CHERN-SIMONS STATE FOR THE NONDIAGONAL BIANCHI TYPE IX MODEL - ART. NO. 083501

The Bianchi type IX mixmaster model is quantized in its nondiagonal fo rm, imposing spatial diffeomorphism, time reparametrization, and Loren tz invariance as constraints on physical state vectors before gauge fi xing. The result turns out to be different from quantizing the diagona l model obtained by gauge fixing already on the classical level. For t he nondiagonal model a generalized nine-dimensional Fourier transforma tion over a suitably chosen manifold connects the representations in m etric variables and in Ashtekar variables. A space of five states in t he metric representation is generated from the single physical Chem-Si mons state in Ashtekar variables by choosing five different integratio n 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 type IX model to the non diagonal case. It is shown that additional discrete (permutation) symm etries of physical states arise in the quantization of the nondiagonal model, which are satisfied by two of the five states connected to the Chern-Simons state. These have the characteristics of a wormhole grou nd state and a Hartle-Hawking ''no-boundary'' state, respectively. We also exhibit a special gauge fixing of the time reparametrization inva riance of the quantized system and define an associated manifestly pos itive scalar product. Then the wormhole ground state is left as the on ly normalizable physical state connected to the Chern-Simons state. [S 0556-2821(98)04316-1].