Lecturer: Bengt E W Nilsson
|
|
- Sharleen Farmer
- 5 years ago
- Views:
Transcription
1 9 4 3 Leturer: Bengt E W Nilsson 8::: Leturer absent, three students present. 8::9: Leturer present, three students. 8::: Six students. 8::8: Five students. Waiting a ouple of minutes to see if more ome. 8:3:: Six students again. 8:4:: So this is the hardore of the lass. 8:4:4: Seven students. Leturer wants to start. 8:5:: Eight students. Today you will see the first sign of one of the really important aspet of string theory: the Virasoro generators. You will see how fantasti it is. Let s reall a little bit what we have done: We had a relativisti string, with an ation: S = T Equation of motion: dτ dσ detγ αβ ) = T dτdσ ẋ x ) ẋ x ) τ P τ µ+ P σ µ=, P τ µ = L ẋ µ = ompliated! We need to simplify this situation not nie). Just to remind you: You have these Poisson brakets [q, p] PB =, and these you turn into [qˆ, pˆ] = i. We want to turn this into string theory, and we need anonial momenta. The above is above what we an do, so we need to simplify it. Turning quantities into operators is alled quantisation, but the proedure is not unique. The real physis is in the quantum mehanis: the lassial situation is a limit of the quantum mehanis. Going in the other diretion is not unique, but we have a guiding proedure quantisation). We use the oordinate freedom in τ and σ to simplify these expressions. ) Stati gauge: x τ, σ) = τ. Neumann boundary onditions the ends of the string move with v = v =. ) ẋ x =. The energy gauge : 3) These we ombine into is now gone. x ± ẋ ) =. Is true for the equations of motion: But there is a quadrati onstraint still, x ) + ẋ =. x = x σ x τ = x ± ẋ ) =. Solving it a new.
2 So what we will do today, that will solve all these problems: 9. Light-one relativisti string Another, even better gauge... this is a very important point, you have to understand why we do this weird step to go to the light one: it is all to avoid the ), an even better gauge is the light-one. Here all relations an be solved expliitly without any square roots. This will give us full insight into the set of independent anonial modes i.e. the full set of pairs p, q)!) To use this light-one gauge we need x ± τ,σ)= x τ,σ) ± x τ, σ) ) Pik the light-one gauge: x + α p + τ τ,σ) = α p + τ open string) losed string) The p + here is not the P that is a funtion of τ and σ, this is the integrated p. Here p + is the entre of mass momentum. It is a non-zero number that we an divide by. We put in the α for dimensional units. Units: From now on we use dimensionless τ and σ. Goal: To show that only the transverse string oordinates x τ, σ), p, q) are the independent variables or modes). Notation: We write n µ x µ = n x = x + using n µ =,,, use the notation for the stati gauge if we take another n µ, n µ =,, Then n xτ,σ)= β α n p)τ, x d τ, σ) plus one pair of, ). Nothing deep here, we an, ).) β = for the open string, β = for the losed string). This is a kind of generalised gauge ondition for τ-parametrisations. Choosing an n breaks Lorentz invariane. 9.: σ-parametrisations Before we used the onstant energy gauge along the string σ-oordinate). P τ, = T ds dσ v ) and the equation of motion τ P τ, = beause P σ =. Now we want to demand τ-independene of the more general quantity n P τ, and also set σ = π, just for onveniene. σ [, π] for all open strings. We also need this n P τ to be onstant along the string, i.e. independent of σ. One way to see that we an always ahieve this for one ombination of the string oordinates, i.e. n xτ,σ), is to note P τµ dσ. σ σ σ) dσ P τµ
3 This is lear from the fat that P τµ ds dσ x, x τ s ). v dσ /dσ is one funtion of σ. n xτ,σ) is one funtion of σ. Choosing one speifi ombination n xτ, σ) we an always let the σ-funtion dσ σ) be the dσ reiproal of n xτ, σ), i.e. n xτ, σ) is independent of both τ and σ. We an put this as follows: n p = πn P τ τ,σ)), p small This is the energy gauge ondition. Chek: n p. π π πn p = π n P τ τ,σ)dσ Equations of motion τ P τµ + σ P σµ =. n P σ is σ-independent. τ n P τ ) + σ n P σ ) = So n P σ is onstant along the string. What is the onstant? For the open string: We want this gauge to generalise the stati gauge. The x µ s involved must have Neumann boundary onditions. This means that n P σ = at the ends, and sine σ-independent we have n P σ = for all σ. For losed strings: Reall n P σ = ẋ x )n ẋ) ẋ n x ) πα ẋ x ) ẋ x ) Also here we have that n P σ = onst, but we do not know if it is =. Let us use the τ gauge ondition: n xτ,σ)=α n p)τ We take a σ derivative on this: n x τ,σ)= Consider ẋ x. n P σ = ẋ x )n ẋ) πα 3
4 Figure. So ẋ x = is possible to hoose at, let s say, σ =. So n P σ = at σ = and being onstant in σ, n P σ = for all σ. Note: The hoie of where on the losed string we set σ = is arbitrary, i.e. there is a remaining freedom of letting σ σ + σ where σ is a onstant. This is a rigid rotation of the oordinate system. It sounds ompletely useless, but it will turn out to be extremely ruial at a later point.) 9.3: Constrains and equations of motion n P σ = for all σ ẋ x = This looks ovariant in spaetime. That s a bit of an illusion. Compare ẋ x =, et. Then Then we dot this into n µ : [τ gauge: n x = β α n p)] P τµ = πα x ) ẋ µ ẋ x ) π n p σ gauge β n P τ = ẋ n ẋ) βα ẋ x ) x = ) ẋ x ) ẋ + x ) = 4
5 Combining it with ẋ x = we get: ẋ ±x ) =. Note: in the stati gauge, x being τ, this beomes ẋ ± x ) =. Also P τµ = πα ẋ µ P σµ = πα x µ ẍ µ x µ = 9.4: Wave equation and mode expansions We do the open ase first: ) Solve x µ =. x µ τ,σ) = f µ τ + σ) + g µ τ σ)) ) Neumann boundary onditions Dirihlet later). σ = : f = g g = f + onstant. x µ τ,σ)= f µ τ + σ) + f µ τ σ)) We let the arbitrary funtion absorb the onstant. 3) σ = π f τ +π)= f τ π) f is π periodi. f u) = f µ + a µ n os nu +b µ n sin nu), a µ n R, b µ n R. n= Integrate: fu)= f µ + f µ u+ A µ n osnu + B µ n sin nu) n= u: τ +σ, τ σ in x µ = fτ +σ)+ fτ σ)) x µ τ,σ)= f µ + f µ τ + n= A n µ os nτ + B n µ sin nτ) osnσ Note π ) P µ = dσp τ π µτ,σ)= dσ πα ẋ µ = π πα f µ = α f µ x µ τ,σ)=x µ +α p µ τ zero modes) + i α µ an e inτ a µ n e inτ) n= os nσ n 5
6 Simplify the notation: n =: α µ α p µ n : α µ n = n µ an, α µ n = n an µ µ ) α n = µ αn Open string x µ τ,σ)=x µ + α µ α τ + i α n n α µ n e inτ os nσ ẋ µ = α n Z 9.5: The use of the light-one gauge We have the onstraint α µ n os nσ e inτ, x µ = i α ẋ ±x ) =. n Z α n µ sin nσ e inτ We need to solve this without getting new square roots. In light-one gauge we avoid this effet! So we pik n µ =,,,, ). x + τ,σ)= β α p + τ p + σ = π β σ dσ P τ+ τ,σ ) n p = πβ n P τ ) Now use this in ẋ ± x ) =. In light-one oordinates, with µ = +,,I, with I =, 3, ) ) ) ẋ ± x ) = ẋ + ± x + ẋ ± x + ẋ I ± x I =, d: Solve for ẋ ±x = ) ẋ + ± x + ) ẋ I ±x I Is ẋ + ± x + a number? Well: x + =, ẋ + = βα p + a positive number). ẋ ± x = ) βα p + ẋ I ± x I Important. We learn: The whole mode expansion of x τ, σ), exept x, an be solved for in terms of x I modes! The independent anonial modes ) x I, p I. ) α n I, n=±, ±, [ I ] I α, α = 6
7 3) x, p +. We want to ompute the mass spetrum: M = p = p + p p I p I Reall ẋ ±x = 4α p + ẋ I ± x I ) = mode exp. = p + n Z L n e inτ ±σ), L n = p Z I I α n p α p L = α p = α α = L p + p Z I α p α I p = α I I α + p I) p= I I α p α p This lassial result M.. M = I α α I p α p p= 7
Lecturer: Bengt E W Nilsson
009 04 8 Lecturer: Bengt E W Nilsson Chapter 3: The closed quantised bosonic string. Generalised τ,σ gauges: n µ. For example n µ =,, 0,, 0).. X ±X ) =0. n x = α n p)τ n p)σ =π 0σ n P τ τ,σ )dσ σ 0, π]
More informationLecture 15 (Nov. 1, 2017)
Leture 5 8.3 Quantum Theor I, Fall 07 74 Leture 5 (Nov., 07 5. Charged Partile in a Uniform Magneti Field Last time, we disussed the quantum mehanis of a harged partile moving in a uniform magneti field
More informationQuantum Mechanics: Wheeler: Physics 6210
Quantum Mehanis: Wheeler: Physis 60 Problems some modified from Sakurai, hapter. W. S..: The Pauli matries, σ i, are a triple of matries, σ, σ i = σ, σ, σ 3 given by σ = σ = σ 3 = i i Let stand for the
More informationVector Field Theory (E&M)
Physis 4 Leture 2 Vetor Field Theory (E&M) Leture 2 Physis 4 Classial Mehanis II Otober 22nd, 2007 We now move from first-order salar field Lagrange densities to the equivalent form for a vetor field.
More information(a p (t)e i p x +a (t)e ip x p
5/29/3 Lecture outline Reading: Zwiebach chapters and. Last time: quantize KG field, φ(t, x) = (a (t)e i x +a (t)e ip x V ). 2Ep H = ( ȧ ȧ(t)+ 2E 2 E pa a) = p > E p a a. P = a a. [a p,a k ] = δ p,k, [a
More informationMOLECULAR ORBITAL THEORY- PART I
5.6 Physial Chemistry Leture #24-25 MOLECULAR ORBITAL THEORY- PART I At this point, we have nearly ompleted our rash-ourse introdution to quantum mehanis and we re finally ready to deal with moleules.
More informationGreen s function for the wave equation
Green s funtion for the wave equation Non-relativisti ase January 2019 1 The wave equations In the Lorentz Gauge, the wave equations for the potentials are (Notes 1 eqns 43 and 44): 1 2 A 2 2 2 A = µ 0
More informationQUANTUM MECHANICS II PHYS 517. Solutions to Problem Set # 1
QUANTUM MECHANICS II PHYS 57 Solutions to Problem Set #. The hamiltonian for a lassial harmoni osillator an be written in many different forms, suh as use ω = k/m H = p m + kx H = P + Q hω a. Find a anonial
More information(a) We desribe physics as a sequence of events labelled by their space time coordinates: x µ = (x 0, x 1, x 2 x 3 ) = (c t, x) (12.
2 Relativity Postulates (a) All inertial observers have the same equations of motion and the same physial laws. Relativity explains how to translate the measurements and events aording to one inertial
More information). In accordance with the Lorentz transformations for the space-time coordinates of the same event, the space coordinates become
Relativity and quantum mehanis: Jorgensen 1 revisited 1. Introdution Bernhard Rothenstein, Politehnia University of Timisoara, Physis Department, Timisoara, Romania. brothenstein@gmail.om Abstrat. We first
More informationGreen s function for the wave equation
Green s funtion for the wave equation Non relativisti ase 1 The wave equations In the Lorentz Gauge, the wave equations for the potentials in Lorentz Gauge Gaussian units are: r 2 A 1 2 A 2 t = 4π 2 j
More informationClassical Trajectories in Rindler Space and Restricted Structure of Phase Space with PT-Symmetric Hamiltonian. Abstract
Classial Trajetories in Rindler Spae and Restrited Struture of Phase Spae with PT-Symmetri Hamiltonian Soma Mitra 1 and Somenath Chakrabarty 2 Department of Physis, Visva-Bharati, Santiniketan 731 235,
More informationRelativistic Dynamics
Chapter 7 Relativisti Dynamis 7.1 General Priniples of Dynamis 7.2 Relativisti Ation As stated in Setion A.2, all of dynamis is derived from the priniple of least ation. Thus it is our hore to find a suitable
More informationPhysics 486. Classical Newton s laws Motion of bodies described in terms of initial conditions by specifying x(t), v(t).
Physis 486 Tony M. Liss Leture 1 Why quantum mehanis? Quantum vs. lassial mehanis: Classial Newton s laws Motion of bodies desribed in terms of initial onditions by speifying x(t), v(t). Hugely suessful
More informationSubject: Introduction to Component Matching and Off-Design Operation % % ( (1) R T % (
16.50 Leture 0 Subjet: Introdution to Component Mathing and Off-Design Operation At this point it is well to reflet on whih of the many parameters we have introdued (like M, τ, τ t, ϑ t, f, et.) are free
More informationLecture Notes 4 MORE DYNAMICS OF NEWTONIAN COSMOLOGY
MASSACHUSETTS INSTITUTE OF TECHNOLOGY Physis Department Physis 8.286: The Early Universe Otober 1, 218 Prof. Alan Guth Leture Notes 4 MORE DYNAMICS OF NEWTONIAN COSMOLOGY THE AGE OF A FLAT UNIVERSE: We
More informationThe Electromagnetic Radiation and Gravity
International Journal of Theoretial and Mathematial Physis 016, 6(3): 93-98 DOI: 10.593/j.ijtmp.0160603.01 The Eletromagneti Radiation and Gravity Bratianu Daniel Str. Teiului Nr. 16, Ploiesti, Romania
More informationThe homopolar generator: an analytical example
The homopolar generator: an analytial example Hendrik van Hees August 7, 214 1 Introdution It is surprising that the homopolar generator, invented in one of Faraday s ingenious experiments in 1831, still
More informationStrauss PDEs 2e: Section Exercise 3 Page 1 of 13. u tt c 2 u xx = cos x. ( 2 t c 2 2 x)u = cos x. v = ( t c x )u
Strauss PDEs e: Setion 3.4 - Exerise 3 Page 1 of 13 Exerise 3 Solve u tt = u xx + os x, u(x, ) = sin x, u t (x, ) = 1 + x. Solution Solution by Operator Fatorization Bring u xx to the other side. Write
More informationDifferential Equations 8/24/2010
Differential Equations A Differential i Equation (DE) is an equation ontaining one or more derivatives of an unknown dependant d variable with respet to (wrt) one or more independent variables. Solution
More informationAdvanced Computational Fluid Dynamics AA215A Lecture 4
Advaned Computational Fluid Dynamis AA5A Leture 4 Antony Jameson Winter Quarter,, Stanford, CA Abstrat Leture 4 overs analysis of the equations of gas dynamis Contents Analysis of the equations of gas
More informationSymplectic Projector and Physical Degrees of Freedom of The Classical Particle
Sympleti Projetor and Physial Degrees of Freedom of The Classial Partile M. A. De Andrade a, M. A. Santos b and I. V. Vanea arxiv:hep-th/0308169v3 7 Sep 2003 a Grupo de Físia Teória, Universidade Católia
More informationThe Dirac Equation in a Gravitational Field
8/28/09, 8:52 PM San Franiso, p. 1 of 7 sarfatti@pabell.net The Dira Equation in a Gravitational Field Jak Sarfatti Einstein s equivalene priniple implies that Newton s gravity fore has no loal objetive
More informationAharonov-Bohm effect. Dan Solomon.
Aharonov-Bohm effet. Dan Solomon. In the figure the magneti field is onfined to a solenoid of radius r 0 and is direted in the z- diretion, out of the paper. The solenoid is surrounded by a barrier that
More informationThe Thomas Precession Factor in Spin-Orbit Interaction
p. The Thomas Preession Fator in Spin-Orbit Interation Herbert Kroemer * Department of Eletrial and Computer Engineering, Uniersity of California, Santa Barbara, CA 9306 The origin of the Thomas fator
More informationPractice Exam 2 Solutions
MASSACHUSETTS INSTITUTE OF TECHNOLOGY Department o Physis Physis 801T Fall Term 004 Problem 1: stati equilibrium Pratie Exam Solutions You are able to hold out your arm in an outstrethed horizontal position
More informationRemark 4.1 Unlike Lyapunov theorems, LaSalle s theorem does not require the function V ( x ) to be positive definite.
Leture Remark 4.1 Unlike Lyapunov theorems, LaSalle s theorem does not require the funtion V ( x ) to be positive definite. ost often, our interest will be to show that x( t) as t. For that we will need
More informationIntro to Nuclear and Particle Physics (5110)
Intro to Nulear and Partile Physis (5110) Marh 7, 009 Relativisti Kinematis 3/7/009 1 Relativisti Kinematis Review! Wherever you studied this before, look at it again, e.g. Tipler (Modern Physis), Hyperphysis
More informationLecture 3 - Lorentz Transformations
Leture - Lorentz Transformations A Puzzle... Example A ruler is positioned perpendiular to a wall. A stik of length L flies by at speed v. It travels in front of the ruler, so that it obsures part of the
More information1 sin 2 r = 1 n 2 sin 2 i
Physis 505 Fall 005 Homework Assignment #11 Solutions Textbook problems: Ch. 7: 7.3, 7.5, 7.8, 7.16 7.3 Two plane semi-infinite slabs of the same uniform, isotropi, nonpermeable, lossless dieletri with
More informationHankel Optimal Model Order Reduction 1
Massahusetts Institute of Tehnology Department of Eletrial Engineering and Computer Siene 6.245: MULTIVARIABLE CONTROL SYSTEMS by A. Megretski Hankel Optimal Model Order Redution 1 This leture overs both
More information12.1 Events at the same proper distance from some event
Chapter 1 Uniform Aeleration 1.1 Events at the same proper distane from some event Consider the set of events that are at a fixed proper distane from some event. Loating the origin of spae-time at this
More informationClassical Field Theory
Preprint typeset in JHEP style - HYPER VERSION Classial Field Theory Gleb Arutyunov a a Institute for Theoretial Physis and Spinoza Institute, Utreht University, 3508 TD Utreht, The Netherlands Abstrat:
More informationSpinning Charged Bodies and the Linearized Kerr Metric. Abstract
Spinning Charged Bodies and the Linearized Kerr Metri J. Franklin Department of Physis, Reed College, Portland, OR 97202, USA. Abstrat The physis of the Kerr metri of general relativity (GR) an be understood
More informationChapter 8 Hypothesis Testing
Leture 5 for BST 63: Statistial Theory II Kui Zhang, Spring Chapter 8 Hypothesis Testing Setion 8 Introdution Definition 8 A hypothesis is a statement about a population parameter Definition 8 The two
More information( ) which is a direct consequence of the relativistic postulate. Its proof does not involve light signals. [8]
The Speed of Light under the Generalized Transformations, Inertial Transformations, Everyday Clok Synhronization and the Lorentz- Einstein Transformations Bernhard Rothenstein Abstrat. Starting with Edwards
More informationProbabilistic Graphical Models
Probabilisti Graphial Models David Sontag New York University Leture 12, April 19, 2012 Aknowledgement: Partially based on slides by Eri Xing at CMU and Andrew MCallum at UMass Amherst David Sontag (NYU)
More informationThe gravitational phenomena without the curved spacetime
The gravitational phenomena without the urved spaetime Mirosław J. Kubiak Abstrat: In this paper was presented a desription of the gravitational phenomena in the new medium, different than the urved spaetime,
More informationLecturer: Bengt E W Nilsson
2009 05 07 Lecturer: Bengt E W Nilsson From the previous lecture: Example 3 Figure 1. Some x µ s will have ND or DN boundary condition half integer mode expansions! Recall also: Half integer mode expansions
More informationGeneration of EM waves
Generation of EM waves Susan Lea Spring 015 1 The Green s funtion In Lorentz gauge, we obtained the wave equation: A 4π J 1 The orresponding Green s funtion for the problem satisfies the simpler differential
More informationRelativity fundamentals explained well (I hope) Walter F. Smith, Haverford College
Relativity fundamentals explained well (I hope) Walter F. Smith, Haverford College 3-14-06 1 Propagation of waves through a medium As you ll reall from last semester, when the speed of sound is measured
More informationStudy of EM waves in Periodic Structures (mathematical details)
Study of EM waves in Periodi Strutures (mathematial details) Massahusetts Institute of Tehnology 6.635 partial leture notes 1 Introdution: periodi media nomenlature 1. The spae domain is defined by a basis,(a
More informationQuantization of the open string on exact plane waves and non-commutative wave fronts
Quantization of the open string on exact plane waves and non-commutative wave fronts F. Ruiz Ruiz (UCM Madrid) Miami 2007, December 13-18 arxiv:0711.2991 [hep-th], with G. Horcajada Motivation On-going
More informationPhys 561 Classical Electrodynamics. Midterm
Phys 56 Classial Eletrodynamis Midterm Taner Akgün Department of Astronomy and Spae Sienes Cornell University Otober 3, Problem An eletri dipole of dipole moment p, fixed in diretion, is loated at a position
More informationMethods of evaluating tests
Methods of evaluating tests Let X,, 1 Xn be i.i.d. Bernoulli( p ). Then 5 j= 1 j ( 5, ) T = X Binomial p. We test 1 H : p vs. 1 1 H : p>. We saw that a LRT is 1 if t k* φ ( x ) =. otherwise (t is the observed
More informationGeneralized Dimensional Analysis
#HUTP-92/A036 7/92 Generalized Dimensional Analysis arxiv:hep-ph/9207278v1 31 Jul 1992 Howard Georgi Lyman Laboratory of Physis Harvard University Cambridge, MA 02138 Abstrat I desribe a version of so-alled
More informationFour-dimensional equation of motion for viscous compressible substance with regard to the acceleration field, pressure field and dissipation field
Four-dimensional equation of motion for visous ompressible substane with regard to the aeleration field, pressure field and dissipation field Sergey G. Fedosin PO box 6488, Sviazeva str. -79, Perm, Russia
More information1 Summary of Electrostatics
1 Summary of Eletrostatis Classial eletrodynamis is a theory of eletri and magneti fields aused by marosopi distributions of eletri harges and urrents. In these letures, we reapitulate the basi onepts
More informationOn the Geometrical Conditions to Determine the Flat Behaviour of the Rotational Curves in Galaxies
On the Geometrial Conditions to Determine the Flat Behaviour of the Rotational Curves in Galaxies Departamento de Físia, Universidade Estadual de Londrina, Londrina, PR, Brazil E-mail: andrenaves@gmail.om
More informationAnnouncements. Lecture 5 Chapter. 2 Special Relativity. The Doppler Effect
Announements HW1: Ch.-0, 6, 36, 41, 46, 50, 51, 55, 58, 63, 65 *** Lab start-u meeting with TA yesterday; useful? *** Lab manual is osted on the ourse web *** Physis Colloquium (Today 3:40m anelled ***
More informationMillennium Relativity Acceleration Composition. The Relativistic Relationship between Acceleration and Uniform Motion
Millennium Relativity Aeleration Composition he Relativisti Relationship between Aeleration and niform Motion Copyright 003 Joseph A. Rybzyk Abstrat he relativisti priniples developed throughout the six
More informationChapter 2: Solution of First order ODE
0 Chapter : Solution of irst order ODE Se. Separable Equations The differential equation of the form that is is alled separable if f = h g; In order to solve it perform the following steps: Rewrite the
More informationComplexity of Regularization RBF Networks
Complexity of Regularization RBF Networks Mark A Kon Department of Mathematis and Statistis Boston University Boston, MA 02215 mkon@buedu Leszek Plaskota Institute of Applied Mathematis University of Warsaw
More informationThe Concept of Mass as Interfering Photons, and the Originating Mechanism of Gravitation D.T. Froedge
The Conept of Mass as Interfering Photons, and the Originating Mehanism of Gravitation D.T. Froedge V04 Formerly Auburn University Phys-dtfroedge@glasgow-ky.om Abstrat For most purposes in physis the onept
More informationCMSC 451: Lecture 9 Greedy Approximation: Set Cover Thursday, Sep 28, 2017
CMSC 451: Leture 9 Greedy Approximation: Set Cover Thursday, Sep 28, 2017 Reading: Chapt 11 of KT and Set 54 of DPV Set Cover: An important lass of optimization problems involves overing a ertain domain,
More information1 Josephson Effect. dx + f f 3 = 0 (1)
Josephson Effet In 96 Brian Josephson, then a year old graduate student, made a remarkable predition that two superondutors separated by a thin insulating barrier should give rise to a spontaneous zero
More informationSQUARE ROOTS AND AND DIRECTIONS
SQUARE ROOS AND AND DIRECIONS We onstrut a lattie-like point set in the Eulidean plane that eluidates the relationship between the loal statistis of the frational parts of n and diretions in a shifted
More informationWhere as discussed previously we interpret solutions to this partial differential equation in the weak sense: b
Consider the pure initial value problem for a homogeneous system of onservation laws with no soure terms in one spae dimension: Where as disussed previously we interpret solutions to this partial differential
More informationPHYS-3301 Lecture 4. Chapter 2. Announcement. Sep. 7, Special Relativity. Course webpage Textbook
Announement Course webage htt://www.hys.ttu.edu/~slee/330/ Textbook PHYS-330 Leture 4 HW (due 9/4 Chater 0, 6, 36, 4, 45, 50, 5, 55, 58 Se. 7, 07 Chater Seial Relativity. Basi Ideas. Consequenes of Einstein
More informationMeasuring & Inducing Neural Activity Using Extracellular Fields I: Inverse systems approach
Measuring & Induing Neural Ativity Using Extraellular Fields I: Inverse systems approah Keith Dillon Department of Eletrial and Computer Engineering University of California San Diego 9500 Gilman Dr. La
More informationWave Propagation through Random Media
Chapter 3. Wave Propagation through Random Media 3. Charateristis of Wave Behavior Sound propagation through random media is the entral part of this investigation. This hapter presents a frame of referene
More informationUniversity of Groningen
University of Groningen Port Hamiltonian Formulation of Infinite Dimensional Systems II. Boundary Control by Interonnetion Mahelli, Alessandro; van der Shaft, Abraham; Melhiorri, Claudio Published in:
More informationThe Unified Geometrical Theory of Fields and Particles
Applied Mathematis, 014, 5, 347-351 Published Online February 014 (http://www.sirp.org/journal/am) http://dx.doi.org/10.436/am.014.53036 The Unified Geometrial Theory of Fields and Partiles Amagh Nduka
More informationParticle-wave symmetry in Quantum Mechanics And Special Relativity Theory
Partile-wave symmetry in Quantum Mehanis And Speial Relativity Theory Author one: XiaoLin Li,Chongqing,China,hidebrain@hotmail.om Corresponding author: XiaoLin Li, Chongqing,China,hidebrain@hotmail.om
More informationELECTROMAGNETIC WAVES
ELECTROMAGNETIC WAVES Now we will study eletromagneti waves in vauum or inside a medium, a dieletri. (A metalli system an also be represented as a dieletri but is more ompliated due to damping or attenuation
More informationLight Cone Gauge Quantization of Strings, Dynamics of D-brane and String dualities.
Light Cone Gauge Quantization of Strings, Dynamics of D-brane and String dualities. Muhammad Ilyas Department of Physics Government College University Lahore, Pakistan Abstract This review aims to show
More informationRelativity in Classical Physics
Relativity in Classial Physis Main Points Introdution Galilean (Newtonian) Relativity Relativity & Eletromagnetism Mihelson-Morley Experiment Introdution The theory of relativity deals with the study of
More informationExercise 1 Classical Bosonic String
Exercise 1 Classical Bosonic String 1. The Relativistic Particle The action describing a free relativistic point particle of mass m moving in a D- dimensional Minkowski spacetime is described by ) 1 S
More informationMetric of Universe The Causes of Red Shift.
Metri of Universe The Causes of Red Shift. ELKIN IGOR. ielkin@yande.ru Annotation Poinare and Einstein supposed that it is pratially impossible to determine one-way speed of light, that s why speed of
More informationA Characterization of Wavelet Convergence in Sobolev Spaces
A Charaterization of Wavelet Convergene in Sobolev Spaes Mark A. Kon 1 oston University Louise Arakelian Raphael Howard University Dediated to Prof. Robert Carroll on the oasion of his 70th birthday. Abstrat
More informationModal Horn Logics Have Interpolation
Modal Horn Logis Have Interpolation Marus Kraht Department of Linguistis, UCLA PO Box 951543 405 Hilgard Avenue Los Angeles, CA 90095-1543 USA kraht@humnet.ula.de Abstrat We shall show that the polymodal
More informationPhysics 523, General Relativity Homework 4 Due Wednesday, 25 th October 2006
Physis 523, General Relativity Homework 4 Due Wednesday, 25 th Otober 2006 Jaob Lewis Bourjaily Problem Reall that the worldline of a ontinuously aelerated observer in flat spae relative to some inertial
More informationThe concept of the general force vector field
The onept of the general fore vetor field Sergey G. Fedosin PO box 61488, Sviazeva str. 22-79, Perm, Russia E-mail: intelli@list.ru A hypothesis is suggested that the lassial eletromagneti and gravitational
More informationEVALUATION OF THE COSMOLOGICAL CONSTANT IN INFLATION WITH A MASSIVE NON-MINIMAL SCALAR FIELD
EVALUATON OF THE OSMOLOGAL ONSTANT N NFLATON WTH A MASSVE NON-MNMAL SALAR FELD JUNG-JENG HUANG Department of Mehanial Engineering, Physis Division Ming hi University of Tehnology, Taishan, New Taipei ity
More informationMost results in this section are stated without proof.
Leture 8 Level 4 v2 he Expliit formula. Most results in this setion are stated without proof. Reall that we have shown that ζ (s has only one pole, a simple one at s =. It has trivial zeros at the negative
More informationOutline 1. Introduction 1.1. Historical Overview 1.2. The Theory 2. The Relativistic String 2.1. Set Up 2.2. The Relativistic Point Particle 2.3. The
Classical String Theory Proseminar in Theoretical Physics David Reutter ETH Zürich April 15, 2013 Outline 1. Introduction 1.1. Historical Overview 1.2. The Theory 2. The Relativistic String 2.1. Set Up
More informationarxiv:gr-qc/ v7 14 Dec 2003
Propagation of light in non-inertial referene frames Vesselin Petkov Siene College, Conordia University 1455 De Maisonneuve Boulevard West Montreal, Quebe, Canada H3G 1M8 vpetkov@alor.onordia.a arxiv:gr-q/9909081v7
More informationProblem 3 : Solution/marking scheme Large Hadron Collider (10 points)
Problem 3 : Solution/marking sheme Large Hadron Collider 10 points) Part A. LHC Aelerator 6 points) A1 0.7 pt) Find the exat expression for the final veloity v of the protons as a funtion of the aelerating
More informationMath 225B: Differential Geometry, Homework 6
ath 225B: Differential Geometry, Homework 6 Ian Coley February 13, 214 Problem 8.7. Let ω be a 1-form on a manifol. Suppose that ω = for every lose urve in. Show that ω is exat. We laim that this onition
More informationThe spatially one-dimensional relativistic Ornstein-Uhlenbeck process in an arbitrary inertial frame
Eur. Phys. J. B 9, 37 47 00 THE EUROPEAN PHYSICAL JOURNAL B EDP Sienes Soietà Italiana di Fisia Springer-Verlag 00 The spatially one-dimensional relativisti Ornstein-Uhlenbek proess in an arbitrary inertial
More informationAngular Distribution of Photoelectrons during Irradiation of Metal Surface by Electromagnetic Waves
Journal of Modern Physis, 0,, 780-786 doi:0436/jmp0809 Published Online August 0 (http://wwwsirporg/journal/jmp) Angular Distribution of Photoeletrons during Irradiation of Metal Surfae by letromagneti
More informationString Theory I Mock Exam
String Theory I Mock Exam Ludwig Maximilians Universität München Prof. Dr. Dieter Lüst 15 th December 2015 16:00 18:00 Name: Student ID no.: E-mail address: Please write down your name and student ID number
More informationA Queueing Model for Call Blending in Call Centers
A Queueing Model for Call Blending in Call Centers Sandjai Bhulai and Ger Koole Vrije Universiteit Amsterdam Faulty of Sienes De Boelelaan 1081a 1081 HV Amsterdam The Netherlands E-mail: {sbhulai, koole}@s.vu.nl
More informationThe Laws of Acceleration
The Laws of Aeleration The Relationships between Time, Veloity, and Rate of Aeleration Copyright 2001 Joseph A. Rybzyk Abstrat Presented is a theory in fundamental theoretial physis that establishes the
More informationarxiv: v1 [physics.class-ph] 14 Dec 2010
Classial relativisti ideal gas in thermodynami equilibrium in a uniformly aelerated referene frame arxiv:11.363v1 [physis.lass-ph] 14 De 1 Domingo J. Louis-Martinez Department of Physis and Astronomy,
More informationSection 3. Interstellar absorption lines. 3.1 Equivalent width
Setion 3 Interstellar absorption lines 3.1 Equivalent width We an study diuse interstellar louds through the absorption lines they produe in the spetra of bakground stars. Beause of the low temperatures
More informationA 4 4 diagonal matrix Schrödinger equation from relativistic total energy with a 2 2 Lorentz invariant solution.
A 4 4 diagonal matrix Shrödinger equation from relativisti total energy with a 2 2 Lorentz invariant solution. Han Geurdes 1 and Koji Nagata 2 1 Geurdes datasiene, 2593 NN, 164, Den Haag, Netherlands E-mail:
More informationA EUCLIDEAN ALTERNATIVE TO MINKOWSKI SPACETIME DIAGRAM.
A EUCLIDEAN ALTERNATIVE TO MINKOWSKI SPACETIME DIAGRAM. S. Kanagaraj Eulidean Relativity s.kana.raj@gmail.om (1 August 009) Abstrat By re-interpreting the speial relativity (SR) postulates based on Eulidean
More informationTime Domain Method of Moments
Time Domain Method of Moments Massahusetts Institute of Tehnology 6.635 leture notes 1 Introdution The Method of Moments (MoM) introdued in the previous leture is widely used for solving integral equations
More information36-720: Log-Linear Models: Three-Way Tables
36-720: Log-Linear Models: Three-Way Tables Brian Junker September 10, 2007 Hierarhial Speifiation of Log-Linear Models Higher-Dimensional Tables Digression: Why Poisson If We Believe Multinomial? Produt
More informationMBS TECHNICAL REPORT 17-02
MBS TECHNICAL REPORT 7-02 On a meaningful axiomati derivation of some relativisti equations Jean-Claude Falmagne University of California, Irvine Abstrat The mathematial expression of a sientifi or geometri
More informationn n=1 (air) n 1 sin 2 r =
Physis 55 Fall 7 Homework Assignment #11 Solutions Textbook problems: Ch. 7: 7.3, 7.4, 7.6, 7.8 7.3 Two plane semi-infinite slabs of the same uniform, isotropi, nonpermeable, lossless dieletri with index
More informationThe Lorenz Transform
The Lorenz Transform Flameno Chuk Keyser Part I The Einstein/Bergmann deriation of the Lorentz Transform I follow the deriation of the Lorentz Transform, following Peter S Bergmann in Introdution to the
More informationThe Concept of the Effective Mass Tensor in GR. The Gravitational Waves
The Conept of the Effetive Mass Tensor in GR The Gravitational Waves Mirosław J. Kubiak Zespół Szkół Tehniznyh, Grudziądz, Poland Abstrat: In the paper [] we presented the onept of the effetive mass tensor
More informationSolutions to Problem Set 1
Eon602: Maro Theory Eonomis, HKU Instrutor: Dr. Yulei Luo September 208 Solutions to Problem Set. [0 points] Consider the following lifetime optimal onsumption-saving problem: v (a 0 ) max f;a t+ g t t
More informationMaximum Entropy and Exponential Families
Maximum Entropy and Exponential Families April 9, 209 Abstrat The goal of this note is to derive the exponential form of probability distribution from more basi onsiderations, in partiular Entropy. It
More informationEECS 120 Signals & Systems University of California, Berkeley: Fall 2005 Gastpar November 16, Solutions to Exam 2
EECS 0 Signals & Systems University of California, Berkeley: Fall 005 Gastpar November 6, 005 Solutions to Exam Last name First name SID You have hour and 45 minutes to omplete this exam. he exam is losed-book
More informationClassical Diamagnetism and the Satellite Paradox
Classial Diamagnetism and the Satellite Paradox 1 Problem Kirk T. MDonald Joseph Henry Laboratories, Prineton University, Prineton, NJ 08544 (November 1, 008) In typial models of lassial diamagnetism (see,
More informationRemarks Around Lorentz Transformation
Remarks Around Lorentz Transformation Arm Boris Nima arm.boris@gmail.om Abstrat After diagonalizing the Lorentz Matrix, we find the frame where the Dira equation is one derivation and we alulate the speed
More informationSome Math of the PAR Problem
Some Math of the PAR Problem Peter Oswald, Jaobs University Bremen http://www.faulty.jaobs-university.de/poswald Summer Shool Math & Comm July 2007 1. Introdution A pilot tone design problem and PAR optimization
More information