Geodesic
Quantum Mechanics in Riemannian Space: Different Approaches to Quantization of the Geodesic Motion Compared
Teoretičeskaâ i matematičeskaâ fizika, Tome 136 (2003) no. 2, pp. 209-230
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We compare different approaches to the construction of the quantum mechanics of a particle in the general Riemannian space and space-time via quantization of motion along geodesic lines. We briefly review different quantization formalisms and the difficulties arising in their application to geodesic motion in a Riemannian configuration space. We then consider canonical, semiclassical (Pauli–De Witt), and Feynman (path-integral) formalisms in more detail and compare the quantum Hamiltonians of a particle arising in these models in the case of a static, topological elementary Riemannian configuration space. This allows selecting a unique ordering rule for the coordinate and momentum operators in the canonical formalism and a unique definition of the path integral that eliminates a part of the arbitrariness involved in the construction of the quantum mechanics of a particle in the Riemannian space. We also propose a geometric explanation of another main problem in quantization, the noninvariance of the quantum Hamiltonian and the path integral under configuration space diffeomorphisms.
Keywords: quantum mechanics, Riemannian space, geodesic motion.
Mots-clés : quantization 
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     author = {\'E. A. Tagirov},
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É. A. Tagirov. Quantum Mechanics in Riemannian Space: Different Approaches to Quantization of the Geodesic Motion Compared. Teoretičeskaâ i matematičeskaâ fizika, Tome 136 (2003) no. 2, pp. 209-230. http://geodesic.mathdoc.fr/item/TMF_2003_136_2_a2/

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