Group averaging and the Ashtekar-Horowitz model

We investigate refined algebraic quantisation of the constrained Hamiltonian system known as the Ashtekar-Horowitz model. We study two versions of this model which are defined on a two-torus and on a cylinder, respectively. The dimension of the physical Hilbert space depends on the topological structure of the model. In particular, we see that for the compact version of the model the representation of the physical observable algebra is irreducible for generic potentials but decomposes into irreducible subrepresentations for certain special potentials. The superselection sectors are related to singularities in the reduced phase space and to the rate of divergence in the formal group averaging integral. For both versions, there is no tunnelling into the classically forbidden region of the unreduced configuration space.

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