Two time physics and Hamiltonian Noether theorem for gauge systems

Transcription

1 Two time physics and Hamiltonian Noether theorem for gauge systems J. A. Nieto,1, L. Ruiz, J. Silvas and V. M. Villanueva,2 Escuela de Ciencias Fisico-Matemáticas Universidad Autónoma de Sinaloa Culiacán, Sinaloa, México Instituto de Física y Matemáticas Universidad Michoacana de San Nicolás de Hidalgo P.O. Box 2-82, Morelia, Michoacán, México Abstract. Motivated by two time physics theory we revisited the Noether theorem for Hamiltonian constrained systems. Our review presents a novel method to show that the gauge transformations are generated by the conserved quantities associated with the first class constraints. Keywords: Noether Theorem; Gauge Systems, Two Time Physics PACS: m, e, q, Ly 1.- INTRODUCTION The purpose of this brief note is to review the main results of Ref. [1]. The key idea in such a reference is to obtain Noether s first and second theorems [2] for gauge systems in the Hamiltonian sector (see Refs. [3]-[6]). Is proved that when the Noether theorem is applied, the conserved quantities can be identified precisely with the first class constraints. Is important to emphasize that these results are derived avoiding the Lagrangian sector in the sense of Gràcia and Pons [7]-[9] construction and focusing completely on the Hamiltonian sector [10]-[14]. In order to motivate our approach we first discuss two examples: two time physics and the Friedberg et al. model [15] (see also Refs. [16]-[17]). And then we proceed to generalize our formalism putting special emphasis in the connection between the first class constraints and the conserved quantities. Two time physics [18]-[23] offers an interesting example for discussing our formalism because in this case the variables q i and p j are unified in just one object x i a, with a = 1,2, where x i 1 qi and x i 2 pi, and consequently in the corresponding action the hidden symmetry Sp(2,R) or SL(2,R) becomes manifest (see Refs. [24]). Thus, we show that our formalism shed some new light on this hidden symmetry. 1 nieto@uas.uasnet.mx 2 vvillanu@ifm.umich.mx 249

2 This work is organized as follows. In Section 2, we discuss two times physics. In Sections 3, we describe the helix model of Friedberg et al. In Section 4, we generalize our procedure Finally, in Section 5 we make some final remarks. 2.- TWO TIME PHYSICS Two time physics is a proposal that has been reconsidered by many theoretical physicist in last years. The main motivation for considering this theory is that can be used to obtain various dynamical systems in one-time physics from the same action, through gauge fixing, providing a more unified point of view of one-time dynamics in a higher dimensional theory which is achieved by introducing new gauge symmetries in order to insure unitarity, causality and absence of ghosts [18]-[23]. The action for two time physics is given by [18] (see also Refs. [19]-[23]) τ ( ) f 1 S = dτ τ i 2 εab ẋ a µ xb ν η µν H T (x a µ ), (1) where the overdot means total derivative with respect to the parameter τ, µ = 0,1,2,3,4, H T denotes the total Hamiltonian and the symbol x µ a (a = 1,2) is defined as x µ 1 = qµ, x µ 2 = pµ. (2) It can be seen that the first term in the action (1), which up to a total derivative is equivalent to q µ p µ, has the manifest Sp(2,R) (or SL(2,R)) invariance. It turns out that the simplest possible choice for H T which maintains the symmetry Sp(2,R) in the massless case is H T = 1 2 λ ab x µ a x ν b η µν, (3) where λ ab = λ ba is a Lagrange multiplier. One could think about an extension to consider the "massive" case as H T = 1 2 λ ab (x a µ xb ν η µν + m 2 ab ), (4) with m 2 11 = R2, m 2 22 = m2 0 and m2 12 = 0, but turns out that the mass term m2 ab breaks the Sp(2, R)-symmetry as shown by considering (4) in the action (1) from where arbitrary variations on λ ab throw out the constraint Ω ab = x a µ xb ν η µν + m 2 ab = 0, (5) which turns out to be first class. In terms of notation (2), expression (5) is equivalent to the following set of equations (see Ref. [24]) 250

3 q µ q µ R 2 = 0, q µ p µ = 0, (6) p µ p µ + m 2 0 = 0, but this set of equations does not hold if q µ and p µ are interchanged. So, if we are interested in maintaining the Sp(2,R)-symmetry for the entire action one must consider m ab = 0, bun this case one can observe thaf η µν corresponds to just one time, that is, if η µν has the signature η µν = diag( 1,1,...,1) then from (7) it follows that p µ is parallel to q µ and this implies that the angular momentum L µν = q µ p ν q ν p µ (7) associated with the Lorentz symmetry of (2) should vanish, which is, of course, an unlikely result. This observation is one of the reasons that support two time physics theory. Thus, if we want to consider a system for which L µν 0, keep the constraints (7) and solve the problem without ghosts it will be consistent only in two times physics (see Refs. [18]-[23]). With these observations at hand, we proceed to consider the total Hamiltonian H T of the form H T = H(x µ a ;τ) λ bc (τ)φ bc (x µ a ;τ), (8) where φ bc (= φ cb ) denotes a generalization of the first class constraint Ω ab (see expression (5)). Now consider the following coordinate transformations δτ = τ (τ) τ, δ x a µ = x a µ (τ ) x a µ (τ) = δx a µ + ẋ a µ δτ, δ λ ab = λ ab (τ ) λ ab (τ) = δλ ab + λ ab δτ, where δx µ a = x µ a (τ) x µ a (τ), δλ ab = λ a(τ) λ a (τ). Note that δ ẋ µ a = δẋ µ a + ẍ µ a δτ. Under transformations (9) the total variation of the action (2) with H T given by (8) is (9) δ S = τ f τ i dτ { [ d 12 dτ ε ab δ x a µ xb νη ] µν δτh T + ε abẋ a µ δxb νη µν } +δτḣ T δ H T (10) but due to action invariance under transformations then (10) can be written as τ f δ S = dτ d τ i dτ δ Λ(x a µ ;τ), (11) where Λ(x µ a ;τ) is an arbitrary function. Now, if one define the quantity Q = Q(x µ a ;τ) as Q = 1 2 εab δ x µ a x ν b η µν δτh T δ Λ (12) 251

4 and observing that δτḣ T δ H T = δh T, then (10) and (12) implies τ { } f d dτ dτ Q + εab ẋ a µ δxb ν η µν δh T = 0. (13) τ i From this expression is not difficult to show that Q is a conserved quantity when equations of motion hold, thas when ẋ a µ = ε ab H T. (14) x bµ Conversely, when equations of motion (14) are not satisfied Q becomes to be the generator of canonical transformations, namely and δx µ a = {x µ a,q} (15) δ H T = Q. (16) t In order to proceed further, observe than terms of the symbol x µ a the Poisson brackets for any canonical functions f (x µ a ) and g(x µ a ) can be written as From this expression it follows that f g { f,g} = ε ab x a µ. (17) x bµ {x µ a,x ν b } = ε abη µν, (18) and for the constraint Ω ab given in (5) is straightforward to check that {Ω ab,ω cd } = C e f abcd Ω e f, (19) which establishes that Ω ab is in fact a first class constraint. The coefficients C e f abcd are called structure constants and are given by C e f abcd = 1 [ 2 εac (δb eδ f d + δ d eδ f b ) + ε ad(δb eδ c f + δc e δ f b ) +ε bc (δ e aδ f d + δ d eδ a f ) + ε bd (δ e aδc f + δc e δa f ) ]. Now consider the quantity Q as (20) Q = ξ ab (τ)ω ab, (21) where ξ ab = ξ ba are infinitesimal parameters. When equations of motion are not satisfied we can use the formulas (15) and (16) to obtain that the transformations generated by the constraint Ω ab are δx µ a = ε ab ξ bc x µ c (22) 252

5 and δλ ab = ξ ab ξ e f λ cd C ab e f cd. (23) We recognize in the expression (22) the infinitesimal transformation associated with the group Sp(2,R) = SL(2,R) with infinitesimal parameter ς c a = ε ab ξ bc. Thus, we have proved thaf the Lagrange multipliers variation δλ ab is given by (23) then the action (1) is invariant under the Sp(2, R) gauge transformation (22). The remarkable facs that Sp(2, R) invariance of the action (1) is generated by the conserved quantity (21) corresponding to the first class constraint Ω ab. 3.- THE FRIEDBERG ET AL. MODEL. The helix model of Friedberg et al. [15] (see also Refs. [16] and [17]) can be described in terms of the fundamental Hamiltonian first order action: S = [ t f dt ẋp x + ẏp y + żp z 1 ( 2 p 2 x + p 2 y + p 2 ) z ] U(x,y) [λ p z + gxp y yp x ], where (x,y,z) and (p x, p y, p z ) stand for the three dimensional coordinates and canonical momenta respectively. In (24) U(x,y) = U ( x 2 + y 2), and λ is a Lagrange multiplier associated with the first class constraint φ = p z + g(xp y yp x ), where g denotes a coupling constant. Is easy to verify that this action is invariant under the infinitesimal gauge transformations δx = αy, δy = +αx, δz = + g 1α, (25) δ p x = α p y, δ p y = +α p x, δ p z = 0, and (24) δλ = + 1 α, (26) g where α = α(t) is a transformation parameter. Note that the variation of the action is exactly zero so there is no need for the surface term. Let us introduce the quantity Q = δx i p i, = p z + g(xp y yp x ). As we can see Q corresponds to the first class constraint of the physical system whose motion is governed by the action (24). (27) 253

6 4.- GENERALIZATION The above two examples provide a motivation for a generalization. For this purpose let us first rewrite the action (1) in the form where t S[q, p;λ α f ] = dt [ q i ] p i H T, (28) H T = H(q, p;t) + λ α (t)φ α (q, p;t) (29) denotes de total Hamiltonian. Our aim is to see the consequences of applying to the action (28) the total variations: δt = t (t) t, δ q i = q i (t ) q i (t) = δq i + q i δt, δ p i = p i (t ) p i (t) = δ p i + ṗ i δt, (30) δ λ α = λ α (t ) λ α (t) = δλ α + λ α δt, where δq i = q i (t) q i (t) and similar expressions hold for δ p i and δλ α. Also δ q i = δ q i + q i δt. Is important to remark that δ q i = d dt δqi but δ q i d dt δ q i. Invariance of the action (28) under total variations implies δ S = t f dtδ [ q i p i H T ] + t f dt dδt dt [ qi p i H T ] = t f dt d dt δ Λ(q, p). where Λ(q, p) is an arbitrary function. Is straightforward to show that, in virtue of definitions of the total variations (30) and defining the quantity Q = Q(q, p;t) as relation (31) leads to t f Let us define the quantities (31) Q = δ q i p i δth T δ Λ, (32) { } d dt dt Q + qi δ p i ṗ i δq i δh T = 0. (33) 254

7 and t f A = dt d dt Q (34) t f B = dt { q i δ p i ṗ i δq i } δh T. (35) Expression (33), offers three different possibilities, namely (i) If A = 0 then (33) implies that B = 0. (ii) If B = 0 then (33) implies that A = 0. (iii) If neither A nor B are zero then (33) establishes that they are related by A+B = 0. The first two cases are well known, and it can be easily seen that both are equivalent, but the third one seems to have passed unnoticed. In order to clarify these observations let us briefly discuss each one of these cases. In the first case, we assume that the quantity Q satisfies the expression which is equivalent to say that A = t f t f Q t f = 0, (36) dt dt d Q = 0, from where it follows that dt{( q i H T )δ p i + ( ṗ i q i H T )δq i δλ α φ α } = 0 (37) and since variations δq i, δ p i and δλ α are arbitrary, then (37) implies the following equations of motion: and q i = H T = { q i,h T }, ṗ i = q i H T = {p i,h T } (38) φ α = 0. (39) Here the symbol { f,g}, for any functions f and g of the canonical variables q i and p i, stands for the usual Poisson bracket, thas { f,g} = f q i g g q i f. (40) In the second case, we assume that the dynamical system satisfies equations of motion (39) and (40). This means that (37) follows, which says that B = 0. Therefore from (33) we see that t f dt d Q = 0. (41) dt Since the interval t f does not have been defined, from (41) we have d dt Q = 0 and therefore we find that Q is a conserved quantity. 255

8 The last possibility arises if we assume that neither (37) nor (41) hold, thas, we assume that A and B are different from zero. We shall show than this case expression (33) implies that Q is the generator of canonical transformations. For this purpose let us first compute dt d Q. Since Q = Q(q, p;t) hence d dt Q = Q q i qi + Q ṗ i + Q t. (42) Thus, for an undefined interval t f, (33) can be rewritten as ω = 0. (43) with the quantity ω defined as ( ) ( ) ( ) Q Q Q ω = q i + δ p i dq i + δq i d p i + t δh T dt. (44) From (44) we observe that ω may admit an interpretation of 1-form. Thus, under usual assumptions (43) implies that ω is an exact form which means that ω = d f, (45) where f is an arbitrary zero-form. We shall assume that f = f (q, p). From (44) and (43) we see that Q t δh T = 0. (46) Considering (46), the expressions (43) and (45) yield ( Q ) ( Q q i + δ p i dq i ) + δq i d p i = 0, (47) where Since, dq i and d p i are 1 form bases then (47) implies and Q = Q + f. (48) δq i = Q = { q i,q } (49) δ p i = Q q i = { p i,q }. (50) Thus, we have shown that up to an arbitrary function f the quantity Q, which is a conserved quantity when the equations of motion are satisfied, is the generator of canonical transformations. 256

9 In order to clarify the meaning of expression (46), we investigate the consequences of invariances under gauge transformations, i.e., we consider the particular case Q = ξ α (t)φ α (q, p;t), (51) where the quantities ξ α (t) are infinitesimal parameters associated with the first class constraints φ α (q, p;t). Moreover; since we are dealing (by assumption) with only first class constraints, we can write (see Refs. [4] and [25]) and {H,φ α } = V β α φ β (52) { φα,φ β } = C γ αβ φ γ, (53) where Vα β and C γ αβ are structure "constants". Then, (46), (50)-(52) lead to ( ) δλ α φ α = ξ α ξ β Vβ α ξ β λ γ Cβγ α φ α, (54) and by considering the constraints φ α (q, p;t) as independent functions this expression implies the result δλ α = ξ α ξ β V α β ξ β λ γ C α βγ, (55) which describes the usual transformations of the Lagrange multipliers λ α under gauge transformations generated by first class constraints (see Refs. [4], [5] and [25]). 5.- FINAL REMARKS In this work we revisited two time physics and the Friedberg et al. model using fundamental constrained Hamiltonian formalism. We proved that our method may reveal hidden symmetries in specific cases. Since in two time physics the phase space has a unified character in the sense that the spacetime and the momentum space are put together at the same level, we found that an application of our formalism in this context requires a generalization of the usual Noether s procedure. Using this generalization we showed that the gauge transformations for the coordinates and momenta also exhibit a unified character. Further, we also proved that the gauge transformations are generated by the conserved quantities associated with the first class constraints. ACKNOWLEDGMENTS We would like to thank I. Bars, J. M. Pons, L. Lusanna, J. L. Lucio-Martinez and H. Villegas for helpful comments. V. M. V. whishes to thank financial support from Universidad Michoacana through Coordinación de Investigación Científica CIC 4.14 and CONACyT project E. 257

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