Quantum gravity

374 Pages·2007·2.424 MB·English

by  Claus Kiefer

INTERNATIONAL SERIES
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S. F. EDWARDS UNIVERSITY OF CAMBRIDGE
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M. REES UNIVERSITY OF CAMBRIDGE
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International Series of Monographs on Physics
136. C.Kiefer:Quantum gravity, Second edition
135. V.Fortov,I.Iakubov,A.Khrapak:Physics of strongly coupled plasma
134. G.Fredrickson:The equilibrium theory of inhomogeneous polymers
133. H.Suhl:Relaxation processes in micromagnetics
132. J.Terning:Modern supersymmetry
131. M.Marin˜o:Chern-Simons theory, matrix models, and topological strings
130. V.Gantmakher:Electrons and disorder in solids
129. W.Barford:Electronic and optical properties of conjugated polymers
128. R.E.Raab,O.L.deLange:Multipole theory in electromagnetism
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125. S.Atzeni,J.Meyer-ter-Vehn:The physics of inertial fusion
123. T.Fujimoto:Plasma spectroscopy
122. K.Fujikawa,H.Suzuki:Path integrals and quantum anomalies
121. T.Giamarchi:Quantum physics in one dimension
120. M.Warner,E.Terentjev:Liquid crystal elastomers
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117. G.Volovik:The Universe in a helium droplet
116. L.Pitaevskii,S.Stringari:Bose-Einstein condensation
115. G.Dissertori,I.G.Knowles,M.Schmelling:Quantum chromodynamics
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111. H.Nishimori:Statistical physics of spin glasses and information processing - an
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108. R.Dobbs:Helium three
107. R.Wigmans:Calorimetry
106. J.Ku¨bler:Theory of itinerant electron magnetism
105. Y.Kuramoto,Y.Kitaoka:Dynamics of heavy electrons
104. D.Bardin,G.Passarino:The Standard Model in the making
103. G.C.Branco,L.Lavoura,J.P.Silva:CP Violation
102. T.C.Choy: Effective medium theory
101. H.Araki:Mathematical theory of quantum fields
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99. L.Mestel:Stellar magnetism
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94. S.Chikazumi:Physics of ferromagnetism
91. R.A.Bertlmann:Anomalies in quantum field theory
90. P.K.Gosh:Ion traps
88. S.L.Adler:Quaternionic quantum mechanics and quantum fields
87. P.S.Joshi:Global aspects in gravitation and cosmology
86. E.R.Pike,S.Sarkar:The quantum theory of radiation
83. P.G.deGennes,J.Prost:The physics of liquid crystals
82. B.H.Bransden,M.R.C.McDowell: Charge exchange and the theory of ion-atom
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73. M.Doi,S.F.Edwards:The theory of polymer dynamics
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69. S.Chandrasekhar:The mathematical theory of black holes
51. C.Møller:The theory of relativity
46. H.E.Stanley:Introduction to phase transitions and critical phenomena
32. A.Abragam:Principles of nuclear magnetism
27. P.A.M.Dirac:Principles of quantum mechanics
23. R.E.Peierls:Quantum theory of solids
Quantum Gravity
Second Edition
CLAUS KIEFER
Institute for Theoretical Physics
University of Cologne
1
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ISBN 978–0–19-921252-1(Hbk)
1 3 5 7 9 10 8 6 4 2
PREFACE TO THE SECOND EDITION
The construction of a consistent theory of quantum gravity continues to be the
major open problem in fundamental physics. The present second edition of my
book is an extended and revised version of the first edition. It contains, in par-
ticular, new sections on asymptotic safety, dynamical triangulation, primordial
black holes, the information-loss problem, loop quantum cosmology, and other
topics. The text has been revised throughout.
I thank all the readers who have commented on various parts of the first
edition or who have pointed out the occurrence of typos. Comments would be
greatly appreciated also for the second edition. They can be sent to my e-mail
address [email protected].
Forhelpfuldiscussionsandcriticalcommentswhilepreparingthesecondedi-
tionIamgratefultoMarkAlbers,AndreiBarvinsky,MartinBojowald,Friedrich
Hehl, Gerhard Kolland, Renate Loll, Hermann Nicolai, Martin Reuter, Barbara
Sandho¨fer, Stefan Theisen, and H.-Dieter Zeh.
LastbutnotleastIwanttothankOxfordUniversityPress,andinparticular
Sonke Adlung, for their efficient cooperation.
Cologne Claus Kiefer
December 2006
v
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PREFACE
The unification ofquantum theorywith Einstein’s theoryof generalrelativity is
perhapsthebiggestopenproblemoftheoreticalphysics.Suchatheoryisnotonly
needed for conceptual reasons, but also for the understanding of fundamental
issues such as the origin of the Universe, the final evaporation of black holes,
and the structure of space and time.
Historically, the oldest approach is the direct quantization of Einstein’s the-
ory of general relativity, an approach which is still being actively pursued. This
includes covariant methods such as path-integral quantization as well as canon-
ical methods like the Wheeler–DeWitt approach or the more recent loop quan-
tization. Although one arrives at a perturbatively non-renormalizable theory,
quantum general relativity can yield physically interesting results in both the
perturbative and the non-perturbative regimes. It casts light, in particular, on
the fundamental nature of space and time.
The secondmain approachis string theory. It encapsulates the idea that the
problem of constructing a viable quantum theory of gravity can only be solved
withinaunificationofallinteractions.Inthisrespect,itgoesfarbeyondquantum
general relativity. From a methodological point of view, however, string theory
does not stand much apart from it. It is a natural extension of perturbative
quantumgravity(fromwhichitinheritstheconceptofagraviton),andmethods
of constrained quantization, which are crucial for canonical quantum gravity,
appear at key stages in the theory.
Whereas there exist excellent textbooks that discuss string theory at great
depth, the present monograph is the first that, to my knowledge, covers quan-
tumgravityinthisbroadsense.Themainpartofthe bookis devotedtogeneral
concepts, the quantization of general relativity, and applications to cosmology
and black holes. String theory is discussed from the point of view of its quan-
tum gravitational aspects and its connection to other approaches. The edifice
of theoretical physics cannot be completed without the conceptual unification
that will be provided by quantum gravity. I hope that my book will convince
my readers of this outstanding problem and encourage them to work on its
solution.
This bookhasgrownoutoflecturesthatI gaveatthe UniversitiesofZu¨rich,
Freiburg, and Cologne between 1990 and 2003. My main intention is to discuss
the general features that a quantum theory of gravity is expected to show and
to give an up-to-date overview of the main approaches. The reader is assumed
to have some familiarity with generalrelativity and quantum field theory.Com-
ments can be sent to my e-mail address [email protected] and are
highly welcome.
vii
viii Preface
It is clear that my book could not have been written in this form without
the influence of many people overthe past 20 years.I am in particular indebted
to H.-Dieter Zeh for encouragingme to enter this field of researchand for many
stimulatingandinspiringinteractions.IalsothankNorbertStraumanninZu¨rich
and Hartmann Ro¨mer in Freiburg for providing me with the excellent working
conditions that gave me the freedom to follow the research I wanted to do.
Many people have read throughdrafts of this book, and their criticalcomments
have helped me to improve various parts. I owe thanks to Julian Barbour, An-
drei Barvinsky, Domenico Giulini, Alexander Kamenshchik, Thomas Mohaupt,
PauloMoniz,AndreasRathke,ThomasThiemann,andH.-DieterZeh.Iamalso
deeply indebted to them and also, for discussions over many years, to Mariusz
Da¸browski,LajosDio´si,PetrHa´j´ıˇcek,ErichJoos,JormaLouko,DavidPolarski,
T. P. Singh, Alexei Starobinsky, and Andreas Wipf. I have actively collabo-
rated with most of the aforementioned theoreticians and I want to take this
opportunityto thank them forthe pleasurethat I couldexperience during these
collaborations.
Cologne Claus Kiefer
January 2004
CONTENTS
1 Why quantum gravity? 1
1.1 Quantum theory and the gravitationalfield 1
1.1.1 Introduction 1
1.1.2 Main motivations for quantizing gravity 3
1.1.3 Relevant scales 4
1.1.4 Quantum mechanics and Newtonian gravity 7
1.1.5 Quantum field theory in curved space–time 14
1.2 Problems of a fundamentally semiclassical theory 15
1.3 Approaches to quantum gravity 22
2 Covariant approaches to quantum gravity 25
2.1 The concept of a graviton 25
2.1.1 Weak gravitationalwaves 25
2.1.2 Gravitons from representations of the Poincar´e
group 30
2.1.3 Quantization of the linear field theory 33
2.2 Path-integralquantization 39
2.2.1 General properties of path integrals 39
2.2.2 The perturbative non-renormalizability 42
2.2.3 Effective action and Feynman rules 47
2.2.4 Semiclassical Einstein equations 55
2.2.5 Asymptotic safety 63
2.2.6 Regge calculus and dynamical triangulation 66
2.3 Quantum supergravity 70
3 Parametrized and relational systems 73
3.1 Particle systems 73
3.1.1 Parametrized non-relativistic particle 73
3.1.2 Some remarks on constrained systems 76
3.1.3 The relativistic particle 78
3.2 The free bosonic string 81
3.3 Parametrizedfield theories 86
3.4 Relational dynamical systems 93
4 Hamiltonian formulation of general relativity 98
4.1 The seventh route to geometrodynamics 98
4.1.1 Principle of path independence 98
4.1.2 Explicit form of generators 100
4.1.3 Geometrodynamics and gauge theories 103
4.2 The 3+1 decomposition of general relativity 105
4.2.1 The canonical variables 106
ix