Refined algebraic quantisation with the triangular subgroup of SL(2,R)

TL;DR

This paper explores refined algebraic quantisation using group averaging for a constrained system with a gauge group related to SL(2,R), revealing the structure of the resulting quantum Hilbert space and its representations.

Contribution

It provides a detailed analysis of group averaging in a specific gauge group setting and compares it with algebraic quantisation, highlighting conditions for nontrivial quantum theories.

Findings

Hilbert space carries a maximally degenerate principal unitary series representation of O(p,q)

Quantum theory is trivial when (p,q) = (1,1), (1,2), (2,1), or when p>1, q>1 and p+q is odd

Group averaging yields a nontrivial representation only under specific conditions on p and q

Abstract

We investigate refined algebraic quantisation with group averaging in a constrained Hamiltonian system whose gauge group is the connected component of the lower triangular subgroup of SL(2,R). The unreduced phase space is T^*R^{p+q} with p>0 and q>0, and the system has a distinguished classical o(p,q) observable algebra. Group averaging with the geometric average of the right and left invariant measures, invariant under the group inverse, yields a Hilbert space that carries a maximally degenerate principal unitary series representation of O(p,q). The representation is nontrivial iff (p,q) is not (1,1), which is also the condition for the classical reduced phase space to be a symplectic manifold up to a singular subset of measure zero. We present a detailed comparison to an algebraic quantisation that imposes the constraints in the sense H_a Psi = 0 and postulates self-adjointness of the o(p,q) observables. Under certain technical assumptions that parallel those of the group averaging theory, this algebraic quantisation gives no quantum theory when (p,q) = (1,2) or (2,1), or when p>1, q>1 and p+q is odd.