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Scale relativity and fractal space-time: applications to quantum physics, cosmology and chaotic systems. (English) Zbl 1080.81525

Summary: The theory of scale relativity is a new approach to the problem of the origin of fundamental scales and of scaling laws in physics, that consists of generalizing Einstein’s principle of relativity (up to now applied to motion laws) to scale transformations. Namely, we redefine space-time resolutions as characterizing the state of scale of the reference system and require that the equations of physics keep their form under resolution transformations (i.e. be scale covariant). We recall in the present review paper how the development of the theory is intrinsically linked to the concept of fractal space-time, and how it allows one to recover quantum mechanics as mechanics on such a non-differentiable space-time, in which the Schrödinger equation is demonstrated as a geodesies equation. We recall that the standard quantum behavior is obtained, however, as a manifestation of a ”Galilean” version of the theory, while the application of the principle of relativity to linear scale laws leads to the construction of a theory of special scale relativity, in which there appears impassable, minimal and maximal scales, invariant under dilations. The theory is then applied to its preferential domains of applications, namely very small and very large length- and time-scales, i.e. high energy physics, cosmology and chaotic systems.

MSC:

81P99 Foundations, quantum information and its processing, quantum axioms, and philosophy
37N99 Applications of dynamical systems
83D05 Relativistic gravitational theories other than Einstein’s, including asymmetric field theories
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[1] Nottale, L., (Fractal Space-Time and Microphysics: Towards a Theory of Scale Relativity (1993), World Scientific: World Scientific Singapore) · Zbl 0789.58003
[2] Nottale, L., Fractals and the quantum theory of space-time, Int. J. Mod. Phys. A, 4, 5047-5117 (1989)
[3] Nottale, L., The theory of scale relativity, Int. J. Mod. Phys. A, 7, 4899-4936 (1992) · Zbl 0954.81501
[4] Feynman, R. P.; Hibbs, A. R., (Quantum Mechanics and Path Integrals (1965), MacGraw-Hill: MacGraw-Hill New York) · Zbl 0176.54902
[5] Mandelbrot, B., (The Fractal Geometry of Nature (1982), Freeman: Freeman San Francisco) · Zbl 0504.28001
[6] Nottale, L.; Schneider, J., Fractals and non standard analysis, J. Math. Phys., 25, 1296-1300 (1984)
[7] Ord, G. N., Fractal space-time: a geometric analogue of relativistic quantum mechanics, J. Phys. A: Math. Gen., 16, 1869-1884 (1983)
[8] El Naschie, M. S.; Rössler, O. E.; Prigogine, I., (Quantum Mechanics, Diffusion and Chaotic Fractals (1995), Elsevier: Elsevier Oxford) · Zbl 0830.58001
[9] Weinberg, S., (Gravitation and Cosmology (1972), Wiley: Wiley New York)
[10] Einstein, A., (The Principle of Relativity (1916), Dover: Dover New York), 109-164
[11] Nottale, L., Scale relativity, fractal space-time and quantum mechanics, Chaos, Solitons & Fractals. (El Naschie, M. S.; Rössler, O. E.; Prigogine, I., Quantum Mechanics, Diffusion and Chaotic Fractals (1995), Elsevier: Elsevier Oxford), 4, 51-388 (1994), Reprinted in · Zbl 0805.58057
[12] Nottale, L., Scale-relativity: from quantum mechanics to chaotic dynamics, Chaos, Solitons & Fractals, 6, 399-410 (1995) · Zbl 0905.58053
[13] Nottale, L., Scale-relativity: first steps toward a field theory, (Alonso, J. Diaz; Paramo, M. Lorente, Relativity in General. Relativity in General, E.R.E. 93 (Spanish Relativity Meeting) (1995), Editions Frontières: Editions Frontières Paris), 121-132 · Zbl 0863.58093
[14] Nottale, L., New formulation of stochastic mechanics: application to chaos, (Benest, D., Chaos and Diffusion in Hamiltonian Systems (1995), Editions Frontières: Editions Frontières Paris) · Zbl 0873.70017
[15] Nottale, L., Scale relativity: many particle Schrödinger equation, (Novak, M. M., Fractal Reviews in the Natural and Applied Sciences. Fractal Reviews in the Natural and Applied Sciences, Proceedings of ‘Fractals’ 95 (1995), Chapman & Hall: Chapman & Hall London), 12-23, (1995)
[16] Nottale, L., Scale relativity and structuration of the universe, (Maurogordato, S.; Balkowski, C., Proceedings of Moriond Meeting (1995), Editions Frontières: Editions Frontières Paris) · Zbl 0970.70013
[17] Nottale, L., Quantization of the universe (1996), Submitted for publication
[18] Tricot, C., (Courbes et Dimensions Fractales (1994), Springer: Springer Paris)
[19] L. Nottale, In preparation.; L. Nottale, In preparation.
[20] Aitchison, I. J.R., (An Informal Introduction to Gauge Field Theories (1982), Cambridge University Press: Cambridge University Press Cambridge) · Zbl 1139.81300
[21] Wilson, K. G., The renormalization group and critical phenomena, Am. J. Phys., 55, 583-600 (1983)
[22] Levy-Leblond, J. M., On more derivation of the Lorentz transformation, Am. J. Phys., 44, 271-277 (1976)
[23] Cohen, E. R.; Taylor, B. N., The 1986 adjustment of the fundamental physical constants, Rev. Mod. Phys., 59, 1121-1148 (1987)
[24] Le Mèhauté, A., (Les Geometries Fractales (1990), Hermès: Hermès Paris) · Zbl 0834.58002
[25] Abbott, L. F.; Wise, M. B., Dimension of a quantum mechanical path, Am. J. Phys., 49, 37-39 (1981)
[26] Nelson, E., Derivation of the Schrödinger equation from Newtonian mechanics, Phys. Rev., 150, 1079-1085 (1966)
[27] Nelson, E., (Quantum Fluctuations (1985), Princeton University Press: Princeton University Press Princeton, NJ) · Zbl 0563.60001
[28] Bjorken, J. D.; Drell, S. D., (Relativistic Quantum Fields (1965), McGraw-Hill: McGraw-Hill New York) · Zbl 0184.54201
[29] Mandelbrot, B. B.; Van Ness, J. W., Fractional Brownian motions, fractional noises and applications, SIAM Rev., 10, 422 (1968) · Zbl 0179.47801
[30] Dohrn, D.; Guerra, F., Phys. Rev., D31, 2521-2524 (1985)
[31] Zastawniak, T., Europhys. Lett., 13, 13-17 (1990)
[32] Serva, M., Ann. Inst. Henri Poincaré-Phys. Théor, 49, 415-432 (1988) · Zbl 0659.60095
[33] L. Nottale and J. C. Pissondes, Submitted for publication (1996).; L. Nottale and J. C. Pissondes, Submitted for publication (1996).
[34] Weyl, H., (The Principle of Relativity (1918), Dover: Dover New York), 201-216
[35] Dirac, P. A.M., (Proc. Roy. Soc. Lond. A, 333 (1973)), 403-418
[36] Gaveau, B.; Jacobson, T.; Kac, M.; Schulman, L. S., Phys. Rev. Lett., 53, 419-422 (1984)
[37] McKeon, D. G.C.; Ord, G. N., Time reversal in stochastic processes and the Dirac equation, Phys. Rev. Lett., 69, 3-4 (1992) · Zbl 0968.81526
[38] Finkel, R. W., Generalized uncertainty relations, Phys. Rev. A, 35, 1486-1489 (1987)
[39] Burkhardt, H.; Legerlehner, F.; Penso, G.; Vergegnassi, C., Uncertainties in the hadronic contribution to the QED vacuum polarization, Z. Phys. C-Particles and Fields, 43, 497-501 (1989)
[40] Abe, F., Evidence for top quark production in pp collisions, Phys. Rev. Lett., 73, 225 (1994)
[41] Novikov, V. A.; Okun, L. B.; Vysotsky, M. I., CERN Preprint-TH.6943/93 (1993)
[42] Langacker, P.; Luo, M.; Mann, A. K., Rev. Mod. Phys., 64, 87-192 (1992)
[43] Weinberg, S., Rev. Mod. Phys., 61, 1 (1989) · Zbl 1129.83361
[44] Carroll, S. M.; Press, W. H.; Turner, E. L., Ann. Rev. Astron. Astrophys, 30, 499 (1992)
[45] Zeldovich, Ya. B., JETP Lett., 6, 316 (1967)
[46] Davis, M.; Peebles, P. J.E., Astrophys. J., 267, 465 (1983)
[47] Vader, J. P.; Sandage, A., Astrophys. J. Lett., 379, L1 (1991)
[48] Campos, A., (Maurogordato, S.; Balkowski, C., Proceedings of Moriond Meeting (1995), Frontières: Frontières Paris)
[49] De Gouveia Dal Pino, E. M., Astrophys. J. Lett., 442, L45 (1995)
[50] Nottale, L., Emergence of structures from chaos, (Lejeune, A.; Perdang, J., Cellular Automata: Prospects in Astrophysical Applications, Université de Liège (1993), World Scientific: World Scientific Singapore), 268-277
[51] Chauvineau, B.; Gay, J.; Nottale, L.; Schumacher, G., Is there a small planet between the Sun and Mercury?, Astron. Astrophys. (1996), Submitted to
[52] Wisdom, J., Urey prize lecture: chaotic dynamics in the Solar System, Icarus, 72, 241 (1987)
[53] Tifft, W. G., Astrophys. J., 262, 44 (1982)
[54] Schneider, S. E.; Salpeter, E. E., Astrophys. J., 385, 32 (1992)
[55] Cocke, W. J., Astrophys. J., 393, 59 (1992)
[56] Tifft, W. G., Astrophys. J., 221, 756 (1978)
[57] Tifft, W. G.; Cocke, W. J., Astrophys. J., 287, 492 (1984)
[58] Guthrie, B. N.; Napier, W. M., Mon. Not. R. Astr. Soc., 253, 533 (1991)
[59] Croasdale, M. R., Astrophys. J., 345, 72 (1989)
[60] Landau, L.; Lifchitz, E., (Mechanics (1968), Mir: Mir Moscow)
[61] Nottale, L., The fractal structure of the quantum space-time, (Heck, A.; Perdang, J. M., Applying Fractals in Astronomy (1991), Springer: Springer New York), 181-200
[62] Scully, M. O.; Englert, B. G.; Walther, H., Quantum optical tests of complementarity, Nature, 351, 111 (1991)
[63] El Naschie, M. S., Young two slits experiment, (El Naschie, M. S.; Rössler, O. E.; Prigogine, I., Quantum Mechanics, Diffusion and Chaotic Fractals (1995), Elsevier: Elsevier Oxford), 191 · Zbl 0801.58040
[64] Boyarsky, A.; Gora, P., A dynamical system model for interference effects and the two-slit experiment of quantum physics, Phys. Lett A, 168, 103-112 (1992)
[65] Ord, G. N., Classical analog of quantum phase, Int. J. Theor. Phys., 31, 1177 (1992) · Zbl 0761.58059
[66] Rössler, O. E., Intra-observer chaos: hidden roots of quantum mechanics?, (El Naschie, M. S.; Rössler, O. E.; Prigogine, I., Quantum Mechanics, Diffusion and Chaotic Fractals (1995), Elsevier: Elsevier Oxford), 105-111
[67] Petrovsky, T.; Prigogine, I., (El Naschie, M. S.; Rössler, O. E.; Prigogine, I., Quantum Mechanics, Diffusion and Chaotic Fractals (1995), Elsevier: Elsevier Oxford), 1-49 · Zbl 0830.58001
[68] Kröger, H., Fractal Wilson loop and confinement in non-compact gauge field theory, Phys. Lett. B, 284, 357 (1992)
[69] Kroger, H.; Lantagne, S.; Moriarty, K. J.M.; Plache, B., Measuring the Hausdorff dimension of quantum mechanical paths, Phys. Lett. A, 199, 299 (1995) · Zbl 1020.81932
[70] Le Méhauté, A.; Héliodore, F.; Cottevieille, D.; Latreille, F., Introduction to wave phenomena and uncertainty in fractal space, (El Naschie, M. S.; Rössler, O. E.; Prigogine, I., Quantum Mechanics, Diffusion and Chaotic Fractals (1995), Elsevier: Elsevier Oxford), 79-92 · Zbl 0798.58063
[71] Dubrulle, B.; Graner, F., Possible statistics of scale invariant systems, J. Phys. (1996), Submitted to
[72] C. Castro. Preprint (1995).; C. Castro. Preprint (1995).
[73] El Naschie, M. S., Iterated function systems, information and the two-slit experiment of quantum mechanics, (El Naschie, M. S.; Rössler, O. E.; Prigogine, I., Quantum Mechanics, Diffusion and Chaotic Fractals (1995), Elsevier: Elsevier Oxford), 185-189
[74] El Naschie, M. S., A note on quantum mechanics, diffusional interference and informions, Chaos, Solitons & Fractals, 5, 881-884 (1995) · Zbl 0900.81007
[75] El Naschie, M. S., Banach-Tarski theorem and Cantorian micro space-time, Chaos, Solitons & Fractals, 5, 1503-1508 (1995) · Zbl 0903.28010
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