Canonical relativistic formulation to calculate the Casimir force four-vector on anisotropic conductor polarizable and magnetizable media

Abstract

A canonical relativistic formulation to calculate the Casimir force four-vector exerted on anisotropic conductor polarizable and magnetizable media is provided. The anisotropic conductor polarizable and magnetizable medium is modeled by a continuum collection of the antisymmetric tensor fields of the second rank and a continuum collection of the vector fields in Minkowski spacetime. The collection of the antisymmetric tensor fields describes the polarization and the magnetization properties of the medium. The collection of the vector fields describes the conductivity property of the medium. The quantum relativistic wave equation of the four-vector potential of the electromagnetic field is solved using an iteration method. According to the conservation principle of the energy–momentum four-vector and using of the energy–momentum tensors of the relativistic dynamical fields, contained in the theory, the Casimir force four-vector on the anisotropic conductor polarizable and magnetizable medium is obtained. The Casimir force four-vector is obtained in terms of the relativistic dynamical fields, contained in the theory and the coupling tensors that couple the electromagnetic field to the anisotropic conductor polarizable and magnetizable medium. The Casimir force four-vector exerted on the medium is calculated in the vacuum state of the total system. As a special case, the formulation is applied to a multilayer medium. The tangential component of the Casimir force exerted on a multilayer medium vanish when the anisotropic conductor polarizable and magnetizable medium is converted to an isotropic one.