Integral formulation of the macroscopic quantum electrodynamics in dispersive dielectric objects

We propose an integral formulation of macroscopic quantum electrodynamics in the Heisenberg picture for linear dispersive dielectric objects of finite size, utilizing the Hopfield-type approach. By expressing the electromagnetic field operators as a function of the polarization density field operator via the retarded Green function for the vacuum, we obtain an integral equation that governs the evolution of the polarization density field operator. This formulation offers significant advantages, as it allows for the direct application of well-established computational techniques from classical electrodynamics to perform quantum electrodynamics computations in open, dispersive, and absorbing environments.