The relativistic Pauli equation

TL;DR

This paper reformulates the Dirac and Pauli equations using 2x2 complex matrices, providing a gamma-matrix-free, more elegant representation of relativistic quantum mechanics for spinning particles.

Contribution

It introduces a matrix-based formulation of the Dirac and Pauli equations, simplifying their structure and avoiding gamma matrices, thus offering a new perspective on relativistic quantum equations.

Findings

Matrix representation of Dirac bispinors simplifies equations.

Derivation of a Klein-Gordon type second order equation.

Lagrangian formulation of the relativistic Pauli equation.

Abstract

After discussing the way that C2 and the algebra of complex 2x2 matrices can be used for the representation of both non-relativistic rotations and Lorentz transformations, we show that Dirac bispinors can be more advantageously represented as 2x2 complex matrices. One can then give the Dirac equation a form for such matrix-valued wave functions that no longer necessitates the introduction of gamma matrices or a choice for their representation. The minimally-coupled Dirac equation for a charged spinning particle in an external electromagnetic field then implies a second order equation in the matrix-valued wave functions that is of Klein-Gordon type and represents the relativistic analogue of the Pauli equation. We conclude by presenting the Lagrangian form for the relativistic Pauli equation.