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

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.

35 pages (standard article format). v4: Version accepted for publication in Annalen der Physik: Redactional improvements and precisions added in Section 2. Footnote added in the Conclusion, with new references. v3: Introduction and Conclusion reinforced. References added. v2: subsection 2.3 added: the Lagrangian and the spin group. Also, added explanations on admissible coefficient changes

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• Four-vector vs. four-scalar representation of the Dirac wave function
• Some remarks on quantum mechanics in a curved spacetime, especially for a Dirac particle
• On the non-uniqueness problem of the covariant Dirac theory and the spin-rotation coupling
• 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
• Defining the space in a general spacetime
• Pauli equation and charged spin-1/2 particle in a weak gravitational field