Appendix A: continuity and nondifferentiability implies scale-divergence
One can demonstrate (Nottale 1993a, 1996a) that the length of a continuous and nowhere differentiable curve is dependent on resolution , and, further, that when , i.e. that this curve is fractal (we used the word "fractal" in this extended meaning throughout this paper). The scale divergence of continuous and almost nowhere differentiable curves is a direct consequence of Lebesgue's theorem, which states that a curve of finite length is almost everywhere differentiable.
Consider indeed a continuous but nondifferentiable function between two points A and A in the Euclidean plane. Since f is non-differentiable, there exists a point A1 of coordinates with , such that is not on the segment A0 A . Then the total length . We can now iterate the argument and find two coordinates and with and , such that . By iteration we finally construct successive approximations of whose lengths increase monotonically when the "resolution" tends to zero. In other words, continuity and nondifferentiability implies a monotonous scale dependence of f on resolution . Now, Lebesgue's theorem states that a curve of finite length is almost everywhere differentiable (see e.g. Tricot 1993).
Therefore, if f is continuous and almost everywhere nondifferentiable, then when the resolution ; namely f is not only scale-dependent, but even scale-divergent. This theorem is also demonstrated in (Nottale 1993a, p.82) by using non-standard analysis.
What about the reverse proposition: Is a continuous function whose
length is scale-divergent between any couple of points (such that
finite), i.e., when
, nondifferentiable? The answer is as follows:
This can be demonstrated by looking at the way the length increases and the slope changes under successive zooms of a constant factor . There are two ways by which the divergence can occur: either by a regular increase of the length (due to the regular appearance of new structures at all scales that continuously change the slope), or by the existence of jumps (in this case, whatever the scale, there will always exist a smaller scale at which the slope will change). The power law corresponds to a continuous length increase, , then to a continuous and regular change of slope when : therefore the function is nondifferentiable in this case. Divergences slower than power laws may correspond to a regular length increase, but with a factor which becomes itself scale-dependent: with when . In this case, some functions can be differentiable, if they are such that new structures indeed appear at all scales (and could then be named "fractal" under the general definition initially given by Mandelbrot 1975 to this word), but these structures become smaller and smaller with decreasing scale, so that a slope can finally be defined in the limit . Some other functions diverging slower than power laws are not differentiable, e.g. if there always exists a scale smaller than any given scale such that an important change of slope occurs: in this case, the slope limit may not exist in the end.
Appendix B: special and generalized scale-relativity
B.1. Special scale relativity
It is well known that the Galileo group of motion is only a degeneration of the more general Lorentz group. The same is true for scale laws. Indeed, if one looks for the general linear scale laws that come under the principle of scale relativity, one finds that they have the structure of the Lorentz group (Nottale 1992). Therefore, in special scale relativity, we have suggested to substitute to the Galilean laws of dilation the more general Lorentzian law (Nottale 1992, 1993a):
This expression is yet uncomplete, since under this form the scale relativity symmetry remains unbroken. Such a law corresponds, at the present epoch, only to the null mass limit. It is expected to apply in a universal way during the very first instants of the Universe. This law assumes that, at very high energy, no static scale and no space or time unit can be defined, so that only pure contractions and dilations have physical meaning. The corresponding physics is a physics of pure numbers. In Eq. (B1), there appears a universal, purely numerical constant , which plays the role of a maximal possible dilation. We have found that the value of is about (Nottale 1993a, 1995d, 1996a): its existence yields an explanation to the Eddington-Dirac large number hypothesis, and connects the cosmological constant to the Planck scale. A more detailed study of these questions will be presented in a forthcoming work (Nottale 1997).
However, the effect of the spontaneous scale symmetry breaking which arises at some scale is to yield a new law in which the invariant is no longer a dilation , but becomes a length-time scale . In other words, there appears in the theory a fundamental scale that plays the role of an unpassable resolution, invariant under dilations (Nottale 1992). Such a scale of length and time is an horizon for scale laws, in a way similar to the status of the velocity of light for motion laws. The new law of composition of dilations and the scale-dimension now write respectively (in the scale-dependent domains, i.e. only below the transition scale in microphysics and beyond it in the cosmological case):
A fractal curvilinear coordinate becomes now scale-dependent in a covariant way, namely . One of the main new feature of special scale relativity with respect to the previous fractal or scale-invariant approaches is that the scale-dimension , which was previously constant, is now explicitly varying with scale (see Fig. 5) and even diverges when resolution tends to the new invariant scales. In the microphysical domain, the invariant length-scale is naturally identified with the Planck scale, , that now becomes impassable and plays the physical role that was previously devoted to the zero point (Nottale 1992, 1993a). The same is true in the cosmological domain, with once again an inversion of the scale laws. We have identified the invariant maximal scale with the scale of the cosmological constant, . The consequences of this new interpretation of the cosmological constant have been considered in (Nottale 1993a, 1995d, 1996a) and will be developed further in a forthcoming work (Nottale 1997).
Note that special scale-relativistic laws (Nottale 1992) have also recently been considered by Dubrulle (1994) and Dubrulle and Graner (1996) for the description of turbulence, with a different interpretation of the variables.
It is also noticeable that recent developments in string theories (Witten 1996) have reached conclusions that are extraordinarily similar to those of scale relativity. One finds that there is a smallest circle in string theory (whose radius is about the Planck length), and that strings are characterized by duality symmetries. Two of these dualities are especially relevant to our approach, since they make already part of it in a natural way. The first is the quantum / classical duality, which we recover in terms of our scale / motion duality. The second is a microscopic / macroscopic duality: it has been found that strings do not distinguish small spacetime scales from large ones, relating them through an inversion. But scale inversion is a transformation which is naturally included in the scale-relativistic framework (see Sect. 2.5), since this is nothing but the symmetric element of the scale group ( in the Galilean case). Therefore it has recently been claimed by Castro (1996) that scale relativity is the right framework in which the newly discovered string structures will take their full physical meaning. The string duality between the small and large scales adds a new argument to our main conclusion: namely, that the laws of physics take again a quantum-like form at very large spacetime scales.
B.2. From scale dynamics to general scale relativity
The whole of our previous discussion indicates to us that the scale invariant behavior corresponds to freedom in the framework of a scale physics. However, in the same way as there exists forces in nature that imply departure from inertial, rectilinear uniform motion, we expect most natural fractal systems to also present distorsions in their scale behavior respectively to pure scale invariance. Such distorsions may be, as a first step, attributed to the effect of a scale "dynamics", i.e. to "scale-forces". (Caution: this is only an analog of "dynamics" which acts on the scale axis, on the internal structures of a given point at this level of description, not in space-time. See Sect. B.3 for first hints about the effects of coupling with space-time displacements). In this case the Lagrange scale-equation takes the form of Newton's equation of dynamics:
where is a "scale-mass", which measures the way the system resists to the scale-force.
B.2.1. Constant scale-force
Let us first consider the case of a constant scale-force. Eq. (B4) writes
where constant. It is easily integrated in terms of the usual parabolic solution (where :
However the physical meaning of this result is not clear under this form. This is due to the fact that, while in the case of motion laws we search for the evolution of the system with time, in the case of scale laws we search for the dependence of the system on resolution, which is the directly measured observable. We find, after redefinition of the integration constants:
The scale dimension becomes a linear function of resolution (the same being then true of the fractal dimension ), and the relation is now parabolic rather than linear as in the standard power-law case. There are several physical situations where, after careful examination of the data, the power-law models were clearly rejected since no constant slope could be defined in the plane. In the several cases where a clear curvature appears in this plane (e.g., turbulence, sand piles,...), the physics could come under such a "scale-dynamical" description. In these cases it might be of the highest interest to identify and study the scale-force responsible for the scale distorsion (i.e., for the deviation to standard scaling).
B.2.2. Harmonic oscillator
Another interesting case of scale-potential is that of a repulsive harmonic oscillator. It is solved as
For it gives the standard Galilean case , but its large-scale behavior is particularly interesting, since it does not permit the existence of resolutions larger than a scale . Such a behavior could provide a model of confinement in QCD (Nottale 1997).
More generally, we shall be led to look for the general non-linear scale laws that satisfy the principle of scale relativity (see also Dubrulle and Graner 1997). As remarked in (Nottale 1994b, 1996a), such a generalized framework implies working in a five-dimensional fractal space-time. The development of such a "general scale-relativity" lies outside the scope of the present paper and will be considered elsewhere (Nottale 1997).
B.3. Scale-motion coupling and gauge invariance
The theory of scale relativity also allows to get new insights about the physical meaning of gauge invariance (Nottale 1994b, 1996a). In the previous scale laws, only scale transformations at a given point were considered. But we must also wonder about what happens to the structures in scale of a scale-dependent object when it is displaced. Consider anyone of these structures, lying at some (relative) resolution (such that , where is the fractal/nonfractal transition) for a given position of the particle. In a displacement of the object, the relativity of scales implies that the resolution at which this given structure appears in the new position will a priori be different from the initial one. In other words, is now a function of the space-time coordinates, , and we expect the occurrence of dilatations of resolutions induced by translations, which read:
where a four-vector must be introduced since is itself a four-vector and a scalar (in the case of a global dilation). This behavior can be expressed in terms of a new scale-covariant derivative:
However, if one wants such a "field" to be physical, it must be defined whatever the initial scale from which we started. Starting from another scale (we consider only Galilean scale-relativity here, see Nottale 1994b, 1996a for the additional implications of special scale-relativity), we get
which depends on the relative "state of scale", . However, if one now considers translation along two different coordinates (or, in an equivalent way, displacement on a closed loop), one may write a commutator relation:
This relation defines a tensor field which, contrarily to , is independent of the initial scale. One recognizes in the analog of an electromagnetic field, in , that of an electromagnetic potential, in e that of the electric charge, and in Eq. (B14) the property of gauge invariance which, in accordance with Weyl's initial ideas (Weyl 1918), recovers its initial status of scale invariance. However, Eq. (B14) represents a progress compared with these early attempts and with the status of gauge invariance in today's physics. Indeed the gauge function, which has, up to now, been considered as arbitrary and devoid of physical meaning, is now identified with the logarithm of internal resolutions. In Weyl's theory, and in its formulation by Dirac (1973), the metric element ds (and consequently the length of any vector) is no longer invariant and can vary from place to place in terms of some (arbitrary) scale factor. Such a theory was excluded by experiment, namely by the existence of universal and unvarying lengths such as the electron Compton length (i.e., by the existence of particle masses). In scale relativity, we are naturally led to introduce two "proper times", the classical one ds which remains invariant, and the fractal one , which is scale-divergent and can then vary from place to place (its variation amounting to a scale transformation of resolution). In Galilean scale-relativity, the fractal dimension of geodesics is , so that the scale-dependence of writes . Therefore we have , and we recover the basic relation of the Weyl-Dirac theory, in the asymptotic high energy domain . Another advantage with respect to Weyl's theory is that we are now allowed to define four different and independent dilations along the four space-time resolutions instead of only one global dilation. The above U(1) field is then expected to be embedded into a larger field, in agreement with the electroweak theory, and the charge e to be one element of a more complicated, "vectorial" charge (Nottale 1997). Moreover, when combined with the Lorentzian structure of dilations of special scale relativity, our interpretation of gauge invariance yields new relations between the charges and the masses of elementary particles (Nottale 1994b, 1996a).
© European Southern Observatory (ESO) 1997
Online publication: April 6, 1998