Błaszak, Maciej; Domański, Ziemowit Canonical quantization of classical mechanics in curvilinear coordinates. Invariant quantization procedure. (English) Zbl 1343.81142 Ann. Phys. 339, 89-108 (2013). 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. Cited in 4 Documents 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 Keywords:quantum mechanics; deformation quantization; canonical transformations; Moyal product; phase space PDFBibTeX XMLCite \textit{M. Błaszak} and \textit{Z. Domański}, Ann. Phys. 339, 89--108 (2013; Zbl 1343.81142) Full Text: DOI arXiv References: [1] Podolsky, B., Phys. Rev., 32, 812-816 (1928) [2] DeWitt, B. S., Phys. Rev., 85, 653-661 (1952) [3] DeWitt, B. S., Rev. Modern Phys., 29, 377-397 (1957) [4] Gervais, J.-L.; Jevicki, A., Nuclear Phys. B, 110, 93-112 (1976) [5] Carter, B., Phys. Rev. D, 16, 3395-3414 (1977) [6] Essén, H., Amer. J. Phys., 46, 983 (1978) [7] Dekker, H., Physica A (Utrecht), 103, 586-596 (1980) [8] Liu, Z. J.; Quian, M., Trans. Amer. Math. Soc., 331, 321-333 (1992) [9] Duval, C.; Ovsienko, V., Selecta Math. (N.S.), 7, 291-320 (2001) [10] Loubon Djounga, S. E., Lett. Math. Phys., 64, 203-212 (2003) [11] Duval, C.; Valent, G., J. Math. Phys., 46, 053516 (2005) [12] Moyal, J. E., Proc. Cambridge Philos. Soc., 45, 99-124 (1949) [13] Bayen, F.; Flato, M.; Frønsdal, C.; Lichnerowicz, A.; Sternheimer, D., Ann. Phys., 111, 61-110 (1978) [14] Bayen, F.; Flato, M.; Frønsdal, C.; Lichnerowicz, A.; Sternheimer, D., Ann. Phys., 111, 111-151 (1978) [15] Dito, G.; Sternheimer, D., (Halbout, G., Deformation Quantization. Deformation Quantization, IRMA Lectures in Mathematics and Theoretical Physics, vol. 1 (2002), Walter de Gruyter: Walter de Gruyter Berlin, New York), 9-54 · Zbl 1014.53054 [16] Gutt, S., (Dito, G.; Sternheimer, D., Conférence Moshé Flato 1999: Quantization, Deformations, and Symmetries. Conférence Moshé Flato 1999: Quantization, Deformations, and Symmetries, Mathematical Physics Studies, vol. 21 (2000), Kluwer Academic Publishers: Kluwer Academic Publishers Netherlands), 217-254 [17] Weinstein, A., Séminaire Bourbaki, Vol. 36, 389-409 (1993-1994), Association des Collaborateurs de Nicolas Bourbaki [18] Curtright, T.; Fairlie, D. B.; Zachos, C., Phys. Rev. D, 58, 25002-25015 (1998) [19] Curtright, T.; Zachos, C., J. Phys. A, 32, 771-779 (1999) [20] Błaszak, M.; Domański, Z., Ann. Phys., 327, 167-211 (2012) [21] Kontsevich, M., Lett. Math. Phys., 66, 157-216 (2003) [22] Gutt, S.; Rawnsley, J., J. Geom. Phys., 29, 347-392 (1999) [24] Błaszak, M.; Domański, Z., Ann. Phys., 331, 70-96 (2013) [25] Benenti, S.; Chanu, C.; Rastelli, G., J. Math. Phys., 43, 5183-5222 (2002) [26] Benenti, S.; Chanu, C.; Rastelli, G., J. Math. Phys., 43, 5223-5253 (2002) 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.