Deparametrization and Path... | Buy

In this book, homogeneous cosmological models whose Hamilton -- Jacobi equation is separable are deparametrized by turning their action functional into that of an ordinary gauge system. Canonical gauges imposed on the gauge system are used to define global phase time in terms of the canonical variables of the minisuperspace. The procedure clearly shows how the geometry of the constraint surface restricts the choice of time. The consequences that this has for path integral quantization are discussed, and the transition amplitude is obtained from relativistic isotropic models, relativistic anisotropic models (Kantowski-Sachs and Taub) and isotropic string cosmologies.

Preface

vii

Introduction

1

(6)

The gravitational field as a constrained Hamiltonian system

7

(10)

Momentum and Hamiltonian constraints

7

(3)

Minisuperspaces as constrained systems

10

(3)

Quantization

13

(4)

Canonical quantization

13

(1)

Path integral quantization

14

(3)

Deparametrization and path integral quantization

17

(28)

The identification of time

17

(4)

Gauge fixation and deparametrization

17

(3)

Topology of the constraint surface: intrinsic and extrinsic time

20

(1)

Gauge-invariant action for a parametrized system

21

(7)

End point terms

22

(2)

Observables and time

24

(3)

Non separable constraints

27

(1)

Path integral

28

(5)

General formalism

28

(3)

The function f and the reduced Hamiltonian. Unitarity

31

(2)

Examples

33

(12)

Feynman propagator for the Klein-Gordon equation

33

(3)

The ideal clock

36

(4)

Transition probability for empty Friedmann-Robertson-Walker universes

40

(5)

Homogeneous relativistic cosmologies

45

(32)

Isotropic universes

45

(16)

A toy model

46

(2)

True degrees of freedom

48

(4)

A more general constraint

52

(6)

Extrinsic time. The closed ``de Sitter'' universe

58

(3)

Comment

61

(1)

Anisotropic universes

61

(16)

The Kantowski-Sachs universe

63

(5)

The Taub universe

68

(6)

Other anisotropic models

74

(3)

String cosmologies

77

(14)

String theory on background fields

77

(3)

String cosmological models

80

(2)

Path integral quantization

82

(9)

Gauge-invariant action

82

(2)

Extrinsic time

84

(2)

Intrinsic time and path integral

86

(2)

Summary

88

(3)

Canonical quantization

91

(14)

Approximate solutions of the Wheeler-DeWitt equation

92

(1)

Gauge fixation and Schrodinger equation for isotropic models

93

(1)

The Taub universe

94

(6)

Standard procedure

94

(1)

Boundary conditions and Schrodinger equation

95

(2)

Wheeler-DeWitt equation with extrinsic time

97

(3)

String cosmologies

100

(5)

Wheeler-DeWitt equation

100

(2)

Schrodinger equation

102

(3)

Discussion

105

(6)

Appendix A Constrained Hamiltonian systems

111

(6)

A.1 Hamiltonian formalism for constrained systems

111

(3)

A.2 Gauge transformations and gauge fixation

114

(3)

Appendix B Path integral and inner product

117

(6)

B.1 Ordinary mechanical systems

117

(1)

B.2 Constrained systems

118

(1)

B.3 Inner product for constrained systems

119

(2)

B.4 Propagator and projected kernel

121

(2)

Appendix C End point terms

123

(6)

Appendix D An extrinsic time for the Taub universe

129

(2)

Appendix E Free-particle constraint for minisuperspaces

131

(2)

Bibliography

133

(4)

Index

137

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