Quantum Field Theory

609 Pages·2019·46.72 MB·English

by  François Gelis

Quantum Field Theory
This modern text combines fundamental principles with advanced topics and recent tech-
niques in a rigorous and self-contained treatment of quantum field theory.
Key features include:
• a review of basic principles, starting with quantum mechanics and special relativity, and
covering elementary aspects of quantum field theory and perturbation theory;
• standard results and tools relevant to many applications, including canonical quantization,
path integrals, non-Abelian gauge theories and the renormalization group;
• advanced topics such as effective field theories, quantum anomalies, stable extended field
configurations, lattice field theory and field theory at a finite temperature or in the strong
field regime;
• two chapters dedicated to new methods for calculating scattering amplitudes (spinor-
helicity, on-shell recursion and generalized unitarity), equipping students with practical
skills for future research; and
• an accessible style featuring numerous worked examples, applications and end-of-chapter
problems.
This is an essential text for graduate students in physics, and an equally excellent reference for
researchers in the field.
Quantum Field Theory
From Basics to Modern Topics
FRANÇOIS GELIS CEA, Saclay
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www.cambridge.org
Information on this title: www.cambridge.org/9781108480901
DOI: 10.1017/9781108691550
©Cambridge University Press 2019
This publication is in copyright. Subject to statutory exception
and to the provisions of relevant collective licensing agreements,
no reproduction of any part may take place without the written
permission of Cambridge University Press.
First published 2019
Printed in the United Kingdom by TJ International Ltd, Padstow Cornwall
A catalogue record for this publication is available from the British Library.
Library of Congress Cataloging-in-Publication Data
Names: Gelis, François, 1972- author.
Title: Quantum field theory : from basics to modern topics / François Gelis
(CEA, Saclay).
Description: Cambridge, United Kingdom ; New York, NY : Cambridge University
Press, 2019. |Includes bibliographical references and index.
Identifiers: LCCN 2019005987 | ISBN 9781108480901 (hardback ; alk. paper) |
ISBN 110848090X (hardback ; alk. paper)
Subjects: LCSH: Quantum field theory.
Classification: LCC QC174.45 .G45 2019 | DDC 530.14/3–dc23
LC record available at https://lccn.loc.gov/2019005987
ISBN 978-1-108-48090-1 Hardback
Additional resources for this publication at www.cambridge.org/gelis.
Cambridge University Press has no responsibility for the persistence or accuracy of
URLs for external or third-party internet websites referred to in this publication
and does not guarantee that any content on such websites is, or will remain,
accurate or appropriate.
To Kanako and Nathan.
In order to improve the mind,
we ought less to learn,
than to contemplate.
RENÉ DESCARTES
Contents
List of Figures and Tables page
xii
Preface xv
Acknowledgments xvii
1 Basics of Quantum Field Theory 1
1.1 Why Quantum Field Theory? 1
1.2 Special Relativity 3
1.3 Free Scalar Fields, Mode Decomposition 7
1.4 Interacting Scalar Particles 14
1.5 From Field Correlations to Reaction Rates 17
1.6 Källen–Lehmann Spectral Representation 24
1.7 Generating Functional 27
Exercises 32
2 Perturbation Theory 35
2.1 Perturbative Expansion and Feynman Rules 35
2.2 Calculation of Loop Integrals 41
2.3 Ultraviolet Divergences and Renormalization 44
2.4 Perturbative Unitarity 53
Exercises 61
3 Quantum Electrodynamics 63
3.1 Spin-1/2 Fields 63
3.2 Spin-1 Fields 70
3.3 Quantum Electrodynamics 74
3.4 Charge Conservation, Ward–Takahashi Identities 80
3.5 Ultraviolet Renormalization 82
3.6 Cutting Rules in QED and Unitarity 86
viii CONTENTS
3.7 Infrared Divergences 88
Exercises 93
4 Spontaneous Symmetry Breaking 95
4.1 Potential Energy Landscape 95
4.2 Conserved Currents and Charges 101
4.3 Spectral Properties 103
4.4 Coleman’s Theorem 105
4.5 Linear Sigma Model 107
4.6 Heisenberg Model of Ferromagnetism 111
Exercises 114
5 Functional Quantization 116
5.1 Path Integral in Quantum Mechanics 116
5.2 Functional Manipulations 120
5.3 Path Integral in Scalar Field Theory 125
5.4 Functional Determinants 127
5.5 Quantum Effective Action 129
5.6 Two-Particle Irreducible Effective Action 138
5.7 Euclidean Path Integral and Statistical Mechanics 144
Exercises 147
6 Path Integrals for Fermions and Photons 150
6.1 Grassmann Variables 150
6.2 Path Integral for Fermions 156
6.3 Path Integral for Photons 157
6.4 Schwinger–Dyson Equations 160
6.5 Quantum Anomalies 162
Exercises 170
7 Non-Abelian Gauge Symmetry 173
7.1 Non-Abelian Lie Groups and Algebras 173
7.2 Yang–Mills Lagrangian 181
7.3 Non-Abelian Gauge Theories 185
7.4 Spontaneous Gauge Symmetry Breaking 190
7.5 θ-term and Strong-CP Problem 194
7.6 Non-Local Gauge Invariant Operators 201
Exercises 209
CONTENTS ix
8 Quantization of Yang–Mills Theory 212
8.1 Naive Quantization of the Gauge Bosons 212
8.2 Gauge Fixing 214
8.3 Fadeev–Popov Quantization and Ghost Fields 215
8.4 Feynman Rules for Non-Abelian Gauge Theories 217
8.5 On-Shell Non-Abelian Ward–Takahashi Identities 221
8.6 Ghosts and Unitarity 223
Exercises 232
9 Renormalization of Gauge Theories 234
9.1 Ultraviolet Power Counting 234
9.2 Symmetries of the Quantum Effective Action 236
9.3 Renormalizability 241
9.4 Background Field Method 245
Exercises 251
10 Renormalization Group 252
10.1 Scale Dependence of Correlation Functions 252
10.2 Correlators Containing Composite Operators 256
10.3 Operator Product Expansion 258
10.4 Example: QCD Corrections to Weak Decays 261
10.5 Non-Perturbative Renormalization Group 267
Exercises 276
11 Effective Field Theories 278
11.1 General Principles of Effective Theories 279
11.2 Example: Fermi Theory of Weak Decays 282
11.3 The Standard Model as an Effective Field Theory 285
11.4 Effective Theories in QCD 291
11.5 EFT of Spontaneous Symmetry-Breaking 300
Exercises 309
12 Quantum Anomalies 312
12.1 Axial Anomalies in a Gauge Background 312
12.2 Generalizations 323
12.3 Wess–Zumino Consistency Conditions 329
12.4 ’t Hooft Anomaly Matching 333
x CONTENTS
12.5 Scale Anomalies 334
Exercises 341
13 Localized Field Configurations 342
13.1 Domain Walls 343
13.2 Monopoles 346
13.3 Instantons 355
13.4 Skyrmions 366
Exercises 370
14 Modern Tools for Tree Amplitudes 372
14.1 Color Decomposition of Gluon Amplitudes 373
14.2 Spinor-Helicity Formalism 379
14.3 Britto–Cachazo–Feng–Witten On-Shell Recursion 388
14.4 Tree-Level Gravitational Amplitudes 398
14.5 Cachazo–Svrcek–Witten Rules 406
Exercises 415
15 Worldline Formalism 418
15.1 Worldline Representation 418
15.2 Quantum Electrodynamics 423
15.3 Schwinger Mechanism 427
15.4 Calculation of One-Loop Amplitudes 430
Exercises 438
16 Lattice Field Theory 440
16.1 Discretization of Bosonic Actions 441
16.2 Lattice Fermions 445
16.3 Hadron Mass Determination on the Lattice 449
16.4 Wilson Loops and Color Confinement 451
16.5 Gauge Fixing on the Lattice 454
16.6 Lattice Hamiltonian 457
16.7 Lattice Worldline Formalism 458
Exercises 462
17 Quantum Field Theory at Finite Temperature 465
17.1 Canonical Thermal Ensemble 465
17.2 Finite-T Perturbation Theory 466