Feynman Path Integral Formulation

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Contents xvii 7 Analytical Lattice Expansion Methods ............................225 7.1 Motivation ................................................225 7.2 Lattice Weak Field Expansion and Transverse-Traceless Modes . . . . 225 7.3 Lattice Diffeomorphism Invariance . . . . . . ......................234 7.4 Strong Coupling Expansion . .................................242 7.5 Gravitational Wilson Loop . . .................................248 7.6 Discrete Gravity in the Large-d Limit..........................260 7.7 Mean Field Theory . ........................................269 8 Numerical Studies ..............................................273 8.1 Nonperturbative Gravity .....................................273 8.2 Observables, Phase Structure and Critical Exponents . . ...........274 8.3 Invariant Local Gravitational Averages . . . ......................276 8.4 Invariant Correlations at Fixed Geodesic Distance . . . . ...........278 8.5 WilsonLinesandStaticPotential .............................281 8.6 ScalingintheVicinityoftheCriticalPoint .....................284 8.7 Physical and Unphysical Phases . .............................285 8.8 Numerical Determination of the Scaling Exponents . . . ...........290 8.9 Renormalization Group and Lattice Continuum Limit . ...........295 8.10 Curvature Scales . . . ........................................300 8.11 Gravitational Condensate . . . .................................302 9 Scale Dependent Gravitational Couplings .........................305 9.1 Renormalization Group and Scale Dependence of G .............305 9.2 EffectiveFieldEquations ...................................305 9.3 Poisson’s Equation and Vacuum Polarization Cloud . . . ...........308 9.4 StaticIsotropicSolution.....................................311 9.5 CosmologicalSolutions .....................................314 9.6 Quantum Gravity and Mach’s Principle . . ......................321 References .........................................................325 Index .............................................................335
Page 2: Quantum Gravitation
Page 8: Prof. Dr. Herbert W. Hamber Univers
Page 14: Preface Almost a century has gone b
Page 18: Preface ix mately leading to the Ei
Page 22: Preface xi thereof) have meaning an
Page 26: Acknowledgements Over the years I h
Page 32: xvi Contents 4 Hamiltonian and Whee
Page 38: 2 1 Continuum Formulation +c∂ ν
Page 42: 4 1 Continuum Formulation which, by
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Page 66: 16 1 Continuum Formulation 1.5 One-
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Page 74: 20 1 Continuum Formulation terms of
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26 1 Continuum Formulation The Weyl
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28 1 Continuum Formulation and the
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30 1 Continuum Formulation consiste
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32 1 Continuum Formulation ∂ω =
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34 1 Continuum Formulation under th
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36 1 Continuum Formulation {Q i α,
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38 1 Continuum Formulation the grav
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40 1 Continuum Formulation β 0 =
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42 1 Continuum Formulation Fig. 1.3
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44 1 Continuum Formulation Similarl
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46 1 Continuum Formulation It is po
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48 1 Continuum Formulation ultravio
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50 1 Continuum Formulation The supe
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52 1 Continuum Formulation allowed
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54 1 Continuum Formulation date no
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56 2 Feynman Path Integral Formulat
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68 3 Gravity in 2 + ε Dimensions L
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70 3 Gravity in 2 + ε Dimensions
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74 3 Gravity in 2 + ε Dimensions g
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76 3 Gravity in 2 + ε Dimensions F
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78 3 Gravity in 2 + ε Dimensions T
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84 3 Gravity in 2 + ε Dimensions I
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86 3 Gravity in 2 + ε Dimensions a
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88 3 Gravity in 2 + ε Dimensions (
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90 3 Gravity in 2 + ε Dimensions T
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92 3 Gravity in 2 + ε Dimensions o
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94 3 Gravity in 2 + ε Dimensions n
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98 3 Gravity in 2 + ε Dimensions N
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Chapter 4 Hamiltonian and Wheeler-D
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4.2 First Order Formulation 105 ṗ
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4.3 Arnowitt-Deser-Misner (ADM) For
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4.3 Arnowitt-Deser-Misner (ADM) For
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4.5 Intrinsic and Extrinsic Curvatu
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4.6 Matter Source Terms 113 One sti
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4.8 Semiclassical Expansion of the
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4.9 Connection with the Feynman Pat
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4.10 Minisuperspace 119 is the inve
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4.10 Minisuperspace 121 H = p a ȧ
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4.11 Solution of Simple Minisupersp
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4.11 Solution of Simple Minisupersp
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4.12 Quantum Hamiltonian for Gauge
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4.13 Lattice Regularized Hamiltonia
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4.14 Lattice Hamiltonian for Quantu
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Chapter 5 Semiclassical Gravity 5.1
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5.1 Cosmological Wavefunctions 143
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5.1 Cosmological Wavefunctions 145
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5.2 Semiclassical Expansion 147 wit
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5.2 Semiclassical Expansion 149 log
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5.2 Semiclassical Expansion 151 ∫
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5.3 Pair Creation in Constant Elect
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5.4 Black Hole Particle Emission 15
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5.4 Black Hole Particle Emission 15
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5.5 Method of In and Out Vacua 159
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5.5 Method of In and Out Vacua 161
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5.6 Complex Periodic Time 163 5.6 C
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5.6 Complex Periodic Time 165 with
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5.8 Quantum Gravity Corrections 167
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Chapter 6 Lattice Regularized Quant
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6.3 Volumes and Angles 171 Fig. 6.2
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6.4 Rotations, Parallel Transports
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6.5 Invariant Lattice Action 179 6.
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6.5 Invariant Lattice Action 181 Th
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6.5 Invariant Lattice Action 183 wh
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6.6 Lattice Diffeomorphism Invarian
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6.7 Lattice Bianchi Identities 187
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6.8 Gravitational Wilson Loop 189 w
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6.9 Lattice Regularized Path Integr
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6.9 Lattice Regularized Path Integr
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6.10 An Elementary Example 195 ∫
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6.10 An Elementary Example 197 wher
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6.11 Lattice Higher Derivative Term
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6.11 Lattice Higher Derivative Term
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6.12 Scalar Matter Fields 203 if an
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6.12 Scalar Matter Fields 205 A ij
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6.12 Scalar Matter Fields 207 defin
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6.13 Invariance Properties of the S
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6.14 Lattice Fermions, Tetrads and
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6.15 Gauge Fields 213 6.15 Gauge Fi
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6.16 Lattice Gravitino 215 and invo
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6.17 Alternate Discrete Formulation
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6.17 Alternate Discrete Formulation
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6.18 Lattice Invariance versus Cont
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6.18 Lattice Invariance versus Cont
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Chapter 7 Analytical Lattice Expans
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7.2 Lattice Weak Field Expansion an
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7.4 Strong Coupling Expansion 243 w
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7.4 Strong Coupling Expansion 245 A
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7.4 Strong Coupling Expansion 247
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7.5 Gravitational Wilson Loop 249 G
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7.5 Gravitational Wilson Loop 251 F
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7.5 Gravitational Wilson Loop 253 I
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7.5 Gravitational Wilson Loop 255 c
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7.5 Gravitational Wilson Loop 257 I
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7.5 Gravitational Wilson Loop 259 T
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7.6 Discrete Gravity in the Large-d
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+O( 1 d 2 ) . (7.141) 7.6 Discrete
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7.7 Mean Field Theory 269 ξ ∼
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7.7 Mean Field Theory 271 The secon
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274 8 Numerical Studies are not aff
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276 8 Numerical Studies 8.3 Invaria
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278 8 Numerical Studies ) Z latt (
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280 8 Numerical Studies Fig. 8.1 Ge
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282 8 Numerical Studies ∫ τ(b)
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284 8 Numerical Studies task, since
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286 8 Numerical Studies important o
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288 8 Numerical Studies Fig. 8.5 A
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290 8 Numerical Studies As a conseq
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292 8 Numerical Studies the scaling
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294 8 Numerical Studies [ χ R (k,L
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296 8 Numerical Studies 10 8 6 1Ν
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298 8 Numerical Studies ξ ξ ξ Fi
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300 8 Numerical Studies guide, the
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302 8 Numerical Studies value for
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Chapter 9 Scale Dependent Gravitati
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9.2 Effective Field Equations 307 (
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9.3 Poisson’s Equation and Vacuum
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9.4 Static Isotropic Solution 311 3
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9.4 Static Isotropic Solution 313 w
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9.5 Cosmological Solutions 315 equa
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9.5 Cosmological Solutions 317 whic
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9.5 Cosmological Solutions 319 One
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9.6 Quantum Gravity and Mach’s Pr
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9.6 Quantum Gravity and Mach’s Pr
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326 References Bern, Z., J. J. Carr
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328 References Fröhlich, J., 1981,
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330 References Kuchař, K., 1992,
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332 References Smolin, L., 2003,
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Index 1/N expansion, 82 2 + ε expa
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Index 337 effective action, 152, 22
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Index 339 lattice supermetric, 135
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Index 341 scalar-graviton vertex, 1
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Feynman Path Integral Formulation

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