Refined Algebraic Quantization: Systems with a single constraint

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This paper explores in some detail a recent proposal (the Rieffel induction/refined algebraic quantization scheme) for the quantization of constrained gauge systems. Below, the focus is on systems with a single constraint and, in this context, on the uniqueness of the construction. While in general the results depend heavily on the choices made for certain auxiliary structures, an additional physical argument leads to a unique result for typical cases. We also discuss the `superselection laws' that result from this scheme and how their existence also depends on the choice of auxiliary structures. Again, when these structures are chosen in a physically motivated way, the resulting superselection laws are physically reasonable.