By Steven Carlip
Publisher: Cambridge University Press
Print Publication Year: 1998
Online Publication Date:December 2009
Online ISBN:9780511564192
Hardback ISBN:9780521564083
Paperback ISBN:9780521545884
Chapter DOI: http://dx.doi.org/10.1017/CBO9780511564192.009
Subjects: Cosmology, Relativity and Gravitation, Theoretical Physics and Mathematical Physics
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The approaches to quantization described in chapters 5–7, although quite different, share one common feature. They are all ‘reduced phase space’ quantizations, quantum theories based on the true physical degrees of freedom of the classical theory.
As we saw in chapter 2, not all of the degrees of freedom that determine the metric in general relativity have physical significance; many are ‘pure gauge’, describing coordinate choices rather than dynamics, and can be eliminated by solving the constraints and factoring out the diffeomorphisms. Indeed, we have seen that in 2+1 dimensions only a finite number of the ‘6 × ∞3’ metric degrees of freedom are physical. In each of the preceding approaches to quantization, our first step was to eliminate the nonphysical degrees of freedom, sometimes explicitly and sometimes indirectly through a clever choice of variables; only then were the remaining degrees of freedom quantized.
An alternative approach, originally developed by Dirac, is to quantize the entire space of degrees of freedom of classical theory, and only then to impose the constraints. In Dirac quantization, states are initially determined from the full classical phase space; in quantum gravity, for instance, they are functionals ψ[gij] of the full spatial metric. The constraints act as operators on this auxiliary Hilbert space, and the physical Hilbert space consists of those states that are annihilated by the constraints, acted on by physical operators that commute with the constraints.
pp. i-vi
pp. vii-x
pp. xi-xiv
1 - Why (2+1)-dimensional gravity? : Read PDF
pp. 1-8
2 - Classical general relativity in 2+1 dimensions : Read PDF
pp. 9-37
3 - A field guide to the (2+1)-dimensional spacetimes : Read PDF
pp. 38-59
4 - Geometric structures and Chern–Simons theory : Read PDF
pp. 60-86
5 - Canonical quantization in reduced phase space : Read PDF
pp. 87-99
6 - The connection representation : Read PDF
pp. 100-116
7 - Operator algebras and loops : Read PDF
pp. 117-130
8 - The Wheeler–DeWitt equation : Read PDF
pp. 131-142
9 - Lorentzian path integrals : Read PDF
pp. 143-162
10 - Euclidean path integrals and quantum cosmology : Read PDF
pp. 163-170
11 - Lattice methods : Read PDF
pp. 171-193
12 - The (2+1)-dimensional black hole : Read PDF
pp. 194-211
pp. 212-216
Appendix A - The topology of manifolds : Read PDF
pp. 217-235
Appendix B - Lorentzian metrics and causal structure : Read PDF
pp. 236-242
Appendix C - Differential geometry and fiber bundles : Read PDF
pp. 243-249
pp. 250-266
pp. 267-276