Journal of Mathematical Sciences

, Volume 141, Issue 4, pp 1399–1406 | Cite as

Quantization of theories with non-Lagrangian equations of motion

  • D. M. Gitman
  • V. G. Kupriyanov


We present an approach to the canonical quantization of systems with non-Lagrangian equations of motion. We first construct an action principle for equivalent first-order equations of motion. Hamiltonization and canonical quantization of the constructed Lagrangian theory is a nontrivial problem, since this theory involves time-dependent constraints. We adapt the general approach to 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 a 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. Bibliography: 23 titles.


Poisson Bracket Action Principle Lagrange Function Hamiltonian Formulation Quantization Scheme 
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  1. 1.
    V. G. Kupriyanov, S. L. Lyakhovich, and A. A. Sharapov, “Deformation quantization of linear dissipative systems,” J. Phys. A, 38, 8039–8051 (2005).CrossRefMathSciNetGoogle Scholar
  2. 2.
    P. A. M. Dirac, “Quantised singularities in the electromagnetic field,” Proc. Roy. Soc. Lond, Ser. A, 133, 60–72 (1931).Google Scholar
  3. 3.
    J. Douglas, “Solution of the inverse problem of the calculus of variations,” Trans. Amer. Math. Soc., 50, No. 71, 71–128 (1941).CrossRefMathSciNetGoogle Scholar
  4. 4.
    V. V. Dodonov, V. I. Man’ko, and V. D. Skarzhinsky, “Arbitrariness in the choice of action and quantization of the given classical equations of motion,” Preprint of the P. N. Lebedev Physical Institute, Moscow (1978).Google Scholar
  5. 5.
    W. Sarlet, “The Helmholtz conditions revisited. A new approach to the inverse problem of Lagrangian dynamics,” J. Phys. A, 15, 1503–1517 (1982).CrossRefMathSciNetGoogle Scholar
  6. 6.
    V. G. Kupriyanov, “Hamiltonian formulation and action principle for the Lorentz-Dirac system,” Int. J. Theor. Phys., 45, 1091–1106 (2006).CrossRefMathSciNetGoogle Scholar
  7. 7.
    P. Havas, “The connection between conservation laws and invariance groups: folklore, fiction, and fact,” Actra. Phys. Aust., 38, 145–167 (1973).Google Scholar
  8. 8.
    R. Santilli, “Necessary and sufficient conditions for the existence of a Lagrangian in field theory. I. Variational approach to self-adjointness for tensorial field equations,” Ann. Physics, 103, 354–408 (1977).CrossRefMathSciNetGoogle Scholar
  9. 9.
    S. Hojman and L. Urrutia, “On the inverse problem of the calculus of variations,” J. Math. Phys., 22, 1896–1903 (1981).CrossRefMathSciNetGoogle Scholar
  10. 10.
    M. Henneaux, “Equations of motion, commutation relations and ambiguities in the Lagrangian formalism,” Ann. Phys., 140, 45–64 (1982).CrossRefMathSciNetGoogle Scholar
  11. 11.
    M. Henneaux and L. Shepley, “Lagrangians for spherically symmetric potentials,” J. Math. Phys., 23, 2101–2107 (1982).CrossRefMathSciNetGoogle Scholar
  12. 12.
    J. Cislo and J. Lopuszanski, “To what extent do the classical equations of motion determine the quantization scheme?,” J. Math. Phys., 42, 5163–5176 (2001).CrossRefMathSciNetGoogle Scholar
  13. 13.
    P. Tempesta, E. Alfinito, R. Leo, and G. Soliani, “Quantum models related to fouled Hamiltonians of the harmonic oscillator,” J. Math. Phys., 43, 3538–3553 (2002).CrossRefMathSciNetGoogle Scholar
  14. 14.
    D. M. Gitman and I. V. Tyutin, Quantization of Fields with Constraints, Springer-Verlag, Berlin (1990).MATHGoogle Scholar
  15. 15.
    D. M. Gitman, S. L. Lyakhovich, M. D. Noskov, and I. V. Tyutin, “Lagrange formulation of the Hamiltonian theory of general type with constraints,” Izv. Vyssh. Uchebn. Zaved. Fiz., 29, No. 3, 104–112 (1986).MathSciNetGoogle Scholar
  16. 16.
    D. M. Gitman and I. V. Tyutin, “Hamiltonization of theories with degenerate coordinates,” Nuclear Phys. B, 630, 509–527 (2002).CrossRefMathSciNetGoogle Scholar
  17. 17.
    F. A. Berezin and M. A. Shubin, The Schrödinger Equation, Kluwer, Dordrecht (1991).MATHGoogle Scholar
  18. 18.
    H. R. Lewis and W. B. Riesenfeld, “An exact quantum theory of the time-dependent harmonic oscillator and of a charged particle in a time-dependent electromagnetic field,” J. Math. Phys., 10, 1458–1473 (1969).CrossRefMathSciNetGoogle Scholar
  19. 19.
    N. A. Chernikov, “The system whose Hamiltonian is a time-dependent quadratic form in coordinates and momenta,” Comm. Joint Inst. Nuclear Res., Dubna (1990).Google Scholar
  20. 20.
    J. K. Kim and S. P. Kim, “One-parameter squeezed Gaussian states of a time-dependent harmonic oscillator and the selection rule for vacuum states,” J. Phys. A, 32, 2711–2718 (1999).CrossRefMathSciNetGoogle Scholar
  21. 21.
    V. V. Dodonov, O. V. Man’ko, and V. I. Man’ko, “Quantum nonstationary oscillator: models and applications,” J. Russian Laser Res., 16, 1–56 (1995).Google Scholar
  22. 22.
    H. Kim and J. Yee, “Time-dependent driven anharmonic oscillator and adiabaticity,” Phys. Rev. A, 66, 032117 (2002).Google Scholar
  23. 23.
    Kyu Hwang Yeon, Chung In Um, and T. F. George, “Time-dependent general quantum quadratic Hamiltonian system,” Phys. Rev. A, 68, 052108 (2003).Google Scholar

Copyright information

© Springer Science+Business Media, Inc. 2007

Authors and Affiliations

  • D. M. Gitman
    • 1
  • V. G. Kupriyanov
    • 1
    • 2
  1. 1.Instituto de FísicaUniversidade de São PauloBrazil
  2. 2.Tomsk State UniversityTomskRussia

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