On the derivation of the Dirac Equation

TL;DR

This paper shows that the fundamental properties of Dirac matrices and the necessity of negative energies can be derived from minimal assumptions about the Dirac equation's solutions, without referencing the Klein-Gordon equation.

Contribution

It introduces a new derivation of Dirac matrices' properties based solely on the existence of positive energy solutions, avoiding traditional methods involving squaring the Hamiltonian.

Findings

Anticommutation relations derived without Klein-Gordon reference

Negative energy solutions are a consequence of minimal assumptions

Trace and determinant properties follow from solution requirements

Abstract

We point out that the anticommutation properties of the Dirac matrices can be derived without squaring the Dirac hamiltonian, that is, without explicit reference to the Klein-Gordon equation. We only require the Dirac equation to admit two linearly independent plane wave solutions with positive energy for all momenta. The necessity of negative energies as well as the trace and determinant properties of the Dirac matrices are also a direct consequence of this simple and minimal requirement.