Arminjon, M.; Reifler, F. A non-uniqueness problem of the Dirac theory in a curved spacetime. (English) Zbl 1223.81110 Ann. Phys. (8) 20, No. 7, 531-551 (2011). 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. Cited in 10 Documents 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 Keywords:relativistic wave equations; Einstein-Maxwell space-times; space-times with fluids; radiation or classical fields; Lagrangian and Hamiltonian approach; quantum fields in curved spacetimes PDFBibTeX XMLCite \textit{M. Arminjon} and \textit{F. Reifler}, Ann. Phys. (8) 20, No. 7, 531--551 (2011; Zbl 1223.81110) Full Text: DOI arXiv References: [1] Brill, Rev. Mod. Phys. 29 pp 465– (1957) · Zbl 0078.43503 · doi:10.1103/RevModPhys.29.465 [2] de Oliveira, Nuovo Cimento 24 pp 672– (1962) · Zbl 0102.21606 · doi:10.1007/BF02816716 [3] Chapman, Am. J. Phys. 44(9) pp 858– (1976) · doi:10.1119/1.10256 [4] Weyl, Z. Phys. 56 pp 330– (1929) · doi:10.1007/BF01339504 [5] Fock, Z. Phys. 57 pp 261– (1929) · doi:10.1007/BF01339714 [6] Soffel, J. Phys. A, Math. Gen. 10 pp 551– (1977) · doi:10.1088/0305-4470/10/4/017 [7] Hehl, Phys. Rev. D 42 pp 2045– (1990) · doi:10.1103/PhysRevD.42.2045 [8] Varjú, Phys. Lett. A 250 pp 263– (1998) · Zbl 0941.81051 · doi:10.1016/S0375-9601(98)00831-7 [9] Obukhov, Phys. Rev. Lett. 86 pp 192– (2001) · doi:10.1103/PhysRevLett.86.192 [10] Arminjon, Phys. Rev. D 74 pp 065017– (2006) · doi:10.1103/PhysRevD.74.065017 [11] Boulanger, Phys. Rev. D 74 pp 125014– (2006) · doi:10.1103/PhysRevD.74.125014 [12] Villalba, J. Math. Phys. 31 pp 715– (1990) · Zbl 0704.53070 · doi:10.1063/1.528799 [13] M.V. Gorbatenko T.M. Gorbatenko [14] Fatibene, J. Math. Phys. 50 pp 053516– (2009) · Zbl 1187.81102 · doi:10.1063/1.3115042 [15] Arminjon, Braz. J. Phys. 38 pp 248– (2008) · doi:10.1590/S0103-97332008000200007 [16] H.A. Bethe R. Jackiw [17] L.H. Ryder [18] Arminjon, Found. Phys. 38 pp 1020– (2008) · Zbl 1161.81339 · doi:10.1007/s10701-008-9249-6 [19] Arminjon, Found. Phys. Lett. 19 pp 225– (2006) · Zbl 1119.83310 · doi:10.1007/s10702-006-0514-7 [20] Arminjon, Braz. J. Phys. 40 pp 242– (2010) · doi:10.1590/S0103-97332010000200020 [21] Leclerc, Class. Quantum Gravity 23 pp 4013– (2006) · Zbl 1096.83003 · doi:10.1088/0264-9381/23/12/001 [22] Mashhoon, Phys. Rev. Lett. 61 pp 2639– (1988) · doi:10.1103/PhysRevLett.61.2639 [23] N.D. Birrell P.C.W. Davies [24] S.A. Fulling [25] Geroch, J. Math. Phys. 9 pp 1739– (1968) · Zbl 0165.29402 · doi:10.1063/1.1664507 [26] Isham, Proc. R. Soc. Lond. A 364 pp 591– (1978) · Zbl 0396.53031 · doi:10.1098/rspa.1978.0219 [27] R. Penrose W. Rindler [28] M. Arminjon F. Reifler [29] Pauli, Ann. Phys. (Berlin) 18(5) pp 337– (1933) · Zbl 0008.03703 · doi:10.1002/andp.19334100402 [30] Pauli, Ann. Inst. Henri Poincaré 6 pp 109– (1936) [31] Audretsch, Int. J. Theor. Phys. 9 pp 323– (1974) · doi:10.1007/BF01811234 [32] Arminjon, Int. J. Geom. Methods Mod. Phys. 8 pp 1– (2011) · Zbl 1213.83024 · doi:10.1142/S0219887811005051 [33] K. Schulten [34] Reifler, Int. J. Theor. Phys. 44 pp 1307– (2005) · Zbl 1100.83021 · doi:10.1007/s10773-005-4688-8 [36] Gorbatenko, Phys. Rev. D 82 pp 104056– (2010) · doi:10.1103/PhysRevD.82.104056 [37] Gorbatenko, Phys. Rev. D 83 pp 105002– (2011) · doi:10.1103/PhysRevD.83.105002 This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.