Scaling, Fractals and Wavelets

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invariance in quantum electrodynamics becomes an invariance under the phase transformations of wave functions and is linked to current conservation using Noether s theorem It is known that this theorem connects fundamental symmetries to the appearance of conservative quantities, which are manifestations of these symmetries (thus the existence of energy results from the uniformity of time, the momentum of space homogenity, etc) In the case of electrodynamics, it appears that the existence of the electric charge itself results from gauge symmetry This fact is apparent in the writing of the Lagrangian which describes Dirac s electronic eld coupled to an electromagnetic eld This Lagrangian is not invariant under the gauge transformation of electromagnetic eld A = A + (x), but becomes invariant, provided it is completed by a local gauge transformation on the phase of the electron wave function, e ie (x) This result can be interpreted by saying that the existence of the electromagnetic eld (and its gauge symmetry) implies that of the electric charge However, although impressive (particularly through its capacity for generalization to non-Abelian gauge theories which includes weak and strong elds and allows description of weak electric elds), this progress in comprehending the nature of the electromagnetic eld and the charge remains incomplete, in our opinion Indeed, the gauge transformation keeps an arbitrary nature The essential point is that no explicit physical meaning is given to function (x): however, this is the conjugate variable of the charge in the electron phase (just as energy is the conjugate of time and momentum of space), so that it is from an understanding of its nature that an authentic comprehension of the nature of charge could arise Moreover, the quantization of charge remains misunderstood within the framework of the current theory However, its conjugate variable still holds the key to this problem The example of angular momentum is clear in this regard: its conjugate quantity is the angle, so that its conservation results from the isotropy of space Moreover, the fact that angle variations cannot exceed 2 implies that the differences in angular momentum are quantized in units of In the same way, we can expect that the existence of limitation on the variable (x), once its nature is elucidated, would imply charge quantization and leads to new quantitative results As we will see, scale relativity makes it possible indeed to make proposals in this direction 14572 Nature of gauge elds Let us consider an electron or any other charged particle In scale relativity, we identify particles with the geodesics of a non-differentiable space-time These paths are characterized by internal (fractal) structures (beyond the Compton scale c = /mc of the particle in rest frame) Now consider any one of these structures (which is de ned only in a relative way), lying at a resolution < c In a displacement of the electron, the relativity of scales will imply the appearance of a eld induced by this displacement

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Scale Relativity, Non-differentiability and Fractal Space-time

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To understand it, we can take as model an aspect of the construction, from the general relativity of motion, of Einstein s gravitation theory In this theory, gravitation is identi ed with the manifestation of the curvature of space-time, which results in vector rotation of geometric origin However, this general rotation of any vector during a translation can result simply from the only generalized relativity of motion Indeed, since space-time is relative, a vector V subjected to a displacement dx cannot remain identical to itself (the reverse would mean absolute space-time) It will thus undergo a rotation, which is written, by using Einstein summation convention on identical lower and upper indices, V = V dx Christoffel symbols , which emerge naturally in this transformation, can then be calculated, while processing this construction, in terms of derivatives of the metric potentials g , which makes it possible to regard them as components of the gravitational eld generalizing Newton s gravitational force Similarly, in the case of fractal electron structures, we expect that a structure, which was initially characterized by a certain scale, jumps to another scale after the electron displacement (if not, the scale space would be absolute, which would be in contradiction with the principle of scale relativity) A dilation eld of resolution induced by the translations is then expected to appear, which is written: e = A x (1437)

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This effect can be described in terms of the introduction of a covariant derivative: eD ln( / ) = e ln( / ) + A (1438)

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Now, this eld of dilation must be de ned irrespective of the initial scale from which we started, ie, whatever the substructure considered Therefore, starting from another scale = (here we take into account, as a rst step, only the Galilean scale relativity law in which the product of two dilations is the standard one), we get during the same translation of the electron: e = A x (1439)

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The two expressions for the potential A are then connected by the relation: A = A + e ln (1440)

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where ln (x) = ln( / ) is the relative scale state (it depends only on the ratio between resolutions and ) which depends now explicitly on the coordinates In this regard, this approach already comes under the framework of general scale relativity and of non-linear scale transformations, since the scale velocity has been rede ned as a rst derivative of the djinn, ln = d ln L/d , so that equation (1440) involves a second-order derivative of fractal coordinate, d2 ln L/dx d

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