Peculiarities of squaring method applied to construct solutions of the Dirac, Majorana, and Weyl equations

TL;DR

This paper investigates the peculiarities of the squaring method for solving Dirac, Majorana, and Weyl equations, revealing the structure of solution spaces and their dependence on matrix representations, with applications to the Casimir effect.

Contribution

It provides a detailed analysis of the squaring method's application to Dirac, Majorana, and Weyl equations, including basis construction and transformation relations, highlighting representation-dependent features.

Findings

The solution space for the Dirac equation is four-dimensional.

Linear transformations relate different solution bases.

Representation choice affects the squaring method's application.

Abstract

It is shown that the known method to solve the Dirac equation by means of the squaring method, when relying on the scalar function of the form \Phi = e^{-i\epsilon t} e^{ik_{1} x} e^{ik_{2} y} \sin (kz + \alpha) leads to a 4-dimensional space of the Dirac solutions. It is shown that so constructed basis is equivalent to the space of the Dirac states relied on the use of quantum numbers k_{1}, k_{2}, \pm k and helicity operator; linear transformations relating these two spaces are found. Application of the squaring method substantially depends on the choice of representation for the Dirac matrices, some features of this are considered. Peculiarities of applying the squaring method in Majorana representation are investigated as well. The constructed bases are relevant to describe the Casimir effect for Dirac and Weyl fields in the domain restricted by two planes.