Refined Algebraic Quantization in the oscillator representation of SL(2,R)

TL;DR

This paper explores Refined Algebraic Quantization with group averaging for a constrained system with SL(2,R), revealing a complex phase space structure and connecting classical and quantum approaches.

Contribution

It demonstrates how RAQ with group averaging can be applied to a system with a non-manifold phase space, connecting classical and quantum quantization methods.

Findings

Decomposition of physical Hilbert space into four sectors

Identification of natural subalgebras of observable algebra

Implementation of group averaging in oscillator representation

Abstract

We investigate Refined Algebraic Quantization (RAQ) with group averaging in a constrained Hamiltonian system with unreduced phase space T^*R^4 and gauge group SL(2,R). The reduced phase space M is connected and contains four mutually disconnected `regular' sectors with topology R x S^1, but these sectors are connected to each other through an exceptional set where M is not a manifold and where M has non-Hausdorff topology. The RAQ physical Hilbert space H_{phys} decomposes as H_{phys} = (direct sum of) H_i, where the four subspaces H_i naturally correspond to the four regular sectors of M. The RAQ observable algebra A_{obs}, represented on H_{phys}, contains natural subalgebras represented on each H_i. The group averaging takes place in the oscillator representation of SL(2,R) on L^2(R^{2,2}), and ensuring convergence requires a subtle choice for the test state space: the classical analogue of this choice is to excise from M the exceptional set while nevertheless retaining information about the connections between the regular sectors. A quantum theory with the Hilbert space H_{phys} and a finitely-generated observable subalgebra of A_{obs} is recovered through both Ashtekar's Algebraic Quantization and Isham's group theoretic quantization.