Noncommutativity of constraining and quantizing: A U(1)-gauge model

After discussing the general question of (non)commutativity of constraining and quantizing, we present a quantum-mechanical model with a U(1)-gauge constraint. The phase spaces before and after the symplectic reduction are topologically nontrivial, and hence the usual canonical quantization has to be modified. We show that Isham quantization can be used in both cases. The groups governing the quantizations are $\mathrm{Sp}(4,\mathbf{R})\simeq \mathrm{SO}\mathrm{}(3,2)\mathrm{}$ for the unreduced and $\mathrm{Sp}(2,\mathbf{R})\simeq \mathrm{SO}\mathrm{}(2,1)\mathrm{}$ for the reduced theory. The quantizations are nonunique and the analysis of the Dirac condition depends on the quantization chosen. In most cases quantization and constraining do not commute; in particular, we may find additional observables if we quantize before reduction.