What Is the Generalized Representation of Dirac Equation in Two Dimensions?

TL;DR

This paper derives the most general form of 2x2 Dirac matrices in 2+1 dimensions, providing a comprehensive framework that encompasses all specific representations used in various physical contexts.

Contribution

It presents the first complete derivation of the general form of Dirac matrices in 2+1 dimensions, including relations among elements and the generalized Lorentz transformation.

Findings

Derived the general form of 2x2 Dirac matrices in 2+1 dimensions.

Established relations among matrix elements and the generalized Lorentz transformation.

Unified various specific representations into a single comprehensive framework.

Abstract

In this work, the general form of $2\times2$ Dirac matrices for 2+1 dimension is found. In order to find this general representation, all relations among the elements of the matrices and matrices themselves are found,and the generalized Lorentz transform matrix is also found under the effect of the general representation of Dirac matrices. As we know, the well known equation of Dirac, $ \left( i\gamma^{\mu}\partial_{\mu}-m\right) \Psi=0 $, is consist of matrices of even dimension known as the general representation of Dirac matrices or Dirac matrices. Our motivation for this study was lack of the general representation of these matrices despite the fact that more than nine decades have been passed since the discovery of this well known equation. Everyone has used a specific representation of this equation according to their need; such as the standard representation known as Dirac-Pauli Representation, Weyl Representation or Majorana representation. In this work, the general form which these matrices can have is found once for all.