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TIIEFRACTALSTRUCTURE of TIIE QUANTUM SPACE-T™E Laurent 13 TIIEFRACTALSTRUCTURE OF TIIEQUANTUM SPACE-T™E Laurent Nottale CNRS. Departement d'Astrophysique Extragalactique et de Cosmologie. Observatoire de Meudon. F-92195 Meudon Cedex. France ABSTRACT. We sum up in this contribution the first results obtained in an attempt at understanding quantum physics in terms of non differential geometrical properties.I) It is proposed that the dependance of physical laws on spatio-temporal resolutions is the concern of a scale relativity theory, which could be achieved using the concept of a fractal space-time. We recall that the Heisenberg relations may be expressed by a universal fractal dimension 2 of all four coordinates of quantum "trajectories", and that such a point particle path has a finite proper angular momentum (spin) precisely in this case D=2. Then we comment on the possibility of a geodesical interpretation of the wave-particle duality. Finally we show that this approach may imply a break down of Newton gravitational law between two masses both smaller than the Planck mass. 14 1. INTRODUCTION The present contribution describes results obtained from an analysis of what may appear as inconsistencies and incompleteness in the present state of fu ndamental physics. We give here a summary of the principles to which we have been led and of some of our main results, which are fully described in Ref. 1. The first remark is that, following the Galileo/Mach/Einstein analysis of motion relativity, the non absolute character of space and space-time appears as an inescapable conclusion.2) The geometry of space-time should depend on its material and energetic content. However present quantum physics assumes space-time to be Minkowskian, i.e. absolute, while moreover the fundamental behaviour and properties of quantum objects are known to be radically at variance with classical properties, from which the Minkowskian space-time was yet derived. The second remark is that Einstein's principle of general relativity ("the laws of physics should apply to systems of reference in any state of motion") is still unachieved. It is now considered, in particular, as not applying to quantum motion. To quote Einstein in this connection, " .. .I am a fierce supporter, not of differential equations, but of the principle of general relativity, whose heuristic strength is essential to us."3) The third remark is that the consequences of one of the radically new behaviour of the quantum world relative to the classical one, i.e. the scale (and/or resolution) dependance of physical laws, may still not have beenfully drawn. Though an essential part of the quantum theory through the so-called measurement theory, this scale dependance has still not be included into the laws of physics themselves, in spite of its clearly recognized universality based on the Heisenberg relations. It is clear that a set of physical measurements takes its complete physical sense, even in the classical domain, only when the measurement resolutions or "errors" have been specified. In quantum physics, the result of a momentum measurement depends explicitely, although in a statistical manner, of the spatial resolution, and the result of an energy measurement depends of the temporal resolution with which the measurement has been performed. We suggestl,4) that this fundamental scale dependance of physics is relevant, as motion does, of a relativity theory. Our proposal is to introduce explicitely the resolution in physical laws, either as a new coordinate, or better as a state of scale of the coordinate system, in the same way as velocity and acceleration describe its state of motion. The axes of such a generalized coordinate system can be viewed as endowed with thickness. In such a frame one would require general covariance of physical equations, not only on motion, but also on scale. The hereabove generalization in the definition of coordinate systems, once assumed universal for a consistent description of physical laws, immediately implies a generalization of the nature of space-time itself. We postulate that the scale 15 dependance of physics in the quantum domain, and more generally the quantum behaviour itself, take their origin into an intrinsic dependance of space-time geometry on resolution. This implies a generalized metric element where the metrics potentials become explicit functions of resolution: (1) The achievement of the hereabove working hypothesis needs the use of an adequate geometrical mathematical tool. We have suggested that the concept of a continuous and self-avoiding fractal space-time might be such an adequate tooi. 1l 2. WHYFRACTALS ? The suggestion that the quantum space-time possesses fractal structure is supported by a lot of converging arguments. First of all, fractals are characterized by an effective and explicit dependance on resolution. Covering a fractal domain of D> topological dimension T and fractal dimension T by balls of radius L1x yields a T-hypervolume measure which diverges when Llx�O as: 2 V(x,L1x) = �(x,L1x).(A!L1x)D-T ( ) A where �(x,Llx) is a finite fractal function and is a fractal I non fractal scale transition. Second, fractals are also characterized by their non-diffe rentiability, while it had be realized by Einstein3 ) (see the hereabove quotation) that giving up differentiability could be the price to be paid in order to apply the principle of general relativity to microphysics, and by Feynman5) that the quantum "trajectories" of particles can be described as continuous but non differentiable curves. The third point is that fractals contain infinities, which may be renormalized in a natural way.6) The correspondence between the renormalization group methods, particularly efficient in the domain of the asymptotic behaviour of quantum field theory, and fractals, has already been pointed out by Callan, as quoted by Mandelbrot7). In fact the fractal approach may come as a completion of the renormalization group theory. 8) Our working hypothesis is that the quantum space-time is a fractal self-avoiding continuum whose geodesics define free particle trajectories. The continuum assumption is a conservative choice allowing to keep a field approach and representation, while the self-avoidance assumption is a necessary condition for such a geodesical interpretation. But because of the non-differentiability of a fractal space, the properties of geodesics will be fundamentally different from those of standard spaces (see Ref.I and Sec.4 hereafter). 16 The fact that fractal trajectories run backward relatively to classical coordinates also impiies a loss of information in the projection from the intrinsic (fractal) to the classical coordinates, which are defined as the result of a smoothing out of the fractal coordinates with balls larger than the fractaVnon fractal transition A, (see hereafter and Refs. 1 and 4 for the quantum interpretation of this transition). We have proposed to describe this behaviour by a "fractal derivative":!) = = (3 as/dx Lj ds;/dx �'(x,L\x).(Affil)D-T ) the sum being performed on all proper time (fractal) intervals found in the classical interval dx . The power D-T is found again because both the topological dimension and the fractal dimension (nearly everywhere9)) are decreased by 1 in the projection . 3.'TIIEFRACTAL DIMENSION OF QUANTUMTRAJECTORIES. We now sum up some of the main results obtained or reviewed in Ref.I. The question of the fractal dimension of a quantum mechanical path in the non relativistic case has been considered by Abbott and WiselO), Campesino-Romeo et alll) , Allen12) and others. The result is a universal value, D=2, the transition from non-fractal (classical, D=T=l) to fractal (quantum, T=l, D=2) occuring around the de Broglie length, Ac.s=filp. Let us recall briefly the argument to show how this result is a direct consequence of the position-momentum Heisenberg relation. The total length travelled in the average during a set of experiments where the successive positions of a particle are measured with a resolution L1x is given by: L (4) � lvl � <lpl>, in the non relativistic case. The classical case <lpl>"'Po>>L1p, i.e. Lix>>Acts (since L1x.L1p,,,fi) yields as expected a length independant from L1x. On the contrary the quantum case L1p<p0 (i.e. Lix<Acts) yields <lpl>"'.1p, so that the length diverges as: (5) which corresponds with T=l and from Eq. (1), to D=2. The relativistic case is more difficult to deal with. A superficial analysis may lead to the conclusion that, because of the limitation v�c, the length will become bounded again for Lix<""Ac=filmc. However one should also account for the radically new physical behaviour which takes place in the quantum relativistic domain, i.e. virtual particle-antiparticle pair creation-annihilation. I have proposed to reinterpret the virtual pairs as a manifestation of the fact that the fractal trajectory is now allowed to run backward in time for time intervals L1t<"'rds=n!E0.1) This is based on the Feynman/Stueckelberg/Wheeler interpretation of antiparticles as particles running backward in time. However here we consider electron-positron virtual pairs to be part 17 of the nature of the electron itself, in agreement with the QED expression for the electron self-energy, which contains all successive Feynman graphs.13) Indeed consider that for a time interval Lit, an energy fluctuation L1Ezn/L1t may give rise to the creation of n e+e- pairs, with L1Ez2nmc2. With the hereabove reinterpretation, the total time elapsed on the fractal trajectory is now given by the sum of the proper times of the (2n+1) particles, i.e.: T = (2n+l)t0 = (E+f1E)t0/mc2 (1HdBILlt)t0 (6) = So we get a temporal non-fractal D=l to D=2 fractal transition around the de Broglie time rd8.l) The argument holds also for the total distance travelled, L=(2n+l)L0= (1+crdBIL1.x)ct0, which demonstrates that the spatial fractaldimension remains D=2 in the relativistic case.
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