Table of Contents
Preface v
Acknowledgments vii
1 Historical overview 1
1.1 Early years 1
1.2 DcWitt's era 1
1.3 Loop quantum gravity 4
1.4 The standard model 5
1.5 GUTs, super gravity and superstrings 7
1.6 Recent developments and future prospects 9
2 Gravitons 13
2.1 The linear field equations 13
2.1.1 The relativistic spin-2 field equation 13
2.1.2 Linearizing Einstein's equations 16
2.1.3 Plane waves 17
2.1.4 Quantization 20
2.1.5 Spin projectors 21
2.2 Four derivative theories 26
2.2.1 Actions 26
2.2.2 Linearized equations 28
2.3 Power counting 30
2.4 Appendix: Topological invariants 33
3 Failure of renormalizability 35
3.1 Divergences in curved spacetime: Scalar field 35
3.1.1 The Euclidean functional integral 36
3.1.2 Functional determinant 37
3.1.3 Zeta function regularization 38
3.1.4 The heat kernel 40
3.2 Generalizations 43
3.2.1 The master formula 43
3.2.2 Effective potential 44
3.2.3 Yang-Mills theory 46
3.2.4 Other fields in an external metric 48
3.3 The gravitational path integral 52
3.4 Perturbations around a general background 55
3.5 One-loop divergences in quantum GR 60
3.6 Two loop divergences in quantum GR 64
3.7 Appendix: Calculations of heat kernel coefficients 66
3.7.1 Terms due to a potential 66
3.7.2 Terms due to a connection 66
3.7.3 Terms due to an external metric 68
4 Other perturbative approaches 71
4.1 Options 71
4.2 Emergent gravity 72
4.3 Higher derivative gravities 73
4.4 Special matter choice: Supergravity 75
4.5 The Effective Field Theory approach 77
4.5.1 The general idea 77
4.5.2 Example: Chiral perturbation theory 79
4.5.3 Gravity 81
4.5.4 The leading corrections to the Newtonian potential 83
5 Technical developments 89
5.1 York decomposition 89
5.2 The Wick rotation revisited 93
5.3 Some Laplace-type operators 98
5.4 Quantum GR in general gauges 100
5.4.1 York-decomposed Hessian 100
5.4.2 The conformal factor problem 102
5.4.3 Gauge fixed Hessian 104
5.4.4 Gauge invariance of the one-loop effective action on-shell 107
5.4.5 Physical gauge 109
5.4.6 Exponential parametrization 110
5.5 Spectral geometry of differentially constrained fields 112
5.5.1 Lichnerowicz Laplacians 113
5.5.2 Boclmer Laplacians 115
5.6 Heat kernels of maximally symmetric spaces 116
5.6.1 Sphere 117
5.6.2 Hyperboloid 121
5.7 Formula for functional traces 123
5.8 Appendix: The Euler-Maclaurin formula 126
6 The functional renormalization group equation 129
6.1 The Effective Average Action (EAA) 129
6.2 The Wetterich equation 132
6.3 Beta functions 134
6.4 Scalar potential interactions 137
6.5 Flow equation for Yang-Mills theory 141
6.5.1 The background ERGE 141
6.5.2 One-loop beta function 143
6.6 Gaussian matter fields coupled to an external metric 146
6.6.1 Type II cutoff 147
6.6.2 Type I cutoff 149
6.6.3 Spectral sum for the Dirac operator 151
6.7 The ERGE for gravity 153
6.8 The Einstein-Hilbert truncation 156
6.8.1 Cutoff of type Ila 157
6.8.2 Cutoff of type la 159
6.8.3 Cutoff of type IIb 161
6.5.1 Cutoff of type Ib 162
6.8.1 Spectrally adjusted cutoff 163
6.9 Appendix: Evaluation of some Q-functionals 166
7 The gravitational fixed point 169
7.1 Non-perturbative renormalizability 169
7.2 Approximation schemes for gravity 174
7.24 Truncations 174
7.2.2 The level expansion 175
7.3 The single-metric Einstein-Hilbert truncation at one loop 177
7.3.1 Without cosmological constant 177
7.3.2 Expanding in the cosmological constant 179
7.3.3 Gauge dependence 182
7.4 Higher derivative gravity at one loop 188
7.4.1 Expansion of the action 188
7.4.2 Gauge fixing 191
7.4.3 Derivation of beta functions 192
7.4.4 Beta functions in d = 4 195
7.5 Topologicallly massive gravity at one loop 198
7.5.1 The quadratic action, gauge fixing and cutoff 199
7.5.2 Evaluation of the beta functions 202
7.5.3 Ascending root cutoff 208
7.5.4 Descending root cutoff 210
7.5.5 Spectrally balanced cutoff 212
7.5.6 Summary 213
7.6 A first peek beyond one loop 215
7.6.1 The background anomalous dimension 215
7.6.2 Level-two Einstein-Hilbert truncation 217
7.6.3 The anomalous dimensions ηh and ηC 218
7.7 Truncation to polynomials in R 223
7.7.1 Hessian and gauge choice 224
7.7.2 Inserting into the ERGE 225
7.7.3 Polynomial truncation 227
7.8 Effect of matter 230
7.8.1 Single-metric, one-loop results 231
7.8.2 Inclusion of anomalous dimensions 234
7.8.3 Fixed points 238
8 The asymptotic safety programme 247
8.1 Overview of the literature 247
8.1.1 Pre-ERGE work 247
8.1.2 The first ten years of ERGE 248
8.1.3 The subsequent ten years 250
8.2 State of the art 256
8.2.1 Finite dimensional truncations 256
8.2.2 Functional truncations 259
8.2.3 Bi-field truncations and the shift Ward identity 260
8.3 Outlook 262
Appendix A Appendix 267
A.1 Units 267
A.2 Notations 268
A.2.1 Conventions 268
A.2.2 Acronyms 270
Bibliography 271
Subject Index 292
Author Index 298
