Scale Relativity and Fractal Space-Time: Applications to Quantum Physics, Cosmology and Chaotic Systems L. NOTTALE CNRS, DAEC, Observatoire de Paris-Meudon, F-92195 Meudon Cedex, France Completed 1 December 1995 Abstract 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 Einsteins principle of relativity (up to now applied to motion laws) to scale transformations. Namely, we redefi ne 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- diff erentiable space-time, in which the Schr odinger equation is demonstrated as a geodesics equation. We recall that the standard quantum behavior is ob- tained, 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 im- passable, 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. Copyright c ?1996 Elsevier Science Ltd Published in: Chaos, Solitons but conversely, we expect the diff erential method to fail when confronted with truly nondiff erentiable or nonintegrable phenomena, namely at very small and very large length scales (i.e., quantum physics and cosmology), and also for chaotic systems seen at very large time scales. The new frontier of physics is, in our opinion, to construct a continuous but nondiff erentiable physics. (We stress the fact, well known to mathematicians, that giving up diff erentiability does not impose giving up continuity). Set in such terms, the project may seem extraordinarily diffi cult. Fortunately, there is a fundamental key which will be of great help in this quest, namely, the concept of scale trans- formations. Indeed, the main consequence of continuity and nondiff erentiability is scale-divergence 1,11. One can demonstrate that the length of a continuous and nowhere-diff erentiable curve is dependent on resolution , and, further, that L() when 0, i.e. that this curve is fractal (in a general meaning). The scale divergence of continuous and almost nowhere-diff erentiable curves is a direct consequence of Lebesgues theorem, which states that a curve of fi nite length is almost everywhere diff erentiable. Consider a continuous but nondiff erentiable function f(x) between two points A0 x0,f(x0) and Ax,f(x ). Since f is non-diff erentiable, there exists a point A1of coordinates x1,f(x1) with x0 x1 L0= L(A0A). We can now iterate the argument and fi nd two coordinates x01and x11with x0 x01 x1 and x1 x11 L1 L0 . By iteration we fi nally construct successive approximations f0,f1,.fn of f(x) whose lengths L0,L1,.Lnincrease monotonically when the resolution r (xx0)2n tends to zero. In other words, continuity and nondiff erentiability implies a monotonous scale dependence of f (see Fig. 1). From Lebesgues theorem (a curve of fi nite length is almost everywhere diff er- entiable, see Ref. 18), one deduces that if f is continuous and almost everywhere nondiff erentiable, then L() when the resolution 0, i.e., f is scale- divergent. This theorem is also demonstrated in Ref. 1, p.82 using non-standard analysis. What about the reverse proposition: Is a continuous function whose length is scale-divergent between any two points (such that xA xB fi nite), i.e., L(r) when r 0, non-diff erentiable? The answer is as follows: If the length diverges as fast as, or faster than, a power law, i.e. L(r) (/r), (i.e. standard fractal behavior), then the function is certainly nondiff erentiable; in the intermediate domain of slower divergences (for example, logarithmic divergence, L(r) ln(/r), etc.), the function may be either diff erentiable or nondiff erentiable 19. It is interesting to see that the standard (self-similar, power-law) fractal be- havior plays a critical role in this theorem: it gives the limiting behavior beyond which non-diff erentiability is ensured. This result is the key for a description of nondiff erentiable processes in terms of diff erential equations: We introduce explicitly the resolutions in the expressions 5 Figure 1: Construction of a non-diff erentiable function by successive dissections. Its length tends to infi nity when the resolution interval tends to zero. of the main physical quantities, and, as a consequence, in the fundamental equa- tions of physics. This means that a physical