Quantum Cosmology and Baby Universes

The following sections are included:

• PREFACE

• CONTENTS

The following sections are included:

• INTRODUCTION

• HAMILTONIAN FORMULATION OF GAUGE THEORIES (PRE-BRST)

• BRST HAMILTONIAN FORMULATION OF GAUGE THEORIES

• DYNAMICS OF GRAVITATIONAL FIELD

• DOES THE HAMILTONIAN VANISH? GENERAL COVARIANCE AS AN “ORDINARY” GAUGE INVARIANCE
  • GENERALLY COVARIANT SYSTEMS
  • TIME AS A CANONICAL VARIABLE. ZERO HAMILTONIAN
    • Parametrized Systems
    • Zero Hamiltonian
    • Parametrization and Explicit Time Dependence
  • TIME REPARAMETRIZATION INVARIANCE
    • Form of Gauge Transformations
    • Must the Hamiltonian be Zero for a Generally Covariant System?
    • Simple Example of a Generally Covariant System with a Nonzero Hamiltonian
  • “TRUE DYNAMICS” VERSUS GAUGE TRANSFORMATIONS
    • Interpretation of the Formalism
    • Reduced Phase Space

• MUST TIME FLOW?
  • GAUGE INDEPENDENCE OF PATH INTEGRAL FOR A PARAMETRIZED SYSTEM ILLUSTRATED. EQUIVALENCE OF THE GAUGES t = τ AND t = 0
    • Reduced Phase Space Transition Amplitude as a Reduced Phase Space Path Integral
    • Canonical Gauge Conditions
    • Gauge t = 0
    • Gauge t α τ

• BRST CHARGE OF GRAVITATIONAL FIELD

• ELEMENTS OF BRST THEORY
  • THE GHOST, YOU'VE COME A LONG WAY BABY
    • Introduction
    • Quantum mechanics, the art of finding and combining simple elementary processes
    • Ghosts necessary to keep elementary processes simple
    • BRST symmetry: ghosts and matter become different components of single geometrical object
  • BRST SYMMETRY IN CLASSICAL MECHANICS
    • Ghosts have role in classical mechanics
    • Gauge invariance and constraints
    • Classical mechanics over Grassmann algebra necessary
    • Higher order structure functions
    • Rank defined. Open algebras
    • Ghosts. Ghost number. BRST generator as generating function for structure functions
    • A belianization of constraints. Existence of Ω
    • Uniqueness of Ω
    • Classical BRST cohomology
  • QUANTUM BRST THEORY
    • States and operators
    • Ghost number
    • BRST invariant states
    • Quantum BRST cohomology
    • Equivalence of the BRST physical subspace with the conventional gauge invariant subspace
    • Action principle
    • Path integral. B-F-V theorem
    • Contact with lagrangian notation
    • BRST invariant boundary conditions

• Acknowledgements

• References

The following sections are included:

• INTRODUCTION

• POST-EVERETT QUANTUM MECHANICS
  • Probability
    • Probabilities in general
    • Probabilities in Quantum Mechanics
  • Decoherent Histories
    • Fine and Coarse Grained Histories
    • Decohering Sets of Coarse Grained Histories
    • No Moment by Moment Definition of Decoherence
  • Prediction, Retrodiction, and History
    • Prediction and Retrodiction
    • The Reconstruction of History
  • Branches (Illustrated by a Pure ρ)
  • Sets of Histories with the Same Probabilities
  • The Origins of Decoherence in Our Universe
    • On What Does Decoherence Depend?
    • Two Slit Model
    • The Caldeira-Leggett Oscillator Model
    • The Evolution of Reduced Density Matrices
  • Towards a Classical Domain
  • The Branch Dependence of Decoherence
  • Measurement
  • The Ideal Measurement Model and the Copenhagen Approximation to Quantum Mechanics
  • Approximate Probabilities Again
  • Complex Adaptive Systems
  • Open Questions

• GENERALIZED QUANTUM MECHANICS
  • General Features
  • Hamiltonian Quantum Mechanics
  • Sum-Over-Histories Quantum Mechanics for Theories with a Time
  • Differences and Equivalences between Hamiltonian and Sum-Over-Histories Quantum Mechanics for Theories with a Time
  • Classical Physics and the Classical Limit of Quantum Mechanics
  • Generalizations of Hamiltonian Quantum Mechanics

• TIME IN QUANTUM MECHANICS
  • Observables on Spacetime Regions
  • The Arrow of Time in Quantum Mechanics
  • Topology in Time
  • The Generality of Sum Over Histories Quantum Mechanics

• THE QUANTUM MECHANICS OF SPACETIME
  • The Problem of Time
    • General Covariance and Time in Hamiltonian Quantum Mechanics
    • The “Marvelous Moment”
  • A Quantum Mechanics for Spacetime
    • What we Need
    • Sum-Over-Histories Quantum Mechanics for Theories Without a Time
    • Sum-Over-Spacetime-Histories Quantum Mechanics
    • Extensions and Contractions
  • The Construction of Sums Over Spacetime Histories
  • Some Open Questions

• PRACTICAL QUANTUM COSMOLOGY
  • The Semiclassical Regime
  • The Semiclassical Approximation to the Quantum Mechanics of a Non-Relativistic Particle
  • Semiclassical Prediction in Quantum Cosmology

• ACKNOWLEDGEMENTS

• REFERENCES

• APPENDIX: BUZZWORDS

The following sections are included:

• INTRODUCTION

• A SIMPLE EXAMPLE

• THE HAMILTONIAN FORMULATION OF GENERAL RELATIVITY

• QUANTIZATION

• MINISUPERSPACE – GENERAL THEORY

• CLASSICAL SPACETIME

• THE WKB APPROXIMATION

• BOUNDARY CONDITION PROPOSALS

• NO-BOUNDARY VS. TUNNELING

• BEYOND MINISUPERSPACE

• VACUUM STATES FROM QUANTUM COSMOLOGY

• SUMMARY

• ACKNOWLEDGEMENTS

• A GUIDE TO THE LITERATURE

• REFERENCES

One of the things I found most mysterious, when I first learnt quantum mechanics, was that one did not deal with the probability density, for finding an electron at the point x. Instead, one worked with a wave function, ψ(x), which was a complex square root of the probability density. Of course, the fact that one deals with amplitudes or wave functions, is fundamental to quantum mechanics, because it allows the possibility of interference: probabilities are necessarily real and positive, so they can only add up. But amplitudes or wave functions, can be negative or complex, so they can cancel each other. Nevertheless, it was not clear to me, why one could describe the state of an electron by a wave function, rather than just by a probability density…

The following sections are included:

• INTRODUCTION

• TOPOLOGY CHANGE AND THIRD QUANTIZATION IN 0+1 DIMENSIONS
  • Third Quantization of Free One-dimensional Universes
  • Third Quantization of Interacting One-Dimensional Universes
  • The Single-Universe Approximation and Dynamical Determination of Coupling Constants
  • The Third Quantized Uncertainty Principle

• THIRD QUANTIZATION IN 3+1 DIMENSIONS
  • The Gauge Invariant Action
  • Relation to Other Formalisms

• PARENT AND BABY UNIVERSES
  • The Hybrid Action
  • Baby Universe Field Operators and Spacetime Couplings

• INSTANTONS–FROM QUANTUM MECHANICS TO QUANTUM GRAVITY
  • Quantum Mechanics
  • Quantum Field Theory
  • Quantum Gravity
  • Axionic Instantons
  • The Small Expansion Parameter

• THE AXION MODEL AND THE INSTANTON APPROXIMATION

• THE COSMOLOGICAL CONSTANT
  • The Hawking-Baum Argument
  • Baby Universes and Coleman's Argument

• ACKNOWLEDGEMENTS

• REFERENCES

In these lectures I would like to review some of the criticisms to the Coleman worm–hole theory of the vanishing cosmological constant. In particular, I would like to focus on the most fundamental assumption that the path integral over topologies defines a probability for the cosmological constant which has the form EXP(A) with A being the Baum–Hawking–Coleman saddle point. Coleman argues that the euclideam path integral over all geometries may be dominated by special configurations which consist of large smooth “spheres” connected by any number of narrow wormholes. Formally summing up such configurations gives a very divergent expression for the path integral…