Quantum electrodynamics of the dispersive dielectric medium–a path integral approach

Abstract

The Feynman path integration method of quantization of classical systems provides a useful alternative to the canonical quantization scheme. In the present paper this method is applied for the first time to be quantization of the electromagnetic field in a linear, inhomogeneous, dispersive and absorptive dielectric medium, described in the framework of the Hopfield model. The medium is assumed to be linear, so that the corresponding functional integrations are of the Gaussian type and can be explicitly performed. With the use of local microscopic action as the starting point, the method allows us to obtain in a simple and natural way the (non-local in time) effective action describing properties of the electromagnetic field in the medium on the macroscopic level. Effective action can be obtained directly by elimination of the matter degrees of freedom, facilitated by functional integration over the matter fields, and leaving the electromagnetic field on the classical level. The macroscopic effective action of the electromagnetic field was further used to obtain an explicit expression for the frequency dependent dielectric constant, fulfilling the Kramers–Kronig relations. The relation between electric and displacement fields is on this level classical, that is it does not contain Langevin quantum noise term, characteristic for the quantum electromagnetic field in a dispersive and absorptive medium. Complete quantization of the model is performed by functional integration over both the matter and the electromagnetic fields. This gives an expression for the generating functional, and for the propagators (Green functions) of the quantized electromagnetic and matter polarization fields. The propagators fulfil Dyson-type equations, which in some cases (e.g. homogeneous medium) can be solved exactly. In the case of a homogeneous dielectric, expressions for the field operators are obtained from the propagators.