Lectures on Non-Perturbative Canonical Gravity

The following sections are included:

• CONTENTS

• PREFACE

• Acknowledgements

• LIST OF SYMBOLS

• CONVENTIONS

A fundamental premise of these lecture notes is that gravitational physics is intimately intertwined with the geometry of space and time. In the classical regime, general relativity captures this idea in an elegant way. Indeed, it is the identification of the gravitational field with the curvature of space-time that has led to the most dramatic predictions of general relativity, such as the existence of black holes and the occurrence of the big bang. This leads one to the viewpoint that a primary goal of any quantum theory of gravity should be to probe what one may call, in absence of a better terminology, the quantum structure of space and time…

The following sections are included:

• Introduction

• Hamiltonian framework

• Quantum theory
  • The program
  • The connection representation
  • The loop representation

• Outlook

• References

In this Part, we shall present in detail the introduction of the new canonical variables for classical general relativity, possibly coupled with matter. The new configuration variable is a complex-valued SO(3) connection . In any solution to the field equations, is a potential for the self dual part of the Weyl curvature. The canonically conjugate momentum, which would be the electric field in Yang-Mills theory, now has the interpretation of a (density weighted) triad…

The following sections are included:

• Introduction

• Palatini action

• Self dual action

• References

The following sections are included:

• Introduction

• Palatini action

• Self dual action

• References

In this chapter, I will show how one can go back and forth between the triads used in chapters 3 and 4 and the SU(2) spinors which will be used in the remaining chapters in this Part. This chapter is “practically oriented” in the sense that the discussion will be focussed on calculations. A conceptual introduction to SU(2) as well as SL(2, ℂ) spinors is given in appendix A…

The following sections are included:

• Introduction

• Kinematical arena

• Extended phase space

• New variables

• References

The following sections are included:

• Introduction

• Constraint algebra

• Dynamics and Hamiltonians

• Discussion

• References

The following sections are included:

• Introduction

• Polynomial form

• References

The following sections are included:

• Introduction

• Self dual action and reality of equations

• Euler-Lagrange equations

• (3+1)-decomposition

• Constraint algebra

• Hamiltonians and reality conditions

• Polynomial matter actions

• Discussion

• References

In this part, we shall use the Hamiltonian framework of Part II as the point of departure for canonical quantization of general relativity.

In chapter 11, we present the general quantization program. The overall procedure is the same as the one proposed by Dirac for quantization of constrained systems. However, the program goes beyond the Dirac theory in that it provides a strategy for finding the inner product on the space of physical states: Choose that inner product with respect to which the operators corresponding to any two classical variables which are complex conjugates of one another become Hermitian adjoints. Thus, far from being a nuisance, the classical “reality condition” are to be thought of as the vehicles that are lead us to the correct inner product. At a certain level, the strategy is an obvious one. However, it is not the one that is generally employed. In Minkowskian field theories, for example, the inner product is chosen by making an appeal to Poincaré invariance. We implement the quantization program in a number of examples with and without constraints and show that it does indeed succeed. Although these examples have only a finite number of degrees of freedom, they all mimick different features of general relativity and are also of considerable interest in their own right. In particular, we show how one can unambiguously quantize a harmonic oscillator in the z, q variables which are analogous to the pair , of general relativity and how the program enables one to quantize parametrized particle systems without having to first single out time…

The following sections are included:

• Introduction

• Simple systems

• (q, z) variables for the oscillator

• Constraints

• Constrained systems: Examples
  • Non-relativistic parametrized particle
  • Relativistic free particle
  • Two oscillators constrained to have a fixed energy difference

• General relativity

• References

The following sections are included:

• Introduction

• Canonical framework

• Bargmann representation

• Mathematical digression

• Self dual representation

• References

The following sections are included:

• Introduction

• Connection representation

• Problem of time

• Two strategies

• Truncated theory

• References

The following sections are included:

• Introduction

• CP problem in Yang-Mills Theory

• CP problem in quantum gravity

• Discussion

• References

The following sections are included:

• Introduction

• Rovelli-Smolin transform

• Loop quantization

• References

The following sections are included:

• Introduction

• Classical loop variables

• The small τ algebra

• The full τ algebra

• Properties of T variables

• References

The following sections are included:

• Introduction

• Quantization ignoring constraints
  • Algebra of quantum operators
  • Representation of the CCR
  • Representation of A
  • Higher order operators
  • Smearing in the quantum theory

• Quantum constraints
  • The diffeomorphism constraint
  • Diffeomorphism invariant states
  • The scalar constraint
  • Solutions to the scalar constraint

• Discussion

• References

The following sections are included:

• Introduction

• Canonical formulation of 2 + 1 gravity

• Quantum theory
  • Connection representation
  • Loop Representation
  • Illustration

• Discussion

• References

The following sections are included:

• Appendix A: SPINORS

• Appendix B: SYMPLECTIC FRAMEWORK

• Appendix C: QUANTUM MECHANICS ON MANIFOLDS

• Appendix D: QUANTIZATION OF CONSTRAINED SYSTEMS

• BIBLIOGRAPHY