Dirac equation: Representation independence and tensor transformation

TL;DR

This paper demonstrates that the Dirac equation's physical predictions are independent of the choice of Dirac matrices and introduces a tensor formulation that extends to non-inertial coordinate systems.

Contribution

It establishes the representation independence of the Dirac equation and introduces a tensor-based formulation compatible with general coordinate systems.

Findings

Current and Hamiltonian spectrum are independent of Dirac matrix representation.

Tensor Dirac theory is physically equivalent to the standard spinor-based theory.

Tensor formulation extends to non-inertial coordinate systems.

Abstract

We define and study the probability current and the Hamiltonian operator for a fully general set of Dirac matrices in a flat spacetime with affine coordinates, by using the Bargmann-Pauli hermitizing matrix. We find that with some weak conditions on the affine coordinates, the current, as well as the spectrum of the Dirac Hamiltonian, thus all of quantum mechanics, are independent of that set. These results allow us to show that the tensor Dirac theory, which transforms the wave function as a spacetime vector and the set of Dirac matrices as a third-order affine tensor, is physically equivalent to the genuine Dirac theory, based on the spinor transformation. The tensor Dirac equation extends immediately to general coordinate systems, thus to non-inertial (e.g. rotating) coordinate systems.