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A non-uniqueness problem of the Dirac theory in a curved spacetime. (English) Zbl 1223.81110

Summary: The Dirac equation in a curved space-time 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 space-time 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.

MSC:

81Q35 Quantum mechanics on special spaces: manifolds, fractals, graphs, lattices
81Q05 Closed and approximate solutions to the Schrödinger, Dirac, Klein-Gordon and other equations of quantum mechanics
81T20 Quantum field theory on curved space or space-time backgrounds
70S05 Lagrangian formalism and Hamiltonian formalism in mechanics of particles and systems
82C22 Interacting particle systems in time-dependent statistical mechanics
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