The Fractal Structure of the Quantum Space-Time

  • Laurent Nottale
Part of the Lecture Notes in Physics Monographs book series (LNPMGR, volume 3)


We report here on the present state of an attempt at understanding micro-physics by basing ourselves on a ‘principle of scale relativity’, according to which the laws of physics should apply to systems of reference whatever their scale. The continuity but non differentiability of quantum mechanical particle paths, the occurrence of infinities in quantum field theories and the universal length and time scale dependence of measurement results implied by Heisenberg’s relations have led us, among other arguments, to suggest the achievement of such a principle by using the mathematical tool of fractals (Nottale 1989). The concept of a continuous and self-avoiding fractal space-time is worked out: arguments are given for generally describing them as families of Riemannian space-times whose curvature tends to infinity when scale approaches zero. We recall some basic results already obtained in this quest: the fractal dimension of quantum trajectories jumps from 1 to 2 for all 4 space-time coordinates, with the fractal/non fractal transition occurring around the de Broglie’s length and time; conversely the Heisenberg relations may be obtained as a consequence of assumed fractal structures; point particles following curves of fractal dimension 2 are naturally endowed with a spin. We develop the interpretation of the wave-particle duality as a property of families of geodesical lines in a fractal space-time and examplify it with the Young’s hole and Einstein-Podolsky-Rosen experiments. Finally some possible consequences concerning gravitation are recalled, with the suggestion that Newton’s law may break down for active gravitational masses smaller than the Planck mass.


Fractal Dimension Fractal Structure Planck Mass Fractal Space Geometric Optic Approximation 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. Abbott, L.F., Wise, M.B. (1981): Amer. J. Phys. 49, 37.CrossRefADSMathSciNetGoogle Scholar
  2. Allen, A.D. (1983): Speculations Sc. Techn. 6, 165.Google Scholar
  3. Aspect, A., Dalibar, J., Roger, G. (1982): Phys. Rev. Letters 49, 1804.CrossRefADSGoogle Scholar
  4. Bjorken, J.D., Drell, S.D. (1964): Relativistic Quantum Mechanics (McGraw-Hill, New York).Google Scholar
  5. Campesino-Romeo, E., D’Olivo, J.C., Socolovsky, M. (1982): Phys. Letters 89A, 321.ADSMathSciNetGoogle Scholar
  6. Cannata, F., Ferrari, L. (1988): Amer. J. Phys. 56, 721.CrossRefADSMathSciNetGoogle Scholar
  7. Einstein, A. (1907): Jahrb. Rad. Elektr. 4, 411Google Scholar
  8. Einstein, A. (1916): Ann. Physik 49, 769.CrossRefADSGoogle Scholar
  9. Einstein, A. (1948): Letter to Pauli, in Quanta (Seuil/CNRS, Paris), p. 249.Google Scholar
  10. Feynman, R., Hibbs, A. (1965): Quantum Mechanics and Path Integrals (McGraw-Hill, New York).zbMATHGoogle Scholar
  11. Feynman, R., Leighton, R.B., Sands, M. (1965): The Feynman Lectures on Physics III (Addison-Wesley, Reading).zbMATHGoogle Scholar
  12. Fuller, F.W., Wheeler, J.A. (1962): Phys. Rev. 128, 919.zbMATHCrossRefADSMathSciNetGoogle Scholar
  13. Gödel, K. (1931): in Le Théorème de Gödel (Seuil, Paris).Google Scholar
  14. Hawking, S. (1978): Nucl. Phys. B 144, 349.CrossRefADSMathSciNetGoogle Scholar
  15. Landau, L., Lifshitz, E. (1972) Relativistic Quantum Theory (Mir, Moscow).Google Scholar
  16. Le Bellac, M. (1988): Des Phénomènes Critiques aux Champs de Jauge (InterEdition/CNRS, Paris).Google Scholar
  17. Mandelbrot, B. (1982a): The Fractal Geometry of Nature (Freeman, San Francisco), p. 331.zbMATHGoogle Scholar
  18. Mandelbrot, B. (1982b): The Fractal Geometry of Nature (Freeman, San Francisco), p. 365.zbMATHGoogle Scholar
  19. Nelson, E. (1966): Phys. Rev. 150, 1079.CrossRefADSGoogle Scholar
  20. Nelson, E. (1977): Bull. Amer. Math. Soc. 83, 1165.zbMATHCrossRefMathSciNetGoogle Scholar
  21. Nelson, E. (1985): Quantum Fluctuations (Princeton Univ. Press, Princeton).zbMATHGoogle Scholar
  22. Nottale, L. (1988a): C.R. Acad. Sc. Paris 306, 341.ADSMathSciNetGoogle Scholar
  23. Nottale, L. (1988b): Ann. Phys. Fr. 13, 223.CrossRefADSGoogle Scholar
  24. Nottale, L. (1989): Int. J. Mod. Phys. A4, 5047.ADSMathSciNetGoogle Scholar
  25. Nottale, L. (1991): submitted.Google Scholar
  26. Nottale, L., Lachaud, Y. (1991): in preparation.Google Scholar
  27. Nottale, L., Schneider, J. (1984): J. Math. Phys. 25, 1296.CrossRefADSMathSciNetGoogle Scholar
  28. Omnes, R. (1988): J. Stat. Phys. 53, 893, 933, 957.zbMATHCrossRefADSMathSciNetGoogle Scholar
  29. Robinson, A. (1961) Proc. Roy. Acad. Sc. Amsterdam A 64, 432.Google Scholar
  30. Sachs, P.K. (1961): Proc. Roy. Soc. London A 264, 309.ADSMathSciNetGoogle Scholar
  31. Sagdeev, R.Z., Usikov, D.A., Zaslovski, G.M. (1998): Non-Linear Physics (Harwood, New York).Google Scholar
  32. Weinberg, S. (1972): Gravitation and Cosmology (Wiley, New York).Google Scholar
  33. Wilson, K.G. (1979): Scientific American 241, 140.CrossRefGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 1991

Authors and Affiliations

  • Laurent Nottale
    • 1
  1. 1.C.N.R.S., Département d’Astrophysique Extragalactique et de CosmologieObservatoire de MeudonMeudon CedexFrance

Personalised recommendations