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

TL;DR

This paper quantizes the non-diagonal Bianchi IX cosmological model, revealing new discrete symmetries, multiple physical states derived from a single Chern-Simons state, and identifying a unique normalizable wormhole ground state.

Contribution

It introduces a quantization approach for the non-diagonal Bianchi IX model, uncovering new symmetries and physical states, and extends previous diagonal model results to the non-diagonal case.

Findings

Five distinct physical states generated from the Chern-Simons state.

Identification of discrete permutation symmetries in the non-diagonal model.

The wormhole ground state is the only normalizable state connected to the Chern-Simons state.

Abstract

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.