Scale relativity, fractal space-time and quantum mechanics. (English) Zbl 0805.58057

Author’s abstract: “This paper describes the present state of an attempt at understanding the quantum behaviour of microphysics in terms of a nondifferentiable space-time continuum having fractal (i.e. scale- dependent) properties. The fundamental principle upon which we rely is that of scale relativity, which generalizes Einstein’s principle of relativity to scale transformations. After having related the fractal and renormalization group approaches, we develop a new version of stochastic quantum mechanics, in which the correspondence principle and the Schrödinger equation are demonstrated by replacing the classical time derivative by a “quantum-covariant” derivative. Then we recall that the principle of scale relativity leads one to generalize the standard “Galilean” laws of scale transformation into a Lorentzian form, in which the Planck length-scale becomes invariant under dilations, and so plays for scale laws the same role as played by the velocity of light for motion laws. We conclude by an application of our new framework to the problem of the mass spectrum of elementary particles.”.
Reviewer: I.Gottlieb (Iaşi)


37N99 Applications of dynamical systems
81P20 Stochastic mechanics (including stochastic electrodynamics)
83A05 Special relativity
37D45 Strange attractors, chaotic dynamics of systems with hyperbolic behavior
Full Text: DOI


[1] Nottale, L.; Schneider, J., Fractals and non standard analysis, J. Math. Phys., 25, 1296-1300 (1984)
[2] Ord, G. N., Fractal space-time: a geometric analogue of relativistic quantum mechanics, J. Phys. A: Math. Gen., 16, 1869-1884 (1983)
[3] Feynman, R. P.; Hibbs, A. R., Quantum Mechanics and Path Integrals (1965), MacGraw-Hill: MacGraw-Hill New York · Zbl 0176.54902
[4] Abbott, L. F.; Wise, M. B., Dimension of a quantum mechanical path, Am. J. Phys., 49, 37-39 (1981)
[5] Campesino-Romeo, E.; D’Olivo, J. C.; Socolovsky, M., Hausdorff dimension for the quantum harmonic oscillator, Phys. Lett, 89A, 321-324 (1982)
[6] Allen, A. D., Fractals and quantum mechanics, Speculations Sci. Tech., 6, 165-170 (1983)
[7] Schweber, S. S., Feynman and the visualization of space-time processes, Rev. Mod. Phys., 58, 449-508 (1986)
[8] Nottale, L., Fractals and the quantum theory of space-time, Int. J. Mod. Phys., A4, 5047-5117 (1989)
[9] Nottale, L., Fractal Space-Time and Microphysics: Towards a Theory of Scale Relativity (1993), World Scientific: World Scientific Singapore · Zbl 0789.58003
[10] Mandelbrot, B., The Fractals Geometry of Nature (1982), San Francisco · Zbl 0504.28001
[11] Nelson, E., Derivation of the Schrödinger equation from Newtonian mechanics, Phys. Rev., 150, 1079-1085 (1966)
[12] Nelson, E., Quantum Fluctuations (1985), Princeton University Press: Princeton University Press Princeton, NJ · Zbl 0563.60001
[13] L. Nottale, Submitted for publication.; L. Nottale, Submitted for publication.
[14] Serva, M., Relativistic stochastic processes associated to Klein-Gordon equation, Ann. Inst. Henri Poincaré-Physique théorique, 49, 415-432 (1988) · Zbl 0659.60095
[15] Zastawniak, T., A relativistic version of Nelson’s stochastic mechanics, Europhys. Lett., 13, 13-17 (1990)
[16] Nottale, L., The theory of scale relativity, Int. J. Mod. Phys., A7, 4899-4936 (1992) · Zbl 0954.81501
[17] Einstein, A., (The Principle of Relativity (1952), Dover: Dover New York), 109-164, English translation in
[18] Einstein, A., (Albert Einstein, Oeuvres choisies, Vol. I (1990), Seuil-CNRS: Seuil-CNRS Paris), 249
[19] Nottale, L., The fractal structure of the quantum space-time, (Heck, A.; Perdang, J. M., Applying Fractals in Astronomy (1991), Springer), 181-200, Lecture Notes in Physics
[20] Sornette, D., Brownian representation of fractal quantum paths, Euro. J. Phys., 11, 334-337 (1990)
[21] El Naschie, M. S., On universal behaviour and statistical mechanics of multidimensional triadic sets, SAMS, 217-225 (1993) · Zbl 0791.58084
[22] Höfer, K. W., Differential geometry on fractal space-time (1991), Preprint University of Freiburg THEP 91/6
[23] Wilson, K. G., The renormalization group and critical phenomena, Rev. Mod. Phys., 55, 583-600 (1983)
[24] Aitchinson, I. J.R., An Informal Introduction to Gauge Field Theories (1982), Cambridge University Press: Cambridge University Press Cambridge · Zbl 1139.81300
[25] Stroyan, K. D.; Luxemburg, W. A.J., Introduction to the Theory of Infinitesimals (1976), Academic Press: Academic Press New York · Zbl 0336.26002
[26] Nelson, E., Bull. Amer. Math. Soc., 83, 1165 (1977) · Zbl 0373.02040
[27] Guerra, F.; Morato, L., Quantization of dynamical systems and stochastic control theory, Phys. Rev., D27, 1774-1786 (1983)
[28] Unruh, W. G.; Zurek, W. H., Reduction of a wave packet in quantum Brownian motion, Phys. Rev., D40, 1071-1094 (1989)
[29] Dohrn, D.; Guerra, F., Compatibility between the Brownian metric and the kinetic metric in Nelson stochastic quantization, Phys. Rev., D31, 2521-2524 (1985)
[30] Gaveau, B.; Jacobson, T.; Kac, M.; Schulman, L. S., Relativistic extension of the analogy between quantum mechanics and Brownian motion, Phys. Rev. Lett., 53, 419-422 (1984)
[31] Levy-Leblond, J. M., On more derivation of the Lorentz transformation, Am. J. Phys., 44, 271-277 (1976)
[32] Cohen, E. R.; Taylor, B. N., The 1986 adjustment of the fundamental physical constants, Rev. Mod. Phys., 59, 1121-1148 (1987)
[33] Le Méhauté, A., Les géométries fractales (1990), Hermès: Hermès Paris · Zbl 0834.58002
[34] Itzykson, C.; Zuber, J. B., Quantum Field Theory (1980), McGraw-Hill: McGraw-Hill New York · Zbl 0453.05035
[35] Nanopoulos, D. V.; Ross, D. A., Limit on the number of flavours in grand unified theories from higher order corrections to fermion masses, Nucl. Phys., B157, 273-284 (1979)
[36] Gasser, J.; Leutwyler, H., Quark masses, Physics Reports, 87, 77-169 (1982)
[37] Marciano, W. J.; Sirlin, A., Precise SU(5) predictions, Phys. Rev. Lett., 46, 163-166 (1981)
[38] Burkhardt, H.; Legerlehner, F.; Penso, G.; Vergegnassi, C., Uncertainties in the hadronic contribution to the QED vacuum polarization, Z. Phys. C-Particles Fields, 43, 497-501 (1989)
[39] Langacker, P.; Luo, M.; Mann, A. K., High precision electroweak experiments: a global search for new physics beyond the Standard Model, Rev. Mod. Phys., 64, 87-192 (1992)
[40] L. Nottale, in preparation.; L. Nottale, in preparation.
[41] L. Nottale, A. Le Méhauté and F. Héliodore, in preparation.; L. Nottale, A. Le Méhauté and F. Héliodore, in preparation.
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