A non-uniqueness problem of the Dirac theory in a curved spacetime
Mayeul Arminjon, Frank Reifler
备注：
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
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.
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Comment on "Quantum phase transitions of Dirac particles in a magnetized rotating curved background: Interplay of geometry, magnetization, and thermodynamics"
Relativistic spin-1/2 particles in curved spacetime are naturally described by Dirac theory, which is a dynamical and Lorentz-invariant field theory. In this work, we propose a non-dynamical fermion theory in 3+1 dimensions dubbed spinor topological field theory, built in terms of a spinor field and a Cartan connection related to de Sitter group. We show that our model gives rise to the Dirac theory in curved spacetime when the local de Sitter gauge invariance of the model breaks down to the Lorentz gauge invariance, providing also a geometric origin to the fermion mass. Finally, we show that …
