Geodesic
Quantum mechanics in~Riemannian spacetime.
Teoretičeskaâ i matematičeskaâ fizika, Tome 84 (1990) no. 3, pp. 419-430

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A study is made of the $c^{-2}$ asymptotics ($c$ is the speed of light) of the theory of a complex scalar field in a general Riemannian spacetime; the field interacts with an external electromagnetic field. In a freely falling (Gaussian normal) frame of reference we obtain a generally covariant analog of the Schrödinger equation for a scalar particle in external gravitational and electromagnetic fields with relativistic corrections of arbitrary order. It is shown that allowance for the geometrical variation in time of the phase-space element leads to a Hamiltonian that is (asymptotically) Hermitian with respect to the standard scalar product, and this provides a basis for the Born interpretation of the corresponding wave functions. 
@article{TMF_1990_84_3_a8,
     author = {\'E. A. Tagirov},
     title = {Quantum mechanics {in~Riemannian} spacetime.},
     journal = {Teoreti\v{c}eska\^a i matemati\v{c}eska\^a fizika},
     pages = {419--430},
     publisher = {mathdoc},
     volume = {84},
     number = {3},
     year = {1990},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/TMF_1990_84_3_a8/}
}
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É. A. Tagirov. Quantum mechanics in~Riemannian spacetime.. Teoretičeskaâ i matematičeskaâ fizika, Tome 84 (1990) no. 3, pp. 419-430. http://geodesic.mathdoc.fr/item/TMF_1990_84_3_a8/