3. Theoretical framework
3.1. Description of a non-differentiable and fractal space-time
Giving up differentiability of the space-time coordinates has three main consequences: (i) the explicit scale-dependence of physical quantities on space-time resolutions, that implies the construction of new fundamental laws of scale; (ii) the multiplication to infinity of the number of geodesics, that suggests jumping to a statistical and probabilistic description; (iii) the breaking of the time symmetry () at the level of the space-time geometry, that implies a "two-valuedness" of velocities which we represent in terms of a complex and non-classical new physics. The aim of the present section is to explain in more detail how the giving up of differentiability leads us to introduce such new structures.
3.1.1. New scale laws
Strictly, the nondifferentiability of the coordinates means that the velocity is no longer defined. However, as recalled in the introduction, the combination of continuity and nondifferentiability implies an explicit scale-dependence of the various physical quantities (Nottale 1993a, 1994a). Therefore the basis of our method consists in replacing the classical velocity by a function that depends explicitly on resolution, . Only is now undefined, while is now defined for any non-zero . Consider indeed the usual expression for the velocity:
In the nondifferentiable case, the limit is undefined. This means that, when dt tends to zero, either the ratio tends to infinity, or it fluctuates without reaching any limit. The solution proposed in scale relativity to this problem is very simple. We replace the differential dt by a scale variable , and we consider now as an explicit function of this variable:
Here is a "fractal function" (see Nottale, 1993a, Chap. 3.8). It is defined modulo some equivalence relation which expresses that the variable has the physical meaning of a resolution: , where is the resolution in f and g which corresponds to the resolution in t (e.g., in the case of a constant fractal dimension D, . This means that we no longer work with the limit , which is anyway devoid of observable physical meaning (since an infinite energy-momentum would be needed to reach it, according to quantum mechanics), and that we replace this limit by a description of the various structures which appear during the zoom process toward the smaller scales. Our tool can be thought of as the theoretical equivalent of what are wavelets in fractal and multifractal data analysis (see e.g. Arneodo et al. 1988, Argoul et al. 1989, Farge et al. 1996).
The advantage of our method is now that, for a given value of the resolution , differentiability in t is recovered. The non-differentiability of a fractal function means that does not exist. But exists for any given value of the resolution , which allows us to recover a differential calculus even when dealing with non-differentiability. However, one should be cautious about the fact that the physical description and the mathematical description are no longer always coincident. Indeed, once given, one can write a mathematical differential equation involving terms like . In such an equation, one can make and then use the standard mathematical methods to solve for it and determine . But it must be understood that this is a purely mathematical intermediate description with no physical counterpart, since for the real system under consideration, the very consideration of an interval changes the function f (such a behavior is experimentally well-known in quantum systems). As a consequence of this analysis, there is a particular subspace of description where the physics and the mathematics coincide, namely, when making the particular choice . We shall work in what follows with such an identification of the time differential and of the new time resolution variable.
A consequence of the new description is that the current equations of physics are now incomplete, since they do not describe the variation of the various physical quantities in scale transformations . The scale-dependence of the velocity suggests that we complete the standard equations of physics by new differential equations of scale.
In order to work out such a completion, let us apply to the velocity and to the differential element (now interpreted as a resolution) the reasoning already touched upon in Sect. 2. The simplest possible equation that one can write for the variation of the velocity in terms of the new scale variable dt is:
i.e., a first order, renormalization-group-like differential equation, written in terms of the dilatation operator
in which the infinitesimal scale-dependence of V is determined by the "field" V itself. The -function here is a priori unknown, but we can use the fact that (in motion-relativistic units) to expand it in terms of a Taylor expansion. We obtain:
where v is a mean velocity and a fractal fluctuation that is explicitly scale-dependent, and where and are chosen such that and .
We recognize here the combination of a typical fractal behavior with fractal dimension D, and of a breaking of the scale symmetry at scale , that plays the role of an upper fractal / nonfractal scale transition (since when and when ). As announced in Sect. 2, the symmetry breaking is not added artificially here to the scale laws, but is obtained as a natural consequence of the scale-relativistic approach, in terms of solutions to Eq. (22). It is now clear from Eq. (23) (see also Fig. 3) that the symmetry breaking comes from a confrontation of the motion behavior (as described by the v component of V) with the scale behavior (as described by the component of V). Their relative sizes determine the scale of the transition (Fig. 3).
Concerning the value of the fractal dimension, recall that plays the role of a critical dimension in the whole theory, (see Nottale 1993a and refs. therein, Nottale 1995a). In this case we find in the asymptotic scaling domain that , in agreement with Feynman and Hibbs (1965).
Let us finally write the expression for the elementary displacement derived from the above value of the velocity. We shall now consider the two inverse cases identified in Sect. 2.5, i.e. not only the case where the scaling domain is at small scales (standard quantum mechanics) but also the case where it lies at large scales, which is the relevant situation in the present paper. In both cases, the elementary displacement dX in the scaling domain can be written under the sum of two terms,
where is a constant. The comparison with Eq. (23) allows to show that the transition scale is therefore (Nottale 1994a) when . In the scaling regime () both terms are relevant, since vanishes in the mean, i.e. , but (left of Fig. 3): we shall see in what follows that the fluctuation, in spite of its vanishing in the mean, plays nevertheless an essential role in the laws of average motion. When applied to atomic and elementary particle physics (microscopic case), we find that the fluctuation becomes dominated at larger scales () by the classical term , and the system becomes classical beyond the de Broglie scale (since is the Compton scale in this case). When applied to the macroscopic case, the situation is different, since: (i) there is a new transition to classical behavior below some smaller scale , in accordance with the solutions of Eq. (12); (ii) the upper transition scale is expected to be pushed to infinity, since the theory will be preferentiably applied to bound systems such that the classical average velocity (hydrogen atom-like systems), while it will a priori be irrelevant for free systems.
Equations (24- 25- 26) can be used to recover a fundamental, well-known formula relating the space-resolution and the time-resolution in the asymptotic domain on a fractal curve (see Fig. 4)
in which the length scale and the time-scale are naturally introduced for dimensional reasons.
In the present series of paper, only the above simplest scale-laws with fractal dimension will be developed. However, as recalled in Sect. 2 and Appendix B, these laws can be identified with "Galilean" approximations of more general scale-relativistic laws in which the fractal dimension becomes itself variable with scale (Nottale 1992, 1993a, 1995d). Such special scale-relativistic laws are expected to apply toward the very small and very large scales (see Appendix B).
3.1.2. Infinity of geodesics
The above description applies to an individual fractal trajectory. However, we are not interested here in the description of fractal trajectories in a space that would remain Euclidean or Riemannian, but in the description of a fractal space and of its geodesics. The trajectories are then fractal as a consequence of the fractality of space itself. This problem is analogous to the jump from flat to curved space-time in Einstein's general relativity. One can work in a curvilinear coordinate system in flat space-time, and this introduces a GR-like metric element, but this apparent new structure is trivial and can be cancelled by coming back to a Cartesian coordinate system; on the contrary the curvature of space-time itself implies structures (described e.g. by the curvature invariants) that are new and irreducible to the flat case, since no coordinate system can be found where they would be cancelled (except locally). The same is true when jumping, as we attempt here, from a differentiable (Riemannian) manifold to a nondifferentiable (non Riemannian) manifold. We expect the appearance of new structures that would be also new and irreducible to the old theory. Two of these new geometric properties will be now described (but it is clear that this is only a minimal description, and that several other features will have to be introduced for a general description of nondifferentiable spacetimes).
One of the geometric consequences that is specific of the nondifferentiability and of the subsequent fractal character of space itself (not only of the trajectories), is that there will be an infinity of fractal geodesics that relate any couple of points in a fractal space (Nottale 1989, 1993a). The above description of an individual fractal trajectory is thus insufficient to account for the properties of motion in a fractal space. This is an important point, since, as recalled in the introduction, our aim here is to recover a physical description of motion and scale laws, even in the microscopic case, by using only the geometric concepts and methods of general relativity (once generalized, using new tools, to the nondifferentiable case). These basic concepts are the geometry of space-time and its geodesics, so that we have suggested (Nottale 1989) that the description of a quantum mechanical particle (including its property of wave-corpuscle duality) could be reduced to the various geometric properties of the ensemble of fractal geodesics of the fractal space-time that correspond to a given state of this "particle" (defined here as a geometric property of a subset of all geodesics). In such an interpretation, we do not have to endow the "particle" with internal properties such as mass, spin or charge, since the "particle" is identified with the geodesics themselves (not with a point mass which would follow them), and since these "internal" properties can be defined as geometric properties of the fractal geodesics themselves. As a consequence, any measurement is interpreted as a sorting out of the geodesics, namely, after a measurement, only the subset of geodesics which share the geometrical property corresponding to the measurement result is remaining (for example, if the "particle" has been observed at a given position with a given resolution, this means that the geodesics which pass through this domain have been selected).
This new interpretation of what are "particles" ensures the validity of the Born axiom and of the Von Neumann axiom (reduction of wave function) of quantum mechanics. This is confirmed by recent numerical simulations by Hermann (1997), that have indeed shown that one can obtain solutions to the Schrödinger equation without using it, directly from the elementary process introduced in scale relativity. Moreover, a many-particle simulation of quantum mechanics has been performed by Ord (1996a, b) in the fractal space-time framework. He finds, in agreement with our own results, that the Schrödinger equation may describe ensembles of classical particles moving on fractal random walk trajectories, so that it has a straighforward microscopic model which is not, however, appropriate for standard quantum mechanics.
This point is also a key to understanding the differences between
the microscopic and macroscopic descriptions, which implies a
fundamental difference of interpretation of the final quantum-like
equations and of their solutions. The two main differences are the
The infinity of geodesics leads us to jump to a statistical
description, i.e., we shall in what follows consider averages on the
set of geodesics, not on an a priori defined probability density as in
stochastic theories. Namely, two kinds of averaging processes are
relevant in our description:
In what follows, the decomposition of dX in terms of a mean, , and a fluctuation respective to the mean, (such that by definition) will be made using both averaging processes. Since all geodesics are assumed to share the same statistical fractal geometric properties, the form of Eqs. (25- 27), is conserved. We stress once again the fact that the various expectations are taken in our theory on the set of geodesics, not on a previously given probability density. The probability density will be introduced as the density of the fluid of geodesics, ensuring by construction the Born interpretation of the theory.
We also recall again that, in the particular domain of application with which we are concerned in the present series of papers (macroscopic large scale systems), two particular features are relevant: (i) a lower transition scale must be introduced, as recalled above and as predicted from Eq. (12); (ii) the average classical velocity must be zero, implying an infinite upper fractal / nonfractal transition (see the discussion in Sect. 5). Remark, however, (see Sects. 3.1.3 and 3.2) that we will be led to introduce two average velocities, a forward one and a backward one , in terms of which the classical average velocity writes . Therefore its vanishing does not mean the vanishing of and individually.
3.1.3. Differential time symmetry breaking
The nondifferentiable nature of space-time implies an even more dramatic consequence, namely, a breaking of local time reflection invariance. Remark that such a discrete symmetry breaking can not be derived from only the fractal or nondifferentiable nature of trajectories, since it is a consequence of the irreducible nondifferentiable nature of space-time itself.
Consider indeed again the definition of the derivative of a given function with respect to time:
The two definitions are equivalent in the differentiable case. One passes from one to the other by the transformation (local time reflection invariance), which is therefore an implicit discrete symmetry of differentiable physics. In the nondifferentiable situation considered here, both definitions fail, since the limits are no longer defined. The scale-relativistic method solves this problem in the following way.
We have attributed to the differential element dt the new meaning of a variable, identified with a time-resolution, as recalled hereabove ("substitution principle"). The passage to the limit is now devoid of physical meaning (since quantum mechanics itself tells us that an infinite momentum and an infinite energy would be necessary to make explicit measurements at zero resolution interval). In our new framework, the physics of the problem is contained in the behavior of the function during the "zoom" operation on . The two functions and are now defined as explicit functions of t and of dt:
When applied to the space variable, we get for each geodesic two velocities that are fractal functions of resolution, and . In order to go back to the classical domain, we first smooth out each geodesic with balls of radius larger than : this defines two classical velocity fields now independent of resolution, and ; then we take the average on the whole set of geodesics. We get two mean velocities and , but after this double averaging process, there is no reason for these two velocities to be equal, contrarily to what happens in the classical, differentiable case.
In summary, while the concept of velocity was classically a one-valued concept, we must introduce, if space-time is nondifferentiable, two velocities instead of one even when going back to the classical domain. Such a two-valuedness of the velocity vector is a new, specific consequence of nondifferentiability that has no classical counterpart (in the sense of differential physics), since it finds its origin in a breaking of the discrete symmetry (). This symmetry was considered self-evident up to now in physics, so that it has not been analysed on the same footing as the other well-known symmetries. It is actually independent from the time reflection symmetry T, even though it is clear that the breaking of this "dt symmetry" implies a breaking of the T symmetry at this level of the description.
Now we have no way to favor rather than . Both choices are equally qualified for the description of the laws of nature. The only solution to this problem is to consider both the forward () and backward () processes together. The number of degrees of freedom is doubled with respect to the classical, differentiable description (6 velocity components instead of 3).
A simple and natural way to account for this doubling of the needed information consists in using complex numbers and the complex product. As we shall recall hereafter, this is the origin of the complex nature of the wave function in quantum mechanics, since the wave function can be identified with the exponential of the complex action that is naturally introduced in such a theory. One can indeed demonstrate (Nottale 1997) that the choice of complex numbers to represent the two-valuedness of the velocity is not an arbitrary choice, since it achieves a covariant description of the new mechanics: namely, it ensures the Euler-Lagrange equations to keep their classical form and allows one not to introduce additional terms in the Schrödinger equation. Note also that the new complex process, as a whole, recovers the fundamental property of microscopic reversibility.
3.2. Scale-covariant derivative
Finally, we can describe (in the scaling domain) the elementary displacement dX for both processes as the sum of a mean, , and a fluctuation about this mean, which is then by definition of zero average, , i.e.:
Consider first the average displacements. The fundamental irreversibility of the description is now apparent in the fact that the average backward and forward velocities are in general different. So mean forward and backward derivatives, and are defined. Once applied to the position vector x, they yield the forward and backward mean velocities, and .
standing for a fundamental parameter that characterizes the new scale law at this simple level of description (see Sect. 4.6for a first generalization). The 's are of mean zero and mutually independent. If one assumes them to be also Gaussian, our process becomes a standard Wiener process. But such an assumption is not necessary in our theory, since only the property (Eq. (32)) will be used in the calculations.
Our main tool now consists of recovering local time reversibility in terms of a new complex process (Nottale 1993a): we combine the forward and backward derivatives in terms of a complex derivative operator
which, when applied to the position vector, yields a complex velocity
The real part V of the complex velocity generalizes the classical velocity, while its imaginary part, U, is a new quantity arising from non-differentiability (since at the classical limit, , so that ).
Equation (32) now allows us to get a general expression for the complex time derivative . Consider a function . Contrarily to what happens in the differentiable case, its total derivative with respect to time contains finite terms up to higher order (Einstein 1905). In the special case of fractal dimension 2, only the second order intervenes. Indeed its total differential writes
Classically the term is infinitesimal, but here its average reduces to , so that the last term of Eq. (35) will amount to a Laplacian thanks to Eq. (32). Then
By inserting these expressions in Eq. (33), we finally obtain the expression for the complex time derivative operator (Nottale 1993a):
The passage from classical (differentiable) mechanics to the new nondifferentiable mechanics can now be implemented by a unique prescription: Replace the standard time derivative by the new complex operator . In other words, this means that plays the role of a scale-covariant derivative (in analogy with Einstein's general relativity where the basic tool consists of replacing by the covariant derivative .
3.3. Scale-covariant mechanics
Let us now give the main steps by which one may generalize classical mechanics using this scale-covariance. We assume that any mechanical system can be characterized by a Lagrange function , from which an action is defined:
Our Lagrange function and action are a priori complex and are obtained from the classical Lagrange function and classical action S precisely from applying the above prescription . The action principle (which is no longer a "least-action principle", since we are now in a complex plane, but remains a "stationary-action principle"), applied on this new action with both ends of the above integral fixed, leads to generalized Euler-Lagrange equations (Nottale 1993a)
which are exactly the equations one would have obtained from applying the scale-covariant derivative () to the classical Euler-Lagrange equations themselves: this result demonstrates the self-consistency of the approach and vindicates the use of complex numbers. Other fundamental results of classical mechanics are also generalized in the same way. In particular, assuming homogeneity of space in the mean leads to defining a generalized complex momentum and a complex energy given by
If one now considers the action as a functional of the upper limit of integration in Eq. (38), the variation of the action from a trajectory to another nearby trajectory, when combined with Eq. (39), yields a generalization of other well -known relations of classical mechanics:
We shall now apply the scale-relativistic approach to various domains of physics which are particularly relevant to astrophysical problems.
© European Southern Observatory (ESO) 1997
Online publication: April 6, 1998