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Astron. Astrophys. 327, 867-889 (1997)

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Scale-relativity and quantization of the universe

I. Theoretical framework

L. Nottale

CNRS. DAEC. Observatoire de Paris-Meudon, F-92195 Meudon Cedex, France

Received 19 November 1996 / Accepted 5 May 1997


The theory of scale relativity extends Einstein's principle of relativity to scale transformations of resolutions. It is based on the giving up of the axiom of differentiability of the space-time continuum. The new framework generalizes the standard theory and includes it as a special case. Three consequences arise from this withdrawal: (i) The geometry of space-time must be fractal, i.e., explicitly resolution-dependent. This allows us to include resolutions in the definition of the state of the reference system, and to require scale-covariance of the equations of physics under scale transformations. (ii) The geodesics of the non-differentiable space-time are themselves fractal and in infinite number. This divergence strongly suggests we undertake a statistical, non-deterministic description. (iii) Time reversibility is broken at the infinitesimal level. This can be described in terms of a two-valuedness of the time derivative, which we account by using complex numbers. We finally combine these three effects by constructing a new tool, the scale-covariant derivative, which transforms classical mechanics into a generalized, quantum-like mechanics.

Scale relativity was initially developed in order to re-found quantum mechanics on first principles (while its present foundation is axiomatic). However, the scale-relativistic approach is expected to apply not only at small scales, but also at very large space- and time-scales, although with a different interpretation. Indeed, we find that the scale symmetry must be broken at two (relative) scales, so that the scale axis is divided in three domains: (i) the quantum, scale-dependent microphysical domain, (ii) the classical, intermediate, scale-independent domain, (iii) but also the macroscopic, cosmological domain which becomes scale-dependent again and may then be described on very large time-scales (beyond a predictability horizon) in terms of a non-deterministic, statistical, quantum-like theory. In the new framework, we definitively give up the hope to predict individual trajectories on very large time scales. This leads us to describing their virtual families in terms of complex probability amplitudes, which are solutions of generalized Schrödinger equations. The squared modulus of these probability amplitudes yields probability densities, whose peaks are interpreted as a tendency for the system to make structures. Since the quantizations in quantum mechanics appear as a direct consequence of the limiting conditions and of the shape of the input field, the theory thus naturally provides self-organization of the system it describes, in connection with its environment.

In the present first paper of this series, we first recall the structure of the scale-relativity theory, then we apply our scale-covariant procedure to various equations of classical physics that are relevant to astrophysical processes, including the equation of motion of a particle in a gravitational field (Newtonian and Einsteinian), in an electromagnetic field, the Euler and Navier-Stokes equations, the rotational motion of solids, dissipative systems, and first hints on field equations themselves. In all these cases, we obtain new generalized Schrödinger equations which allow quantized solutions. In scale-relativity therefore, the underlying fractal geometry of space-time plays the role of a universal structuring "field". In subsequent papers of this series, we shall derive the solutions of our equations, then show that several new theoretical predictions can be made, and that they can be successfully checked by an analysis of the observational data.

Key words: relativity – gravitation – chaos – hydrodynamics – cosmology: theory

Send offprint requests to: L. Nottale


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© European Southern Observatory (ESO) 1997

Online publication: April 6, 1998