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Canonical quantization of classical mechanics in curvilinear coordinates. Invariant quantization procedure. (English) Zbl 1343.81142

Summary: In the paper is presented an invariant quantization procedure of classical mechanics on the phase space over flat configuration space. Then, the passage to an operator representation of quantum mechanics in a Hilbert space over configuration space is derived. An explicit form of position and momentum operators as well as their appropriate ordering in arbitrary curvilinear coordinates is demonstrated. Finally, the extension of presented formalism onto non-flat case and related ambiguities of the process of quantization are discussed.

MSC:

81S05 Commutation relations and statistics as related to quantum mechanics (general)
53D55 Deformation quantization, star products
81Q35 Quantum mechanics on special spaces: manifolds, fractals, graphs, lattices
53C80 Applications of global differential geometry to the sciences
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