Abstract
A canonical relativistic quantization of the electromagnetic field is introduced in the presence of an anisotropic conductor magneto-dielectric medium in a standard way in the Gupta–Bleuler framework. The medium is modeled by a continuum collection of the vector fields and a continuum collection of the antisymmetric tensor fields of the second rank in Minkowski space–time. The collection of vector fields describes the conductivity property of the medium and the collection of antisymmetric tensor fields describes the polarization and the magnetization properties of the medium. The conservation law of the total electric charges, induced in the anisotropic conductor magneto-dielectric medium, is deduced using the antisymmetry conditions imposed on the coupling tensors that couple the electromagnetic field to the medium. Two relativistic covariant constitutive relations for the anisotropic conductor magneto-dielectric medium are obtained. The constitutive relations relate the antisymmetric electric–magnetic polarization tensor field of the medium and the free electric current density four-vector, induced in the medium, to the strength tensor of the electromagnetic field, separately. It is shown that for a homogeneous anisotropic medium the susceptibility tensor of the medium satisfies the Kramers–Kronig relations. Also it is shown that for a homogeneous anisotropic medium the real and imaginary parts of the conductivity tensor of the medium satisfy the Kramers–Kronig relations and a relation other than the Kramers–Kronig relations.
Highlights
There is a quantization method of the electromagnetic field in the presence of dispersive and absorbing dielectric media known as the Green function method.[1,2,3,4,5,6,7] In this quantization scheme the causal constitutive relation of the dielectric medium is written in frequency domain
In order to quantize electromagnetic field in the presence of an anisotropic conductor polarizable and magnetizable medium in a canonical relativistic language, following the two previous sections, we model the medium by a continuum set of antisymmetric tensor fields Y αβ(ω, x) and a continuum set of vector fields Xν(ω, x) in Minkowski space–time each is labeled by a positive real parameter ω
The antisymmetric electric–magnetic polarization tensor of the anisotropic magneto-dielectric medium was defined in terms of the antisymmetric tensor fields modeling the medium and the coupling tensors that couple the electromagnetic field to the anisotropic magneto-dielectric medium
Summary
There is a quantization method of the electromagnetic field in the presence of dispersive and absorbing dielectric media known as the Green function method.[1,2,3,4,5,6,7] In this quantization scheme the causal constitutive relation of the dielectric medium is written in frequency domain. The noise electric and magnetic polarization densities of the magneto-dielectric medium are obtained in terms of the coupling tensors that couple the electromagnetic field to the medium and the harmonic oscillators fields modeling the medium at t = 0 This nonrelativistic canonical quantization scheme of the electromagnetic field has been generalized in presence of absorbing bi-anisotropic magneto-dielectric media.[56] In this case, four continuum sets of the coupling tensors of the second rank, in usual three-dimensional space, are applied in the interaction part of the Lagrangian of the total system. The electromagnetic field together the anisotropic conductor magneto-dielectric medium is quantized in a canonical relativistic standard way using the Gupta–Bleuler method.[60] The total noise electric current density four-vector induced in anisotropic conductor magneto-dielectric medium is written in terms of the ladder operators of the total system and the coupling tensors that couple the electromagnetic field to the medium.
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