The Dirac Equation

Abstract

The Dirac equation with a fixed energy and the Vekua equation describing pseudoanalytic functions both are first-order elliptic systems, and it would be quite natural to expect a deep interrelation between their theories, especially in the case when all potentials and wave functions in the Dirac equation depend on two space variables only. Nevertheless there has not been much work done in this direction¹ due to the fact that traditional matrix representations of the Dirac operator do not allow us to visualize a relation between the Dirac equation in the two-dimensional case and the Vekua equation. Written using the traditional matrix formalism, the Dirac equation is a system of four complex equations which does not decouple in a two-dimensional situation but decouples in the one-dimensional case only.