  We investigate refined algebraic quantisation within a family of classically
equivalent constrained Hamiltonian systems that are related to each other by
rescaling a momentum-type constraint. The quantum constraint is implemented by
a rigging map that is motivated by group averaging but has a resolution finer
than what can be peeled off from the formally divergent contributions to the
averaging integral. Three cases emerge, depending on the asymptotics of the
rescaling 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. Prospects of generalising the analysis
to systems with several constraints are discussed.
