Electromagnetic space-time crystals. I. Fundamental solution of the Dirac equation

Abstract

The fundamental solution of the Dirac equation for an electron in an electromagnetic field with harmonic dependence on space-time coordinates is obtained. The field is composed of three standing plane harmonic waves with mutually orthogonal phase planes and the same frequency. Each standing wave consists of two eigenwaves with different complex amplitudes and opposite directions of propagation. The fundamental solution is obtained in the form of the projection operator defining the subspace of solutions to the Dirac equation.

Electromagnetic space-time crystals. II. Fractal computational approach

“…. , k − 1 and n belongs to the subset L ss [(0, 2), s] containing m. This conclusion follows immediately from the recurrent relations presented in [6].…”

“…Due to the specific Fourier spectrum [6] of the 4d periodic electromagnetic field of ESTC, in the current series of papers we use indexing of Fourier components of the wave function and many other mathematical objects by points n = (n 1 , n 2 , n 3 , n 4 ) of the integer lattice L with even values of the sum n 1 + n 2 + n 3 + n 4 . The fundamental solution S and the projection operator P of the infinite system of equations under study are defined as follows [6]…”

“…In other words, each amplitude c(n) enters in 13 different matrix equations of the infinite system. The fundamental solution of this system is obtained in the previous paper [6] by a recurrent process. It is expressed in terms of an infinite series of projection operators.…”

“…In this paper we present a fractal computational approach to calculating the fundamental solution of the Dirac equation describing the motion of an electron in an electromagnetic field with four-dimensional (4d) periodicity (electromagnetic space-time crystal, or ESTC) [6]. The electromagnetic field is composed of three standing plane harmonic waves with mutually orthogonal phase planes and the same frequency.…”

“…It is expressed in terms of an infinite series of projection operators. This process begins with the selection of an infinite subsystem consisting from independent equations and the calculation of the projection operators ρ 0 (n) = P (n), n ∈ F 0 ⊂ L, which uniquely define the fundamental solutions of these equations [6]. At each new step of the recurrent process, we add another infinite set of mutually independent equations (MIE) which, however, are related with some of the equations introduces at the previous steps.…”