Rescaling constraints, BRST methods, and refined algebraic quantisation

We investigate the canonical BRST–quantisation and refined algebraic quantisation within a family of classically equivalent constrained Hamiltonian systems that are related to each other by rescaling constraints with nonconstant functions on the configuration space. The quantum constraints are implemented by a rigging map that is motivated by a BRST version of group averaging. Two systems are considered. In the first one we avoid topological built–in complications by considering R^4 as phase space, on which a single constraint, linear in momentum is defined and rescaled. Here, the rigging map has a resolution finer than what can be extracted from the formally divergent contributions to the group averaging integral. Three cases emerge, depending on the asymptotics of the scaling function: (i) quantisation is equivalent to that with identity scaling; (ii) quantisation fails, owing to nonexistence of self–adjoint extensions of the constraint operator; (iii) a quantisation ambiguity arises from the self–adjoint extension of the constraint operator, and the resolution of this purely quantum mechanical ambiguity determines the superselection structure of the physical Hilbert space. The second system we consider is a generalisation of the aforementioned model, two constraints linear in momenta are defined on the phase space R^6 and their rescalings are analysed. With a suitable choice of a parametric family of scaling functions, we turn the unscaled abelian gauge algebra either into an algebra of constraints that (1) keeps the abelian property, or, (2) has a nonunimodular behaviour with gauge invariant structure functions, or, (3) contains structure functions depending on the full configuration space. For cases (1) and (2), we show that the BRST version of group averaging defines a proper rigging map in refined algebraic quantisation. In particular, quantisation case (2) becomes the first example known to the author where structure functions in the algebra of constraints are successfully handled in refined algebraic quantisation. Prospects of generalising the analysis to case (3) are discussed