Φ-Ψ model for electrodynamics in dielectric media: exact quantisation in the Heisenberg representation

We investigate the quantisation in the Heisenberg representation of a model which represents a simplification of the Hopfield model for dielectric media, where the electromagnetic field is replaced by a scalar field ϕ and the role of the polarisation field is played by a further scalar field ψ. The model, which is quadratic in the fields, is still characterised by a non-trivial physical content, as the physical particles correspond to the polaritons of the standard Hopfield model of condensed matter physics. Causality is also taken into account and a discussion of the standard interaction representation is also considered.