The Chern-Simons state for the non-diagonal Bianchi IX model

Details The Chern-Simons state for the non-diagonal Bianchi IX model The Chern-Simons state for the non-diagonal Bianchi IX model

1998-02-03

arxiv.org/abs/gr-qc/9802009v1

Phys.Rev. D58 (1998) 083501

Astronomy and Astrophysics

Astrophysics

General Relativity and Quantum Cosmology

29 pages, Revtex, 1 figure included, submitted to Phys. Rev. D

Scientific paper

10.1103/PhysRevD.58.083501

The Bianchi IX mixmaster model is quantized in its non-diagonal form, imposing spatial diffeomorphism, time reparametrization and Lorentz invariance as constraints on physical state vectors before gauge-fixing. The result turns out to be different from quantizing the diagonal model obtained by gauge-fixing already on the classical level. For the non-diagonal model a generalized 9-dimensional Fourier transformation over a suitably chosen manifold connects the representations in metric variables and in Ashtekar variables. A space of five states in the metric representation is generated from the single physical Chern-Simons state in Ashtekar variables by choosing five different integration manifolds, which cannot be deformed into each other. For the case of a positive cosmological constant $\Lambda$ we extend our previous study of these five states for the diagonal Bianchi IX model to the non-diagonal case. It is shown that additional discrete (permutation) symmetries of physical states arise in the quantization of the non-diagonal model, which are satisfied by two of the five states connected to the Chern-Simons state. These have the characteristics of a wormhole groundstate and a Hartle-Hawking `no-boundary' state, respectively. We also exhibit a special gauge-fixing of the time reparametrization invariance of the quantized system and define an associated manifestly positive scalar product. Then the wormhole ground state is left as the only normalizable physical state connected to the Chern-Simons state.

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