Canonical Quantization of Electromagnetic Field in the Presence of Nonlinear Anisotropic Magnetodielectric Medium with Spatial-Temporal Dispersion

Modeling a nonlinear anisotropic magnetodielectric medium with spatial-temporal dispersion by two continuum collections of three dimensional harmonic oscillators, a fully canonical quantization of the electromagnetic field is demonstrated in the presence of such a medium. Some coupling tensors of various ranks are introduced that couple the magnetodielectric medium with the electromagnetic field. The polarization and magnetization fields of the medium are defined in terms of the coupling tensors and the oscillators modeling the medium. The electric and magnetic susceptibility tensors of the medium are obtained in terms of the coupling tensors. It is shown that the electric field satisfy an integral equation in frequency domain. The integral equation is solved by an iteration method and the electric field is found up to an arbitrary accuracy.     

CONFORMALLY COVARIANT THEORY OF SECOND RANK ANTISYMMETRIC TENSOR FIELD

The conformally covariant field equation on second rank antisymmetric tensor is derived and its conformally covariant energy-momentum tensor is also obtained.     

The double duals and gauges of the Lanczos tensor

It is shown that: i) the Weyl tensor can be expressed in terms of the sum of a tensor and its double dual, where the tensor is constructed from the covariant derivatives of the Lanczos tensor, ii) a similar expression does not exist for the Riemann tensor in electromagnetic theory, iii) the electromagnetic field cannot be identified with the differential gauge freedom of the Lanczos tensor, iv) the symmetries of Einstein Maxwell theory and the Lanczos tensor do not prohibit the identification of the electromagnetic field with the algebraic gauge freedom of the Lanczos tensor, these symmetries require a differential equation relating the electromagnetic field tensor to the algebraic gauge vector and this is given.     

Equations of electrodynamics in a rotating solid dielectric

The equations of electrodynamics in a rotating isotropic homogeneous dielectric are obtained in a covariant form in coordinates of a reference frame that accompanies the rotation of the dielectric. It is found from these equations, which have variable coefficients, that the medium of the rotating dielectric is anisotropic and inhomogeneous. To derive tensors of the electromagnetic field in a rotating reference frame (RRF), the fields and inductions of a virtual inertial reference frame (IRF) that instantaneously accompanies the motion of one of the points of the dielectric are used twice. Initially, using instantaneous local relations, they are expressed in terms of real fields and inductions of the rotating medium, and then they are transformed into fields and inductions of a stationary IRF, in which they are used as components of the tensors of the electromagnetic field. Thus, the electromagnetic field tensors in the IRF are determined taking into account a priori unknown real inhomogeneous permittivity \(\bar \varepsilon \) and permeability \(\bar \varepsilon \) of the rotating medium. At the final stage, the tensors in the RRF are obtained by transformation rules for covariant and contravariant tensor components in accordance with known analytical relationships of fixed and rotating coordinates. The properties of modes of a rotating ring resonator in the form of homogeneous TE waves that travel along and against the direction of rotation and, in particular, their normal frequencies are examined. The contribution of inhomogeneous properties of the medium of a rotating dielectric to the difference between the normal frequencies of the counterpropagating waves (to the Sagnac effect) is determined. In a solid material with known elastic and striction characteristics, its density and dielectric permittivity depend on the radial coordinate. These dependences are caused by the action of the centrifugal force and changes in the polarization and magnetization of the medium because of the rotational motion of charged particles. The coordinate dependences of permittivity \(\bar \varepsilon \) and permeability \(\bar \varepsilon \) make additional contributions to the inhomogeneous properties of the medium of the rotating dielectric and to the Sagnac effect.     

Einstein-aether theory with a Maxwell field: General formalism

We extend the Einstein-aether theory to include the Maxwell field in a nontrivial manner by taking into account its interaction with the time-like unit vector field characterizing the aether. We also include a generic matter term. We present a model with a Lagrangian that includes cross-terms linear and quadratic in the Maxwell tensor, linear and quadratic in the covariant derivative of the aether velocity four-vector, linear in its second covariant derivative and in the Riemann tensor. We decompose these terms with respect to the irreducible parts of the covariant derivative of the aether velocity, namely, the acceleration four-vector, the shear and vorticity tensors, and the expansion scalar. Furthermore, we discuss the influence of an aether non-uniform motion on the polarization and magnetization of the matter in such an aether environment, as well as on its dielectric and magnetic properties. The total self-consistent system of equations for the electromagnetic and the gravitational fields, and the dynamic equations for the unit vector aether field are obtained. Possible applications of this system are discussed. Based on the principles of effective field theories, we display in an appendix all the terms up to fourth order in derivative operators that can be considered in a Lagrangian that includes the metric, the electromagnetic and the aether fields.     

On the evolution equations for a self-gravitating charged scalar field

We consider a complex scalar field minimally coupled to gravity and to a U(1) gauge symmetry and we construct of a first order symmetric hyperbolic evolution system for the Einstein-Maxwell-Klein-Gordon system. Our analysis is based on a $1+3$ tetrad formalism which makes use of the components of the Weyl tensor as one of the unknowns. In order to ensure the symmetric hyperbolicity of the evolution equations, implied by the Bianchi identity, we introduce a tensor of rank 3 corresponding to the covariant derivative of the Faraday tensor, and two tensors of rank 2 for the covariant derivative of the vector potential and the scalar field.     

Covariant extensions and the nonsymmetric unified field

The problem of generally covariant extension of Lorentz invariant field equations, by means of covariant derivatives extracted from the nonsymmetric unified field, is considered. It is shown that the contracted curvature tensor can be expressed in terms of a covariant gauge derivative which contains the gauge derivative corresponding to minimal coupling, if the universal constantp, characterizing the nonsymmetric theory, is fixed in terms of Planck's constant and the elementary quantum of charge. By this choice the spinor representation of the linear connection becomes closely related to the spinor affinity used by Infeld and Van Der Waerden in their generally covariant formulation of Dirac's equation.     

Asymptotic analysis of ultra-relativistic charge

This article offers a new approach for analysing the dynamic behaviour of distributions of charged particles in an electromagnetic field. After discussing the limitations inherent in the Lorentz-Dirac equation for a single point particle a simple model is proposed for a charged continuum interacting self-consistently with the Maxwell field in vacuo. The model is developed using intrinsic tensor field theory and exploits to the full the symmetry and light-cone structure of Minkowski spacetime. This permits the construction of a regular stress-energy tensor whose vanishing divergence determines a system of non-linear partial differential equations for the velocity and self-fields of accelerated charge. Within this covariant framework a particular perturbation scheme is motivated by an exact class of solutions to this system describing the evolution of a charged fluid under the combined effects of both self and external electromagnetic fields. The scheme yields an asymptotic approximation in terms of inhomogeneous linear equations for the self-consistent Maxwell field, charge current and time-like velocity field of the charged fluid and is defined as an ultra-relativistic configuration. To facilitate comparisons with existing accounts of beam dynamics an appendix translates the tensor formulation of the perturbation scheme into the language involving electric and magnetic fields observed in a laboratory (inertial) frame.     

Nonlinear gravito-electromagnetism

There is a non-linear and covariant electromagnetic analogy for gravity, in which the full Bianchi identities are Maxwell-type equations for the free gravitational field, encoded in the Weyl tensor. This tensor gravito-electromagnetism is based on a covariant generalization of spatial vector algebra and calculus to spatial tensor fields, and includes all non-linear effects from the gravitational field and matter sources. The non-linear vacuum Bianchi equations are invariant under spatial duality rotation of the gravito-electric and gravito-magnetic tensor fields. The super-energy density and super-Poynting vector of the gravitational field are natural duality invariants, and satisfy a super-energy conservation equation.     

Metric energy-momentum tensor for polarizable particles in an electormagnetic field

Within the covariant Lagrange formalism and the relativistic theory of continuous media, the metric energy-momentum tensor is obtained for spin polarizable particles interacting with an electromagnetic field. An equation of motion of the polarizable particles with a spin of 1/2 in an external electromagnetic field is derived. __________ Translated from Izvestiya Vysshikh Uchebnykh Zavedenii, Fizika, No. 12, pp. 21–29, December, 2006.     

Covariant Wigner function approach to the relativistic charged gas in a strong magnetic field: I. Electron gas in thermal equilibrium

The relativistic quantum electron gas embedded in a strong magnetic field is studied by calculating its covariant Wigner function in thermal equilibrium. Previous results obtained earlier by Canuto and Chiu are then recovered in a unified way. The polarization tensor is calculated with the use of a covariant quantum BGK equation. Also the lifetime of the neutron in such a medium is calculated for the sake of illustration of the usefulness of the covariant Wigner function.     

Invariance under conformal coordinate and point transformations

Coordinate and point transformations are studied in the context of conformal symmetry. When invariance requirements on arbitrary rank tensors are involved in both contexts, the similitudes and differences in transformation laws and invariance conditions are analysed in connection with those on tensor densities of weight W. Physically interesting tensors like the metric tensor, the electromagnetic field and the energy-momentum tensor are specifically examined. Some remarks on scalar fields and densities are added.     

Group theory of the massless spin 2 field and gravitation

The simplest manifestly covariant unitary representation of the Poincaré group for zero mass and spin 2 is constructed. This representation is carried by fourth rank tensors which satisfy the equations of the Riemann curvature tensor in the linearized theory of gravitation in vacuo. In particular, the requirement of unitarity implies the Bianchi identities.     

An invariant theory of the constitutive parameters of a plasma and a ferrite

Under the influence of a constant magnetic field, the electric property of a plasma and the magnetic property of a ferrite are anisotropic. In this paper, the general coordinatefree invariant forms of the dielectric tensor of a plasma and the permeability tensor of a ferrite are obtained. The tensors are expressed explicitly as a sum of three tensors: a unit tensor, a symmetric tensor and an antisymmetric tensor, each of which is weighted by different constants. The symmetric and antisymmetric tensors are related to the unit vector of the constant magnetic field. The invariant forms in terms of the sum of the projectors of the tensors are also derived. When a Cartesian coordinate system is introduced, the invariant forms are easily reduced to the commonly used matrix representations. The invariant forms clearly show the effects of the constant magnetic field on the anisotropies of the media. Moreover, they effectuate and simplify the deduction of the general solutions of problems involving wave propagation and excitation in plasma and ferrite and thus facilitate interpretations of the final results.     

Microscopic derivation of hydrodynamic balance equations in the presence of an electromagnetic field

Using field-theoretic methods we derive balance equations for a charged fluid in an external electromagnetic field the effects of which are included by minimal coupling. An infinite hierarchy of balance equations for tensor operators is derived. A macroscopic velocity field is introduced by a unitary transformation on the field operators. Suitable statistical averages in the local equilibrium approximation yield macroscopic balance equations. The significance of new terms is discussed.     

Differential operators in terms of Clebsch-Gordon (C-G) coefficients and the wave equation of massless tensor fields

The quantum theory of angular momentum affords a treatment of tensors and vectors in a spherical basis. By using this theory we define the tensor differential operators: divergence, curl and gradient which act on a tensor of any rank, in terms of C-G coefficients. With these definitions we obtain a matrix representation and useful properties for those operators. An interesting application of this formalism is to find the wave equation of a tensor of any rank in a linear theory. This provides a new common way to look at the wave equations associated with both Maxwell's equations and the Maxwell-like equations for the linearized Weyl curvature tensor in gravitoelectromagnetism describing gravitational radiation on a Minkowski spacetime background.     

Covariant electrodynamics in a medium,II

The energy-momentum and angular momentum emission rates for an arbitrarily moving charge (whose speed is less than that of light in the medium) in a uniform transparent medium are calculated in manifestly covariant form. The calculations are executed for three types of stress tensor: Minkowski, Abraham, and Marx. Among other things it is found that the energy-momentum emission rates for the latter two tensors are equal and differ from that of the former. Further, the angular momentum emission rates for all three tensors are found to be equal. Only for the Marx tensor is this rate independent of the orientation of the associated asymptotic space-like surface.     

Group representation theory in relativistic electrodynamics

Invariant expansions of the electromagnetic field tensor F and its covariant derivative ℬF in the presence of gravity are obtained on the basis of the classical problem of elementary orthogonal group representation theory regarding the expansion of the tensor product of representations in terms of irreducible components. Vladimirsk State Pedagogical University. Translated from Izvestiya Vysshikh Uchebnykh Zavedenii, Fizika, No. 5, pp. 90–93, May, 1996.