Geodesic
Quantization of theories with non-Lagrangian equations of motion
Zapiski Nauchnykh Seminarov POMI, Representation theory, dynamical systems, combinatorial and algoritmic methods. Part XIV, Tome 331 (2006), pp. 30-42 Cet article a éte moissonné depuis la source Math-Net.Ru

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We present an approach to the canonical quantization of systems with non-Lagrangian equations of motion. We first construct an action principle for an equivalent first-order equations of motion. A hamiltonization and canonical quantization of the constructed Lagrangian theory is a non-trivial problem, since this theory involves time-dependent constraints. We adopt the general approach of hamiltonization and canonical quantization for such theories (Gitman, Tyutin, 1990) to the case under consideration. There exists an ambiguity (not reduced to a total time derivative) in associating a Lagrange function with the given set of equations. We give a complete description of this ambiguity. It is remarkable that the quantization scheme developed in the case under consideration provides arguments in favor of fixing this ambiguity. Finally, as an example, we consider the canonical quantization of a general quadratic theory. 
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D. M. Gitman; V. G. Kupriyanov. Quantization of theories with non-Lagrangian equations of motion. Zapiski Nauchnykh Seminarov POMI, Representation theory, dynamical systems, combinatorial and algoritmic methods. Part XIV, Tome 331 (2006), pp. 30-42. http://geodesic.mathdoc.fr/item/ZNSL_2006_331_a3/

[1] V. G. Kupriyanov, S. L. Lyakhovich and A. A. Sharapov, J. Phys. A: Math. Gen., 38 (2005), 8039 | DOI | MR | Zbl

[2] P. A. M. Dirac, Proc. Roy. Soc., A133 (1931), 60 | DOI

[3] J. Douglas, Trans. Amer. Math. Soc, 50:71 (1941), 71–87 | DOI | MR

[4] V. V. Dodonov, V. I. Man'ko, and V. D. Skarzhinsky, Arbitrariness in the choice of action and quantization of the given classical eqations of motion, Preprint of P. N. Lebedev Physical Institute, 1978 | MR

[5] W. Sarlet, J. Phys. A: Math. Gen., 15 (1982), 1503 | DOI | MR | Zbl

[6] V. G. Kupriyanov, Int. J. Theor. Phys., 45:6 (2006), 1129–1144 | DOI | MR | Zbl

[7] P. Havas, Actra. Phys. Aust., 38 (1973), 145

[8] R. Santilli, Ann. Phys. (NY), 103 (1977), 354–359 | DOI | MR

[9] S. Hojman, L. Urrutia, J. Math. Phys., 22 (1981) | MR

[10] M. Henneaux, Ann. Phys., 140 (1982), 45 | DOI | MR | Zbl

[11] M. Henneaux, L. Sheplley, J. Math. Phys., 23 (1982), 2101 | DOI | MR | Zbl

[12] J. Cislo, J. Lopuzanski, J. Math. Phys., 42 (2001) | DOI | MR | Zbl

[13] P. Tempesta, E. Alfinito, R. Leo, G. Soliani, J. Math. Phys., 43 (2002), 3583 | DOI | MR

[14] D. M. Gitman, I. V. Tyutin, Quantization of Fields with Constraints, Springer-Verlag, 1990 | MR | Zbl

[15] D. M. Gitman, S. L. Lyakhovich, M. D. Noskov, I. V. Tyutin, Izv. VUZov, 29:3 (1986), 104–112 | MR

[16] D. M. Gitman, I. V. Tyutin, Nucl. Phys. B, 630:3 (2002), 509 | DOI | MR | Zbl

[17] F. A. Berezin, M. A. Shubin, Schrödinger Equation, Kluwer Publ., New York, Amsterdam, 1991 | MR | Zbl

[18] H. R. Lewis, W. B. Riesenfeld, J. Math. Phys., 10 (1969), 1458 | DOI

[19] N. A. Chernikov, “The system whose hamiltonian is a time-dependent quadratic form in coordinates and momenta”, Communications for the Joint Institute for Nuclear Research (Dubna, 1990) | MR

[20] H. Kim, J. Yae, J. Phys. A: Math. Gen., 32 (1999), 2711 | DOI | Zbl

[21] V. V. Dodonov, O. V. Man'ko, V. I. Man'ko, J. Russian Laser Research, 16 (1995), 1 | DOI

[22] H. Kim, J. Yae, Phys. Rev., A66 (2002), 032117

[23] Kyu Hwang Yeon, ChungIn UM, T. F. George, Phys. Rev., A68 (2003), 052108 | MR