Lectures of Sidney Coleman on Quantum Field Theory (Foreword by David Kaiser)

915 Pages·2019·61.03 MB·English

by  Bryan Gin-ge Chen, David Derbes, David Griffiths, Brian Hill, Richard Sohn, Yuan-Sen Ting

Published by
World Scientific Publishing Co. Pte. Ltd.
5 Toh Tuck Link, Singapore 596224
USA office: 27 Warren Street, Suite 401-402, Hackensack, NJ 07601
UK office: 57 Shelton Street, Covent Garden, London WC2H 9HE
Library of Congress Cataloging-in-Publication Data
Names: Coleman, Sidney, 1937–2007. | Chen, Bryan Gin-ge, 1985– editor.
Title: Lectures of Sidney Coleman on quantum field theory / [edited by]
Bryan Gin-ge Chen (Leiden University, Netherlands) [and five others].
Description: New Jersey : World Scientific, 2018. | Includes bibliographical
references and index.
Identifiers: LCCN 2018041457| ISBN 9789814632539 (hardcover : alk. paper) |
ISBN 9789814635509 (pbk. : alk. paper)
Subjects: LCSH: Quantum field theory.
Classification: LCC QC174.46 .C65 2018 | DDC 530.14/3--dc23
LC record available at https://lccn.loc.gov/2018041457
British Library Cataloguing-in-Publication Data
A catalogue record for this book is available from the British Library.
Copyright © 2019 by World Scientific Publishing Co. Pte. Ltd.
All rights reserved. This book, or parts thereof, may not be reproduced in any form or by any means, electronic or
mechanical, including photocopying, recording or any information storage and retrieval system now known or to
be invented, without written permission from the publisher.
For photocopying of material in this volume, please pay a copying fee through the Copyright Clearance Center,
Inc., 222 Rosewood Drive, Danvers, MA 01923, USA. In this case permission to photocopy is not required from
the publisher.
For any available supplementary material, please visit
https://www.worldscientific.com/worldscibooks/10.1142/9371#t=suppl
Printed in Singapore
to Diana Coleman
and for all of Sidney’s students—past, present, and future
Contents
Foreword
Preface
Frequently cited references
Index of useful formulae
A note on the problems
1Adding special relativity to quantum mechanics
1.1Introductory remarks
1.2Theory of a single free, spinless particle of mass µ
1.3Determination of the position operator X
2The simplest many-particle theory
2.1First steps in describing a many-particle state
2.2Occupation number representation
2.3Operator formalism and the harmonic oscillator
2.4The operator formalism applied to Fock space
3Constructing a scalar quantum field
3.1Ensuring relativistic causality
3.2Conditions to be satisfied by a scalar quantum field
3.3The explicit form of the scalar quantum field
3.4Turning the argument around: the free scalar field as the fundamental object
3.5A hint of things to come
Problems 1
Solutions 1
4The method of the missing box
4.1Classical particle mechanics
4.2Quantum particle mechanics
4.3Classical field theory
4.4Quantum field theory
4.5Normal ordering
5Symmetries and conservation laws I. Spacetime symmetries
5.1Symmetries and conservation laws in classical particle mechanics
5.2Extension to quantum particle mechanics
5.3Extension to field theory
5.4Conserved currents are not uniquely defined
5.5Calculation of currents from spacetime translations
5.6Lorentz transformations, angular momentum and something else
Problems 2
Solutions 2
6Symmetries and conservation laws II. Internal symmetries
6.1Continuous symmetries
6.2Lorentz transformation properties of the charges
6.3Discrete symmetries
7Introduction to perturbation theory and scattering
7.1The Schrödinger and Heisenberg pictures
7.2The interaction picture
7.3Dyson’s formula
7.4Scattering and the S-matrix
Problems 3
Solutions 3
8Perturbation theory I. Wick diagrams
8.1Three model field theories
8.2Wick’s theorem
8.3Dyson’s formula expressed in Wick diagrams
8.4Connected and disconnected Wick diagrams
8.5The exact solution of Model 1
Problems 4
Solutions 4
9Perturbation theory II. Divergences and counterterms
9.1The need for a counterterm in Model 2
9.2Evaluating the S matrix in Model 2
9.3Computing the Model 2 ground state energy
9.4The ground state wave function in Model 2
9.5An infrared divergence
Problems 5
Solutions 5
10Mass renormalization and Feynman diagrams
10.1Mass renormalization in Model 3
10.2Feynman rules in Model 3
10.3Feynman diagrams in Model 3 to order g2
10.4O(g2) nucleon–nucleon scattering in Model 3
11Scattering I. Mandelstam variables, CPT and phase space
11.1Nucleon–antinucleon scattering
11.2Nucleon–meson scattering and meson pair creation
11.3Crossing symmetry and CPT invariance
11.4Phase space and the S matrix
12Scattering II. Applications
12.1Decay processes
12.2Differential cross-section for a two-particle initial state
12.3The density of final states for two particles
12.4The Optical Theorem
12.5The density of final states for three particles
12.6A question and a preview
Problems 6
Solutions 6
13Green’s functions and Heisenberg fields
13.1The graphical definition of
13.2The generating functional Z[ρ] for G(n)(x)
i
13.3Scattering without an adiabatic function
13.4Green’s functions in the Heisenberg picture
13.5Constructing in and out states
Problems 7
Solutions 7
14The LSZ formalism
14.1Two-particle states
14.2The proof of the LSZ formula
14.3Model 3 revisited
14.4Guessing the Feynman rules for a derivative interaction
Problems 8
Solutions 8
15Renormalization I. Determination of counterterms
15.1The perturbative determination of A
15.2The Källén-Lehmann spectral representation
15.3The renormalized meson propagator
15.4The meson self-energy to O(g2)
15.5A table of integrals for one loop
16Renormalization II. Generalization and extension
16.1The meson self-energy to O(g2), completed
16.2Feynman parametrization for multiloop graphs
16.3Coupling constant renormalization
16.4Are all quantum field theories renormalizable?
Problems 9
Solutions 9
17Unstable particles
17.1Calculating the propagator for µ > 2m
17.2The Breit–Wigner formula
17.3A first look at the exponential decay law
17.4Obtaining the decay law by stationary phase approximation
18Representations of the Lorentz Group
18.1Defining the problem: Lorentz transformations in general
18.2Irreducible representations of the rotation group
18.3Irreducible representations of the Lorentz group
18.4Properties of the SO(3) representations D(s)
18.5Properties of the SO(3,1) representations D(s+, s−)
Problems 10
Solutions 10
19The Dirac Equation I. Constructing a Lagrangian
19.1Building vectors out of spinors
19.2A Lagrangian for Weyl spinors
19.3The Weyl equation
19.4The Dirac equation
20The Dirac Equation II. Solutions
20.1The Dirac basis
20.2Plane wave solutions
20.3Pauli’s theorem
20.4The γ matrices
20.5Bilinear spinor products
20.6Orthogonality and completeness
Problems 11
Solutions 11
21The Dirac Equation III. Quantization and Feynman Rules
21.1Canonical quantization of the Dirac field
21.2Wick’s theorem for Fermi fields
21.3Calculating the Dirac propagator
21.4An example: Nucleon–meson scattering
21.5The Feynman rules for theories involving fermions
21.6Summing and averaging over spin states
Problems 12
Solutions 12
22CPT and Fermi fields
22.1Parity and Fermi fields
22.2The Majorana representation
22.3Charge conjugation and Fermi fields
22.4PT invariance and Fermi fields
22.5The CPT theorem and Fermi fields
23Renormalization of spin-½ theories
23.1Lessons from Model 3
23.2The renormalized Dirac propagator
23.3The spectral representation of
23.4The nucleon self-energy
23.5The renormalized coupling constant
Problems 13
Solutions 13
24Isospin
24.1Field theoretic constraints on coupling constants
24.2The nucleon and pion as isospin multiplets
24.3Experimental consequences of isospin conservation
24.4Hypercharge and G-parity
25Coping with infinities: regularization and renormalization
25.1Regularization
25.2The BPHZ algorithm
25.3Applying the algorithm
25.4Survey of renormalizable theories for spin 0 and spin ½
Problems 14
Solutions 14
26Vector fields
26.1The free real vector field
26.2The Proca equation and its solutions
26.3Canonical quantization of the Proca field
26.4The limit µ → 0: a simple physical consequence
26.5Feynman rules for a real massive vector field
27Electromagnetic interactions and minimal coupling
27.1Gauge invariance and conserved currents
27.2The minimal coupling prescription
27.3Technical problems
Problems 15
Solutions 15
28Functional integration and Feynman rules
28.1First steps with functional integrals
28.2Functional integrals in field theory
28.3The Euclidean Z [J] for a free theory
0
28.4The Euclidean Z[J] for an interacting field theory
28.5Feynman rules from functional integrals
28.6The functional integral for massive vector mesons
29Extending the methods of functional integrals
29.1Functional integration for Fermi fields
29.2Derivative interactions via functional integrals
29.3Ghost fields
29.4The Hamiltonian form of the generating functional
29.5How to eliminate constrained variables
29.6Functional integrals for QED with massive photons
Problems 16
Solutions 16
30Electrodynamics with a massive photon
30.1Obtaining the Feynman rules for scalar electrodynamics
30.2The Feynman rules for massive photon electrodynamics
30.3Some low order computations in spinor electrodynamics
30.4Quantizing massless electrodynamics with functional integrals
31The Faddeev–Popov prescription
31.1The prescription in a finite number of dimensions
31.2Extending the prescription to a gauge field theory
31.3Applying the prescription to QED
31.4Equivalence of the Faddeev–Popov prescription and canonical quantization
31.5Revisiting the massive vector theory
31.6A first look at renormalization in QED
Problems 17
Solutions 17
32Generating functionals and Green’s functions
32.1The loop expansion
32.2The generating functional for 1PI Green’s functions
32.3Connecting statistical mechanics with quantum field theory
32.4Quantum electrodynamics in a covariant gauge
33The renormalization of QED
33.1Counterterms and gauge invariance
33.2Counterterms in QED with a massive photon
33.3Gauge-invariant cutoffs
33.4The Ward identity and Green’s functions
33.5The Ward identity and counterterms
Problems 18
Solutions 18
34Two famous results in QED
34.1Coulomb’s Law
34.2The electron’s anomalous magnetic moment in quantum mechanics
34.3The electron’s anomalous magnetic moment in QED
35Confronting experiment with QED
35.1Higher order contributions to the electron’s magnetic moment
35.2The anomalous magnetic moment of the muon
35.3A low-energy theorem
35.4Photon-induced corrections to strong interaction processes (via symmetries)
Problems 19
Solutions 19
36Introducing SU(3)
36.1Decays of the η
36.2An informal historical introduction to SU(3)
36.3Tensor methods for SU(n)
36.4Applying tensor methods in SU(2)
36.5Tensor representations of SU(3)