Advanced Concepts in Quantum Field Theory: With Exercises

158 Pages·2020·3.457 MB·English

by  James M. Cline

SPRINGER BRIEFS IN PHYSICS
James M. Cline
Advanced
Concepts in
Quantum
Field Theory
With Exercises
123
SpringerBriefs in Physics
Series Editors
BalasubramanianAnanthanarayan,CentreforHighEnergyPhysics,IndianInstitute
of Science, Bangalore, India
EgorBabaev,PhysicsDepartment,UniversityofMassachusettsAmherst,Amherst,
MA, USA
MalcolmBremer,HHWillsPhysicsLaboratory,UniversityofBristol,Bristol,UK
Xavier Calmet, Department of Physics and Astronomy, University of Sussex,
Brighton, UK
FrancescaDiLodovico,DepartmentofPhysics,QueenMaryUniversityofLondon,
London, UK
Pablo D. Esquinazi, Institute for Experimental Physics II, University of Leipzig,
Leipzig, Germany
Maarten Hoogerland, University of Auckland, Auckland, New Zealand
Eric Le Ru, School of Chemical and Physical Sciences, Victoria University
of Wellington, Kelburn, Wellington, New Zealand
Dario Narducci, University of Milano-Bicocca, Milan, Italy
James Overduin, Towson University, Towson, MD, USA
Vesselin Petkov, Montreal, QC, Canada
Stefan Theisen, Max-Planck-Institut für Gravitationsphysik, Golm, Germany
Charles H.-T. Wang, Department of Physics, The University of Aberdeen,
Aberdeen, UK
James D. Wells, Physics Department, University of Michigan, Ann Arbor, MI,
USA
Andrew Whitaker, Department of Physics and Astronomy, Queen's University
Belfast, Belfast, UK
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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
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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
J.M.Cline,AdvancedConceptsinQuantumFieldTheory,
SpringerBriefsinPhysics,https://doi.org/10.1007/978-3-030-56168-0_1