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 �
(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. Hence the transition from quantum non relativistic to quantum relativistic may be accounted for by a purely temporal transition. The length increase is now compensated by a time increase, which results into the relativistic bound v.$'c. Finally we get a Lorentz covariant scheme with a unique spatio-temporal transition from D=l to D=2 around }.Jl=nfpµ, µ=0 to 3. This result may also be obtained by the inverse argument: Starting from the hypothesis of underlying fractal structures, the Heisenberg relations may indeed be found under the same conditions)) One of the interesting consequences of this interpretation is that it precisely accounts for the Zitterbewegung, this oscillatory motion of the center of mass of an electron resulting from the Dirac equation. This effect is known to be the result of interactions between the negative energy and positive energy solutions of the Dirac equation. Though it indeed disappears if one keeps only the positive energy solutions to describe an electron, this does not yield a satisfactory solution to the problem, since such a positive energy electron would be completely delocalizedl4). Conversely one may set the localization assumed (or measured) for the electron, L1x=ciJ.t=nc!E, and then derive the relative rate of positive and negative energy solutions. One gets:
P JP+ = [pc/(E+mc2)]2 = (E-mc2)!(E+mc2) (7)
Now in the fractal model, for each classical time interval we have 2n+ 1 segments, n+ 1 running forward and n running backward, so that with E=(2n+I)mc2, one gets P-IP+ =n/(n+ 1 )=(E-mc2)/(E+mc2), i.e. exactly the QED result. This relation of the Zitterbewegung to the fractal approach willbe developed in a forthcoming paper. 8)
GEODESICAL INTERPRETATION OF THEWA VE-PARTICLE DUALITY. 4.
Let us define the de Broglie length and time r) as geometrical structures of (.:l, the fractal trajectory of a "particle": They are identified to the fractal/non fractal 18
transition. Then the various classical quantities may be expressed in term of these two geometrical quantities:
(8)
This means that we do not have to endow the point particle with mass, energy, momentum or velocity, but instead that these properties may be reduced to geometrical structures of the resolution space. Hence the energy-momentum tensor writes in terms of the de Broglie periods and of the Planck length Ap (with Xi(S) being fourfractal functions defining the trajectory of particle i):
G Tµv = -.-Ap .A.p 8'[x-X;(S)].(dtfd We have also demonstrated that the quantum spin itself may also be obtained as a proper angular momentum of fractal trajectories having precisely fractal dimension 2.1) While for D<2 and D>2 the proper angular momenta of fractal curves are respectively 0 and infinite, it becomes finite for the strict value D=2 and then owns all the properties of an internal quantum number. Let us now comment on the wave-particle duality. A fractal space-time is characterized by an infinity of obstacles at all scales and by returns and eddies also at an scales and for all 4 coordinates. As a consequence one expects that an infinity of geodesics will connect any two space-time events, so that the properties of physical objects on a fractal space-time will result in a mixing of individual properties (one particular geodesic) and collective properties (those of families of geodesics), in agreement with the wave-particle duality. While one may admit, without any logical inconsistency, that a particle follows one particular geodesic (as indicated by individual measurements of well defined particles), one must admit at the same time that all geodesics are equiprobable, so that any prediction can be only of statistical nature, since applying to a family of geodesics. So in such a frame we get the possibility, at least in principle, to have both a space-time which would be stricly determined by its material and energetic content, and non deterministic particle trajectories. But here the statistical behaviour would not be the primeval stone on which physical laws are based, but instead a consequence of the (fractal) nature of space-time. One result supporting this interpretation is that indeed beams of geodesics may have a wave nature described by a SchrOdinger-like equation in a Riemannian space time, even at the geometric optics approximation,l,15) A light beam in general relativity is described at the geometric optics approximation by a congruence of null geodesics, the equations of which have been written by Sachs.16) The cross sectional area A of the light beam may be subjected on its way to three infinitesimal 19 deformations, expansion 8.d(J):::d-i4./-1A, rotation Q.dOJ=.dW and shear. Considering only the shearless case, when setting l/f=-i4.e iW the beam propagation equation writes, in term of an affine parameter and of Ricci tensor and wave vector ki: 1) m R;j (10) which has exactly the form of the one-dimensional Schri:idingerequation. So a family of geodesics, even at the geometric optics approximation, possesses the equivalent of a quantum phase, interpreted as the beam rotation and of a probability of presence, identified to thebeam cross sectional area. 5. POSSIBLEIMPLICATIONS. Let us consider a simple model of metrics accounting for the constraints which have been hereabove obtained: ds2= .;2(t,L1t) (l r 2 c2 dt2 df2 (11) - /L1t) - in which .; is some finite fractal function (with .;=1 for .1t>-r), -r=Pilmc2 is the de Broglie time of a system of inertial mass m and d/2 is a fractal spatial element which will not be more detailed here. This form of g00 has been chosen for the following reasons: It yields fractal dimension 2 for �t<<-r, the Minkowski limit g00=l for �t�oo, and a singularity g00=0 for .1t=-r, thus accounting for creation/annihilation of particles (in analogy to the cosmological primeval singularity). Consider the domain .1t>>-r, g00 may be expanded as g00=l-2-rl.1t. Let us now identify .1t rlc to the = lifetime of an exchanged virtual boson. One gets: g00 = l-2h!mcr (12) This result holds for one particle of mass m, but also for a complex system of total mass Lm, thanks to the universality of the de Broglie wave nature of any physical system . However one may remark that the wave properties of a system keep a physical sense only if one may, at least in a "gedanken experiment", measure them (i.e. by a diffraction or interference experiment). But when the total mass m becomes larger than the Planck mass mp=(hc!G)I/2,,,,2 10-s g, its Compton length becomes smaller than its Schwarzschild radius rs""Gm/c2, thus becoming unmeasurable, not only for technological limitation, but mainly for a profound physical limitation, since it enters into a black hole horizon.I) Conversely for m (13) As an observable consequence, the Newton law should break down between two dust particles both having mass smaller than the Planck mass.I) This could be checked in a space experiment: For example two 2.10-5 g dust grains of density d=l8 have a radius of 0.064 Their expected Newtonian free fall time for an initial relative distance mm. 0.24 would be "'10 mn. We expect the falling time to become smaller than the mm Newtonian time for Such an experiment would be at the limit of the present m<"'frtp. possibilities, needing a high control of vessel gravity gradients and of electric charges, since the electric force would be equal to the gravitational one for only mpxmp a-1,,,12xl2 elementary electric charges. The hereabove calculation is a very rough one including only temporal terms, in which the mass considered was assumed to remain a point mass. We intend to attempt its spatio-temporal generalization to the extended case, which should lead us to the generalization of the Planck mass transition to a critical density transition. 8) Then astrophysical and cosmological implications, concerning e.g. the formation of structures, are expected to be derived in such a way that the hereabove suggestion of a transition between a quantum and a gravitationnal regime should be falsifiable. REFERENCES I. Nottale, L., 1989, Int. J. Mod. Phys. A4, 5047 2. Einstein, A., 1916, Annalen der Physik 49,769 3. Einstein, A., Letter to Pauli, 1948, in Albert Einstein, Quanta, Seuil/CNRS, p.249 4. Nottale, L., 1988, C.R. Acad. Sci. Paris 306, 341 5. Feynman, R. & Hibbs, A., Quantum mechanics and path integrals (McGraw-Hill, NY, 1965) 6. Nottale, L., & Schneider, J., J. Math. Phys. 25, 1296 7. Mandelbrot , B., The fractal geometry of nature (Freeman, San Francisco, 1982) p.331 8. Nottale, L., in preparation 9. Mandelbrot, B., The fractal geometryof nature (Freeman, San Francisco, 1982) p.365 10. Abbott, L.F. & Wise, M.B., 1981, Am. J. Phys. 49, 37 11. Campesino-Romeo, E., D'Olivo, J.C. & Socolovsky, M.,1982, Phys. Lett. 89A, 321 12. Allen, A.D., 1983, Speculations in Science and Technology 6,165 13. Landau, L. & Lifshitz, E., Relativistic Quantum Theory, (Mir, Moscow, 1972) 14. Bjorken, J.D. & Drell, S.D., Relativistic Quantum Mechanics, (McGraw-Hill, NY, 1964) 15. Nottale, L., 1988, Ann. Phys. Fr. 13, 223 16. Sachs, P.K., 1961, Proc. Roy. Soc. 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