A simpler solution of the non-uniqueness problem of the covariant Dirac theory

TL;DR

The paper proposes a simplified method to resolve the non-uniqueness issues in the covariant Dirac theory by restricting the gamma field to constant gauge transformations within a specific metric form, ensuring uniqueness across all reference frames.

Contribution

It introduces a new approach that restricts the gamma field to constant gauge transformations, solving non-uniqueness problems in covariant Dirac theory for metrics in a space-isotropic diagonal form.

Findings

The proposed method ensures the uniqueness of the Hamiltonian and energy operators.

Using the diagonal tetrad in a space-isotropic metric resolves non-uniqueness issues.

The approach is applicable within the first-quantized framework and for specific metric forms.

Abstract

Although the standard generally-covariant Dirac equation is unique in a topologically simple spacetime, it has been shown that it leads to non-uniqueness problems for the Hamiltonian and energy operators, including the non-uniqueness of the energy spectrum. These problems should be solved by restricting the choice of the Dirac gamma field in a consistent way. Recently, we proposed to impose the value of the rotation rate of the tetrad field. This is not necessarily easy to implement and works only in a given reference frame. Here, we propose that the gamma field should change only by constant gauge transformations. To get that situation, we are naturally led to assume that the metric can be put in a space-isotropic diagonal form. When this is the case, it distinguishes a preferred reference frame. We show that by defining the gamma field from the "diagonal tetrad" in a chart in which the metric has that form, the uniqueness problems are solved at once for all reference frames. We discuss the physical relevance of the metric considered and our restriction to first-quantized theory.