Vladimir Shtelen

On Galilean invariance and nonlinearity in electrodynamics and quantum mechanics

Abstract Recent experimental results such as those on slow light heighten interest in nonlinear Maxwell theories. We obtain Galilei covariant equations for electromagnetism by allowing special nonlinearities in the constitutive equations only, keeping Maxwells equations unchanged. Combining these with linear or nonlinear Schrodinger equations, e.g., as proposed by Doebner and Goldin, yields a consistent, nonlinear, Galilean Schrodinger–Maxwell electrodynamics.

New classes of Schrödinger equations equivalent to the free particle equation through non-local transformations

We introduce new classes of Schrodinger equations with time-dependent potentials which are transformable to the free particle equation through non-local transformations. These non-local transformations arise when considering the potential systems of the Schrodinger equation. Explicit formulae are given for the potentials and the corresponding solutions related to the solutions of the free particle equation.

Generalizations of Yang-Mills theory with nonlinear constitutive equations

We generalize classical Yang–Mills theory by extending nonlinear constitutive equations for Maxwell fields to non-Abelian gauge groups. Such theories may or may not be Lagrangian. We obtain conditions on the constitutive equations specifying the Lagrangian case, of which recently discussed non-Abelian Born–Infeld theories are particular examples. Some models in our class possess nontrivial Galilean (c → ∞) limits; we determine when such limits exist and obtain them explicitly.

On gauge transformations of Bäcklund type and higher order nonlinear Schrödinger equations

We introduce a new, more general type of nonlinear gauge transformation in nonrelativistic quantum mechanics that involves derivatives of the wave function and belongs to the class of Backlund transformations. These transformations satisfy certain reasonable, previously proposed requirements for gauge transformations. Their application to the Schrodinger equation results in higher order partial differential equations. As an example, we derive a general family of sixth-order nonlinear Schrodinger equations, closed under our nonlinear gauge group. We also introduce a new gauge invariant current σ=ρ∇Δ ln ρ, where ρ=ψψ. We derive gauge invariant quantities, and characterize the subclass of the sixth-order equations that is gauge equivalent to the free Schrodinger equation. We relate our development to nonlinear equations studied by Doebner and Goldin, and by Puszkarz.

Generalizations of nonlinear and supersymmetric classical electrodynamics

We first write down a very general description of nonlinear classical electrodynamics, making use of generalized constitutive equations and constitutive tensors. Our approach includes non-Lagrangian as well as Lagrangian theories, which allows for electromagnetic fields in the widest possible variety of media (anisotropic, piroelectric, chiral and ferromagnetic) and accommodates the incorporation of nonlocal effects. We formulate electric–magnetic duality in terms of the constitutive tensors. We then propose a supersymmetric version of the general constitutive equations, in a superfield approach.

Lagrangian and non-Lagrangian approaches to electrodynamics including supersymmetry

We take a general approach to nonlinear electrodynamics that includes non‐Lagrangian as well as Lagrangian theories. We introduce the constitutive tensors which, together with Maxwells equations, describe nonlinear electrodynamics in an extremely general way. We show how this approach specializes to particular cases that were previously considered, and indicate how it generalizes to supersymmetric electrodynamics.

Conformal Inversion and Maxwell Field Invariants in Four- and Six-dimensional Spacetimes

Conformally compactified (3+1)-dimensional Minkowski spacetime may be identified with the projective light cone in (4+2)-dimensional spacetime. In the latter spacetime the special conformal group acts via rotations and boosts, and conformal inversion acts via reflection in a single coordinate. Hexaspherical coordinates facilitate dimensional reduction of Maxwell electromagnetic field strength tensors to (3+1) from (4+2) dimensions. Here we focus on the operation of conformal inversion in different coordinatizations, and write some useful equations. We then write a conformal invariant and a pseudo-invariant in terms of field strengths; the pseudo-invariant in (4 + 2) dimensions takes a new form. Our results advance the study of general nonlinear conformal-invariant electrodynamics based on nonlinear constitutive equations.

Resonance Broadening Theory of Farley-Buneman Turbulence in the Auroral E-Region

The conventional theory of resonance broadening for a two-species plasma in a magnetic field is revised, and applied to an ionospheric turbulence case. The assumptions made in the conventional theory of resonance broadening have, in the past, led to replacing the frequency ω by ω + ik 2 D ∗ in the resonant part of the linear dielectric function to obtain the nonlinear dielectric function. Where D ∗ is an anomalous diffusion coefficient due solely to wave scattering of the particle orbits. We show that in general these assumptions are not valid, and consequently the straightforward substitution of frequencies is not legitimate. We remedy these problems and derive expressions for the time-dependent components of the diffusion tensor. The improved resonance broadening theory is developed in the context of an ionospheric problem, namely that of the Farley-Buneman turbulence in the auroral E-region. A kinetic description of the electrons is used. A general expression for the nonlinear dielectric function is derived in the special case where no parallel electric field is present, and the differences with the conventional dispersion relation are discussed.A concept of asymptotic symmetry is introduced which is based on a definition of symmetry as a reducibility property relative to a corresponding invariant ansatz. It is shown that the nonlocal Lorentz invariance of the free-particle Schrödinger equation, discovered by Fushchych and Segeda in 1977, can be extended to Galilei-invariant equations for free particles with arbitrary spin and, with our definition of asymptotic symmetry, to many nonlinear Schrödinger equations. An important class of solutions of the free Schrödinger equation with improved smoothing properties is obtained.