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

5. Discussion and conclusion

One of the most important consequences of the quantum-like nature of the world at large time and/or length-scales unveiled by scale-relativity is the profound unity of structures that it implies between very different scales. This unity is a consequence:
(i) of the universality of the structuring "force", (i.e. of the fractal geometry of space-time),
(ii) of that of physics, as manifested in the Lagrangian / Hamiltonian formalism, i.e., in the underlying symplectic structure of physics,
(iii) and of the recovered prevalence of the space-time description.

It was already apparent in standard quantum mechanics, in the fact that several features of the solutions to a given Schrödinger equation can be detailed before the precise form of the potential is specified. The structures come in large part from the quantum terms themselves (which we have interpreted as a manifestation of the fractal space-time geometry) and from the matching and limiting conditions for the wave function.

Now, as can be seen on the various generalized Schrödinger equations written throughout the present paper, these quantum terms are in common to all them. Moreover, these equations describe different systems that must be matched together at different scales in the real world: the matching conditions will then imply a unity and a continuity of the structures observed at these different scales. This prediction has already been fairly well verified for various gravitational potentials from the scale of star radii ( km) to extragalactic scales ( 100 Mpc) (Nottale 1996a, b, Nottale, Schumacher & Gay 1996, Nottale, Schumacher & Lefèvre 1997). We shall in future work investigate whether it also applies to systems whose structure do not depend on the gravitational field only, but also on magnetic fields, pressure and dissipative terms (e.g., stellar interiors, stellar atmospheres...).

Before concluding, we want to stress once again the difference between the application of our theory to standard quantum mechanics (at small scales) and to very large time-scale phenomena as studied in the present series of papers. In the case of quantum mechanics (Nottale 1993a), our fundamental assumption is that space-time itself is continuous but non-differentiable, then fractal without any lower limit. The complete withdrawal of the hypothesis of differentiability is necessary if we want the theory not to be a hidden parameter one and to agree with Bell's theorem and the undeterminism of quantum paths. Moreover, we also give up the concept of elementary particle as being something which would own internal properties such as mass, charge or spin, since we are able to recover these properties from the geometric structures of fractal geodesics of the nondifferentiable space-time. Particles, with their wave-corpuscle duality, are identified with the geodesics themselves (Nottale 1989, 1993a, 1996a), i.e., with the shortest lines of topological dimension 1 (singularizing also the topological dimension 2 would lead to string theories, see Castro 1996).

On the contrary, in the application to chaos and fractal space beyond the predictability horizon, we know from the beginning (i) that non-differentiability is only a large time-scale approximation (), and that when going back to small time-resolution we recover differentiable, predictable classical trajectories; and (ii) that the geodesics are indeed trajectories followed by extended bodies. This motivates the use of some of the quantum mechanical tools (probability amplitude, Schrödinger-like equations) but not its whole interpretation, concerning in particular measurement theory, in agreement with the recent construction by Ord (1996a, b) of a microscopic model of the Schrödinger equation in the fractal space-time / random walk framework.

Recall also that the application of the scale-relativity theory to the macrophysical domain implies a different interpretation of our construction respectively to the microphysical domain for yet another reason. In the macroscopic case indeed, the transition to classical physics is toward the small scales, while no upper limit is expected to the scaling behavior at large scales. This can be achieved provided our theory applies only to a "fully quantum" system, i.e., a system for which the mean classical velocity is zero (such as the hydrogen atom in microphysics). Indeed, the upper transition from quantum (fractal) laws to classical (non fractal) laws is given by the equivalent de Broglie length, , which is sent to infinity when .

The scale relativity theory shares some common features with other approaches, even if it also differs from them on essential points. A first related approach is Nelson's (1966, 1985) stochastic mechanics, in which particles are described in terms of a diffusion, Brownian-like process, but with a Newtonian rather than Langevin dynamics. Nelson obtains a complex Schrödinger equation as a combination of real equations, namely a Newton equation of dynamics in which the form of the acceleration is postulated, and two backward and forward Fokker-Planck equations. (Note that this implies that Nelson's diffusion is not a standard diffusion process, since his backward Fokker-Planck equation is a time-reversed forward Kolmogorov equation, which is therefore incompatible with the standard backward Kolmogorov equation, see e.g. Welch 1970). Nelson's theory has been used by Albeverio et al. (1983) and Blanchard (1984) to obtain models of the protoplanetary nebula.

Contrarily to such diffusion approaches and to standard quantum mechanics itself, the scale-relativity theory is not statistical in its essence. In scale relativity, the fractal space-time could be completely "determined", so that the undeterminism of trajectories is not set as a founding stone of the theory, but as a consequence of the nondifferentiability of space-time. This is clear from the fact that we do not use Fokker-Planck equations, but only the equation of dynamics, properly made scale-covariant.

The implications of this difference between the two approches are very important. The diffusion approach is expected to apply only in fluid-like or many-body environments. On the contrary, the structuring "field" in our theory being the underlying fractal geometry of space-time itself, we predict that there is a universal tendency of nature to make structures, even for two-body problems, and that these structures must be themselves related together in a universal way. This prediction has been already verified in a remarkable way for gravitational structures (Nottale 1993a, b, 1995b, 1996a, b; Nottale, Schumacher & Gay 1997).

Recall also that it is now known that Nelson's stochastic mechanics is in contradiction with standard quantum mechanics concerning multitime correlations (Grabert et al 1979, Wang & Liang 1993). The source of the disagreement comes precisely from the Brownian motion interpretation of Nelson's theory, leading to the use of the Fokker-Planck equations, and from the wave function reduction. Once again the fact that we do not use the Fokker-Planck equations reveals itself as an essential features of our theory, since it allows our theory not to come under the Wang & Liang argument. Once we have jumped to the quantum tool (i.e., when we pass from our representation in terms of and to the equivalent representation in terms of ) we know by construction that the representation is complete, (i.e. we recover the quantum equations without any additional constraint) so that the identity of predictions of standard quantum mechanics and of scale relativity is ensured in the microphysical domain (at energies where Galilean scale-laws hold). Another related approach is that of Petrovsky and Prigogine (1996), who attempt to extend classical dynamics by formulating it on the statistical level. They also give up individual trajectories and jump to a non-deterministic and irreversible description. The difference with our own approach is that they keep classical probabilities and irreversibility central to the theory without invoking an explicit scale dependence. In contrast, in scale relativity, the description is fundamentally irreversible (in terms of the elementary displacements on fractal geodesics), but this is not an axiom so much as a consequence of giving up differentiability. Moreover, we recover a reversible description in terms of our complex representation (i.e., of the quantum mechanical tool) which combines the forward and backward process: in other words, irreversibility is at the origin of our complex formulation, but it becomes hidden in the formalism, even though it reappears through the wave function collapse.

Finally, an important point to emphasize once again is also that, in scale-relativity, we really deal with a fractal space, not only with fractal trajectories in a space that could remain flat or curved. This is apparent in our trajectory equation, which is written (in the absence of an external field) in terms of a scale-covariant geodesics equation, which takes the form of the free, Galilean equation of motion . This is the equation for rectilinear uniform motion. It means that the particle goes straight ahead in its proper coordinate system swept along in the fractal space-time, and that its structure, which looks fractal when seen from an exterior reference frame, comes from the very fractal geometry of space-time itself.

In the papers of this series to follow, we shall enter in more detail into our theoretical predictions by looking at the solutions of our equations for different fields, with particular attention given in paper II on gravitational structures, then we shall compare these predictions to observational data (paper III). We shall see that the theory allows us to explain several misunderstood facts concerning gravitational structures at all scales, it allows us to make new predictions (see already Nottale 1996a, b, c, 1997, Nottale et al 1997), and it also opens new domains of investigation, concerning in particular the open question of a more complete description of the "field" of fractal fluctuations.

© European Southern Observatory (ESO) 1997

Online publication: April 6, 1998
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