Book contents
- Frontmatter
- Contents
- Preface
- Acknowledgements
- 1 Introduction
- 2 Quantum cosmology
- 3 Hamiltonian supergravity and canonical quantization
- 4 The quantum amplitude
- 5 Supersymmetric mini-superspace models
- 6 Supersymmetric quantum wormhole states
- 7 Ashtekar variables
- 8 Further developments
- 9 Conclusion
- References
- Index
2 - Quantum cosmology
Published online by Cambridge University Press: 30 October 2009
- Frontmatter
- Contents
- Preface
- Acknowledgements
- 1 Introduction
- 2 Quantum cosmology
- 3 Hamiltonian supergravity and canonical quantization
- 4 The quantum amplitude
- 5 Supersymmetric mini-superspace models
- 6 Supersymmetric quantum wormhole states
- 7 Ashtekar variables
- 8 Further developments
- 9 Conclusion
- References
- Index
Summary
Introduction
Before embarking on the full theory of N = 1 supergravity in the following chapters, it is necessary to review some of what is known about quantum cosmology based on general relativity, possibly coupled to spin-0 or spin-1/2 (non-supersymmetric) matter. The ideas presented in this chapter, based to a considerable extent but not exclusively on Hamiltonian methods, will recur throughout the book. Perhaps the main underlying idea is that there is an analogy between the classical dynamics of a point particle with position x and that of a three-geometry hij(x). The theory of point-particle dynamics, when written in parametrized form [Kuchař 1981] and cast into Hamiltonian form, and the theory of general relativity, again in Hamiltonian form, bear a strong resemblance. In the Hamiltonian form of general relativity, hij(x) can be taken to be the ‘coordinate’ variable, corresponding to x in particle dynamics. In section 2.2, for parametrized particle dynamics, it is shown following [Kuchař 1981] how a constraint arises classically in the Hamiltonian theory, which, when quantized, gives the appropriate Schrödinger or wave equation for the quantum wave function ψ(x, t). As described in subsequent sections, the quantization of the analogous constraint in general relativity gives the Wheeler–DeWitt equation [DeWitt 1967, Wheeler 1968], a second-order functional differential equation for the wave function Ψ[hij(x)], which contains all the information in quantum gravity, if only one could solve and interpret it.
The Hamiltonian form of general relativity is derived from the Einstein–Hilbert Lagrangian in section 2.3.
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- Information
- Supersymmetric Quantum Cosmology , pp. 9 - 85Publisher: Cambridge University PressPrint publication year: 1996