Electric field-based quantization of the gauge invariant Proca theory

We consider the gauge invariant version of the Proca theory, where besides the real vector field there is also the real scalar field. We quantize the theory such that the commutator of the scalar field operator and the electric field operator is given by a predefined three-dimensional vector field, say $\mathcal{E}$ up to a global prefactor. This happens when the field operators of the gauge invariant Proca theory satisfy the proper gauge constraint. In particular, we show that $\mathcal{E}$ given by the classical Coulomb field leads to the Coulomb gauge constraint making the vector field operator divergenceless. We also show that physically unreadable gauge constraints can have a strikingly simple $\mathcal{E}$-representation in our formalism. This leads to the discussion of Debye, Yukawa, etc. gauges. In general terms, we explore the mapping between classical vector fields and gauge constraints imposed on the operators of the studied theory