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The invariant d de nes a proper djinn , which means that, although the effective fractal dimension, given by D = 1 + according to (1434), became variable, the fractal dimension remained constant in the proper reference system However, we can also note that the fractal dimension now tends to in nity when the resolution interval tends to the Planck scale While going to increasingly small resolutions, a fractal dimension will thus successively pass the values 2, 3, 4, which would make it possible to cover a surface, then space, then space-time using a single coordinate It is thus possible to de ne a Minkowskian space-time-djinn requiring, in adequate fractal reference systems, only two dimensions on very small scales By tending towards large resolutions, the space-time-djinn metric signature (+, , , , ) sees its fth dimension vary less and less to become almost constant on scales currently accessible to accelerators (see [NOT 96a, Figure 4]) It nally vanishes beyond the Compton scale of the system under consideration, which is identi ed with the fractal to non-fractal transition in rest frame At this scale the temporal metric coef cient also changes sign, which generates the traditional Minkowskian space-time of metric signature (+, , , ) 1457 Generalized scale relativity and scale-motion coupling This is a vast eld of study We saw how we could introduce non-linear scale transformations and a scale dynamics This approach is, however, only a rst step towards a deeper entirely geometric level in which scale forces are but manifestations of the fractal and non-differentiable geometry This level of

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description also implies taking resolutions into consideration, which would in turn depend on space and time variables The rst aspect leads to the new concept of scale eld, which corresponds to a distortion in scale space compared with usual self-similar laws [NOT 97b] It can also be represented in terms of curved scale space It is intended that this approach will be developed in more detail in future research The second aspect, of which we now point out some of the principal results, leads to a new interpretation of gauge invariance and thus gauge elds themselves This in turn proves the existence of general relations between mass scale and coupling constants (generalized charge) in particle physics [NOT 96a] One of these relations makes it possible, as we will see, to predict the value of the electron mass theoretically (considered as primarily of electromagnetic origin, in this approach), as a function of its charge Lastly, to be complete, let us point out that even these two levels are only transitory stages from the perspective of the theory we intend to build A more comprehensive version will deal with motion and scales on the same footing and thus see the principle of scale relativity and motion uni ed into a single principle This will be done by working in a 5D space-time-djinn provided with a metric, in which all the transformations between the reference points identify with rotations: in the planes (xy, yz, zx), they are ordinary rotations of 3D space; in the planes (xt, yt, zt) they are motion effects (which are reduced to Lorentz boosts when the space-time-djinn is reduced to 4D space time on macroscopic scales); nally, four rotations in the planes (x , y , z , t ) identify with changes of space-time resolutions 14571 A reminder about gauge invariance At the outset, let us recall brie y the nature of the problem set by gauge invariance in current physics This problem already appears in traditional electromagnetic theory This theory, starting from experimental constraints, has led to the introduction of a four-vector potential, A , then of a tensorial eld given by the derivative of the potential, F = A A However, Maxwell eld equations (contrary to what occurs in Einstein s general relativity for motion in a gravitational eld) are not enough to characterize the motion of a charge in an electromagnetic eld It is necessary to add the expression for the Lorentz force, which is written in 4D form f = (e/c)F u , where u is the four-velocity It is seen that only the elds intervene in this and not the potentials This implies that the motion will be unaffected by any transformation of potentials which leave the elds invariant It is obviously the case, if we add to the four-potential the gradient of any function of coordinates: A = A + (x, y, z, t) This transformation is called, following Weyl, gauge transformation and the invariance law, which results from it is the gauge invariance What was apparently only a simple latitude left in the choice of the potentials takes within the quantum mechanics framework a deeper meaning Indeed, gauge

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