Path integrals and anomalies in curved space

398 Pages·2006·1.899 MB·English

by  Bastianelli, Fiorenzo, Van Nieuwenhuizen, Peter

Path Integrals and Anomalies in Curved Space
Path integrals provide a powerful method for describing quantum phenomena, first
introduced in physics by Dirac and Feynman. This book introduces the quantum
mechanics of particles that move in curved space by employing the path integral
method, and uses this formalism to compute anomalies in quantum field theories.
The authors start by deriving path integrals for particles moving in curved space
(one-dimensional nonlinear sigma models), and their supersymmetric generalizations.
Coherent states are used for fermionic particles. They then discuss the regularization
and renormalization schemes essential to constructing and computing these path inte-
grals.
In the second part of the book, the authors apply these methods to discuss and
calculateanomaliesinquantumfieldtheories,withexternalgravitationaland/or(non)
abelian gauge fields. Anomalies constitute one of the most important aspects of quan-
tum field theory; requiring that there are no anomalies is an enormous constraint in
the search for physical theories of elementary particles, quantum gravity and string
theories. In particular, the authors include explicit calculations of the gravitational
anomalies, reviewing the seminal work of Alvarez-Gaum´e and Witten in an original
way, and their own work on trace anomalies.
Fiorenzo Bastianelli is Professor of Theoretical Physics at the Department
of Physics, University of Bologna. Following his Ph.D., from SUNY at Stony Brook in
NewYork,hehasworkedatITP,StockholmUniversity,andattheNielsBohrInstitute,
University of Copenhagen. In 1995 he moved to Italy, initially at the University of
Modena, and since 1999 at the University of Bologna. His research interests include
quantum field theory and string theory.
Peter van Nieuwenhuizen is a Distinguished Professor at the C. N. Yang
Institute of SUNY, Stony Brook. After his Ph.D. from Utrecht University he worked
at CERN (Geneva), the Ecole Normale Superieure (Paris) and Brandeis University
(Mass.) In 1975 he moved to Stony Brook, where he co-discovered supergravity in
1976,forwhichhereceivedtheDiracawardin1993,andtheHeinemannprizein2005.
Hisresearchinterestsareinsupersymmetry,supergravity,quantumfieldtheory,modern
general relativity and string theory, about which he has published over 300 articles in
scientific journals.
CAMBRIDGE MONOGRAPHS ON
MATHEMATICAL PHYSICS
General editors: P. V. Landshoff, D. R. Nelson, S. Weinberg
S.J.AarsethGravitationalN-BodySimulations
J.Ambj/orn,B.DurhuusandT.JonssonQuantumGeometry:AStatisticalFieldTheory
Approach
A.M.AnileRelativisticFluidsandMagneto-Fluids
J.A.deAzc´arrageandJ.M.IzquierdoLieGroups,LieAlgebras,CohomologyandSome
ApplicationsinPhysics†
O.Babelon,D.BernardandM.TalonIntroductiontoClassicalIntegrableSystems
F.BastianelliandP.vanNieuwenhuizenPathIntegralsandAnomaliesinCurvedSpace
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J.BernsteinKineticTheoryintheExpandingUniverse
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N.D.BirrellandP.C.W.DaviesQuantumFieldsinCurvedSpace
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V.N.GribovTheTheoryofComplexAngularMomenta
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Path Integrals and Anomalies in
Curved Space
FIORENZO BASTIANELLI
Universita` degli Studi,
Bologna, Italy
PETER VAN NIEUWENHUIZEN
State University of New York at Stony Brook
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© Cambridge University Press 2006
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without the written permission of Cambridge University Press.
First published in print format 2006
isbn-13 978-0-511-22112-5 eBook (NetLibrary)
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guaranteethatanycontentonsuchwebsitesis,orwillremain,accurateorappropriate.
To Bryce S. DeWitt (1923–2004)
who pioneered quantum mechanics in curved space

Contents
Preface xi
Part I Path integrals for quantum mechanics in
curved space 1
1 Introduction to path integrals 3
1.1 The simplest case: a particle in flat space 5
1.2 Quantum mechanical path integrals in curved space
require regularization 9
1.3 Power counting and divergences 18
1.4 Brief summary of the three regularization schemes 23
1.5 A brief history of path integrals 27
2 Time slicing 33
2.1 Configuration-space path integrals for bosons from
time slicing 34
2.2 The phase-space path integral and Matthews’ theorem 60
2.3 Path integrals for Dirac fermions 73
2.4 Path integrals for Majorana fermions 82
2.5 Direct evaluation of the transition element to order β 86
2.6 Two-loop path integral evaluation of the transition
element to order β 98
3 Mode regularization 108
3.1 Mode regularization in configuration space 109
3.2 The two-loop amplitude and the counterterm V 116
MR
3.3 Calculation of Feynman graphs in mode regularization 123
vii
viii Contents
4 Dimensional regularization 127
4.1 Dimensional regularization in configuration space 128
4.2 Two-loop transition amplitude and the counterterm V 133
DR
4.3 Calculation of Feynman graphs in dimensional
regularization 134
4.4 Path integrals for fermions 136
Part II Applications to anomalies 145
5 Introduction to anomalies 147
5.1 The simplest case: anomalies in two dimensions 149
5.2 The Fujikawa method 163
5.3 How to calculate anomalies using quantum mechanics 167
5.4 A brief history of anomalies 177
6 Chiral anomalies from susy quantum
mechanics 185
6.1 The abelian chiral anomaly for spin-1 fields coupled to
2
gravity in 4k dimensions 185
6.2 The abelian chiral anomaly for spin-1 fields coupled to
2
the Yang–Mills fields in 2k dimensions 200
6.3 Lorentz anomalies for chiral spin-1 fields coupled to
2
gravity in 4k+2 dimensions 211
6.4 Mixed Lorentz and non-abelian gauge anomalies for
chiral spin-1 fields coupled to gravity and Yang–Mills
2
fields in 2k dimensions 219
6.5 The abelian chiral anomaly for spin-3 fields coupled to
2
gravity in 4k dimensions 223
6.6 Lorentz anomalies for chiral spin-3 fields coupled to
2
gravity in 4k+2 dimensions 231
6.7 Lorentz anomalies for self-dual antisymmetric tensor fields
coupled to gravity in 4k+2 dimensions 237
6.8 Cancellation of gravitational anomalies in IIB
supergravity 249
6.9 Cancellation of anomalies in N = 1 supergravity 252
6.10 The SO(16)×SO(16) string 273
6.11 Index theorems and path integrals 277
7 Trace anomalies from ordinary and susy
quantum mechanics 285
7.1 Trace anomalies for scalar fields in two and
four dimensions 286
Contents ix
7.2 Trace anomalies for spin-1 fields in two and
2
four dimensions 293
7.3 Trace anomalies for a vector field in four dimensions 298
7.4 String-inspired approach to trace anomalies 303
8 Conclusions and summary 312
Appendices 320
A Riemann curvatures 320
B Weyl ordering of bosonic operators 325
C Weyl ordering of fermionic operators 331
D Nonlinear susy sigma models and d = 1 superspace 336
E Nonlinear susy sigma models for internal symmetries 348
F Gauge anomalies for exceptional groups 352
References 366
Index 377