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Scaling, Fractals and Wavelets
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If we consider a translation along two different coordinates (or, in an equivalent way, displacement on a closed loop), we may write a commutator relation: e( D D ) ln = ( A A ) (1441)
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This relation de nes a tensor eld F = A A , which, unlike A , is independent of the initial scale from where we started We recognize in F the analog of an electromagnetic eld, in A that of an electromagnetic potential, in e that of electric charge and in equation (1440) the property of gauge invariance which, in accordance with Weyl s initial ideas and their development by Dirac [DIR 73], recovers its initial status of scale invariance However, equation (1440) represents progress compared with these early attempts and with the status of gauge invariance in today s physics Indeed, the gauge function (x, y, z, t) which intervenes in the standard formulation of gauge invariance, A = A + e and which has, up to now, been considered as arbitrary, is identi ed with the logarithm of internal resolutions, = ln (x, y, z, t) Another advantage with respect to Weyl s theory is that we are now allowed to de ne four different and independent dilations along the four space-time resolutions instead of only one global dilation Therefore, we expect that the eld above (which corresponds to a group U(1) of electromagnetic eld type) is embedded into a larger eld, in accordance with the electroweak theory and grand uni cation attempts In the same way, we expect that the charge e is an element of a more complicated, vectorial charge These early remarks have now developed into a full theory non-Abelian gauge elds [NOT 06], in which the main tools and results of Yang-Mills theories can be recovered as a manifestation of fractal geometry Moreover, this generalized approach makes it possible to suggest a new and more completely uni ed preliminary version of electroweak theory [NOT 00b], in which the Higgs boson mass can be predicted theoretically (we nd mH = 2mW = 11373 006 GeV, where mW is the W gauge boson mass) Moreover, our interpretation of gauge invariance yields new insights about the nature of the electric charge and, when it is combined with the Lorentzian structure of dilations of special scale-relativity, it makes it possible to obtain new relations between the charges and the masses of elementary particles [NOT 94, NOT 96a], as recalled in what follows 14573 Nature of the charges In gauge transformation A = A , the wave function of an electron of charge e becomes: = eie (1442)
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Scale Relativity, Non-differentiability and Fractal Space-time
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In this expression, the essential role played by the gauge function is clear It is the conjugate variable of the electric charge, in the same way as position, time and angle are conjugate variables of momentum, energy and angular momentum, (respectively) in the expressions of the action and/or the quantum phase of a free particle, = i(px Et + )/ Our knowledge of what constitutes energy, momentum and angular momentum comes from our understanding of the nature of space, time, angles and their symmetry (translations and rotations), using Noether s theorem Conversely, the fact that we still do not really know what an electric charge is, despite all the development of gauge theories comes, in our view, from the fact that the gauge function is considered devoid of physical meaning We have interpreted in the previous section the gauge transformation as a scale transformation of resolution, , ln = ln( / ) In such an interpretation, the speci c property that characterizes a charged particle is the explicit scale dependence of its action and therefore of its wave function in function of resolution The result is that the electron s wave function is written: = ei
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