Lecture 16 March 29, 2010

Size: px
Start display at page:

Download "Lecture 16 March 29, 2010"

Transcription

1 Lecture 16 March 29, 2010 We know Maxwell s equations the Lorentz force. Why more theory? Newton = = Hamiltonian = Quantum Mechanics Elegance! Beauty! Gauge Fields = Non-Abelian Gauge Theory = Stard Model Anyway, let s look for s actions.

2 for a Motion is x(t). Action A = L( x, x, t) dt. Hamilton: actual path extremizes the action. Doesn t look Lorentz invariant, but all observers must agree (after suitable Lorentz transformation). So A should be a scalar. Start with a free particle. What could action be? Can t depend on x, for translation invariance. What property of path through space-time can we use? How about proper length? A = mc 2 dτ = mc dx µ dx µ = mc U α U α dτ = mc 2 1 u 2 c 2 dt.

3 So the is L( x, u, t) = mc 2/ γ( u) = mc 2 1 u 2 c 2. Note L is not an invariant, but Ldt γl are. Canonical Momentum (in 3-D language) ( P ) i = L u i = mu i 1 u 2 c 2 = ( p ) i, as we previously explored. Euler-Lagrange: d L L = 0, dt u i x i gives p i = constant, as x i is an ignorable coordinate. So this is correct for a free particle.

4 L for particle in a field What if charge q in an external field? Can depend on x µ, but only through the fields dependence on it. Can involve U α = dx α /dτ, but need it in combination as a scalar. Could use A α or F αβ, but U α U β F αβ 0, so only possibility linear in fields is γl int = q c U αa α, = L int = qφ + q u A, c with usual electrostatic vector potentials. Note first term looks like -PE as expected (as L = T V often). So the full lagrangian for the particle is L( x, u, t) = mc 2 1 u 2 c 2 + q c u A( x, t) qφ( x, t), the canonical momentum becomes m u q P = L/ u = + A( x, 1 u 2 c t) = p + q A, c c 2 not just the ordinary momentum p = mγ u.

5 The equations of motion are now d dt L u i }{{} P i L x i = dp i dt + q c d dt A i }{{} ( Ai t + u j j A i q c u j i A j + q i Φ = dp i dt + q A i c t + q iφ + q c (u j j A i u j i A j ) ( d p = 0 = dt + q d ) A c dt + q Φ q ( ) c u A d p dt = q E + q c u B so we see that this gives us the correct Lorentz force equation. ) i

6 The Hamiltonian What is the Hamiltonian? H = P u L, but reexpressed in terms of P rather than u. As u = p/mγ(u) = p m 1 u 2 /c 2 = u = mγ(u) = c p p 2 + m 2 c 2, p 2 + m 2 c 2 /c. Then we need to substitute p P qa/c. Thus ( ) P P qa/c + m 2 c 2 H = mγ(u) ( ) 2 P qa/c + m 2 c 2 ( ) q P qa/c A cmγ(u) + qφ = = + qφ mγ(u) (cp qa) 2 + m 2 c 4 + qφ. Note H is the total energy, the kinetic energy p 0 c + eφ, so this just verifies (p 0 ) 2 p 2 = m 2 c 2.

7 This L still doesn t have dynamical E&M fields - we will come to that later. First Recall from Classical Mechanics: Slowly varying perturbation on an integrable system with cyclic action-angle variables: action is adiabatic invariant. Apply this to motion transverse to uniform static magnetic field. Action J = P d r is an invariant. Need to use canonical momentum P = p + qa/c, not just p = mγ v. So J = mγ v d r + q A d r. c We have circular motion 1 with v = ω B r. 1 Note J12.38 says d v/dt = v ω B = ω B v, which explains the unexpected minus sign.

8 So the first term in J is mγ v d r = 2π 0 mγω B a 2 dθ = 2πmγω B a 2. As mγ ω B = qb/c, this is just 2qΦ B /c, where Φ B is the magnetic flux through the orbit. The second term in J, so q c A d r = q c S A = q c J = qφ B /c = q c Bπa2 = π c q S n B = q c Φ B, p 2 B is an adiabatic invariant, as are Ba 2 p2 B. These are conserved if B varies slowly compared to the gyroradius of the particle s motion.

9 So p 2 /B may be constant. In a purely magnetic field, speed γ are constant, but the transverse speed v B, while v 2 = v 2 + v2 is constant. So if particle drifts into a region of stronger B, v 2 may grow to use up all of v 2, v will vanish reverse. This is a magnetic mirror. Field lines converge where field gets strong, B so Lorentz force has v a component pushing F particle back into the weaker field region. This is called a magnetic mirror or magnetic bottle. Note that those particles with negligible v will not get confined.

10 treating x µ as dynamical The mc 2 1 u 2 /c 2 certainly doesn t look like a covariant formulation, we treated it as a functional to determine x(t), which is certainly not a covariant way of saying things. On the other h mc dx µ dx µ = mc η µν dx µ dx ν is a very covariant way of looking at the action, but what do we vary? All of m µ? or only the spatial part? Note that if we think of x µ (λ) as a parameterized path, we may write the action dx A = mc η µ µν dλ dx ν dλ dλ, think of varying the function x µ (λ) look for an extremum in the usual way. This gives d η µν dxν dλ = 0, dλ dx η µ dx ν µν dλ dλ

11 or dx µ dλ = dx Cµ η µ dx ν µν dλ dλ. Doesn t determine dxµ dλ! Though it looks like four equations, it is really only three, for contracting it with itself gives η µν dx µ dλ dx ν dλ = dx µ dx ν C2 η µν dλ dλ, which does nothing to determine dxν dλ but only that C 2 = 1. This should not be surprising. The path length doesn t depend on how it is parameterized, so any change x µ (λ) x µ (σ(λ)) will not change A, as long as σ(λ) is monotone.

12 Inability to predict the future is a sign of gauge invariance, though in this case it is not the gauge invariance we are used to for E&M. Here it is not a serious problem, because we can choose to use proper time as our parameter, providing the additional equation η µν dx µ dτ dx ν dτ = c2, = dxµ dτ = 1 m pµ = constant.

13 Action for particles in fields A int = q c U µ A µ dτ = q c dx µ dτ A µ dτ = q c A τ dx µ? The last expression is clearly covariant, the penultimate one gives the for the parameterized path x L = mc η α x β αβ λ λ q c A x α α λ with action Ldλ. P α = L xα λ = mc xα λ dx η µ dx ν µν dλ dλ + q c A α m x α λ τ τ + q c A α, Remember in Euler-Lagrange d/dλ is a stream derivative, so d dτ A α = U µ A α x µ.

14 The Euler-Lagrange equations are or m d dτ U α + q c x µ τ d dτ P α = L x α A α x µ = + q c A β x β x α τ, m d dτ U α = q ( c U β Aβ x α A ) α x β = q c F αβu β.

15 Canonical Momentum Canonical Momentum P α = L xα λ = mu α + q c A α, where we have required our parameter λ to be c times the proper time. Note that the canonical momentum is constrained: ( P α q c A α ) ( P α q c Aα) = m 2 U α U α = m 2 c 2. which we found before as P 0 = H/c. Minimum substitution principle: To introduce electromagnetism for a particle, take a free particle replace p α P α := p α q A/c.

16 Dynamics of fields requires a density, a function of the fields 2, say φ i ( x, t). Euler-Lagrange becomes L µ ( φ i / x µ ) L = 0. φ i What are our fundamental fields? L(φ i, m uφ i, x ν ) will give second order differential equations, not Maxwell in F. But we know F = da, so second order in A µ is what we want. We have already seen particle action requires (q/c)a µ dx µ for a single charge. That is, each charge q i at x i contributes to L q i Φ( x i ) + q i c u i A( x i ). 2 Never done dynamics of fields? Need to read up, e.g. shapiro/507/gettext.shtml look at chapter 8 (or get book9 2.pdf from the same location).

17 For many charges, L int = i = 1 c ( q i Φ( x i ) 1 ) c q i u i A( x i, t) ( d 3 x ρ( x)φ( x) 1 ) J( x) c A( x) d 3 xa α ( x)j α ( x). This will give us the J µ on the right h side of the Euler equation from varying A µ, but we need something to give the left h side of Maxwell s equation, which should be linear in F, so we need a quadratic piece in L, Lorentz invariant with a total of two derivatives on A µ s. Let s try L = 1 16π F µν F µν 1 c J µa µ, where it is understood that F µν sts for µ A ν ν A µ is not an independent field.

18 The only contribution to L/ A µ, (taken with ν A µ fixed) is the J µ /c from the interaction term. We have so F µν ( Aρ x σ ) = δ σ µδ ρ ν δ σ ν δ ρ µ, L ( ) = 1 Aρ 4π F ρσ, x σ the full Euler-Lagrange equation is or 1 4π σf σµ + 1 c J µ = 0, σ F σµ = 4π c J µ. Thus we have derived Maxwell s equations (as df = 0 is automatic as F := da).

Dynamics of Relativistic Particles and EM Fields

Dynamics of Relativistic Particles and EM Fields October 7, 2008 1 1 J.D.Jackson, Classical Electrodynamics, 3rd Edition, Chapter 12 Lagrangian Hamiltonian for a Relativistic Charged Particle The equations of motion [ d p dt = e E + u ] c B de dt = e

More information

We would like to give a Lagrangian formulation of electrodynamics.

We would like to give a Lagrangian formulation of electrodynamics. Chapter 7 Lagrangian Formulation of Electrodynamics We would like to give a Lagrangian formulation of electrodynamics. Using Lagrangians to describe dynamics has a number of advantages It is a exceedingly

More information

Electric and Magnetic Forces in Lagrangian and Hamiltonian Formalism

Electric and Magnetic Forces in Lagrangian and Hamiltonian Formalism Electric and Magnetic Forces in Lagrangian and Hamiltonian Formalism Benjamin Hornberger 1/26/1 Phy 55, Classical Electrodynamics, Prof. Goldhaber Lecture notes from Oct. 26, 21 Lecture held by Prof. Weisberger

More information

Physics 452 Lecture 33: A Particle in an E&M Field

Physics 452 Lecture 33: A Particle in an E&M Field Physics 452 Lecture 33: A Particle in an E&M Field J. Peatross In lectures 31 and 32, we considered the Klein-Gordon equation for a free particle. We would like to add a potential to the equation (since

More information

Lecture: Lorentz Invariant Dynamics

Lecture: Lorentz Invariant Dynamics Chapter 5 Lecture: Lorentz Invariant Dynamics In the preceding chapter we introduced the Minkowski metric and covariance with respect to Lorentz transformations between inertial systems. This was shown

More information

Lectures on basic plasma physics: Hamiltonian mechanics of charged particle motion

Lectures on basic plasma physics: Hamiltonian mechanics of charged particle motion Lectures on basic plasma physics: Hamiltonian mechanics of charged particle motion Department of applied physics, Aalto University March 8, 2016 Hamiltonian versus Newtonian mechanics Newtonian mechanics:

More information

Physics 411 Lecture 22. E&M and Sources. Lecture 22. Physics 411 Classical Mechanics II

Physics 411 Lecture 22. E&M and Sources. Lecture 22. Physics 411 Classical Mechanics II Physics 411 Lecture 22 E&M and Sources Lecture 22 Physics 411 Classical Mechanics II October 24th, 2007 E&M is a good place to begin talking about sources, since we already know the answer from Maxwell

More information

Continuity Equations and the Energy-Momentum Tensor

Continuity Equations and the Energy-Momentum Tensor Physics 4 Lecture 8 Continuity Equations and the Energy-Momentum Tensor Lecture 8 Physics 4 Classical Mechanics II October 8th, 007 We have finished the definition of Lagrange density for a generic space-time

More information

FYS 3120: Classical Mechanics and Electrodynamics

FYS 3120: Classical Mechanics and Electrodynamics FYS 3120: Classical Mechanics and Electrodynamics Formula Collection Spring semester 2014 1 Analytical Mechanics The Lagrangian L = L(q, q, t), (1) is a function of the generalized coordinates q = {q i

More information

Mechanics Physics 151

Mechanics Physics 151 Mechanics Physics 151 Lecture 15 Special Relativity (Chapter 7) What We Did Last Time Defined Lorentz transformation Linear transformation of 4-vectors that conserve the length in Minkowski space Derived

More information

SPECIAL RELATIVITY AND ELECTROMAGNETISM

SPECIAL RELATIVITY AND ELECTROMAGNETISM SPECIAL RELATIVITY AND ELECTROMAGNETISM MATH 460, SECTION 500 The following problems (composed by Professor P.B. Yasskin) will lead you through the construction of the theory of electromagnetism in special

More information

Math. 460, Sec. 500 Fall, Special Relativity and Electromagnetism

Math. 460, Sec. 500 Fall, Special Relativity and Electromagnetism Math. 460, Sec. 500 Fall, 2011 Special Relativity and Electromagnetism The following problems (composed by Professor P. B. Yasskin) will lead you through the construction of the theory of electromagnetism

More information

Vectors in Special Relativity

Vectors in Special Relativity Chapter 2 Vectors in Special Relativity 2.1 Four - vectors A four - vector is a quantity with four components which changes like spacetime coordinates under a coordinate transformation. We will write the

More information

Problem 1, Lorentz transformations of electric and magnetic

Problem 1, Lorentz transformations of electric and magnetic Problem 1, Lorentz transformations of electric and magnetic fields We have that where, F µν = F µ ν = L µ µ Lν ν F µν, 0 B 3 B 2 ie 1 B 3 0 B 1 ie 2 B 2 B 1 0 ie 3 ie 2 ie 2 ie 3 0. Note that we use the

More information

Relativistic Quantum Mechanics

Relativistic Quantum Mechanics Physics 342 Lecture 34 Relativistic Quantum Mechanics Lecture 34 Physics 342 Quantum Mechanics I Wednesday, April 30th, 2008 We know that the Schrödinger equation logically replaces Newton s second law

More information

Overthrows a basic assumption of classical physics - that lengths and time intervals are absolute quantities, i.e., the same for all observes.

Overthrows a basic assumption of classical physics - that lengths and time intervals are absolute quantities, i.e., the same for all observes. Relativistic Electrodynamics An inertial frame = coordinate system where Newton's 1st law of motion - the law of inertia - is true. An inertial frame moves with constant velocity with respect to any other

More information

General Relativity (225A) Fall 2013 Assignment 2 Solutions

General Relativity (225A) Fall 2013 Assignment 2 Solutions University of California at San Diego Department of Physics Prof. John McGreevy General Relativity 5A) Fall 13 Assignment Solutions Posted October 3, 13 Due Monday, October 15, 13 1. Special relativity

More information

Mechanics Physics 151

Mechanics Physics 151 Mechanics Physics 151 Lecture 4 Continuous Systems and Fields (Chapter 13) What We Did Last Time Built Lagrangian formalism for continuous system Lagrangian L Lagrange s equation = L dxdydz Derived simple

More information

Orbital Motion in Schwarzschild Geometry

Orbital Motion in Schwarzschild Geometry Physics 4 Lecture 29 Orbital Motion in Schwarzschild Geometry Lecture 29 Physics 4 Classical Mechanics II November 9th, 2007 We have seen, through the study of the weak field solutions of Einstein s equation

More information

2.3 Calculus of variations

2.3 Calculus of variations 2.3 Calculus of variations 2.3.1 Euler-Lagrange equation The action functional S[x(t)] = L(x, ẋ, t) dt (2.3.1) which maps a curve x(t) to a number, can be expanded in a Taylor series { S[x(t) + δx(t)]

More information

Relativistic Mechanics

Relativistic Mechanics Physics 411 Lecture 9 Relativistic Mechanics Lecture 9 Physics 411 Classical Mechanics II September 17th, 2007 We have developed some tensor language to describe familiar physics we reviewed orbital motion

More information

Maxwell s equations. based on S-54. electric field charge density. current density

Maxwell s equations. based on S-54. electric field charge density. current density Maxwell s equations based on S-54 Our next task is to find a quantum field theory description of spin-1 particles, e.g. photons. Classical electrodynamics is governed by Maxwell s equations: electric field

More information

Lecture Notes on Electromagnetism

Lecture Notes on Electromagnetism Lecture Notes on Electromagnetism Abstract. The contents of this text is based on the class notes on Electromagnetism for the PH412 course by Prof. Ananda Dasgupta, IISER Kolkata. Contents Chapter 1. Introduction

More information

1 Lagrangian for a continuous system

1 Lagrangian for a continuous system Lagrangian for a continuous system Let s start with an example from mechanics to get the big idea. The physical system of interest is a string of length and mass per unit length fixed at both ends, and

More information

PROBLEM SET 1 SOLUTIONS

PROBLEM SET 1 SOLUTIONS MASSACHUSETTS INSTITUTE OF TECHNOLOGY Physics Department 8.323: Relativistic Quantum Field Theory I Prof.Alan Guth February 29, 2008 PROBLEM SET 1 SOLUTIONS Problem 1: The energy-momentum tensor for source-free

More information

As usual, these notes are intended for use by class participants only, and are not for circulation. Week 7: Lectures 13, 14.

As usual, these notes are intended for use by class participants only, and are not for circulation. Week 7: Lectures 13, 14. As usual, these notes are intended for use by class participants only, and are not for circulation. Week 7: Lectures 13, 14 Majorana spinors March 15, 2012 So far, we have only considered massless, two-component

More information

We begin our discussion of special relativity with a power point presentation, available on the website.

We begin our discussion of special relativity with a power point presentation, available on the website. Special Relativity We begin our discussion of special relativity with a power point presentation, available on the website.. Spacetime From the power point presentation, you know that spacetime is a four

More information

Lecturer: Bengt E W Nilsson

Lecturer: 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 information

221A Lecture Notes Electromagnetic Couplings

221A Lecture Notes Electromagnetic Couplings 221A Lecture Notes Electromagnetic Couplings 1 Classical Mechanics The coupling of the electromagnetic field with a charged point particle of charge e is given by a term in the action (MKSA system) S int

More information

Linearized Gravity Return to Linearized Field Equations

Linearized Gravity Return to Linearized Field Equations Physics 411 Lecture 28 Linearized Gravity Lecture 28 Physics 411 Classical Mechanics II November 7th, 2007 We have seen, in disguised form, the equations of linearized gravity. Now we will pick a gauge

More information

3 Parallel transport and geodesics

3 Parallel transport and geodesics 3 Parallel transport and geodesics 3.1 Differentiation along a curve As a prelude to parallel transport we consider another form of differentiation: differentiation along a curve. A curve is a parametrized

More information

INTRODUCTION TO GENERAL RELATIVITY AND COSMOLOGY

INTRODUCTION TO GENERAL RELATIVITY AND COSMOLOGY INTRODUCTION TO GENERAL RELATIVITY AND COSMOLOGY Living script Astro 405/505 ISU Fall 2004 Dirk Pützfeld Iowa State University 2004 Last update: 9th December 2004 Foreword This material was prepared by

More information

NTNU Trondheim, Institutt for fysikk

NTNU Trondheim, Institutt for fysikk NTNU Trondheim, Institutt for fysikk Examination for FY3464 Quantum Field Theory I Contact: Michael Kachelrieß, tel. 998971 Allowed tools: mathematical tables 1. Spin zero. Consider a real, scalar field

More information

Quantum Field Theory Notes. Ryan D. Reece

Quantum Field Theory Notes. Ryan D. Reece Quantum Field Theory Notes Ryan D. Reece November 27, 2007 Chapter 1 Preliminaries 1.1 Overview of Special Relativity 1.1.1 Lorentz Boosts Searches in the later part 19th century for the coordinate transformation

More information

4-Vector Notation. Chris Clark September 5, 2006

4-Vector Notation. Chris Clark September 5, 2006 4-Vector Notation Chris Clark September 5, 2006 1 Lorentz Transformations We will assume that the reader is familiar with the Lorentz Transformations for a boost in the x direction x = γ(x vt) ȳ = y x

More information

General Relativity and Cosmology Mock exam

General Relativity and Cosmology Mock exam Physikalisches Institut Mock Exam Universität Bonn 29. June 2011 Theoretische Physik SS 2011 General Relativity and Cosmology Mock exam Priv. Doz. Dr. S. Förste Exercise 1: Overview Give short answers

More information

Chapter 13. Local Symmetry

Chapter 13. Local Symmetry Chapter 13 Local Symmetry So far, we have discussed symmetries of the quantum mechanical states. A state is a global (non-local) object describing an amplitude everywhere in space. In relativistic physics,

More information

Maxwell s equations. electric field charge density. current density

Maxwell s equations. electric field charge density. current density Maxwell s equations based on S-54 Our next task is to find a quantum field theory description of spin-1 particles, e.g. photons. Classical electrodynamics is governed by Maxwell s equations: electric field

More information

Week 1. 1 The relativistic point particle. 1.1 Classical dynamics. Reading material from the books. Zwiebach, Chapter 5 and chapter 11

Week 1. 1 The relativistic point particle. 1.1 Classical dynamics. Reading material from the books. Zwiebach, Chapter 5 and chapter 11 Week 1 1 The relativistic point particle Reading material from the books Zwiebach, Chapter 5 and chapter 11 Polchinski, Chapter 1 Becker, Becker, Schwartz, Chapter 2 1.1 Classical dynamics The first thing

More information

Vector Fields. It is standard to define F µν = µ ϕ ν ν ϕ µ, so that the action may be written compactly as

Vector Fields. It is standard to define F µν = µ ϕ ν ν ϕ µ, so that the action may be written compactly as Vector Fields The most general Poincaré-invariant local quadratic action for a vector field with no more than first derivatives on the fields (ensuring that classical evolution is determined based on the

More information

On Fluid Maxwell Equations

On Fluid Maxwell Equations 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

More information

Gravitation: Special Relativity

Gravitation: Special Relativity An Introduction to General Relativity Center for Relativistic Astrophysics School of Physics Georgia Institute of Technology Notes based on textbook: Spacetime and Geometry by S.M. Carroll Spring 2013

More information

Gauge Theory of Gravitation: Electro-Gravity Mixing

Gauge Theory of Gravitation: Electro-Gravity Mixing Gauge Theory of Gravitation: Electro-Gravity Mixing E. Sánchez-Sastre 1,2, V. Aldaya 1,3 1 Instituto de Astrofisica de Andalucía, Granada, Spain 2 Email: sastre@iaa.es, es-sastre@hotmail.com 3 Email: valdaya@iaa.es

More information

B(r) = µ 0a 2 J r 2ρ 2

B(r) = µ 0a 2 J r 2ρ 2 28 S8 Covariant Electromagnetism: Problems Questions marked with an asterisk are more difficult.. Eliminate B instead of H from the standard Maxwell equations. Show that the effective source terms are

More information

Relativistic Dynamics

Relativistic Dynamics Chapter 4 Relativistic Dynamics The most important example of a relativistic particle moving in a potential is a charged particle, say an electron, moving in an electromagnetic field, which might be that

More information

Curves in the configuration space Q or in the velocity phase space Ω satisfying the Euler-Lagrange (EL) equations,

Curves in the configuration space Q or in the velocity phase space Ω satisfying the Euler-Lagrange (EL) equations, Physics 6010, Fall 2010 Hamiltonian Formalism: Hamilton s equations. Conservation laws. Reduction. Poisson Brackets. Relevant Sections in Text: 8.1 8.3, 9.5 The Hamiltonian Formalism We now return to formal

More information

In deriving this we ve used the fact that the specific angular momentum

In deriving this we ve used the fact that the specific angular momentum Equation of Motion and Geodesics So far we ve talked about how to represent curved spacetime using a metric, and what quantities are conserved. Now let s see how test particles move in such a spacetime.

More information

E = φ 1 A The dynamics of a particle with mass m and charge q is determined by the Hamiltonian

E = φ 1 A The dynamics of a particle with mass m and charge q is determined by the Hamiltonian Lecture 9 Relevant sections in text: 2.6 Charged particle in an electromagnetic field We now turn to another extremely important example of quantum dynamics. Let us describe a non-relativistic particle

More information

Problem 1: Lagrangians and Conserved Quantities. Consider the following action for a particle of mass m moving in one dimension

Problem 1: Lagrangians and Conserved Quantities. Consider the following action for a particle of mass m moving in one dimension 105A Practice Final Solutions March 13, 01 William Kelly Problem 1: Lagrangians and Conserved Quantities Consider the following action for a particle of mass m moving in one dimension S = dtl = mc dt 1

More information

General Relativity (j µ and T µν for point particles) van Nieuwenhuizen, Spring 2018

General Relativity (j µ and T µν for point particles) van Nieuwenhuizen, Spring 2018 Consistency of conservation laws in SR and GR General Relativity j µ and for point particles van Nieuwenhuizen, Spring 2018 1 Introduction The Einstein equations for matter coupled to gravity read Einstein

More information

Solution to Problem Set 4

Solution to Problem Set 4 Solution to Problem Set 4 October 017 Pb 1. 0 pts. There are many ways of doing this problem but the easiest would be â α =â ˆD(α) 0 = â exp ( αâ α â ) 0 = â e α α/ e αâ 0 = α + α e α α/ e αâ 0 = α + α

More information

Einstein Toolkit Workshop. Joshua Faber Apr

Einstein Toolkit Workshop. Joshua Faber Apr Einstein Toolkit Workshop Joshua Faber Apr 05 2012 Outline Space, time, and special relativity The metric tensor and geometry Curvature Geodesics Einstein s equations The Stress-energy tensor 3+1 formalisms

More information

Scalar Fields and Gauge

Scalar Fields and Gauge Physics 411 Lecture 23 Scalar Fields and Gauge Lecture 23 Physics 411 Classical Mechanics II October 26th, 2007 We will discuss the use of multiple fields to expand our notion of symmetries and conservation.

More information

Lectures April 29, May

Lectures April 29, May Lectures 25-26 April 29, May 4 2010 Electromagnetism controls most of physics from the atomic to the planetary scale, we have spent nearly a year exploring the concrete consequences of Maxwell s equations

More information

The Accelerator Hamiltonian in a Straight Coordinate System

The Accelerator Hamiltonian in a Straight Coordinate System Hamiltoninan Dynamics for Particle Accelerators, Lecture 2 The Accelerator Hamiltonian in a Straight Coordinate System Andy Wolski University of Liverpool, and the Cockcroft Institute, Daresbury, UK. Given

More information

Scott Hughes 12 May Massachusetts Institute of Technology Department of Physics Spring 2005

Scott Hughes 12 May Massachusetts Institute of Technology Department of Physics Spring 2005 Scott Hughes 12 May 2005 24.1 Gravity? Massachusetts Institute of Technology Department of Physics 8.022 Spring 2005 Lecture 24: A (very) brief introduction to general relativity. The Coulomb interaction

More information

Quantum Field Theory I Examination questions will be composed from those below and from questions in the textbook and previous exams

Quantum Field Theory I Examination questions will be composed from those below and from questions in the textbook and previous exams Quantum Field Theory I Examination questions will be composed from those below and from questions in the textbook and previous exams III. Quantization of constrained systems and Maxwell s theory 1. The

More information

GRAVITATION F10. Lecture Maxwell s Equations in Curved Space-Time 1.1. Recall that Maxwell equations in Lorentz covariant form are.

GRAVITATION F10. Lecture Maxwell s Equations in Curved Space-Time 1.1. Recall that Maxwell equations in Lorentz covariant form are. GRAVITATION F0 S. G. RAJEEV Lecture. Maxwell s Equations in Curved Space-Time.. Recall that Maxwell equations in Lorentz covariant form are. µ F µν = j ν, F µν = µ A ν ν A µ... They follow from the variational

More information

ν ηˆαˆβ The inverse transformation matrices are computed similarly:

ν ηˆαˆβ The inverse transformation matrices are computed similarly: Orthonormal Tetrads Let s now return to a subject we ve mentioned a few times: shifting to a locally Minkowski frame. In general, you want to take a metric that looks like g αβ and shift into a frame such

More information

Physics 209 Fall 2002 Notes 5 Thomas Precession

Physics 209 Fall 2002 Notes 5 Thomas Precession Physics 209 Fall 2002 Notes 5 Thomas Precession Jackson s discussion of Thomas precession is based on Thomas s original treatment, and on the later paper by Bargmann, Michel, and Telegdi. The alternative

More information

Introduction to particle physics Lecture 9: Gauge invariance

Introduction to particle physics Lecture 9: Gauge invariance Introduction to particle physics Lecture 9: Gauge invariance Frank Krauss IPPP Durham U Durham, Epiphany term 2010 1 / 17 Outline 1 Symmetries 2 Classical gauge invariance 3 Phase invariance 4 Generalised

More information

Übungen zu RT2 SS (4) Show that (any) contraction of a (p, q) - tensor results in a (p 1, q 1) - tensor.

Übungen zu RT2 SS (4) Show that (any) contraction of a (p, q) - tensor results in a (p 1, q 1) - tensor. Übungen zu RT2 SS 2010 (1) Show that the tensor field g µν (x) = η µν is invariant under Poincaré transformations, i.e. x µ x µ = L µ νx ν + c µ, where L µ ν is a constant matrix subject to L µ ρl ν ση

More information

MSci EXAMINATION. Date: XX th May, Time: 14:30-17:00

MSci EXAMINATION. Date: XX th May, Time: 14:30-17:00 MSci EXAMINATION PHY-415 (MSci 4242 Relativistic Waves and Quantum Fields Time Allowed: 2 hours 30 minutes Date: XX th May, 2010 Time: 14:30-17:00 Instructions: Answer THREE QUESTIONS only. Each question

More information

THE QFT NOTES 5. Badis Ydri Department of Physics, Faculty of Sciences, Annaba University, Annaba, Algeria. December 9, 2011

THE QFT NOTES 5. Badis Ydri Department of Physics, Faculty of Sciences, Annaba University, Annaba, Algeria. December 9, 2011 THE QFT NOTES 5 Badis Ydri Department of Physics, Faculty of Sciences, Annaba University, Annaba, Algeria. December 9, 2011 Contents 1 The Electromagnetic Field 2 1.1 Covariant Formulation of Classical

More information

Gravitation: Gravitation

Gravitation: Gravitation An Introduction to General Relativity Center for Relativistic Astrophysics School of Physics Georgia Institute of Technology Notes based on textbook: Spacetime and Geometry by S.M. Carroll Spring 2013

More information

Chapter 10 Operators of the scalar Klein Gordon field. from my book: Understanding Relativistic Quantum Field Theory.

Chapter 10 Operators of the scalar Klein Gordon field. from my book: Understanding Relativistic Quantum Field Theory. Chapter 10 Operators of the scalar Klein Gordon field from my book: Understanding Relativistic Quantum Field Theory Hans de Vries November 11, 2008 2 Chapter Contents 10 Operators of the scalar Klein Gordon

More information

Lagrangian. µ = 0 0 E x E y E z 1 E x 0 B z B y 2 E y B z 0 B x 3 E z B y B x 0. field tensor. ν =

Lagrangian. µ = 0 0 E x E y E z 1 E x 0 B z B y 2 E y B z 0 B x 3 E z B y B x 0. field tensor. ν = Lagrangian L = 1 4 F µνf µν j µ A µ where F µν = µ A ν ν A µ = F νµ. F µν = ν = 0 1 2 3 µ = 0 0 E x E y E z 1 E x 0 B z B y 2 E y B z 0 B x 3 E z B y B x 0 field tensor. Note that F µν = g µρ F ρσ g σν

More information

Special classical solutions: Solitons

Special classical solutions: Solitons Special classical solutions: Solitons by Suresh Govindarajan, Department of Physics, IIT Madras September 25, 2014 The Lagrangian density for a single scalar field is given by L = 1 2 µφ µ φ Uφ), 1) where

More information

Energy, Momentum, and Lagrangians

Energy, Momentum, and Lagrangians Energy, Momentum, and Lagrangians Lecture 10 1 The Poynting vector Previously an expression for the flow of energy in terms of the fields was developed. Note the following uses the MKS system of units

More information

Fig. 1. On a sphere, geodesics are simply great circles (minimum distance). From

Fig. 1. On a sphere, geodesics are simply great circles (minimum distance). From Equation of Motion and Geodesics The equation of motion in Newtonian dynamics is F = m a, so for a given mass and force the acceleration is a = F /m. If we generalize to spacetime, we would therefore expect

More information

2 Canonical quantization

2 Canonical quantization Phys540.nb 7 Canonical quantization.1. Lagrangian mechanics and canonical quantization Q: How do we quantize a general system?.1.1.lagrangian Lagrangian mechanics is a reformulation of classical mechanics.

More information

The Lorentz Force and Energy-Momentum for Off-Shell Electromagnetism

The Lorentz Force and Energy-Momentum for Off-Shell Electromagnetism TAUP 1824-90 The Lorentz Force and Energy-Momentum for Off-Shell Electromagnetism M.C. Land 1 and L.P. Horwitz 2 School of Physics and Astronomy Raymond and Beverly Sackler Faculty of Exact Sciences Tel

More information

The Particle-Field Hamiltonian

The Particle-Field Hamiltonian The Particle-Field Hamiltonian For a fundamental understanding of the interaction of a particle with the electromagnetic field we need to know the total energy of the system consisting of particle and

More information

Caltech Ph106 Fall 2001

Caltech Ph106 Fall 2001 Caltech h106 Fall 2001 ath for physicists: differential forms Disclaimer: this is a first draft, so a few signs might be off. 1 Basic properties Differential forms come up in various parts of theoretical

More information

A873: Cosmology Course Notes. II. General Relativity

A873: Cosmology Course Notes. II. General Relativity II. General Relativity Suggested Readings on this Section (All Optional) For a quick mathematical introduction to GR, try Chapter 1 of Peacock. For a brilliant historical treatment of relativity (special

More information

Hamilton-Jacobi Formulation of A Non-Abelian Yang-Mills Theories

Hamilton-Jacobi Formulation of A Non-Abelian Yang-Mills Theories EJTP 5, No. 17 (2008) 65 72 Electronic Journal of Theoretical Physics Hamilton-Jacobi Formulation of A Non-Abelian Yang-Mills Theories W. I. Eshraim and N. I. Farahat Department of Physics Islamic University

More information

Lecture notes for FYS610 Many particle Quantum Mechanics

Lecture notes for FYS610 Many particle Quantum Mechanics UNIVERSITETET I STAVANGER Institutt for matematikk og naturvitenskap Lecture notes for FYS610 Many particle Quantum Mechanics Note 20, 19.4 2017 Additions and comments to Quantum Field Theory and the Standard

More information

Physics 214 Examples of four-vectors Winter 2017

Physics 214 Examples of four-vectors Winter 2017 Physics 214 Examples of four-vectors Winter 2017 1. The velocity four-vector The velocity four-vector of a particle is defined by: u µ = dxµ dτ = γc; γ v), 1) where dτ = γ 1 is the differential proper

More information

Lecture 20: Effective field theory for the Bose- Hubbard model

Lecture 20: Effective field theory for the Bose- Hubbard model Lecture 20: Effective field theory for the Bose- Hubbard model In the previous lecture, we have sketched the expected phase diagram of the Bose-Hubbard model, and introduced a mean-field treatment that

More information

Retarded Potentials and Radiation

Retarded Potentials and Radiation Retarded Potentials and Radiation No, this isn t about potentials that were held back a grade :). Retarded potentials are needed because at a given location in space, a particle feels the fields or potentials

More information

Preliminaries: what you need to know

Preliminaries: what you need to know January 7, 2014 Preliminaries: what you need to know Asaf Pe er 1 Quantum field theory (QFT) is the theoretical framework that forms the basis for the modern description of sub-atomic particles and their

More information

Classical Field Theory

Classical Field Theory Classical Field Theory Asaf Pe er 1 January 12, 2016 We begin by discussing various aspects of classical fields. We will cover only the bare minimum ground necessary before turning to the quantum theory,

More information

Concepts in Theoretical Physics

Concepts in Theoretical Physics Concepts in Theoretical Physics Lecture 1: The Principle of Least Action David Tong Newtonian Mechanics You've all done a course on Newtonian mechanics so you know how to calculate the way things move.

More information

Heisenberg-Euler effective lagrangians

Heisenberg-Euler effective lagrangians Heisenberg-Euler effective lagrangians Appunti per il corso di Fisica eorica 7/8 3.5.8 Fiorenzo Bastianelli We derive here effective lagrangians for the electromagnetic field induced by a loop of charged

More information

Special Theory of Relativity

Special Theory of Relativity June 17, 2008 1 1 J.D.Jackson, Classical Electrodynamics, 3rd Edition, Chapter 11 Introduction Einstein s theory of special relativity is based on the assumption (which might be a deep-rooted superstition

More information

Lecture 4. Alexey Boyarsky. October 6, 2015

Lecture 4. Alexey Boyarsky. October 6, 2015 Lecture 4 Alexey Boyarsky October 6, 2015 1 Conservation laws and symmetries 1.1 Ignorable Coordinates During the motion of a mechanical system, the 2s quantities q i and q i, (i = 1, 2,..., s) which specify

More information

Übungen zur Elektrodynamik (T3)

Übungen zur Elektrodynamik (T3) Arnold Sommerfeld Center Ludwig Maximilians Universität München Prof. Dr. Ivo Sachs SoSe 17 Übungen zur Elektrodynamik T3) Lösungen zum Übungsblatt 6 1 Lorentz Force The equations of motion for the trajectory

More information

The principle of equivalence and its consequences.

The principle of equivalence and its consequences. The principle of equivalence and its consequences. Asaf Pe er 1 January 28, 2014 This part of the course is based on Refs. [1], [2] and [3]. 1. Introduction We now turn our attention to the physics of

More information

Path Integral Quantization of the Electromagnetic Field Coupled to A Spinor

Path Integral Quantization of the Electromagnetic Field Coupled to A Spinor EJTP 6, No. 22 (2009) 189 196 Electronic Journal of Theoretical Physics Path Integral Quantization of the Electromagnetic Field Coupled to A Spinor Walaa. I. Eshraim and Nasser. I. Farahat Department of

More information

Exercise 1 Classical Bosonic String

Exercise 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 information

Lecture 9: RR-sector and D-branes

Lecture 9: RR-sector and D-branes Lecture 9: RR-sector and D-branes José D. Edelstein University of Santiago de Compostela STRING THEORY Santiago de Compostela, March 6, 2013 José D. Edelstein (USC) Lecture 9: RR-sector and D-branes 6-mar-2013

More information

Übungen zur Elektrodynamik (T3)

Übungen zur Elektrodynamik (T3) Arnold Sommerfeld Center Ludwig Maximilians Universität München Prof. Dr. Ivo Sachs SoSe 08 Übungen zur Elektrodynamik (T3) Lösungen zum Übungsblatt 7 Lorentz Force Calculate dx µ and ds explicitly in

More information

Special Relativity - QMII - Mechina

Special Relativity - QMII - Mechina Special Relativity - QMII - Mechina 2016-17 Daniel Aloni Disclaimer This notes should not replace a course in special relativity, but should serve as a reminder. I tried to cover as many important topics

More information

752 Final. April 16, Fadeev Popov Ghosts and Non-Abelian Gauge Fields. Tim Wendler BYU Physics and Astronomy. The standard model Lagrangian

752 Final. April 16, Fadeev Popov Ghosts and Non-Abelian Gauge Fields. Tim Wendler BYU Physics and Astronomy. The standard model Lagrangian 752 Final April 16, 2010 Tim Wendler BYU Physics and Astronomy Fadeev Popov Ghosts and Non-Abelian Gauge Fields The standard model Lagrangian L SM = L Y M + L W D + L Y u + L H The rst term, the Yang Mills

More information

Physics on Curved Spaces 2

Physics on Curved Spaces 2 Physics on Curved Spaces 2 November 29, 2017 2 Based on: General Relativity M.P.Hobson, G. Efstahiou and A.N. Lasenby, Cambridge 2006 (Chapter 6 ) Gravity E. Poisson, C.M. Will, Cambridge 2014 Physics

More information

Chapter 1. Hamilton s mechanics. 1.1 Path integrals

Chapter 1. Hamilton s mechanics. 1.1 Path integrals Chapter 1 Hamilton s mechanics William Rowan Hamilton was an Irish physicist/mathematician from Dublin. Born in 1806, he basically invented modern mechanics in his 60 years and laid the groundwork for

More information

Lecturer: Bengt E W Nilsson

Lecturer: Bengt E W Nilsson 9 3 19 Lecturer: Bengt E W Nilsson Last time: Relativistic physics in any dimension. Light-cone coordinates, light-cone stuff. Extra dimensions compact extra dimensions (here we talked about fundamental

More information

d 2 Area i K i0 ν 0 (S.2) when the integral is taken over the whole space, hence the second eq. (1.12).

d 2 Area i K i0 ν 0 (S.2) when the integral is taken over the whole space, hence the second eq. (1.12). PHY 396 K. Solutions for prolem set #. Prolem 1a: Let T µν = λ K λµ ν. Regardless of the specific form of the K λµ ν φ, φ tensor, its antisymmetry with respect to its first two indices K λµ ν K µλ ν implies

More information

Physics on Curved Spaces 2

Physics on Curved Spaces 2 Physics on Curved Spaces 2 May 11, 2017 2 Based on: General Relativity M.P.Hobson, G. Efstahiou and A.N. Lasenby, Cambridge 2006 (Chapter 6 ) Gravity E. Poisson, C.M. Will, Cambridge 2014 Physics on Curved

More information

Lecture 5: Sept. 19, 2013 First Applications of Noether s Theorem. 1 Translation Invariance. Last Latexed: September 18, 2013 at 14:24 1

Lecture 5: Sept. 19, 2013 First Applications of Noether s Theorem. 1 Translation Invariance. Last Latexed: September 18, 2013 at 14:24 1 Last Latexed: September 18, 2013 at 14:24 1 Lecture 5: Sept. 19, 2013 First Applications of Noether s Theorem Copyright c 2005 by Joel A. Shapiro Now it is time to use the very powerful though abstract

More information