All the hereabove arguments indicate that one must give up the absolute Minkowskian space-time postulated in the current quantum theory, and replace it by a space-time which is relative to its material and energetic content and explicitly dependent on scale. Three mathematical methods may be considered to achieve such a program. The first one is geometric: the concept of fractal37,38 refers to objects or sets which are indeed scale-dependent. It must be generalized to that of fractal space,37 while the concept of scale invariance must be extended to that of scale covariance. One may also look for an algebraic tool: the renormalization group is thus very well adapted, but must also be generalized in order to satisfy scale covariance. A third method (which will not be considered here) could be to work in the framework of the conformal group, owing to the fact that it already contains dilatations in its transformations. As a consequence, we expect space-time to be described by a metric element based on generalized, explicitly scale-dependent, metric potentials
gmn = gmn(t,x,y,z;Dt,Dx,Dy,Dz).33
Let us specify the physical meaning of this proposal. The concept of space-time allows one to think of all the positions and instants taken together as a whole. Space-time may be viewed as the set of all events and of the transformations between them. But to the set of all events, xm = -infinity to +infinity, (m = 0 to 3), we add the set of all possible resolutions, ln(Dxm) = - infinity to +infinity. Let us call this set "zoom". Hence the geometrical frame in which it will be attempted to work is, strictly speaking, a "zoom-space-time". This means that geometrical structures may be looked for, not only in space-time, but also in the zoom dimension (see [57] Chapter 4). However, as remarked in the Relativity of Scales section, resolutions and space-time variables do not play an identical role. The "space-time-zoom" is equivalent to phase space rather than an extension of space-time.
The important point to be understood by the reader, since it underlies our whole methods and results, is that we call for a profound change of mentality in the physical approach to the problem of scales. One must give up the "reductionist" view of perfect points whose small scale organization would give rise to the large scale one. One must even go beyond the view of a physics where several particular scales are relevant. We claim that a genuine physics of scale can be constructed only in a frame of thought where all scales in Nature would be simultaneously considered, i.e., when placing ourselves in a continuum of scales. In such a perspective, the standard coordinates themselves lose their physical meaning, and should be replaced by fractal coordinates which are explicitly scale dependent, X=X(s,e) (see [57] Chapters 3 and 5).
The geometrical properties and structures of the microphysical space-time remain to be built in details. As stated above, this book reviews an approach of this problem where it is proposed that, on account of its inferred dependence (and divergence) on resolution, one of the main properties of such a geometry would be its fractal character. We recall that we have previously proposed54 that the concept of fractal should be applied not only to sets or objects embedded in Euclidean space, but to a whole space (more generally space-time) considered in an intrinsic way, i.e., for which curvilinear coordinates, metrics elements, geodesics etc... should be defined. Such an approach may be related to Le Méhauté's,56 who was able to describe new electromagnetic properties arising in fractal media, by using the mathematical tool of non-integer integro-differentiation.
We indeed think that the concept of fractal space-time allows one to revise the conclusion that the quantum mechanical behaviour cannot be derived from a geometrical theory. Several mathematical properties of fractals go in the right direction, e.g.:
* One of the main characteristics of the fractal geometry is its dependence on resolution. Thus it offers a natural way of actualization of the hereabove suggested extension of the principle of relativity, by the use of a spatio-temporal description.
* It was realized by Feynman34,35 that a particle path in quantum mechanics may be described as a continuous and non-differentiable curve, while non-differentiability is one of the properties of fractals. More precisely, a particle trajectory (of topological dimension 1) in nonrelativistic quantum mechanics may be characterized by a fractal dimension 2 when the resolution becomes smaller than its de Broglie length (see [36] and [57] Chapter 4). This result, being derived from the Heisenberg relation and equivalent to it (see hereafter), is of a universal character. So, in the same way that general relativity attributes to space-time the universal property of curvature of trajectories, our suggestion is to attribute to quantum space-time the universal property of fractalization owned by quantum mechanical trajectories. The implications of this proposal will be specified throughout this book.
* Infinite numbers arise naturally on fractals, and the occurence of infinite quantities is one of the difficulties of current quantum physics. It is remarkable that the infinities which appeared in quantum electrodynamics precisely concern physical quantities like masses (i.e., self-energy) or charges, which are fundamental invariants built from space-time and quantum phase symmetries. One may wonder whether the need for renormalization comes from the lack of account of the irreducible space-time infinities which would be a part of the nature of a fractal space-time.
* Extending the general relativistic approach, the particles in a fractal space-time are expected to follow the "geodesical" lines. But the absence of derivative, the folding and the infinite number of obstacles at all scales due to the fractal structure allow one to infer that an infinity of geodesics will exist between any two points, so that only statistical predictions will be allowed (see [57] Sec. 5.5).
Additional examples of the adequation of fractals and quantum mechanical properties will be reviewed in this approach. We are led throughout this work by the postulate that microphysical space-time is a self-avoiding fractal continuum of topological dimension 4. Fractal 4- coordinates are assumed to be defined on this fractal space-time. (Some ways to deal with their infinite character and with their non-differentiability are proposed in [57] Chapter 3). These fractal coordinates correspond to the ideal case of infinite resolution, Dxm = 0. Then the various supersystems of coordinates which correspond to finite resolution will be obtained by smoothing them with "4-balls" (Dt,Dx,Dy,Dz). The classical coordinates (which are independent of resolutions) result from the same smoothing process, but with balls larger than some transitional values lm, corresponding to the fractal/nonfractal transition, which we identify with the quantum/classical transition. Note also that the physical being to be used in order to fit with the "zoom-space-time" idea is not only the fractal itself (i.e. the final result of a fractalization process), but mainly the set of all its approximations for all possible values of space-time resolution. Some of its other properties will gradually emerge, while attempts will be made to express the main quantum mechanics results in terms of geometrical fractal structures.
In particular, our application of the principle of scale relativity to the question of the fractal dimension of quantum paths leads us to the conclusion that the constant fractal dimension D=2 obtained from standard quantum physics is only a "large" scale approximation. This Brownian-motion like fractal dimension (see [57] Sec. 5.6) may also be interpreted in terms of a constant anomalous dimension d=1. But this constant value corresponds to "Galilean" scale laws, while the requirement of scale covariance leads us to the conclusion that the correct renormalization group for space-time must be a Lorentz group (see [57] Chapter 6).
Already the new structure of space-time revealed by special motion relativity at the beginning of the century underlies in an inescapable way the Riemannian structure of Einstein's general relativity: Space-time is locally Minkowskian, so that all attempts at constructing a Riemannian theory of gravitation were condemned to fail in the absence of special relativity constraints. In the same way (assuming that the whole approach is correct) if a full theory of scale relativity is to be developed one day in terms of fractal space-time, we think that such a theory will be forced to incorporate in its description the new structure of space-time which is described in [57] Chapter 6: a space-time where the perfect zero point has disappeared from concepts having physical meanings, whose fractal dimension is not constant but scale-dependent and whose local invariance group is Lorentz's, for motion as well as scale transformations.
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Quoted from: Nottale L., 1993, "Fractal Space-Time and Microphysics: Toward of Theory of Scale Relativity", Chap. 2 (© World Scientific, 1993).
Sunday, September 23, 2007
On the Nature of Quantum Space-Time
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3 comments:
Laurent Nottale is one of the most original theoretical physicists ever to contemplate quantum gravity and relativistic cosmology. His approach is unique because the tools which he used are unique. Laurent used fractal geometry, by far the most general geometry that exists. As recounted by Mohamed El Naschie quoting a passage from a paper published recently by Prof. T.N. Palmer in the Proceedings of the Royal Society “quantum mechanics is blind to fractals”. This short quotation encapsulates it all. El Naschie said it reminds him of Einstein’s famous quotation “God doesn’t place dice”. If quantum mechanics is blind to fractals so is quantum field theory and almost everything else based on quantum mechanics. The bad news is at its very deep roots, nature is fractal. Consequently without fractal geometry we could not have a deep understanding of anything as deep as particle physics. That is where the secret of the success of Nottale’s theory as well as the theory of Garnet Ord and Mohamed El Naschie. Alas the establishment is not yet ready to acknowledge this fact. As far as I know only one member of the establishment Gerard ‘tHooft half heartedly acknowledged the fiasco of quantum field theory. You will probably say Palmer, an Oxford professor, discovered the vital importance of being a fractal for quantum mechanics and he is a Fellow of the Royal Society and thus the establishment. This is not completely correct. Although Palmer studied physics in Oxford and Cambridge, most of his professional work is in meteorology. That may explain the fact that he had the courage to try something else and to look deep at the approach using fractals. I have no doubt that the future will belong to fractal spacetime, E-infinity theory and scale relativity.
Nottale is surely an original researcher. Why else would they attack him so viciously. It is not because of his association with Garnet Ord or Mohamed El Naschie. It is because of his originality and his daring to question conventional wisdom. Besides how would the establishment explain the experimental confirmation of the golden mean as the basis of quantum mechanics. Richard Feynmann, Garnet Ord, Nottale and El Naschie were the first to discover fractals for quantum mechanics. By using golden mean fractals better known as random Cantor sets, El Naschie went one step further providing a means for performing complex computation with unheard of simplicity and without using a computer. This is the advantage of the golden mean binary. The golden mean in quantum mechanics is now a fact. It is an experimental fact and no amount of defamation and distortion will be able to blind true researchers from using the work of Nottale and his colleagues.
A few days ago I was accosted by an article entitled A multiverse of probabilities published in Physics World by the author Ben Freivogel (see p.40). I claim that the basic idea of the multiverse proposal is fractal. More precisely the idea is implicit in the work of Laurent Nottale and Garnet Ord and explicit in the work of M.S. El Naschie. If the readers could bear with me I would like to make this assertion plausible. A fractal implies infinity. In its simplest form it is the infinite ability of magnifying and zooming exactly as explained in the excellent World Scientific book of Nottale. A four dimensional fractal implies infinite numbers of concentric four dimensional spaces. Whether you think of the resulting structure as one Cantorian spacetime as presented by El Naschie or alternatively think of the structure as an infinite amount of four spacetimes connected together, it is only a matter of semantics. The way El Naschie adds probabilities together in an infinite series implies an infinite number of universes constituting a multiverse albeit an infinite one. The mathematics is very clear here. The simplest way to think of it is to put a four dimensional cube insider another four dimensional cube and so on ad infinitum. When you do the sums correctly which resembles a continued fracture, then you find that the final dimension is four plus the golden mean to the power three. E-infinity implies multiverse. The correctly weighted multiverse is a fractal spacetime with an expectation value for the topological dimension exactly equal four and an expectation value for the Hausdorff dimension exactly equal to four plus the golden mean to the power three. Some have suggested a connection to a Hilbert cube for instance Prof. Ji-Huan He from Shanghai. It is interesting to realize that there is now real experimental verification for the preceding theory. Finding the golden mean in quantum mechanics in the Helmholtz Centre in Berlin is a major discovery with incalculable consequences for the further development of fundamental physics. Nottale and Garnet Ord exactly as Richard Feynmann had the right haunch. Fractal spacetime is the answer. This is also obvious from the work of Tim Palmer which was published in the Royal Society a few months ago. Of course the establishment is not amused. Never the less science is not about being amused or not. Science is about being right. The New Scientist seems to have sensed the change of tide. Its newest edition carries the title Touching the multiverse, first hint that it really exists, Vol. 205, No. 2750. To be candid the idea is not that brand new. Feynmann’s path integral is the first version of this fast breaking idea. Everet’s multiuniverses theory which was championed by Murray Gellman is another version. However the discovery of the golden mean in the laboratory as a basis for quantum mechanics puts the whole thing in a completely new perspective. We are not philosophizing or theoretizing. The golden mean and thus El Naschie’s E-infinity theory is not a mere theory any more. You could say Nottale, Ord and El Naschie following Feynman discovered the real building blocks of quantum spacetime. These building blocks for which Gerard ‘tHooft searched for a long time are elementary random Cantor sets with a golden mean as a Hausdorff dimension. Similar qualitative ideas not using the quantitative golden mean approach is due to Fay Dowker and are called partially ordered sets. To go deeper than that in this theory will take us too far. I just wanted to give the unbiased reader a taste of the deep meaning of Nottale’s theory of fractal spacetime and what it really implies.
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