Dirac equation: Representation independence and tensor transformation

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.

28 pages, standard LaTeX. v3: matches version accepted in the Brazilian Journal of Physics: minor wording improvements, refs updated. v2: Intro and Conclusion improved (novelty more emphasized). Uniqueness and positive definiteness extended to any admissible affine coordinates. 10 new refs

References in corpus (2)

• Basic quantum mechanics for three Dirac equations in a curved spacetime
• Covariant differentiation of spinors for a general affine connection

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