Field Theoretic Models of the Dispersive Dielectric Medium

The purpose of this paper is a description of the dispersive dielectric medium, both linear and nonlinear, from first principles using the field theoretic methods based on the Feynman path integrals over classical trajectories. The main idea is to use notion of effective fields, in the present case the electromagnetic field modified by presence of a polarizable medium. Interaction of the field with the medium on the microscopic level is described by a modified Hopfield Lagrangian containing terms corresponding to the electromagnetic field, the matter polarization field modelled by harmonic oscillators with some resonance frequency and other matter fields describing the degrees of freedom responsible for absorption in the medium (reservoir fields). The polarization field is coupled both to the electric field and the reservoir fields. Effective theory is obtained by elimination of the matter degrees of freedom which is achieved by functional integration over all matter fields. For a linear medium all calculations can be done exactly leading to the effective Lagrangian from which, among others, an expression for frequency dependent dielectric constant can be extracted. Explicit form of the dielectric constant depends on the way by which the polarization field couples to the reservoir fields. In particular, uniform coupling to all reservoir modes gives the standard Lorentz oscillator model, and for any type of coupling the Lorentz form of dielectric constant is retrieved for frequencies close to the resonance. For weak damping the dispersion ω(k) is little sensitive to the form of coupling leading to polariton modes not different from those of the Lorentz model. It is also outlined briefly how the functional integration method could be used to description of nonlinear effects in the medium.