Arbitrary gauge quantization of light-matter theories with time-dependent constraints

We provide a general framework for the quantization of light-matter theories with time-dependent holonomic constraints. Unless time dependence is present from the outset at the Lagrangian level, different gauges generally produce nonequivalent canonical theories. The irrotational gauge is defined as the one that also yields a correct theory when time dependence is introduced at the Hamiltonian level. Our framework unifies examples of such gauges found in existing literature. In particular, we show that for describing time-dependent light-matter interactions the Coulomb gauge is not generally irrotational, so it does not enjoy any special status.