Non-uniqueness of the Dirac theory in a curved spacetime

TL;DR

This paper investigates the non-uniqueness of the Dirac equation in curved spacetime, showing that different choices of coefficients lead to non-equivalent Hamiltonian and energy operators, affecting the spectrum.

Contribution

It demonstrates that the Dirac equation's coefficient choices in curved spacetime generally produce non-unique Hamiltonian and energy operators, highlighting a fundamental ambiguity.

Findings

Most coefficient changes do not yield equivalent operators.

The Dirac energy spectrum is not unique.

Conditions for equivalence depend on initial data and point-dependent matrices.

Abstract

We summarize a recent work on the subject title. 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. In this paper, we focus on the standard version of the gravitational Dirac equation, but the non-uniqueness applies also to our alternative versions. We find that the changes which lead to an equivalent operator H, or respectively to an equivalent operator E, are determined by initial data, or respectively have to make some point-dependent antihermitian matrix vanish. Thus, the vast majority of the possible coefficient changes lead neither to an equivalent operator H, nor to an equivalent operator E, whence a lack of uniqueness. We show that even the Dirac energy spectrum is not unique.