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

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.

32 pages in 12pt article. V4: Matches exactly with published version: Theorem 1 corrected (physical relevance unchanged)

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• Should there be a spin-rotation coupling for a Dirac particle?
• On the Hamiltonian and energy operators in a curved spacetime, especially for a Dirac particle
• On continuum dynamics and the electromagnetic field in the scalar ether theory of gravitation
• On the definition of energy for a continuum, its conservation laws, and the energy-momentum tensor