Pre-Quantum Mechanics. Introduction to Models with Hidden Variables

  • Jean Gréa
Chapter
Part of the Mathematical Physics and Applied Mathematics book series (MPAM, volume 1)

Abstract

Within the context of formalisms of hidden variable type, we consider the models used to describe mechanical systems before the introduction of the quantum model. We give an account of the characteristics of the theoretical models and their relationships with experimental methodology. We then study in succession the models of analytical, pre-ergodic, ergodic, stochastic, statistical and thermodynamic mechanics. At each stage, the physical hypothesis is enunciated by postulate corresponding to the type of description of the reality of the model. Starting from this postulate, the physical propositions which are meaningful for the model under consideration are defined and their logical structure is indicated. It is then found that on passing from one level of description to another, we can obtain successively Boolean lattices embedded in lattices of continuous geometric type, which are themselves embedded in Boolean lattices. It is therefore possible to envisage a more detailed description than that given by the quantum lattice, and to construct it by analogy.

Keywords

Hilbert Space Hide Variable Borel Function Quantum Model Modular Lattice 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.
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Bibliography

  1. 1.
    Anderson, B. O.: Principles of Relativity Physics (Chapter 1 ), Academic Press, New York, 1967.Google Scholar
  2. 2.
    Araki, H. and Yanase, M. M.: Phys. Rev. 120, 622 (1960).MathSciNetADSMATHCrossRefGoogle Scholar
  3. 3.
    Amesov, W. B.: Acta Math. 118, 95 (1967).MathSciNetCrossRefGoogle Scholar
  4. 4.
    Arnold, V. I. and Avez, A.: Problèmes ergodiques de la mécanique classique, Gauthier-Villars, Paris, 1967.Google Scholar
  5. 5.
    Bachelard, G.: L’activité de la physique rationaliste (chap. 2, parag. 7 ), Presses Univ. de France, Paris, 1966.Google Scholar
  6. (a).
    Birkhoff, G.: Lattice Theory, Am. Math. Soc., fasc. XXV, 1967, p. 282.Google Scholar
  7. (b).
    Birkhoff, G.: Bull. Am. Math. Soc. 50, 764 (1944).MathSciNetMATHCrossRefGoogle Scholar
  8. (c).
    Birkhoff, G.: Lanice Theory, Am. Math. Soc., fasc. XXV, 1967, p. 260.Google Scholar
  9. (d).
    Birkhoff, G.: Dynamical Systems, Am. Math. Soc., Publ. No. I X, 1927.Google Scholar
  10. 6b.
    Boccara, N.: Les principes de la thermodynamique classique, Presses Univ. de France, 1968.Google Scholar
  11. 7.
    Cameron, R. H. and Martin, W. T.: Ann. Math. 48 385 (1947).MathSciNetMATHCrossRefGoogle Scholar
  12. 8.
    Chandrasekhar, S.: Rev. Mod. Phys. 15, 16 (1943).ADSCrossRefGoogle Scholar
  13. 9.
    Cherry, T. M.: Proc. Cambr. Phil. Soc. 22, 287 (1925).ADSCrossRefGoogle Scholar
  14. 10.
    Collins, R. E.: Phys. Rev. 183, 1081 (1969).ADSMATHCrossRefGoogle Scholar
  15. 11.
    Doob, J. L.: Stochastic Processes, Wiley and Sons, New York, 1964, p. 74.Google Scholar
  16. 12.
    Dunford, N. and Schwartz, J. T.: Linear Operators, Vol. 1, Interscience Publ., New York, 1964, p. 62.Google Scholar
  17. 13.
    Einstein, A., Rosen, N., and Podolsky, B.: Phys. Rev. 47, 777 (1935).ADSMATHCrossRefGoogle Scholar
  18. 14.
    Fischer, R. (Sir): Le plan d’expérience, Editions du C.N.R.S., Paris, 1961, No. 110.Google Scholar
  19. 15.
    Gréa, J.: Thèse (1974), Lyon, p. 44.Google Scholar
  20. 16.
    Griffiths, R. G.: J. Math. Phys. 6, 1447 (1965).ADSCrossRefGoogle Scholar
  21. 17.
    Hagiara, Y.: Celestial Mechanics, Vol. 1, MIT Press, Cambridge, 1970, p. 305.Google Scholar
  22. 18.(a)
    Halmos, P. R.: Lectures on Ergodic Theory, Chelsea Publ., New York, 1958.Google Scholar
  23. (b).
    Halmos, P. R.: Measure Theory, Von Nostrand Co., Amsterdam, 1965.Google Scholar
  24. (c).
    Halmos, P. R.: Bull. Am. Math. Soc., 55, No. 11 (1949).MathSciNetGoogle Scholar
  25. 19.
    Hemmer, P. Chr.: Dynamic and Stochastic Types of Motion in the Linear Chain, Ph.D. Thesis, Univ. Trondheim, Norvège, 1959.Google Scholar
  26. 20.(a)
    Hope, E.: Ergoden Theorie, Springer-Verlag, Berlin, 1937.Google Scholar
  27. (b).
    Hope, E.: J. Math. Phys. 13, 51 (1934).Google Scholar
  28. 21.(a)
    Khinchin, A. I.: Mathematical Foundations of Statistical Mechanics, Dover Publ., New York, 1949.MATHGoogle Scholar
  29. (b).
    Khinchin, A. I.: Idem, réf. (21 a), p. 52.Google Scholar
  30. (c).
    Khinchin, A. I.: Mathematical Foundations of Quantum Statistics, Dover Publ., New York, 1960.MATHGoogle Scholar
  31. 22.
    Koopman, B. O. and Von Neumann, J.: Proc. Nat. Acad. Sci. 18, 255 (1932).ADSCrossRefGoogle Scholar
  32. 23.
    Koopman, B. O.: Proc. Nat. Acad. Sci. 17, 315 (1931).ADSCrossRefGoogle Scholar
  33. Trans. Am. Math. 39, 399 (1936).CrossRefGoogle Scholar
  34. 24.
    Kubo, R.: Thermodynamics, North-Holland Publ., Amsterdam, 1968, p. 136.Google Scholar
  35. 25.
    MacLaren, M. D.: Notes on Axioms for Quantum Mechanics, Rapport ANL-7065.Google Scholar
  36. 26.
    Landau, L. and Lifchitz, Y.: Physique statistique, Editions MIR, Moscou, 1967, parag. 113–114.Google Scholar
  37. 27.
    Lewis, R. M.: Arch. Rat. Mec. Anal. 5, 355 (1960).MATHCrossRefGoogle Scholar
  38. 28.
    Mackey, G. W.: Mathematical Foundations of Quantum Mechanics, Benjamin Inc., Londres, 1963.Google Scholar
  39. 29.
    Mielnik, B.: Comm. Math. Phys. 15, 1 (1969).MathSciNetADSMATHCrossRefGoogle Scholar
  40. 30.
    Misra, B.: Nuovo Cimento 47, 841 (1967).ADSMATHCrossRefGoogle Scholar
  41. 31.
    Nikodym, O. M.: The Mathematical Apparatus for Quantum Mechanics, Springer-Verlag, Berlin, 1966.Google Scholar
  42. 32.(a)
    a). Von Neumann, J. V.: Ann. Math. 102, 110 (1929).Google Scholar
  43. (b).
    Von Neumann, J. V.: Proc. Nat. Acad. Sci. 18, 70 (1932).ADSCrossRefGoogle Scholar
  44. 33.
    Onofri, E. and Pauri, M.: J. Math. Phys. 14, 1106 (1973).MathSciNetADSCrossRefGoogle Scholar
  45. 34.
    Penrose, O.: Foundations of Statistical Mechanics, Pergamon Press, New York, 1970.MATHGoogle Scholar
  46. 35.(a)
    Piron, C.: Règles de supersélection continue, Inst. Phys. Théor., Genève. 37, 439 (1964).MathSciNetMATHGoogle Scholar
  47. (b).
    Piron, C.: Helv. Phys. Acta 37, 439 (1964).MathSciNetMATHGoogle Scholar
  48. (c).
    Piron, C.: Helv. Phys. Acta 104. 887 (1963).Google Scholar
  49. 36.
    Piron, C. and Jauch: Helv. Phys. Acta 36 887 (1963).MathSciNetGoogle Scholar
  50. 37a.
    Ruelle, D.: Statistical Mechanics, Benjamin Inc., Londres, 1969.Google Scholar
  51. b.
    Sikorski, R.: Vol. II, Boolean Algebras, Springer-Verlag, Berlin, 1964, parag. 38.Google Scholar
  52. 38.
    Siegel, C. L. and Moser, J. K.: Lectures on Celestial Mechanics, Springer-Verlag, Berlin, 1971.MATHGoogle Scholar
  53. 39.(a)
    Szasz, G.: Théorie des treillis, chap. X, parag. 64Google Scholar
  54. (b).
    Szasz, G.: Théorie des treillis, chap. II, parag. 13Google Scholar
  55. (c).
    Szasz, G.: Théorie des Treillis, pp. 129–130, Théor. 62; Dunod, Paris, 1971.Google Scholar
  56. 40.
    Wang, M. C. and Uhlenbeck, G. E.: Rev. Mod. Phys. 17, 323 (1945).MathSciNetADSMATHCrossRefGoogle Scholar
  57. 41.
    Wiener, N.: Non Linear Problems in Random Theory, MIT Press, Cambridge, 1958.Google Scholar
  58. 42.
    Wiener, N. and Siegel, A.: Phys. Rev. 91, 1551 (1953).MathSciNetADSMATHCrossRefGoogle Scholar
  59. 43.
    Zierler, N.: Pac. J. Math. II, 1152 (1961).MathSciNetGoogle Scholar

Copyright information

© D. Reidel Publishing Company, Dordrecht, Holland 1976

Authors and Affiliations

  • Jean Gréa
    • 1
  1. 1.Institut de Physique NucléaireUniversité Claude BernardLyonFrance

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