Table Of ContentSPRINGER BRIEFS IN PHYSICS
James M. Cline
Advanced
Concepts in
Quantum
Field Theory
With Exercises
123
SpringerBriefs in Physics
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James M. Cline
Advanced Concepts
in Quantum Field Theory
With Exercises
123
James M.Cline
Department ofPhysics
McGill University
Montreal,QC, Canada
ISSN 2191-5423 ISSN 2191-5431 (electronic)
SpringerBriefs inPhysics
ISBN978-3-030-56167-3 ISBN978-3-030-56168-0 (eBook)
https://doi.org/10.1007/978-3-030-56168-0
©TheAuthor(s),underexclusivelicensetoSpringerNatureSwitzerlandAG2020
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Foreword
QuantumFieldTheory(QFT)isthelanguageinwhichNature’sLawsattheirmost
fundamentallevelarewritten,andthismakesitasimportantasubjectforascientist
to learn as is the learning of English for a student of Shakespeare or French to a
student of Voltaire. This is why many good QFT texts have been written to help
novices find their way around what is a vast and rich subject.
But real fluency is required to appreciate masterpieces written in any language,
anditisequallytruethatafullappreciationofNature’sLawsrequiresgoingbeyond
an introductory treatment of QFT methods. Indeed, there is so much accumulated
wisdom about QFT that learningit is alifelong task that never reallyends.
Thatiswherethisbookcomesin.Thisbookisbasedongraduatelecturesgiven
at McGill University by the author, who is a world expert in the subject. It goes
beyondtheusualintroductorytopicsfoundinmanytextbooksanddivesdeeperinto
important grottos of knowledge—such as the nitty-gritty of renormalization and a
discussion of non-perturbative issues like tunnelling and theta vacua innonabelian
gaugetheories.Thesetopicshavebothpracticalandconceptualvalueandareoften
not discussed in introductory textbooks.
Mostimportantly,thebookischock-filledwithusefulexercises.Simplyreading
abookaboutasubjectasvastasthisdoesn’treallymoveanyone’sknowledgeofit
beyond a level good only for cocktail-party conversation. Developing a practi-
tioner’sunderstandingrequiresusingQFTtoolstocalculatesomething,butfinding
a calculation that is both instructive and not insuperably difficult is hard, even for
oldhands.Abookasloadedwithinsightasthisoneisagifttothereader.Butwhen
it is also crammed full of well-chosen exercises, it becomes a gold mine, particu-
larlywhentheexerciseshavebeentestedanddebuggedbygenerationsofstudents
in the forge of their own learning.
This book is an invitation to an intellectual feast. It comes with many courses
and is prepared by a five-star chef. Bon appetit!
Hamilton, ON, USA Clifford P. Burgess
June 2020
v
Preface
These notes represent the second half of a year-long quantum field theory course
thatwasgivenatMcGillUniversity.Itassumesthereaderunderstandsthebasicsof
fieldtheoryatthetreelevel.Istartwiththeloopexpansioninscalarfieldtheoryto
illustrate the procedure of renormalization, and then extend this to QED and other
gauge theories. My goal is to introduce the most important concepts and devel-
opmentsinQFT,withoutnecessarilytreatingthemallindepth,butallowingyouto
learn the basic ideas. The topics to be covered include:
(cid:129) Perturbation theory: the loop expansion; regularization; dimensional regular-
ization; Wick rotation; momentum cutoff; k/4 theory; renormalization; renor-
malization group equation; Wilsonian viewpoint; the epsilon expansion;
relevant, irrelevant and marginal operators; Callan-Symanzik equation; running
couplings; beta function; anomalous dimensions; IR and UV fixed points;
asymptotic freedom; triviality; Landau pole
(cid:129) The effective action: generating functional; connected diagrams;
one-particle-irreducible diagrams; Legendre transform
(cid:129) Gauge theories: QED; QCD; anomalies; gauge invariance and unitarity; gauge
fixing; Faddeev-Popov procedure; ghosts; unitary gauge; covariant gauges;
Ward Identities; BRS transformation; vacuum structure of QCD; instantons;
tunnelling; theta vacuua; superselection sectors; the strong CP problem.
IwouldliketothankBalasubramanianAnanthanarayan(IISc,Bangalore)forhis
encouragement to publish these notes.
Montreal, Canada James M. Cline
vii
Contents
1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
2 The Loop Expansion. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7
3 The Feynman Rules . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11
4 Evaluation of Diagrams; Regularization . . . . . . . . . . . . . . . . . . . . . 13
5 Renormalization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17
6 Running Couplings and the Renormalization Group . . . . . . . . . . . 29
7 Other Regulators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37
8 Fixed Points and Asymptotic Freedom . . . . . . . . . . . . . . . . . . . . . . 45
9 The Quantum Effective Action . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51
10 Fermions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65
11 The Axial Anomaly. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79
12 Abelian Gauge Theories: QED . . . . . . . . . . . . . . . . . . . . . . . . . . . . 87
12.1 Applications of QED . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 100
13 Nonabelian Gauge Theories . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 107
13.1 Group Theory for SU(N). . . . . . . . . . . . . . . . . . . . . . . . . . . . . 107
13.2 Yang-Mills Lagrangian with Fermions . . . . . . . . . . . . . . . . . . . 109
13.3 Unitarity of the S-Matrix. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113
13.4 Loop Structure of Non Abelian Gauge Theory . . . . . . . . . . . . . 117
13.5 Beta Function of Yang-Mills Theory . . . . . . . . . . . . . . . . . . . . 121
13.6 Heuristic Explanation of Asymptotic Freedom . . . . . . . . . . . . . 124
ix
x Contents
14 Nonperturbative Aspects of SU(N) Gauge Theory. . . . . . . . . . . . . . 131
14.1 Instantons. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 133
14.2 Winding Number . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 137
14.3 Tunneling Between n-Vacuua . . . . . . . . . . . . . . . . . . . . . . . . . 140
14.4 Theta Vacua. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 142
14.5 Physical Significance of h . . . . . . . . . . . . . . . . . . . . . . . . . . . . 143
15 Exercises. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 145
References.... .... .... .... ..... .... .... .... .... .... ..... .... 153
Chapter 1
Introduction
Physics,likeallsciences,isbaseduponexperimentalobservations.It’sthereforea
goodthingtoremindourselves:whatarethemajorexperimentalobservablesrelevant
forparticlephysics?Thesearethemasses,lifetimes,andscatteringcrosssectionsof
particles.Polesofpropagators,andscatteringanddecayamplitudesarethequantities
whicharerelatedtotheseobservables:
1
poleof ↔(mass)2, (1.1)
p2−m2
Ta→bc ↔decayrate, (1.2)
Tab→cd ↔scatteringcrosssection. (1.3)
Theseobservables,whicharecomponentsoftheS-matrix(scatteringmatrix)arethe
maingoalsofcomputationinquantumfieldtheory.Tobeprecise,T isthetransition
matrix,whichisrelatedtotheS-matrixby
S =1 +(2π)4iδ(4)(p − p )T . (1.4)
fi fi f i fi
InpreviouscourseyoulearnedabouttheconnectionbetweenGreen’sfunctions
andamplitudes.Therecipe,knownastheLSZreductionprocedure(afterLehmann,
SzymanzikandZimmerman)[1,2],isthefollowing.Foraphysicalprocessinvolving
nincomingandm outgoingparticles,computethecorrespondingGreen’sfunction.
Let’sconsiderascalarfieldtheoryforsimplicity:
G(n+m)(x1,...,xn,y1,...,ym)=(cid:4)0out|T∗[φ(x1)···φ(xn)φ(y1)...φ(ym)]|0in(cid:6).
(1.5)
The blob represents the possibly complicated physics occurring in the scattering
region, while the lines represent the free propagation of the particles as they are
travelingtoorfromthescatteringregion.ItisusefultogotoFourierspace:
©TheAuthor(s),underexclusivelicensetoSpringerNatureSwitzerlandAG2020 1
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