A special irreducible matrix representation of the real Clifford algebra C(3,1)

TL;DR

This paper introduces a symmetric real representation of 4x4 Dirac matrices within the Clifford algebra C(3,1), leveraging geometric insights from isoclinic 2-planes, which could simplify quantum physics calculations.

Contribution

It presents a new symmetric Majorana representation of Dirac matrices for C(3,1), based on geometric properties of isoclinic 2-planes, with potential for generalization.

Findings

A compact formula for transformed Pauli matrices is derived.

The representation simplifies certain quantum physics calculations.

The approach is based on geometric invariants of Clifford algebras.

Abstract

4x4 Dirac (gamma) matrices (irreducible matrix representations of the Clifford algebras C(3,1), C(1,3), C(4,0)) are an essential part of many calculations in quantum physics. Although the final physical results do not depend on the applied representation of the Dirac matrices (e.g. due to the invariance of traces of products of Dirac matrices), the appropriate choice of the representation used may facilitate the analysis. The present paper introduces a particularly symmetric real representation of 4x4 Dirac matrices (Majorana representation) which may prove useful in the future. As a byproduct, a compact formula for (transformed) Pauli matrices is found. The consideration is based on the role played by isoclinic 2-planes in the geometry of the real Clifford algebra C(3,0) which provide an invariant geometric frame for it. It can be generalized to larger Clifford algebras.