Superselection sectors in the Ashtekar-Horowitz-Boulware model

TL;DR

This paper explores the algebraic quantization of a simplified constrained system, revealing how superselection sectors relate to phase space singularities and potential-specific irreducibility, with implications for tunnelling phenomena.

Contribution

It provides a detailed analysis of superselection sectors and the structure of the physical Hilbert space in a simplified quantum gravity model.

Findings

Physical Hilbert space dimension is finite and relates to phase space volume.

Representation of observable algebra is irreducible for generic potentials.

Superselection sectors correspond to phase space singularities and potential-specific features.

Abstract

We investigate refined algebraic quantisation of the constrained Hamiltonian system introduced by Boulware as a simplified version of the Ashtekar-Horowitz model. The dimension of the physical Hilbert space is finite and asymptotes in the semiclassical limit to 1/(2\pi\hbar) times the volume of the reduced phase space. 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. There is no tunnelling into the classically forbidden region of the unreduced configuration space, but there can be tunnelling between disconnected components of the classically allowed region.