of motion (9) is rewritten as Equations of Electromagnetism t v + (v )v + h = 0, (9) t h + v h + a 2 v = 0, (10) t ω + (ω v) = 0. (11) t v + ω v + ( 1 2 v2 + h) = 0. (12) In electromagnetism, Maxwell s equations for electric field E em and magnetic field H em are E em + c 1 t H em = 0, H em = 0, H em c 1 t E em = J e, E em = q e, (13) 1 From the thermodynamics, dh = (1/ρ) dp + T ds where T is the temperature. If ds = 0, we have dh = (1/ρ) dp. 2 The vorticity equation (11) is also obtained by taking curl of (12). 2

365 where q e = 4π ρ e and J e = (4π/c)j e with ρ e and j e being the charge density and current density vector respectively, and c the light velocity. The vector fields E em and H em can be defined in terms of a vector potential A and a scalar potential ϕ (e) by E em = c 1 t A ϕ (e), H em = A. (14) Using these definitions, the above Maxwell equations require that the two fields A and ϕ (e) satisfy the following equations (Landau & Lifshitz (1975), Chap.8): 2.3 Analogy in wave property t ϕ (e) + c A = 0 (Lorentz condition), (15) ( 2 t c 2 2 ) A = c 2 J e, ( 2 t c 2 2 ) ϕ (e) = c 2 q e. (16) Linearizing (9) by neglecting (v )v, and linearizing (10) by neglecting v h and replacing a with a constant value a 0, we have t v + h = 0, t h + a 2 0 v = 0. (17) Eliminating v from the two equations, we obtain the wave equation ( t 2 a0 2 2 )h = 0 for sound waves. Using it, we obtain the wave equation for v as well. Thus, we have ( 2 t a )h = 0, ( 2 t a ) v = 0. (18) It is remarkable that we have a close analogy between the two systems of fluid and electromagnetism. In vacuum space where q e = 0, J e = 0, the wave equations (16) reduce to ( 2 t c 2 2 ) ϕ (e) = 0, ( 2 t c 2 2 ) A = 0. (19) It is seen that c (light speed) a 0 (sound speed). Notable feature of the sund wave equations (18) is non-dispersive. Namely, the dispersion relation for waves of wave number k and frequency ω is given by ω 2 = a 2 0 k2, and the phase speed ω/k is equal to a 0 independent of the wave length 2π/k. The same is true for the equation (19) of the electromagnetic wave. Note that this is closely related to the system of Maxwell equations. In addition, the second equation of (17), obtained from the continuity equation by linearization, is analogous to the Lorentz condition (15) by the correspondence, (c, A, ϕ (e) ) (a 0, a 0 v, h). This implies possibility of formulation of fluid Maxwell equations for which the vector potential is played by v (except for the coefficient a 0, abbreviated in the present formulation for simplicity) and the scalar potential played by h. According to this finding, let us introduce two fields defined by E = t v h, H = v. Consider the following transformations: v = v + f, h = h t f. The vector fields E and H defined by v and h are unchanged, namely we have E = E and H = H. Thus in fluid flows, there exists the same gauge invariance as that in the electromagnetism. 3 Fluid Maxwell Equations Based on the following definition of the two vector fields E and H (analogous to (14)), E = t v h, H = v, (20) Fluid Maxwell Equations can be derived from the fluid equations (9) (11) as follows: (A) H = 0, (B) E + t H = 0, (21) (C) E = q, (D) a 2 0 H t E = J, (22) where q [(v )v], J 2 t v + t h + a 2 0 ( v), (23) 3

366 and a 0 is a constant (the sound speed in undisturbed state). From the calculus t (C) + div(d) operated on the two equations of (22), we have the charge conservation: t q + divj = 0. Using (9) and the definition E = t v h, the fluid-electric field E is given by Hence, the charge density q is given by 3.1 Derivation E = (v )v = ω v + ( 1 2 v2 ). (24) q = E = div [ (v )v ]. (25) Derivation of the fluid Maxwell equations (21) and (22) is carried out as follows. (a) Equation (A) is deduced immediately from the definition of H = v = ω. (b) Equation (B) is an identity obtained from the definition (20). Moreover, if the expression (24) is substituted to E and ω to H, then the equation (B) reduces to the vorticity equation (11). (c) Equation (C) is just div [Eq.(24)] with q defined by (23). (d) Equation (D) can be derived in the following way. Applying t to E = t v h, we obtain t E 2 t v = t h, Adding the term a0 2 H = a2 0 ( v) on both sides, this can be rearranged as follows: a 2 0 H t E = J, J = 2 t v + t h + a 2 0 ( v). which is nothing but the equation (D). The vector J can be given another expression by using t h = (v )h a 2 v from the continuity equation (10): J = 2 t v + a 2 0 ( v) ( a 2 v ) ( (v )h ). This can be rewritten as J = ( 2 t a )v+j, and J = a 2 0 ( v) ( a 2 v ) (v h), where the following identity is used: 4 Equation of Sound Wave ( v) = ( v) + 2 v, (26) Suppose that a localized flow is generated at an initial instant in otherwise uniform state at rest, where undisturbed values of the pressure, density and enthalpy are respectively p 0, ρ 0 and h 0. Equation of sound wave is derived from the system of fluid Maxwell equations (A) (D) as follows. Differentiating Eq.(D) with respect to t, and eliminating t H by using (B), we obtain 2 t E + a 2 0 ( E) = t J. (27) The second term on the left can be rewritten by using the identity (26), with v replaced by E. Then the equation (27) reduces to ( t 2 a0 2 2 )(E + t v) = a0 2 ( E) t J, (28) J = ( (a 2 0 a 2 ) v ) (v h) a0 2 ˆQ, ˆQ = (1 â 2 ) v a0 2 (v )h, (29) where â = a/a 0. We have E + t v = h from (20), and also t J = a0 2 t ˆQ from (29). Therefore, we can integrate (28) spatially, since all the terms are of the form of gradient of scalar fields. Dividing (28) with a0 2 and integrating it, we obtain the following wave equation: (a t 2 ) h = S(x, t), S(x, t) E + t ˆQ, (30) where h h h 0 = (p p 0 )/ρ. Thus, the vectorial form of wave equation (28) has been reduced to the equation for a scalar field h (see the first of (18)). The term S(x, t) is a source of the wave. 4

367 Using (24) and (25), we obtain an explicit form of the first E of the source S as E = div(ω v) v2. The first term div(ω v) implies that the motion of ω generates sound waves. This is the source term of the Vortex sound (Kambe 2010c), and contribution from the second term v2 vanishes in an ideal fluid in which total kinetic energy 1 2 v2 d 3 x is conserved. Mach number of the source flow is defined by M = v /a 0, then the second term t ˆQ of S is O(M 2 ), namely, higher order if M is small enough. 5 Equation of motion of a test particle in a flow field Analogy between fluid mechanics and electromagnetism is also found in the equation of motion of a test particle in a flow field as well. Suppose that a test particle of mass m is moving in a flow field v(x, t), which is unsteady, rotational and compressible. The size of the particle and its velocity are assumed to be so small that its influence on the background velocity field v(x, t) is negligible, namely the velocity field v is regarded as independent of the position and velocity of the particle. The particle velocity is defined by u(t) relative to the fluid velocity v. Then, the total particle velocity is u + v. In this circumstance, the i-th component of total momentum P i associated with the test particle moving in the flow field is expressed by the sum: P i = mu i + m ik u k, 3 and the equation of motion of the particle is given by d dt P = m E + m u H m ϕ g, (31) (Kambe 2010b), where E and H take the same expressions as those of (20), and P = (P i ), P i = mu i + m ik u k, and ϕ g = gz. Obviously, the equation (31) is analogous to the equation of motion of a charged particle in an electromagnetism of electric field E em and magnetic field H em : d dt (mv e) = e E em + (e/c) v e H em m Φ g, (32) where v e is the velocity of a charged particle, and c the light velocity. Rewriting the second term of (31) as (m/a 0 ) u (a 0 H), and comparing the first two terms on the right of (31) and (32), it is found that there is correspondence: e m, E em E, and H em a 0 H. 6 Summary It is shown that there exist similarities between electromagnetism and fluid mechanics. The correspondence betwenn them permits formulation of a system of fluid Maxwell equations. It is found that the sound wave in the fluid is analogous to the electromagnetic wave, in the sense that phase speeds of both waves are independent of wave lengths, i.e. non-dispersive. Another analogy between fluid mechanics and electromagnetism is presented for the equations of motion of a test particle in flow field and in electromagnetic field. References: Kambe T (2008) Variational formulation of ideal fluid flows according to gauge principle, Fluid Dyn. Res. (40), Kambe T (2010a) Geometrical Theory of Dynamical Systems and Fluid Flows (Rev. ed), Chap.7, World Scientific. Kambe T (2010b) A new formulation of equations of compressible fluids by analogy with Maxwell s equations, Fluid Dyn. Res. (42), Kambe T (2010c) Vortex sound with special reference to vortex rings: theory, computer simulations, and experiments, aeroacoustics (9), no.1 & 2, Landau LD and Lifshitz EM (1987) Fluid Mechanics (2nd ed), Pergamon Press. Landau LD and Lifshitz EM (1975) The Classical Theory of Fields (4th ed), Pergamon Press. 3 According to the hydrodynamic theory (e.g. Landau & Lifshitz (1987)) when a solid particle moves through the fluid (at rest), the fluid energy induced by the relative particle motion of velocity u = (u i ) is expressed in the form 1 2 m iku i u k by using the mass tensor m ik. Additional fluid momentum induced by the particle motion is given by m ik u k. 5

368 FFP Udine November 2011 DISCIPLINARY KNOTS AND LEARNING PROBLEMS IN WAVES PHYSICS Simone Di Renzone 1, Serena Frati 2, Vera Montalbano, Phys. Department, University of Siena Abstract An investigation on student understanding of waves is performed during an optional laboratory realized in informal extracurricular way with few, interested and talented pupils. The background and smart intuitions of students rendered the learning path very dynamic and ambitious. The activities started by investigating the basic properties of waves by means of a Shive wave machine. In order to make quantitative observed phenomena, the students used a camcorder and series of measures were obtained from the captured images. By checking the resulting data, it arose some learning difficulties especially in activities related to the laboratory. This experience was the starting point for a further analysis on disciplinary knots and learning problems in the physics of waves in order to elaborate a teaching-learning proposal on this topic. 1. Introduction Wave phenomena are everywhere. Everything can vibrate. There are oscillations and waves in water, ropes and springs. There are sound waves and electromagnetic waves. Even more important in physics is the wave phenomenon of quantum mechanics. When and how can it make sense to use the same word, wave, for all these disparate phenomena? What is it that they all have in common? A first answer lies in the mathematics of wave phenomena. Periodic behaviour of any kind, one might argue, leads to similar mathematics. There is a more physical answer to the questions. If it is possible to recognize deep similarities in different physical phenomenology, then it is likely that we can describe them by mean of the same mathematical tools. In order to introduce interested and talented students on this intriguing field in which wave phenomena can be described most insightfully, we designed an optional laboratory within National Plan for Science Degree 3. In the following, we describe methodological choices made in planning an extracurricular learning path on waves and oscillations and some examples of activities. In particular, we outline the advantages of introducing mechanical waves by using the Shive wave machine (Shive 1959). Many laboratory activities can be proposed in which students explore waves behaviour in qualitative way, guess what can happen and suddenly test their hypothesis. Furthermore, we present same disciplinary knots that arise usually in empirical investigation, according to the Model of Educational Reconstruction (Duit 1997, 2007). Finally, we describe learning problems on fundamental topics which arose in the laboratory. Further analysis on disciplinary knots is needed for a new teaching-learning proposal. 2. An extracurricular learning path on waves In the last years, many Italian Universities are involved in National Plan for Science Degree. The PLS guidelines are the following: orienting to Science Degree by means of training laboratory as a method not as a place student must become the main character of learning joint planning by teachers and university definition and focus on PLS laboratories. There are several types of PLS laboratories. Laboratories which approach the discipline and develop vocations, self-assessment laboratories for improving the standard required by graduate courses and deepening laboratory for motivated and talented students. We proposed two deepening laboratories for selected students 4 titled Waves and energy and Sound and surroundings. The laboratories are optional and the activities take place in Physics teacher enrolled in Master In Educational Innovation in Physics and Orienting - University of Udine teacher enrolled in Master In Educational Innovation in Physics and Orienting - University of Udine Piano nazionale Lauree Scientifiche, i.e. PLS. Coming from third class of Liceo Scientifico Aldi Grosseto, followed by their teacher G. Gargani

369 FFP Udine November 2011 Department. We are meeting students for 3 hours almost every month and planning to continue for last 3 years of high school. We decided that for the first year both laboratories had the same introductory activities on waves physics and all students works together. We planned activities by focusing on conceptual issues such as characterization of the oscillatory motion and energy aspects vs. characterization of wave energy and energy transport methodological issues in order to propose a complementary experience compared to what was done in class. Students ended their learning path on waves and sounds in class before the laboratory starts. Moreover, their class made an instruction trip to our department and perform a standard laboratory experience on diffraction and interference. 3. Shive wave machine We chose to begin with a series of activities performed by student by mean of a Shive wave machine, showed in Fig. 1.a. Figure 1: a) Shive wave machine, b) wave tank This wave machine, developed by Dr John Shive at Bell Labs in 50, consists of a set of equallyspaced horizontal rods attached to a square wire spine. Displacing a rod on one of the ends will cause a wave to propagate across the machine. Torsion waves of the core wire translate into transverse waves. Table 1 shows the main advantages and disadvantages of Shive wave machine vs. wave tank, another educational device very common in school. Shive Wave Machine Easy student interaction Measure λ, T, v Superposition principle Reflection study Energy considerations Easy study of stationary waves and resonance One-dimensional wave Lack of study of refraction, diffraction and interference Table 1: Shive wave machine vs. wave tank Limit of application Wave Tank Surface waves Measure λ, T, v Reflection and refraction study Interference and diffraction Lack of information on energy Poor interaction with student Measures were obtained by using a camcorder and extracted from the captured images. Students used Shive wave machine for studying the following wave aspects: dependence on space-time of waves impulsive and periodic waves wavelength and frequency energy transfer

370 FFP Udine November 2011 speed of propagation superposition principle reflection and transmission For example, by coupling core wires of two Shive wave machine with different rod lengths, an impulsive wave can be reflected and transmitted through the discontinuity (two examples are given in Fig.2). Students can measure from captured images all wave amplitudes and speeds and verify that energy is conserved. Figure 2: a) a pulse from the left (top) and transmitted and reflected pulses (below), b) a pulse from the right (top) and transmitted and reflected pulses. 4. Further activities in laboratory After using Shive wave machine, sound waves were studied by using a microphone and an oscilloscope (Fig. 3.a). Further verification of the principle of superimposition was made. Also beats and patterns of periodic beats (Moiré fringes) were studied. Interference was studied for sound waves in order to stimulate reflection around similarities and differences between different kinds of waves (longitudinal vs. transverse, three-dimensional vs. two-dimensional, and so on). Fig. 3.b shows a schematic layout of interference experiment. Figure 3: a) characterizing a sound waves by using an oscilloscope, b) acoustic interference This experiment is complementary to interference with laser light and allows measuring the speed of sound and discussing problems and connected applications of interference in acoustics. In order to study resonance, we started from a mechanical system with proper modes which can be forced by induction. A ceramic magnet is hanging to a string suspended over an electromagnet.

371 FFP Udine November 2011 Students can vary the frequency of a square wave that power the electromagnet, they can measure the amplitude value varying the frequency of the forcing term. The next steps will be to study resonance with the Shive wave machine, in acoustic and optical resonant cavities. 5. Some disciplinary knots in wave physics Starting from our previous empirical investigation in teaching waves, we designed the learning path on waves described in previous sections in order to use it as a pilot study in an optional laboratory performed with few, interested and talented students, i. e. in the best conditions for teaching and deepening topics in physics. We focused our attention on topics in which the main difficulties in learning usually appear. We have considered the following conceptual knots: Waves as function of several variables; this usually is an hidden trouble. Even brilliant students can use for long times functions of one and several variables without any real understanding of deep difference. Energy transport is essential in order to distinguish waves from other periodic phenomena and comprehend many applications. Superposition principle is a fundamental concept. In wave physics, many phenomena can be clarified and some unexpected behavior can be explained by applying it. Analogy in waves phenomena; difficulties in this area are very common and reported (Podolefsky 2007a, b), especially in recognizing the same behaviour in different context such as total reflection, diffraction, beats, interference and so on. Resonance; despite it is a relevant phenomenon which runs through almost every branch of physics, many students have never studied it. Yet, resonance is one of the most striking and unexpected phenomenon in all physics and it easy to observe but difficult to understand. 6. Learning problems Despite excellent boundary conditions (motivated and talented students), despite optimal behavior and relationship between instructors and pupils (we both have really fun in making this laboratory), we have encountered learning problems on fundamental topics. The first learning problem arose when we requested graphics for Shive wave machine position of one rod vs. time (position fixed) and position of rods vs. space (time fixed), in order to outline the dependences from space and time variables and put them in relation with measurable physical quantities. Home students task was to extract all data from video captured by themselves. We had outlined the importance of calibrating the video for time and space measures by giving examples and hints. Students prepared two graphics with 3 experimental points with no errors, the points were 0, max, 0, i.e. they measured only the maximum wave amplitude and the period or the wavelength, guessed the nodal point must be 0 and make graphics by drawing one half period, like shown in Fig. 4. Figure 4: A drawing like that shown was done on a large sheet of graph paper The next time students presented graphics with about seven points and uncertainties of about 15% or 20% We were astonished. With camcorder, the data can have uncertainties of few percent.

372 FFP Udine November 2011 After a brief discussion, we discovered that videos captured in the previous lab are useless for time calibration because digits on clock display are too small and they had forgot to measure a length in the devices that can be used as calibration. Thus, they adopted the following strategy: measuring lengths on the computer display with a ruler and measuring time by a chronometer. You can imagine what precision could have measuring lengths on a computer display... On the other side, these students are very clever in executing tasks in which they have received instruction as in a laboratory at school where usually a worksheet is given. In our opinion, they are still missing the main purpose of a measure: obtain the maximum of information from nature with the given devices. Furthermore, how can they understand deeply the difference between a one dimensional wave and a function encountered in a math lesson, or between one dimensional wave and two-dimensional or space waves or surface waves, if they are not able to construct graphics that represent completely a physical situation starting from data? 7. Conclusions We proposed and are testing a learning path in wave physics. Some disciplinary knots were identified and laboratory activities were developed in order to help understanding in learning processes. Testing these activities in an optional laboratory with high school student can be considered the first step in order to develop a designed-based research learning path (Hake 2008). These laboratories seemed to be very successful but in the meantime students showed serious lacking in fundamental topics, especially in collecting properly and in using correctly the experimental data in order to describe physical system. These lacks are the basis of observed learning problems and remains a real trouble for any further progress in knowledge. It is crucial to analyze deeply the teaching-learning processes in undergraduate school in order to avoid, or at least, minimize these kind of learning problems. References Duit R (2007), Science Education Research Internationally: Conceptions, Research Methods, Domain of Research, EJMSTE, 3, 3-15 Duit R., Komerek M., Wilbers J. (1997), Studies on Educational Reconstruction of Chaos Theory, Research in Science Education, 27 (3), Hake, R.R. (2008), Design-Based Research in Physics Education Research: A Review," in Kelly, Lesh, & Baek, Podolefsky N. S., Finkelstein N. D. (2007a), Salience of Representations and Analogies in Physics, AIP Conf. Proc. 951, pp , 2007 PHYSICS EDUCATION RESEARCH CONFERENCE Podolefsky N. S., Finkelstein N. D. (2007b), Analogical scaffolding and the learning of abstract ideas in physics: An example from eletromagnetic waves, Phys. Rev. ST - Phys. Educ. Res. 3, Shive J. N. (1959), Similarities of Wave Behavior, Archives-Similarities-of-Wave-Behavior, accessed 2011 October.