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

TL;DR

This paper derives the fundamental solution of the Dirac equation for an electron in a complex electromagnetic field composed of three standing waves, advancing theoretical understanding of electron behavior in structured space-time fields.

Contribution

It provides the first explicit fundamental solution of the Dirac equation in a space-time crystal formed by three standing electromagnetic waves.

Findings

Fundamental solution expressed as a projection operator.

Describes electron solutions in a harmonic space-time electromagnetic field.

Advances theoretical framework for Dirac equation in structured fields.

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.