Electromagnetic space-time crystals. II. Fractal computational approach

Abstract

A fractal approach to numerical analysis of electromagnetic space-time crystals, created by three standing plane harmonic waves with mutually orthogonal phase planes and the same frequency, is presented. Finite models of electromagnetic crystals are introduced, which make possible to obtain various approximate solutions of the Dirac equation. A criterion for evaluating accuracy of these approximate solutions is suggested.

Localized solutions of the Dirac equation in free space and electromagnetic space-time crystals

“…To construct localized solutions of the Dirac equation in the ESTCs or free space, it is necessary to calculate first the basis wave functions Ψ (7) specified by a set of four-dimensional vectors Q, = (q, iq 4 ). To attain these ends in the general case of 4D-ESTCs one can use the solutions and techniques presented in [7][8][9][10][11]. It is shown in Sec.…”

“…The fundamental solution S has been expressed in terms of an infinite series of projection operators calculated by a recurrent process based on a fractal approach. This technique has been detailed and applied to various ESTCs in [8][9][10][11][12].…”

“…( 5). It provides a convenient fitness criterion to accurately compare various approximate solutions of this equation [9][10][11][12]. If R[Ψ] ≪ 1, the function Ψ may be treated as a reasonable approximation to the exact solution for which R[Ψ] = 0, and the smaller is R[Ψ], the more accurate is the approximation.…”

“…In [7][8][9][10][11][12], we presented the fundamental solution of the Dirac equation for the ESTC created by six plane waves with the same frequency ω 0 and the four-dimensional wave vectors…”

Dirac electron in a chiral space-time crystal created by counterpropagating circularly polarized plane electromagnetic waves

“…The motion of electrons in natural crystals is described by the Schrödinger equation with a periodic electrostatic scalar potential. Electromagnetic fields with periodic dependence on space-time coordinates can be treated by analogy with the crystals of solid-state physics, so it is natural to refer to these field lattices as electromagnetic space-time crystals (ESTCs) [1][2][3][4][5][6]. In this context, the idea of a space-time crystal was first presented in [1] and the electron wave functions for the ESTC, created by two linearly polarized plane waves, were calculated by using the first-order perturbation theory for the Schrödinger-Stueckelberg equation.…”