Scale Relativity, Non-differentiability and Fractal Space-time

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length is nevertheless in nite in the limit x 0 Such a function may therefore be almost everywhere differentiable, and in the same time be characterized by a genuine fractal law of scale-dependence of its length, ie by a power law divergence characterized by a fractal dimension DF The same reasoning may be applied to other types of divergences, such as logarithmic, exponential, etc Therefore, when the divergence is inhomogenous, an in nite curve may be either differentiable or non-differentiable whatever its divergence mode, namely, this means that there is no inverse theorem in this case When it is applied to physics, this result means that a fractal behavior may result from the action of singularities (in in nite number even though forming a subset of zero measure) in a space or space-time that nevertheless remains almost everywhere differentiable (such as for example Riemannian manifolds in Einstein s general relativity) This comes in support of Mandelbrot s view about the origin of fractals, which are known to be extremely frequent in many natural phenomena that yet seem to be well described by standard differential equations: this could come from the existence of singularities in differentiable physics (see eg [MAN 82], 11) However, the viewpoint of scale relativity theory is more radical, since the main problem we aim at solving in its framework is not the (albeit very interesting) question of the origin of fractals, but the issue of the foundation of the quantum theory and of gauge elds from geometric rst principles As we shall recall, a fractal space-time is not suf cient to reach this goal (speci cally concerning the emergence of complex numbers) We need to work in the framework of non-differentiable manifolds, which are indeed fractal (ie scale-divergent) as has been shown above However, the fractality is not central in this context, and it mainly appears as a derived (and very useful) geometric property of such continuous non-differentiable manifolds 1433 Description of non-differentiable process by differential equations This result is the key for enabling a description of non-differentiable processes in terms of differential equations We introduce explicitly the resolutions in the expressions of the main physical quantities and, as a consequence, in the fundamental equations of physics This means that a quantity f , usually expressed in terms of space-time variables x, ie, f = f (x), must now be described as also depending on resolutions , ie, f = f (x, ) In other words, rather than considering only the strictly non-differentiable mathematical object f (x), we shall consider its various approximations obtained from smoothing or averaging it at various resolutions :

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f (x, ) =

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(x, y, ) f (x + y) dy

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(142)

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Scaling, Fractals and Wavelets

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where (x, y, ) is a smoothing function centered on x, for example, a Gaussian function of standard error More generally, we can use wavelet transformations based on a lter that is not necessarily conservative Such a point of view is particularly well-adapted to applications in physics: any real measurement is always performed at nite resolution (see [NOT 93] for additional comments on this point) In this framework, f (x) becomes the limit for 0 of the family of functions f (x), ie, in other words, of the function of two variables f (x, ) However, whereas f (x, 0) is non-differentiable (in the sense of the non-existence of the limit df /dx when tends to zero), f (x, ), which we call a fractal function (and which is, in fact, de ned using a class of equivalence that takes into account the fact that is a resolution, see [NOT 93]), is now differentiable for all = 0 The problem of physically describing of the various processes where such a function f intervenes is now posed differently In standard differentiable physics, it amounts to nding differential equations involving the derivatives of f with respect to space-time coordinates, ie, f / x, 2 f / x2 , namely, the derivatives which intervene in laws of displacement and motion The integro-differential method amounts to performing such a local description of space-time elementary displacements, of their effect on quantum physics and then integrating in order to obtain the large scale properties of the system under consideration Such a method has often been called reductionist and it was indeed adapted to traditional problems where no new information appears at different scales The situation is completely different for systems characterized by a fractal geometry and/or non-differentiability Such behaviors are found towards very small and very large scales, but also, more generally, in chaotic and/or turbulent systems and probably in basically all living systems In these cases, new, original information exists at different scales and the project to reduce the behavior of a system to one scale (in general, to a large scale) from its description at another scale (in general, the smallest possible scale, x 0) seems to lose its meaning and to become hopeless Our suggestion consists precisely of giving up such a hope and introducing a new frame of thought where all scales co-exist simultaneously inside a unique scale-space, and are connected together using scale differential equations acting in this scale-space Indeed, in non-differentiable physics, f (x)/ x = f (x, 0)/ x no longer exists However, physics of the given process will be completely described provided we succeed in knowing f (x, ), which is differentiable (for x and ) for all nite values of Such a function of two variables (which is written more precisely, to be complete, as f [x( ), )]) can be the solution of differential equations involving f (x, )/ x but also f (x, )/ ln More generally, with non-linear laws, the equations of physics take the form of second-order differential equations, which will

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