M. Arminjon and F. Reifler
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. The authors 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. The authors 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. The authors 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.