**Unformatted text preview: **Praise for the first edition
“Quantum field theory is an extraordinarily beautiful subject, but it can be an intimidating
one. The profound and deeply physical concepts it embodies can get lost, to the beginner,
amidst its technicalities. In this book, Zee imparts the wisdom of an experienced and
remarkably creative practitioner in a user-friendly style. I wish something like it had been
available when I was a student.”
—Frank Wilczek, Massachusetts Institute of Technology
“Finally! Zee has written a ground-breaking quantum field theory text based on the course
I made him teach when I chaired the Princeton physics department. With utmost clarity
he gives the eager student a light-hearted and easy-going introduction to the multifaceted
wonders of quantum field theory. I wish I had this book when I taught the subject.”
—Marvin L. Goldberger, President, Emeritus, California Institute of Technology
“This book is filled with charming explanations that students will find beneficial.”
—Ed Witten, Institute for Advanced Study
“This book is perhaps the most user-friendly introductory text to the essentials of quantum
field theory and its many modern applications. With his physically intuitive approach,
Professor Zee makes a serious topic more reachable for beginners, reducing the conceptual
barrier while preserving enough mathematical details necessary for a firm grasp of the
subject.”
—Bei Lok Hu, University of Maryland
“Like the famous Feynman Lectures on Physics, this book has the flavor of a good
blackboard lecture. Zee presents technical details, but only insofar as they serve the larger
purpose of giving insight into quantum field theory and bringing out its beauty.”
—Stephen M. Barr, University of Delaware
“This is a fantastic book—exciting, amusing, unique, and very valuable.”
—Clifford V. Johnson, University of Durham
“Tony Zee explains quantum field theory with a clear and engaging style. For budding or
seasoned condensed matter physicists alike, he shows us that field theory is a nourishing
nut to be cracked and savored.”
—Matthew P. A. Fisher, Kavli Institute for Theoretical Physics
“I was so engrossed that I spent all of Saturday and Sunday one weekend absorbing half
the book, to my wife’s dismay. Zee has a talent for explaining the most abstruse and
arcane concepts in an elegant way, using the minimum number of equations (the jokes
and anecdotes help). . . . I wish this were available when I was a graduate student. Buy
the book, keep it by your bed, and relish the insights delivered with such flair and grace.”
—N. P. Ong, Princeton University What readers are saying “Funny, chatty, physical: QFT education transformed!! This text stands apart from others
in so many ways that it’s difficult to list them all. . . . The exposition is breezy and chatty.
The text is never boring to read, and is at times very, very funny. Puns and jokes abound,
as do anecdotes. . . . A book which is much easier, and more fun, to read than any of the
others. Zee’s skills as a popular physics writer have been used to excellent effect in writing
this textbook. . . . Wholeheartedly recommended.”
—M. Haque
“A readable, and rereadable instant classic on QFT. . . . At an introductory level, this type
of book—with its pedagogical (and often very funny) narrative—is priceless. [It] is full
of fantastic insights akin to reading the Feynman lectures. I have since used QFT in a
Nutshell as a review for [my] year-long course covering all of Peskin and Schroder, and
have been pleasantly surprised at how Zee is able to preemptively answer many of the
open questions that eluded me during my course. . . . I value QFT in a Nutshell the same
way I do the Feynman lectures. . . . It’s a text to teach an understanding of physics.”
—Flip Tanedo
“One of those books a person interested in theoretical physics simply must own! A real
scientific masterpiece. I bought it at the time I was a physics sophomore and that was the
best choice I could have made. It was this book that triggered my interest in quantum field
theory and crystallized my dreams of becoming a theoretical physicist. . . . The main goal
of the book is to make the reader gain real intuition in the field. Amazing . . . amusing . . .
real fun. What also distinguishes this book from others dealing with a similar subject
is that it is written like a tale. . . . I feel enormously fortunate to have come across this
book at the beginning of my adventure with theoretical physics. . . . Definitely the best
quantum field theory book I have ever read.”
—Anonymous
“I have used Quantum Field Theory in a Nutshell as the primary text. . . . I am immensely
pleased with the book, and recommend it highly. . . . Don’t let the ‘damn the torpedoes,
full steam ahead’ approach scare you off. Once you get used to seeing the physics quickly,
I think you will find the experience very satisfying intellectually.”
—Jim Napolitano
“This is undoubtedly the best book I have ever read about the subject. Zee does a fantastic
job of explaining quantum field theory, in a way I have never seen before, and I have
read most of the other books on this topic. If you are looking for quantum field theory
explanations that are clear, precise, concise, intuitive, and fun to read—this is the book
for you.”
—Anonymous “One of the most artistic and deepest books ever written on quantum field theory.
Amazing . . . extremely pleasant . . . a lot of very deep and illuminating remarks. . . . I
recommend the book by Zee to everybody who wants to get a clear idea what good physics
is about.”
—Slava Mukhanov
“Perfect for learning field theory on your own—by far the clearest and easiest to follow
book I’ve found on the subject.”
—Ian Z. Lovejoy
“A beautifully written introduction to the modern view of fields . . . breezy and
enchanting, leading to exceptional clarity without sacrificing depth, breadth, or rigor
of content. . . . [It] passes my test of true greatness: I wish it had been the first book on
this topic that I had found.”
—Jeffrey D. Scargle
“A breeze of fresh air . . . a real literary gem which will be useful for students who make
their first steps in this difficult subject and an enjoyable treat for experts, who will find
new and deep insights. Indeed, the Nutshell is like a bright light source shining among
tall and heavy trees—the many more formal books that exist—and helps seeing the forest
as a whole! . . . I have been practicing QFT during the past two decades and with all my
experience I was thrilled with enjoyment when I read some of the sections.”
—Joshua Feinberg
“This text not only teaches up-to-date quantum field theory, but also tells readers how
research is actually done and shows them how to think about physics. [It teaches things
that] people usually say ‘cannot be learned from books.’ [It is] in the same style as Fearful
Symmetry and Einstein’s Universe. All three books . . . are classics.”
—Yu Shi
“I belong to the [group of ] enthusiastic laymen having enough curiosity and insistence . . .
but lacking the mastery of advanced math and physics. . . . I really could not see the forest
for the trees. But at long last I got this book!”
—Makay Attila
“More fun than any other QFT book I have read. The comparisons to Feynman’s
writings made by several of the reviewers seem quite apt. . . . His enthusiasm is quite
infectious. . . . I doubt that any other book will spark your interest like this one does.”
—Stephen Wandzura
“I’m having a blast reading this book. It’s both deep and entertaining; this is a rare breed,
indeed. I usually prefer the more formal style (big Landau fan), but I have to say that when
Zee has the talent to present things his way, it’s a definite plus.”
—Pierre Jouvelot “Required reading for QFT: [it] heralds the introduction of a book on quantum field theory
that you can sit down and read. My professor’s lectures made much more sense as I
followed along in this book, because concepts were actually EXPLAINED, not just worked
out.”
—Alexander Scott
Not your father’s quantum field theory text: I particularly appreciate that things are
motivated physically before their mathematical articulation. . . . Most especially though,
the author’s ‘heuristic’ descriptions are the best I have read anywhere. From them alone
the essential ideas become crystal clear.”
—Dan Dill Q uantum Field Theory in a Nutshell This page intentionally left blank Q uantum Field Theory in a Nutshell
SECOND EDITION A. Zee P R I N C E T O N U N I V E R S I T Y P R E S S . P R I N C E T O N A N D O X F O R D Copyright © 2010 by Princeton University Press
Published by Princeton University Press, 41 William Street,
Princeton, New Jersey 08540
In the United Kingdom: Princeton University Press,
6 Oxford Street, Woodstock, Oxfordshire OX20 1TW
All Rights Reserved
Library of Congress Cataloging-in-Publication Data
Zee, A.
Quantum field theory in a nutshell / A. Zee.—2nd ed.
p. cm.
Includes bibliographical references and index.
ISBN 978-0-691-14034-6 (hardcover : alk. paper) 1. Quantum field theory. I. Title.
QC174.45.Z44 2010
2009015469
530.14 3—dc22
British Library Cataloging-in-Publication Data is available
This book has been composed in Scala LF with ZzTEX
by Princeton Editorial Associates, Inc., Scottsdale, Arizona
Printed on acid-free paper.
press.princeton.edu
Printed in the United States of America
10 9 8 7 6 5 4 3 2 1 To my parents,
who valued education above all else This page intentionally left blank Contents Preface to the First Edition xv Preface to the Second Edition xix Convention, Notation, and Units xxv I Part I: Motivation and Foundation
I.1
I.2
I.3
I.4
I.5
I.6
I.7
I.8
I.9
I.10
I.11
I.12 Who Needs It?
Path Integral Formulation of Quantum Physics
From Mattress to Field
From Field to Particle to Force
Coulomb and Newton: Repulsion and Attraction
Inverse Square Law and the Floating 3-Brane
Feynman Diagrams
Quantizing Canonically
Disturbing the Vacuum
Symmetry
Field Theory in Curved Spacetime
Field Theory Redux 3
7
17
26
32
40
43
61
70
76
81
88 II Part II: Dirac and the Spinor
II.1
II.2
II.3
II.4 The Dirac Equation
Quantizing the Dirac Field
Lorentz Group and Weyl Spinors
Spin-Statistics Connection 93
107
114
120 xii | Contents
II.5
II.6
II.7
II.8 Vacuum Energy, Grassmann Integrals, and Feynman Diagrams
for Fermions
Electron Scattering and Gauge Invariance
Diagrammatic Proof of Gauge Invariance
Photon-Electron Scattering and Crossing 123
132
144
152 III Part III: Renormalization and Gauge Invariance
III.1
III.2
III.3
III.4
III.5
III.6
III.7
III.8 Cutting Off Our Ignorance
Renormalizable versus Nonrenormalizable
Counterterms and Physical Perturbation Theory
Gauge Invariance: A Photon Can Find No Rest
Field Theory without Relativity
The Magnetic Moment of the Electron
Polarizing the Vacuum and Renormalizing the Charge
Becoming Imaginary and Conserving Probability 161
169
173
182
190
194
200
207 IV Part IV: Symmetry and Symmetry Breaking
IV.1
IV.2
IV.3
IV.4
IV.5
IV.6
IV.7 Symmetry Breaking
The Pion as a Nambu-Goldstone Boson
Effective Potential
Magnetic Monopole
Nonabelian Gauge Theory
The Anderson-Higgs Mechanism
Chiral Anomaly 223
231
237
245
253
263
270 V Part V: Field Theory and Collective Phenomena
V.1
V.2
V.3
V.4
V.5
V.6
V.7 Superfluids
Euclid, Boltzmann, Hawking, and Field Theory at Finite Temperature
Landau-Ginzburg Theory of Critical Phenomena
Superconductivity
Peierls Instability
Solitons
Vortices, Monopoles, and Instantons 283
287
292
295
298
302
306 VI Part VI: Field Theory and Condensed Matter
VI.1
VI.2 Fractional Statistics, Chern-Simons Term, and Topological
Field Theory
Quantum Hall Fluids 315
322 Contents | xiii
VI.3
VI.4
VI.5
VI.6
VI.7
VI.8 Duality
The σ Models as Effective Field Theories
Ferromagnets and Antiferromagnets
Surface Growth and Field Theory
Disorder: Replicas and Grassmannian Symmetry
Renormalization Group Flow as a Natural Concept in High Energy
and Condensed Matter Physics 331
340
344
347
350 356 VII Part VII: Grand Unification
VII.1
VII.2
VII.3
VII.4
VII.5
VII.6
VII.7 Quantizing Yang-Mills Theory and Lattice Gauge Theory
Electroweak Unification
Quantum Chromodynamics
Large N Expansion
Grand Unification
Protons Are Not Forever
SO(10) Unification 371
379
385
394
407
413
421 VIII Part VIII: Gravity and Beyond
VIII.1 Gravity as a Field Theory and the Kaluza-Klein Picture
VIII.2 The Cosmological Constant Problem and the Cosmic Coincidence
Problems
VIII.3 Effective Field Theory Approach to Understanding Nature
VIII.4 Supersymmetry: A Very Brief Introduction
VIII.5 A Glimpse of String Theory as a 2-Dimensional Field Theory 433 Closing Words 473 448
452
461
469 N Part N
N.1
N.2
N.3
N.4 Gravitational Waves and Effective Field Theory
Gluon Scattering in Pure Yang-Mills Theory
Subterranean Connections in Gauge Theories
Is Einstein Gravity Secretly the Square of Yang-Mills Theory? 479
483
497
513 More Closing Words 521 Appendix A: Gaussian Integration and the Central Identity of Quantum
Field Theory 523 Appendix B: A Brief Review of Group Theory 525 xiv | Contents
Appendix C: Feynman Rules 534 Appendix D: Various Identities and Feynman Integrals 538 Appendix E: Dotted and Undotted Indices and the Majorana Spinor 541 Solutions to Selected Exercises 545 Further Reading 559 Index 563 Preface to the First Edition As a student, I was rearing at the bit, after a course on quantum mechanics, to learn
quantum field theory, but the books on the subject all seemed so formidable. Fortunately,
I came across a little book by Mandl on field theory, which gave me a taste of the subject
enabling me to go on and tackle the more substantive texts. I have since learned that other
physicists of my generation had similar good experiences with Mandl.
In the last three decades or so, quantum field theory has veritably exploded and Mandl
would be hopelessly out of date to recommend to a student now. Thus I thought of writing
a book on the essentials of modern quantum field theory addressed to the bright and eager
student who has just completed a course on quantum mechanics and who is impatient to
start tackling quantum field theory.
I envisaged a relatively thin book, thin at least in comparison with the many weighty
tomes on the subject. I envisaged the style to be breezy and colloquial, and the choice
of topics to be idiosyncratic, certainly not encyclopedic. I envisaged having many short
chapters, keeping each chapter “bite-sized.”
The challenge in writing this book is to keep it thin and accessible while at the same
time introducing as many modern topics as possible. A tough balancing act! In the end,
I had to be unrepentantly idiosyncratic in what I chose to cover. Note to the prospective
book reviewer: You can always criticize the book for leaving out your favorite topics. I do
not apologize in any way, shape, or form. My motto in this regard (and in life as well),
taken from the Ricky Nelson song “Garden Party,” is “You can’t please everyone so you
gotta please yourself.”
This book differs from other quantum field theory books that have come out in recent
years in several respects.
I want to get across the important point that the usefulness of quantum field theory is far
from limited to high energy physics, a misleading impression my generation of theoretical
physicists were inculcated with and which amazingly enough some recent textbooks on xvi | Preface to the First Edition
quantum field theory (all written by high energy physicists) continue to foster. For instance,
the study of driven surface growth provides a particularly clear, transparent, and physical
example of the importance of the renormalization group in quantum field theory. Instead
of being entangled in all sorts of conceptual irrelevancies such as divergences, we have
the obviously physical notion of changing the ruler used to measure the fluctuating
surface. Other examples include random matrix theory and Chern-Simons gauge theory
in quantum Hall fluids. I hope that condensed matter theory students will find this book
helpful in getting a first taste of quantum field theory. The book is divided into eight parts,1
with two devoted more or less exclusively to condensed matter physics.
I try to give the reader at least a brief glimpse into contemporary developments, for
example, just enough of a taste of string theory to whet the appetite. This book is perhaps
also exceptional in incorporating gravity from the beginning. Some topics are treated quite
differently than in traditional texts. I introduce the Faddeev-Popov method to quantize
electromagnetism and the language of differential forms to develop Yang-Mills theory, for
example.
The emphasis is resoundingly on the conceptual rather than the computational. The
only calculation I carry out in all its gory details is that of the magnetic moment of the
electron. Throughout, specific examples rather than heavy abstract formalism will be
favored. Instead of dealing with the most general case, I always opt for the simplest.
I had to struggle constantly between clarity and wordiness. In trying to anticipate and to
minimize what would confuse the reader, I often find that I have to belabor certain points
more than what I would like.
I tried to avoid the dreaded phrase “It can be shown that . . . ” as much as possible.
Otherwise, I could have written a much thinner book than this! There are indeed thinner
books on quantum field theory: I looked at a couple and discovered that they hardly explain
anything. I must confess that I have an almost insatiable desire to explain.
As the manuscript grew, the list of topics that I reluctantly had to drop also kept growing.
So many beautiful results, but so little space! It almost makes me ill to think about all the
stuff (bosonization, instanton, conformal field theory, etc., etc.) I had to leave out. As one
colleague remarked, the nutshell is turning into a coconut shell!
Shelley Glashow once described the genesis of physical theories: “Tapestries are made
by many artisans working together. The contributions of separate workers cannot be
discerned in the completed work, and the loose and false threads have been covered over.” I
regret that other than giving a few tidbits here and there I could not go into the fascinating
history of quantum field theory, with all its defeats and triumphs. On those occasions
when I refer to original papers I suffer from that disconcerting quirk of human psychology
of tending to favor my own more than decorum might have allowed. I certainly did not
attempt a true bibliography. 1 Murray Gell-Mann used to talk about the eightfold way to wisdom and salvation in Buddhism (M. Gell-Mann
and Y. Ne’eman, The Eightfold Way). Readers familiar with contemporary Chinese literature would know that the
celestial dragon has eight parts. Preface to the First Edition | xvii
The genesis of this book goes back to the quantum field theory course I taught as a
beginning assistant professor at Princeton University. I had the enormous good fortune
of having Ed Witten as my teaching assistant and grader. Ed produced lucidly written
solutions to the homework problems I assigned, to the extent that the next year I went
to the chairman to ask “What is wrong with the TA I have this year? He is not half as
good as the guy last year!” Some colleagues asked me to write up my notes for a much
needed text (those were the exciting times when gauge theories, asymptotic freedom,
and scores of topics not to be found in any texts all had to be learned somehow) but a
wiser senior colleague convinced me that it might spell disaster for my research career.
Decades later, the time has come. I particularly thank Murph Goldberger for urging me
to turn what expository talents I have from writing popular books to writing textbooks. It
is also a pleasure to say a word in memory of the late Sam Treiman, teacher, colleague,
and collaborator, who as a member of the editorial board of Princeton University Press
persuaded me to commit to this project. I regret that my slow pace in finishing the book
depri...

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