Group averaging in the (p,q) oscillator representation of SL(2,R)

TL;DR

This paper explores group averaging in a simplified model of general relativity with SL(2,R) gauge symmetry, revealing new insights into the structure of the physical Hilbert space and observable algebra in quantum constrained systems.

Contribution

It demonstrates the application of refined algebraic quantisation with group averaging in a finite-dimensional model, highlighting cases with nontrivial observable representations and convergence to indefinite forms.

Findings

Physical Hilbert space with nontrivial o(p,q) representation for p>1, q>1

First example of group averaging converging to an indefinite sesquilinear form

Analysis of the (p,q) oscillator representation in quantum gravity models

Abstract

We investigate refined algebraic quantisation with group averaging in a finite-dimensional constrained Hamiltonian system that provides a simplified model of general relativity. The classical theory has gauge group SL(2,R) and a distinguished o(p,q) observable algebra. The gauge group of the quantum theory is the double cover of SL(2,R), and its representation on the auxiliary Hilbert space is isomorphic to the (p,q) oscillator representation. When p>1, q>1 and p+q == 0 (mod 2), we obtain a physical Hilbert space with a nontrivial representation of the o(p,q) quantum observable algebra. For p=q=1, the system provides the first example known to us where group averaging converges to an indefinite sesquilinear form.