A non-uniqueness problem of the Dirac theory in a curved spacetime

TL;DR

This paper investigates the non-uniqueness of the Dirac equation's energy spectrum in curved spacetime, showing that different coefficient choices lead to non-equivalent Hamiltonians and spectra, raising foundational issues in quantum gravity.

Contribution

It demonstrates that the Dirac energy spectrum in curved spacetime is non-unique due to the dependence on coefficient fields, a problem not previously fully characterized.

Findings

Most coefficient changes do not produce equivalent Hamiltonians or energy operators.

The Dirac energy spectrum is not unique in curved spacetime.

The non-uniqueness applies across different formulations of the Dirac equation.

Abstract

The Dirac equation in a curved spacetime depends on a field of coefficients (essentially the Dirac matrices), for which a continuum of different choices are possible. We study the conditions under which a change of the coefficient fields leads to an equivalent Hamiltonian operator H, or to an equivalent energy operator E. We do that for the standard version of the gravitational Dirac equation, and for two alternative equations based on the tensor representation of the Dirac fields. The latter equations may be defined when the spacetime is four-dimensional, noncompact, and admits a spinor structure. We find that, for each among the three versions of the equation, the vast majority of the possible coefficient changes do not lead to an equivalent operator H, nor to an equivalent operator E, whence a lack of uniqueness. In particular, we prove that the Dirac energy spectrum is not unique. This non-uniqueness of the energy spectrum comes from an effect of the choice of coefficients, and applies in any given coordinates.