Refined algebraic quantisation in a system with nonconstant gauge invariant structure functions

TL;DR

This paper extends refined algebraic quantisation to systems with multiple constraints and nonconstant gauge structure functions, including cases with nonunimodular gauge algebras, advancing the quantisation of complex constrained systems.

Contribution

It develops the first refined algebraic quantisation approach for systems with multiple constraints and nonconstant structure functions, including nonunimodular gauge algebras.

Findings

Successfully quantised systems with abelian and nonunimodular gauge algebras.

Handled open algebra cases with non-gauge invariant structure functions.

Provided explicit rescaling functions for constraints in complex gauge structures.

Abstract

In a previous work [J. Math. Phys. 52, 123504 (2011)], refined algebraic quantisation (RAQ) within a family of classically equivalent constrained Hamiltonian systems that are related to each other by rescaling one momentum-type constraint was investigated. In the present work, the first steps to generalise this analysis to cases where more constraints occur are developed. The system under consideration contains two momentum-type constraints, originally abelian, where rescalings of these constraints by a non-vanishing function of the coordinates are allowed. These rescalings induce structure functions at the level of the gauge algebra. Providing a specific parametrised family of real-valued scaling functions, the implementation of the corresponding rescaled quantum momentum-type constraints is performed using RAQ when the gauge algebra: (i) remains abelian and (ii) undergoes into an algebra of a nonunimodular group with nonconstant gauge invariant structure functions. Case (ii) becomes the first example known to the author where an open algebra is handled in refined algebraic quantisation. Challenging issues that arise in the presence of non-gauge invariant structure functions are also addressed.