Cette Note esquisse une construction mathématique simple et naturelle du caractère probabiliste de la mécanique quantique. Elle utilise l'analyse non standard et repose sur l'interprétation due à Feynman, mise en avant dans certaines approches fractales, du principe d'incertitude de Heisenberg, c'est-à-dire des fluctuations quantiques. On aboutit ainsi à des équations différentielles stochastiques, comme dans la mécanique stochastique de Nelson, découlant de marches aléatoires infinitésimales.
This Note is sketching a simple and natural mathematical construction for explaining the probabilistic nature of quantum mechanics. It employs nonstandard analysis and is based on Feynman's interpretation of the Heisenberg uncertainty principle, i.e., of the quantum fluctuations, which was brought to the forefront in some fractal approaches. It results, as in Nelson's stochastic mechanics, in stochastic differential equations which are deduced from infinitesimal random walks.
Accepté le :
Publié le :
@article{CRMATH_2007__344_10_663_0,
author = {Fliess, Michel},
title = {Probabilit\'es et fluctuations quantiques},
journal = {Comptes Rendus. Math\'ematique},
pages = {663--668},
publisher = {Elsevier},
volume = {344},
number = {10},
year = {2007},
doi = {10.1016/j.crma.2007.04.001},
language = {fr},
url = {http://www.numdam.org/articles/10.1016/j.crma.2007.04.001/}
}
TY - JOUR AU - Fliess, Michel TI - Probabilités et fluctuations quantiques JO - Comptes Rendus. Mathématique PY - 2007 SP - 663 EP - 668 VL - 344 IS - 10 PB - Elsevier UR - http://www.numdam.org/articles/10.1016/j.crma.2007.04.001/ DO - 10.1016/j.crma.2007.04.001 LA - fr ID - CRMATH_2007__344_10_663_0 ER -
Fliess, Michel. Probabilités et fluctuations quantiques. Comptes Rendus. Mathématique, Tome 344 (2007) no. 10, pp. 663-668. doi : 10.1016/j.crma.2007.04.001. http://www.numdam.org/articles/10.1016/j.crma.2007.04.001/
[1] Nonstandard Methods in Stochastic Analysis and Mathematical Physics, Academic Press, Orlando, FL, 1986
[2] Euclidean quantum mechanics: analytical approach, Ann. Inst. H. Poincaré Phys. Théor., Volume 50 (1989), pp. 259-308
[3] Entropy, time-irreversibility and the Schrödinger equation in a primarily discrete spacetime, J. Phys. A: Math. Gen., Volume 38 (2005), pp. 2835-2847
[4] E. Benoît, Diffusions discrètes et mécanique stochastique, Prépubli. Lab. Math. J. Dieudonné, Université de Nice, 1989
[5] Random walks and stochastic differential equations (Diener, F.; Diener, M., eds.), Nonstandard Analysis in Practice, Springer, Berlin, 1995, pp. 71-90
[6] L'aveuglante proximité du réel, Flammarion, Paris, 1998
[7] Mathematical and Physical Aspects of Stochastic Mechanics, Lecture Notes in Physics, vol. 281, Springer, Berlin, 1987
[8] Introduction to Random Time and Quantum Randomness, World Scientific, Singapour, 2003
[9] Plongement stochastique des systèmes lagrangiens, C. R. Acad. Sci. Paris, Ser. I, Volume 342 (2006), pp. 333-336
[10] Théorème de Noether stochastique, C. R. Acad. Sci. Paris, Ser. I, Volume 344 (2007), pp. 259-264
[11] M. Davidson, Stochastic mechanics, trace dynamics, and differential space – a synthesis, Prépublication, 2006. Accessible sur | arXiv
[12] Eine wahrscheinlichkeitstheoretische Begründung und Interpretation der Quantenmechanik, Z. Phys., Volume 132 (1952), pp. 81-106
[13] Space–time approach to non-relativistic quantum mechanics, Rev. Modern Phys., Volume 22 (1948), pp. 367-387
[14] Quantum Mechanics and Path Integrals, McGraw-Hill, New York, 1965
[15] Temps-fréquence, Hermès, Paris, 1998
[16] Analyse non standard du bruit, C. R. Acad. Sci. Paris, Ser. I, Volume 342 (2006), pp. 797-802
[17] M. Fliess, Une approche intrinsèque des fluctuations quantiques en mécanique stochastique, Manuscrit, 2006. Accessible sur http://hal.inria.fr/inria-00118460
[18] A new look at the derivation of the Schrödinger equation from Newtonian mechanics, Ann. Physik, Volume 12 (2003), pp. 371-403
[19] Hyperfinite quantum random walks, Chaos Solitons Fractals, Volume 7 (1996), pp. 669-679
[20] Quantization of dynamical systems and stochastic control theory, Phys. Rev. D, Volume 27 (1983), pp. 1774-1786
[21] Fractal geometry in quantum mechanics, field theory and spin systems, Phys. Rep., Volume 323 (2000), pp. 81-181
[22] Derivation of the Schrödinger equation from Newtonian mechanics, Phys. Rev., Volume 150 (1966), pp. 1079-1085
[23] Dynamical Theories of Brownian Motion, Princeton University Press, Princeton, NJ, 1967 (2e éd., datant de 2001, accessible sur http://www.math.princeton.edu/%7Enelson/books/bmotion.pdf)
[24] Internal set theory, Bull. Amer. Math. Soc., Volume 83 (1977), pp. 1165-1198
[25] Quantum Fluctuations, Princeton University Press, Princeton, NJ, 1985 (Accessible sur http://www.math.princeton.edu/%7Enelson/books/qf.pdf)
[26] Radically Elementary Probability Theory, Princeton University Press, Princeton, NJ, 1987 (Accessible sur http://www.math.princeton.edu/%7Enelson/books/rept.pdf)
[27] Fractal Space–Time and Microphysics, World Scientific, Singapour, 1993
[28] Random walks, continuum limits, and Schrödinger's equation, Phys. Rev. A, Volume 54 (1996), pp. 3772-3778
[29] Entwined pairs and Schrödinger's equation, Ann. Physics, Volume 308 (2003), pp. 478-492
[30] Relativistic Brownian motion and gravity as an eikonal approximation to a quantum evolution equation, Foundat. Phys., Volume 35 (2005), pp. 1181-1203
[31] Stochastic mechanics and the Feynman integral, J. Math. Phys., Volume 41 (2000), pp. 6060-6078
[32] The Quantum Dice: An Introduction to Stochastic Electrodynamics, Kluwer, Dordrecht, 1996
[33] Non-Standard Analysis, North-Holland, Amsterdam, 1974
[34] Matrix models and non-local hidden variables theories (Elitzur, A.; Dolev, S.; Kolenda, N., eds.), Quo Vadis Quantum Mechanics, Springer, Berlin, 2005, pp. 121-152
[35] L. Smolin, Could quantum mechanics be an approximation to another theory?, Prépublication, 2006. Accessible sur | arXiv
Cité par Sources :
