Abstract
We give a review of some works where it is shown that certain quantum-like features are exhibited by classical systems. Two kinds of problems are considered. The first one concerns the specific heat of crystals (the so called Fermi–Pasta–Ulam problem), where a glassy behavior is observed, and the energy distribution is found to be of Planck-like type. The second kind of problems concerns the self-interaction of a charged particle with the electromagnetic field, where an analog of the tunnel effect is proven to exist, and moreover some nonlocal effects are exhibited, leading to a natural hidden variable theory which violates Bell's inequalities.
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REFERENCES
L. Galgani and A. Scotti, Phys. Rev. Lett. 28, 1173 (1972).
L. Galgani and A. Scotti, “Recent progress in classical nonlinear dynamics,” Rivista Nuovo Cimento 2, 189 (1972).
E. Fermi, J. Pasta, and S. Ulam, Los Alamos Report, No. LA-1940 (1955); later published in E. Fermi, Collected Papers (University of Chicago Press, Chicago, 1965), and Lect. Appl. Math. 15, 143 (1974).
L. Galgani, C. Angaroni, L. Forti, A. Giorgilli, and F. Guerra, Phys. Lett. A 139, 221 (1989).
D. Bambusi and L. Galgani, Ann. Inst. H. Poincaré, Phys. Théor. 58, 155–171 (1993).
D. Bambusi and D. Noja, Lett. Math. Phys. 37, 449 (1996).
D. Noja and A. Posilicano, Ann. Inst. H. Poincaré, Phys. Théor. 71, 425 (1999).
A. Carati, Found. Phys. 28, 843–853 (1998); see also Math. Rev. 2000a:78006.
A. Carati, P. Delzanno, L. Galgani, and J. Sassarini, Nonlinearity 8, 65–76 (1995).
A. Carati and L. Galgani, Nonlinearity 6, 905–914 (1993).
A. Carati and L. Galgani, Nuovo Cimento B 114, 489–500 (1999).
J. O. Hirschfelder, C. H. Curtiss, and R. B. Bird, Molecular Theory of Gases and Liquids (Wiley, New York, 1965).
L. D. Landau and E. M. Lifshitz, The Classical Theory of Fields (Pergamon, Oxford, 1962).
A. Carati and L. Galgani, Physica A 280, 106–114 (2000).
A. Carati and L. Galgani, “Einstein's nonconventional conception of the photon, and the modern theory of dynamical system,” in Chance in Physics, D. Dürr, G. Ghirardi, and N. Zanghi, eds. (Springer, Berlin, to appear).
G. Gallavotti, Statistical Mechanics: A Short Treatise (Springer, Berlin, 1999).
E. Fermi, Nuovo Cimento 26, 105 (1923).
G. Benettin, G. Ferrari, L. Galgani, and A. Giorgilli, Nuovo Cimento B 72, 137 (1982).
F. M. Izrailev and B. V. Chirikov, Sov. Phys. Dokl. 11, 30 (1966).
L. Boltzmann, Nature 51, 413 (1895). L. Boltzmann, Lectures on Gas Theory (University of California Press, 1966), Sec. 45.
G. Benettin, L. Galgani, and A. Giorgilli, Nature 311, 444 (1984).
J. Jäckle, Rep. Prog. Phys. 49, 171–231 (1986). J. Ja ckle, Physica A 162, 377–404 (1990).
A. Carati and L. Galgani, J. Stat. Phys. 94, 859 (1999).
G. Benettin, L. Galgani, and A. Giorgilli, Comm. Math. Phys. 121, 557 (1989).
L. Galgani, A. Giorgilli, A. Martinoli, and S. Vanzini, Physica D 59, 334–348 (1992).
G. Benettin, L. Galgani, and A. Giorgilli, Phys. Lett. A 120, 23 (1987).
A. Carati and L. Galgani, Phys. Rev. E 61, 4791 (2000).
M. Planck, Verh. D. Phys. Ges. 2 (1900); reprinted in H. Kangro, Planck's Original Papers in Quantum Physics (Taylor 6 Francis, London, 1972).
A. Einstein, Phys. Z. 10, 185 (1909).
A. Einstein, Contribution to the 1911 Solvay Conference, in The Collected Papers of A. Einstein (Princeton University Press, Princeton, 1993), Vol. 3, No. 26.
O. Baldan and G. Benettin, J. Stat. Phys. 62, 201 (1991). G. Benettin, A. Carati, and P. Sempio, J. Stat. Phys. 73, 175 (1993).
G. Benettin, A. Carati, and G. Gallavotti, Nonlinearity 10, 479 (1997).
A. Ponno, L. Galgani, and F. Guerra, Phys. Rev. E 61, 7081 (2000).
F. Bonechi and S. De Bievre, “Exponental mixing and log h time scales in quantized hyperbolic maps on the torus,” mparc 99–381.
D. Bambusi, S. Graffi, and T. Paul, Asymptotic Analysis 21, 149–160 (1999).
L. Galgani, in Non-Linear Evolution and Chaotic Phenomena, G. Gallavotti and P. F. Zweifel, eds. (NATO ASI Series R71B, Vol. 176) (Plenum, New York, 1988).
H. Poincarè, J. Phys. Thè or. Appl. 5, 5–34 (1912), in Oeuvres IX, pp. 626–653.
Physics at the British Association, Nature 92, 304–309 (1913).
P. P. Ewald, Bericht über die Tagung der British Association in Birmingham (10-17 September), Phys. Z. 14, 1297 (1913); see especially p. 1298.
N. O. Birge and S. R. Nagel, Phys. Rev. Lett. 54, 2674 (1985). N. O. Birge, Phys. Rev. B 34, 1631 (1986).
A. Einstein, Ann. der Phys. 22, 180 (1907).
C. Cercignani, L. Galgani, and A. Scotti, Phys. Lett. A 38, 403 (1972).
L. Galgani, in Stochastic Processes in Classical and Quantum Systems, S. Albeverio, G. Casati, and D. Merlini, eds., pp. 269–277 (Lecture Notes in Physics, Vol. 262) (Springer, Berlin, 1986).
M. Abraham, Ann. Phys. (Leipzig) 10, 105 (1903).
P. A. M. Dirac, Proc. Royal Soc. (London) A 167, 148–168 (1938).
P. A. M. Dirac, Ann. Inst. Poincaré9, 13 (1938).
R. P. Feynman, The Feynman Lectures on Physics, Vol. 2 (Addison-Wesley, Reading, 1964).
S. Coleman and R. E. Norton, Phys. Rev. 125, 1422–1428 (1962).
M. Bertini, D. Noja, and A. Posilicano, “Quantum electrodynamics of point particles in the dipole approximation,” in preparation.
E. Nelson, Quantum Fluctuations (Princeton University Press, Princeton, 1985).
E. Nelson, in Stochastic Processes in Classical and Quantum Systems (Ascona, 1985) (Lecture Notes in Physics, Vol. 262) (Springer, Berlin, 1986), pp. 438–469. E. Nelson, in E cole d'E te de Probabilite s de Saint-Flour XVXVII (1985-1987) (Lecture Notes in Mathematics, Vol. 1362) (Springer, Berlin, 1988), pp. 427–450.
B. Ruf and P. N. Srikanth, Rev. Math. Phys., in print.
J. K. Hale and A. P. Stokes, J. Math. Phys. 3, 70 (1962).
J. S. Bell, “Einstein Podolsky Rosen Experiments,” in Proceedings, Frontier Problems in High-Energy Physics, Pisa, 1976; reprinted in J. S. Bell, Speakable and Unspeakable in Quantum Mechanics (Cambridge University Press, Cambridge, 1987); see note 24.
M. Esfeld, Stud. Hist. Phil. Mod. Phys. B 30, 155 (1999).
E. C. G. Stueckelberg, Helv. Phys. Acta 14, 588–594 (1941). R. P. Feynman, Phys. Rev. 74, 939 (1948).
C. J. Eliezer, Proc. Cambr. Phil. Soc. 39, 173 (1943).
A. Carati, “An extension of Eliezer's theorem on the Abraham Lorentz Dirac equation,” in preparation.
J. De Luca, Phys. Rev. Lett. 80, 680 (1998).
G. 't Hooft, Class. Quantum Grav. 16, 3263–3279 (1999).
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Carati, A., Galgani, L. Theory of Dynamical Systems and the Relations Between Classical and Quantum Mechanics. Foundations of Physics 31, 69–87 (2001). https://doi.org/10.1023/A:1004103921290
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DOI: https://doi.org/10.1023/A:1004103921290