Foundations of Physics

, Volume 48, Issue 11, pp 1648–1667 | Cite as

From Classical to Quantum Models: The Regularising Rôle of Integrals, Symmetry and Probabilities

  • Jean-Pierre GazeauEmail author


In physics, one is often misled in thinking that the mathematical model of a system is part of or is that system itself. Think of expressions commonly used in physics like “point” particle, motion “on the line”, “smooth” observables, wave function, and even “going to infinity”, without forgetting perplexing phrases like “classical world” versus “quantum world”.... On the other hand, when a mathematical model becomes really inoperative in regard with correct predictions, one is forced to replace it with a new one. It is precisely what happened with the emergence of quantum physics. Classical models were (progressively) superseded by quantum ones through quantization prescriptions. These procedures appear often as ad hoc recipes. In the present paper, well defined quantizations, based on integral calculus and Weyl–Heisenberg symmetry, are described in simple terms through one of the most basic examples of mechanics. Starting from (quasi-) probability distribution(s) on the Euclidean plane viewed as the phase space for the motion of a point particle on the line, i.e., its classical model, we will show how to build corresponding quantum model(s) and associated probabilities (e.g. Husimi) or quasi-probabilities (e.g. Wigner) distributions. We highlight the regularizing rôle of such procedures with the familiar example of the motion of a particle with a variable mass and submitted to a step potential.



The author is indebted to the Centro Brasileiro de Pesquisas Físicas (Rio de Janeiro) and CNPq Agency (Brazil), and the Institute for Research in Fundamental Sciences (IPM, Tehran) for financial support. He also thanks the CBPF and the IPM for hospitality. He is grateful to Evaldo M.F. Curado (CBPF) for valuable comments on the content of this work.


  1. 1.
    Inönü, E., Wigner, E.: Representations of the Galilei group. Nuovo Cimento 9, 705–718 (1952)MathSciNetCrossRefGoogle Scholar
  2. 2.
    Wightman, A.S.: On the localizibility of quantum mechanical systems. Rev. Mod. Phys. 34, 845–872 (1962)ADSMathSciNetCrossRefGoogle Scholar
  3. 3.
    Berezin, F.A.: Quantization. Mathematics of the USSR-Izvestiya 8(5), 1109–1165 (1974)ADSCrossRefGoogle Scholar
  4. 4.
    Berezin, F.A.: General concept of quantization. Commun. Math. Phys. 40, 153–174 (1975)ADSMathSciNetCrossRefGoogle Scholar
  5. 5.
    Ali, S.T., Engliš, M.: Quantization methods: a guide for physicists and analysts. Rev. Math. Phys. 17, 391 (2005)MathSciNetCrossRefGoogle Scholar
  6. 6.
    Landsman, N.P.: Between classical and quantum. In: Earman, J., Butterfield, J. (eds.) Philosophy of Physics, Handbook of the Philosophy of Science, vol. 2. Elsevier, Amsterdam (2006)Google Scholar
  7. 7.
    Combescure, M., Robert, D.: Coherent States and Applications in Mathematical Physics, Theoretical and Mathematical Physics. Springer, Dordrecht (2012)zbMATHGoogle Scholar
  8. 8.
    de Gosson, M.: Born-Jordan Quantization, Fundamental Theories of Physics, vol. 182. Springer, Cham (2016)CrossRefGoogle Scholar
  9. 9.
    Bergeron, H., Gazeau, J.P., Youssef, A.: Are the Weyl and coherent state descriptions physically equivalent? Phys. Lett. A 377, 598–605 (2013)ADSMathSciNetCrossRefGoogle Scholar
  10. 10.
    Barut, A.O., Ra̧czka, R.: Theory of Group Representations and Applications. PWN, Warszawa (1977)Google Scholar
  11. 11.
    von Neumann, J.: Die eindeutigkeit der Schröderschen Operatoren. Math. Ann. 104, 570–578 (1931)MathSciNetCrossRefGoogle Scholar
  12. 12.
    von Neumann, J.: Mathematical foundations of quantum mechanics. Princeton University Press, Princeton (1955)zbMATHGoogle Scholar
  13. 13.
    Perelomov, A.M.: Generalized Coherent States and Their Applications. Springer, Berlin (1986)CrossRefGoogle Scholar
  14. 14.
    Bergeron, H., Gazeau, J.-P.: Integral quantizations with two basic examples. Ann. Phys. (NY) 344, 43–68 (2014)ADSMathSciNetCrossRefGoogle Scholar
  15. 15.
    Bergeron, H., Curado, E.M.F., Gazeau, J.-P., Rodrigues, Ligia M.C.S.: Weyl-Heisenberg integral quantization(s): a compendium (2017). arXiv:1703.08443 [quant-ph]
  16. 16.
    Cohen, L.: Generalized phase-space distribution functions. J. Math. Phys. 7, 781–786 (1966)ADSMathSciNetCrossRefGoogle Scholar
  17. 17.
    Cohen, L.: The Weyl Operator and Its Generalization. Pseudo-Differential Operators: Theory and Applications, vol. 9. Birkhaüser, Basel (2013)CrossRefGoogle Scholar
  18. 18.
    Agarwal, B.S., Wolf, E.: Calculus for functions of noncommuting operators and general phase-space methods in quantum mechanics. Phys. Rev. D 2, 2161, (I), 2187 (II), 2206 (III) (1970)Google Scholar
  19. 19.
    Akhiezer, N.I., Glazman, I.M.: Theory of Linear Operators in Hilbert Space. Pitman, Boston (1981)zbMATHGoogle Scholar
  20. 20.
    Lévy-Leblond, J.-M.: The pedagogical role and epistemological significance of group theory in quantum mechanics. Riv. Nuovo Cimento 4, 99–143 (1974)MathSciNetCrossRefGoogle Scholar
  21. 21.
    Lévy-Leblond, J.M.: Elementary quantum models with position-dependent mass. Eur. J. Phys. 13, 215–218 (1992)CrossRefGoogle Scholar
  22. 22.
    Lévy-Leblond, J.M.: Position-dependent effective mass and Galilean invariance. Phys. Rev. A 52, 1845–1849 (1995)ADSMathSciNetCrossRefGoogle Scholar
  23. 23.
    Cordero, E., de Gosson, M. Nicola, F.: On the invertibility of Born-Jordan quantization. arXiv:1507.00144 [math.FA]
  24. 24.
    Abramowitz, M., Stegun, I.A. (eds.): Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, 9th printing. Dover, New York (1972)zbMATHGoogle Scholar
  25. 25.
    Gazeau, J.-P., Murenzi, R.: Covariant affine integral quantization(s). J. Math. Phys. 57, 052102 (2016)ADSMathSciNetCrossRefGoogle Scholar
  26. 26.
    Almeida, C.R., Bergeron, H., Gazeau, J.-P., Scardua, A.C.: Three examples of quantum dynamics on the half-line with smooth bouncing. Ann. Phys. 392, 206–228 (2018)ADSMathSciNetCrossRefGoogle Scholar
  27. 27.
    Fresneda, R., Gazeau, J.-P., Noguera, D.: Quantum localisation on the circle. J. Math. Phys. 59, 052105 (2018)ADSMathSciNetCrossRefGoogle Scholar
  28. 28.
    Gazeau, J.-P., Koide, T., Murenzi, R.: More quantum repulsive effect in rotating frame. EPL 118, 50004 (2017)ADSCrossRefGoogle Scholar
  29. 29.
    Gazeau, J.-P., Koide, T.: Quantum motion on the half-line from Weyl-Heisenberg integral quantization (in preparation)Google Scholar

Copyright information

© Springer Science+Business Media, LLC, part of Springer Nature 2018

Authors and Affiliations

  1. 1.APC, UMR 7164Univ Paris Diderot, Sorbonne Paris CitéParisFrance
  2. 2.Centro Brasileiro de Pesquisas FísicasRio de JaneiroBrazil

Personalised recommendations