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.
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Bibliography
Anderson, B. O.: Principles of Relativity Physics (Chapter 1 ), Academic Press, New York, 1967.
Araki, H. and Yanase, M. M.: Phys. Rev. 120, 622 (1960).
Amesov, W. B.: Acta Math. 118, 95 (1967).
Arnold, V. I. and Avez, A.: Problèmes ergodiques de la mécanique classique, Gauthier-Villars, Paris, 1967.
Bachelard, G.: L’activité de la physique rationaliste (chap. 2, parag. 7 ), Presses Univ. de France, Paris, 1966.
Birkhoff, G.: Lattice Theory, Am. Math. Soc., fasc. XXV, 1967, p. 282.
Birkhoff, G.: Bull. Am. Math. Soc. 50, 764 (1944).
Birkhoff, G.: Lanice Theory, Am. Math. Soc., fasc. XXV, 1967, p. 260.
Birkhoff, G.: Dynamical Systems, Am. Math. Soc., Publ. No. I X, 1927.
Boccara, N.: Les principes de la thermodynamique classique, Presses Univ. de France, 1968.
Cameron, R. H. and Martin, W. T.: Ann. Math. 48 385 (1947).
Chandrasekhar, S.: Rev. Mod. Phys. 15, 16 (1943).
Cherry, T. M.: Proc. Cambr. Phil. Soc. 22, 287 (1925).
Collins, R. E.: Phys. Rev. 183, 1081 (1969).
Doob, J. L.: Stochastic Processes, Wiley and Sons, New York, 1964, p. 74.
Dunford, N. and Schwartz, J. T.: Linear Operators, Vol. 1, Interscience Publ., New York, 1964, p. 62.
Einstein, A., Rosen, N., and Podolsky, B.: Phys. Rev. 47, 777 (1935).
Fischer, R. (Sir): Le plan d’expérience, Editions du C.N.R.S., Paris, 1961, No. 110.
Gréa, J.: Thèse (1974), Lyon, p. 44.
Griffiths, R. G.: J. Math. Phys. 6, 1447 (1965).
Hagiara, Y.: Celestial Mechanics, Vol. 1, MIT Press, Cambridge, 1970, p. 305.
Halmos, P. R.: Lectures on Ergodic Theory, Chelsea Publ., New York, 1958.
Halmos, P. R.: Measure Theory, Von Nostrand Co., Amsterdam, 1965.
Halmos, P. R.: Bull. Am. Math. Soc., 55, No. 11 (1949).
Hemmer, P. Chr.: Dynamic and Stochastic Types of Motion in the Linear Chain, Ph.D. Thesis, Univ. Trondheim, Norvège, 1959.
Hope, E.: Ergoden Theorie, Springer-Verlag, Berlin, 1937.
Hope, E.: J. Math. Phys. 13, 51 (1934).
Khinchin, A. I.: Mathematical Foundations of Statistical Mechanics, Dover Publ., New York, 1949.
Khinchin, A. I.: Idem, réf. (21 a), p. 52.
Khinchin, A. I.: Mathematical Foundations of Quantum Statistics, Dover Publ., New York, 1960.
Koopman, B. O. and Von Neumann, J.: Proc. Nat. Acad. Sci. 18, 255 (1932).
Koopman, B. O.: Proc. Nat. Acad. Sci. 17, 315 (1931).
Trans. Am. Math. 39, 399 (1936).
Kubo, R.: Thermodynamics, North-Holland Publ., Amsterdam, 1968, p. 136.
MacLaren, M. D.: Notes on Axioms for Quantum Mechanics, Rapport ANL-7065.
Landau, L. and Lifchitz, Y.: Physique statistique, Editions MIR, Moscou, 1967, parag. 113–114.
Lewis, R. M.: Arch. Rat. Mec. Anal. 5, 355 (1960).
Mackey, G. W.: Mathematical Foundations of Quantum Mechanics, Benjamin Inc., Londres, 1963.
Mielnik, B.: Comm. Math. Phys. 15, 1 (1969).
Misra, B.: Nuovo Cimento 47, 841 (1967).
Nikodym, O. M.: The Mathematical Apparatus for Quantum Mechanics, Springer-Verlag, Berlin, 1966.
a). Von Neumann, J. V.: Ann. Math. 102, 110 (1929).
Von Neumann, J. V.: Proc. Nat. Acad. Sci. 18, 70 (1932).
Onofri, E. and Pauri, M.: J. Math. Phys. 14, 1106 (1973).
Penrose, O.: Foundations of Statistical Mechanics, Pergamon Press, New York, 1970.
Piron, C.: Règles de supersélection continue, Inst. Phys. Théor., Genève. 37, 439 (1964).
Piron, C.: Helv. Phys. Acta 37, 439 (1964).
Piron, C.: Helv. Phys. Acta 104. 887 (1963).
Piron, C. and Jauch: Helv. Phys. Acta 36 887 (1963).
Ruelle, D.: Statistical Mechanics, Benjamin Inc., Londres, 1969.
Sikorski, R.: Vol. II, Boolean Algebras, Springer-Verlag, Berlin, 1964, parag. 38.
Siegel, C. L. and Moser, J. K.: Lectures on Celestial Mechanics, Springer-Verlag, Berlin, 1971.
Szasz, G.: Théorie des treillis, chap. X, parag. 64
Szasz, G.: Théorie des treillis, chap. II, parag. 13
Szasz, G.: Théorie des Treillis, pp. 129–130, Théor. 62; Dunod, Paris, 1971.
Wang, M. C. and Uhlenbeck, G. E.: Rev. Mod. Phys. 17, 323 (1945).
Wiener, N.: Non Linear Problems in Random Theory, MIT Press, Cambridge, 1958.
Wiener, N. and Siegel, A.: Phys. Rev. 91, 1551 (1953).
Zierler, N.: Pac. J. Math. II, 1152 (1961).
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Gréa, J. (1976). Pre-Quantum Mechanics. Introduction to Models with Hidden Variables. In: Flato, M., Maric, Z., Milojevic, A., Sternheimer, D., Vigier, J.P. (eds) Quantum Mechanics, Determinism, Causality, and Particles. Mathematical Physics and Applied Mathematics, vol 1. Springer, Dordrecht. https://doi.org/10.1007/978-94-010-1440-3_6
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DOI: https://doi.org/10.1007/978-94-010-1440-3_6
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