Geodesic
General-covariant quantum mechanics in Riemannian space-time III.~Dirac's particle
Teoretičeskaâ i matematičeskaâ fizika, Tome 106 (1996) no. 1, pp. 122-132

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A general covariant analog of the standard non-relativistic Quantum Mechanics with relativistic corrections in normal geodesic frames in the general Riemannian space-time is constructed for the Dirac particle. Not only the Pauli equation with hermitean hamiltonian and the pre-Hilbert structure of space of its solutions but also the matrix elements of hermitean operators of momentum, (curvilinear) spatial coordinates and spin of the particle are deduced as general-covariant asymptotic approximation in $c^{-2}$, $c$ being the velocity of light, to their naturally determined general-relativistic pre-images. It is shown that the hamiltonian in the Pauli equation originated by the Dirac equation is unitary equivalent to the operator of energy, originated by the metric energy-momentum tensor of the spinor field. Commutation and other properties of the observables connected with the considered change of geometrical background of Quantum Mechanics are briefly discussed. 
@article{TMF_1996_106_1_a9,
     author = {\'E. A. Tagirov},
     title = {General-covariant quantum mechanics  in {Riemannian} space-time {III.~Dirac's} particle},
     journal = {Teoreti\v{c}eska\^a i matemati\v{c}eska\^a fizika},
     pages = {122--132},
     publisher = {mathdoc},
     volume = {106},
     number = {1},
     year = {1996},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/TMF_1996_106_1_a9/}
}
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É. A. Tagirov. General-covariant quantum mechanics  in Riemannian space-time III.~Dirac's particle. Teoretičeskaâ i matematičeskaâ fizika, Tome 106 (1996) no. 1, pp. 122-132. http://geodesic.mathdoc.fr/item/TMF_1996_106_1_a9/