The results reported in this html page have been published in the following references:

"Scale-Relativity: First Steps toward a Field Theory".

"Scale Relativity and Fractal Space-Time: Application to Quantum Physics, Cosmology and Chaotic systems".

Nature of the electromagnetic field

The theory of scale relativity allows one to get new insights about the nature of the electromagnetic field and about the physical meaning of gauge invariance [8,6], [11, Appendix B3]. Up to now, only scale transformations at a given point were considered. But we must also wonder about what happens to the scale structures of a scale-dependent object (e.g., an elctron) when it is displaced. Consider anyone of these structures, lying at some (relative) resolution e (such that e < l, where l is the fractal /nonfractal transition) for a given position of the particle. Under a translation, the relativity of scales implies that the resolution at which this given structure appears in the new position will a priori be different from the initial one. In other words, e is now a function of the space-time coordinates, e = e(x,t), and we expect the occurrence of dilatations of resolutions induced by translations, which read:

e de /e = - A_m dx ^m , (16)

where a four-vector A_m must be introduced since dx ^m is itself a four-vector and dlne is a scalar (in the case of a global dilation). This behavior can be expressed in terms of a new scale-covariant derivative:

e D_m ln(l/e) = e d_m ln(l/e) + A_m . (17)

However, if one wants the "field" A_m to be physical, it must be defined whatever the initial scale from which we started. Starting from another scale e' = re (we consider only Galilean scale-relativity here), we get e de' /e' = -A'_m dx^m, so that we obtain:

A'_m = A_m + e  d_m lnr , (18)

which depends only on the relative "state of scale", V = lnr = ln(e/e'), not on the individual scales, as expected from the principle of scale relativity. Now, if one considers translation along two different coordinates (or, in an equivalent way, displacement on a closed loop), one may write a commutator relation which defines a tensor field F_mn = d_m A_n - d_n A_m . This field is, contrarily to A_m, independent of the scale factor. One recognizes in F_mn the analog of an electromagnetic field, in A_m that of an electromagnetic potential, in e that of the electric charge, and in Eq. 18 the property of gauge invariance which, in accordance with Weyl's initial ideas and their development by Dirac [22,23], recovers its initial status of scale invariance. However, Eq. 18 represents a progress compared with these early attempts and with the status of gauge invariance in today's physics. Indeed, the gauge function, c(x,y,z,t), of the standard expression of gauge invariance, A'_m = A_m - d_m c, which has, up to now, been considered as arbitrary, is now identified with the logarithm of internal resolutions, lnr(x,y,z,t) .

Another advantage with respect to Weyl's theory is that we are now allowed to define four different and independent dilations along the four space-time resolutions instead of only one global dilation. The above U(1) field is then expected to be embedded into a larger field (in accordance with the electroweak theory) and the charge e to be one element of a more complicated, "vectorial" charge.

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 allows one to obtain new relations between the charges and the masses of elementary particles [ 6,8], as recalled in what follows (Eqs. 37 and 40, Figs. 1 and 2).

Nature of the electric charge

In a gauge transformation A_m = A_0m - d_m c, the wave function of an electron of charge e becomes:

y = y₀ e^{i e c} . (19)

In this expression, the essential role played by the so-called "arbitrary" gauge function is clear. It is the variable conjugate to 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 for the action and/or the quantum phase of a free particle, q = (px - Et + sj). Our knowledge of what are energy, momentum and angular momentum comes from our understanding of the nature of space, time, angles and their symmetry, via Noether's theorem. Conversely, the fact that we still do not really know what is an electric charge despite all the development of gauge theories, comes, in our point of view, from the fact that the gauge function c is considered devoid of physical meaning.

We have reinterpreted in the previous section the gauge transformation as a scale transformation of resolution, e₀ -> e, V = lnr = ln (e₀/e). In such an interpretation, the specific property that characterizes a charged particle is the explicit scale-dependence on resolution of its action, then of its wave function. The net result is that the electron wave function writes

y = y₀ exp [i (e²/h c) V] . (20)

Since, by definition (in the system of units where the permittivity of vacuum is 1),

e² = 4p a h c , (21)

Eq. (20) becomes,

y = y₀ e^{i 4paV} . (22)

This result allowed us to suggest a solution to the problem of the nature of the electric charge (and also yields new mass-charge relations, see hereafter). Indeed, considering now the wave function of the electron as an explicitly resolution-dependent function, we can write the scale differential equation of which y is solution as:

-i h dy /d(eV /c) = e y . (23)

We recognize in -i h d /d( e lnr / c) a dilatation operator D. Equation (23) can then be read as an eigenvalue equation:

D y = e y .  (24)

In such a framework, the electric charge is understood as the conservative quantity that comes from the new scale symmetry, namely, the uniformity of the resolution "dimension" lne .

Mass of the electron

In the previous Section, we have suggested to elucidate the nature of the electric charge as being the eigenvalue of the dilation operator corresponding to resolution transformations. We have written the wave function of a charged particle under the form:

y' = e^{i 4pa ln(l/e)} y . (34)

In the Galilean case such a relation leads to no new result, since ln(l/e) is unlimited. But in our special scale-relativistic framework [7], [3, chapt.6], scale laws become Lorentzian below the scale l , then ln(l/e) becomes limited by C = ln(l/L_P), where L_P is the Planck length-scale. This implies a quantization of the charge which amounts to the relation 4paC = 2k p, i.e.:

a C = k / 2 . (35)

where k is integer. Introducing now a running mass scale m and its corresponding running length scale r, we have C = ln(r / L_P) = ln(m_P/m), while a = a(r) is also running. (Here m_P is the Planck mass). Equation (35) is nothing but a new general relation between masses and charges (i.e., running couplings), and can then now be solved for r, i.e., for the mass scale and its corresponding Compton length-scale that satisfies it.

The first "object" to which one can try to apply such a relation is the electron itself. However, we know from the electroweak theory that the electric charge is only a residual of a more general, high energy electroweak coupling. One can define an inverse electromagnetic coupling a₀^-1 from the high energy U(1) and SU(2) couplings:

a₀^-1 = (3/8) a₂^-1 + (5/8) a₁^-1 . (36)

This running coupling is such that a₀ = a₁ = a₂ at unification scale and is related to the fine structure constant at Z scale by the relation a = 3a₀/8. It is a₀ rather than the fine structure constant which must be used in Eq.(35), since the low energy fine structure constant is actually a residual (in which only massless photons act) of the more complex high energy electroweak interaction (in which the four U(1)_Y and SU(2) bosons intervene). Neglecting as a first step threshold effects, we find that Eq.(35) is verified within 0.2% with k = 2 [6,8]. Namely, we predict that:

(8a_e/ 3)  ln(m_P/m_e)  =  1,   (37)

where a_e is the fine structure cosntant, m_e is the electron mass and m_P = (hc /G)^1/2 is the Planck mass. Their experimental values yield respectivelyC_e = ln(m_P/m_e) = 51.528(1) and (3/8)a^-1 = 51.388... (see Figs. 1 and 2).

Figure 1. Variation in function of length-scale r of the running "electromagnetic" inverse coupling a₂^-1 + (5/3) a₁^-1 , from the Planck scale (P) to the electron scale (e), and of 8C(r)/3, where C(r) = ln(r/L_P) is a running scale relativity "constant" (the factor 8/3 takes its origin in the electroweak symmetry breaking). The two curves cross very precisely at the experimental value of the Compton scale of the electron (thus yielding its mass), as theoretically expected in the scale-relativity theory. The plotted coupling identifies with the inverse running fine structure constant a^-1 between the electron and Z scales. Its variation depends on the full mass spectrum of elementary particles.

Figure 2. Enlarged view (by a factor of about 50) of the region around the electron energy scale in Fig. 1, showing the precise crossing of the asymptotic a^-1(r) and of 8C(r)/ 3 at the experimental value of the electron mass.

The agreement is made even better if one accounts for the fact that the measured fine structure constant (at Bohr scale) differs from the limit of its asymptotic behavior, and that the same is true of the running electron mass (see Fig. 3).

Indeed, from the calculation of radiative corrections to the Coulomb law, one finds an asymptotic behavior (i.e., valid for r << l_e, where l_e= h / m_e c is the Compton scale of the electron) [36], [3, chapt. 6.2]:

a(r) = a_e {1 + (2a_e/3p) [ ln (l_e/r) - (g + 5/6) ] }   (38)

where a_e  is the low energy fine structure constant, such that (a_e)^-1 = 137.035990(6) and g = 0.577... is the Euler constant. The asymptotic running self-energy of the electron is given by [37], [3, chapt. 6.2]:

m(r) = m_e {1 + (3a_e/2p) [ ln (l_e/r) + 1/4)}   (39)

Figure 3. Enlarged view (by a new factor of about 5 in alpha) of the region around the electron energy scale in Figs. 1 and 2, now showing the threshold effect around the Compton scale of the electron. The true running inverse coupling a^-1(r) becomes smaller than its asymptotic , high energy value, when it approaches the electron mass scale, and it finally tends toward the (low energy) fine structure constant. The value -1/4 is the zero-point of the running electron self-energy (Eq.39). One finds that 8C_e / 3 is equal, within a relative error of 10^-4, to a_as^-1 at this scale (Eq.40).

We now identify the scale relativity constant C= ln (m_P/m_e) with (3/8 times) the value of the asymptotic running fine structure constant at the scale where the asymptotic self-energy of the electron equals its mass. One obtains the relation:

ln (m_P/m_e) = (3/8) a_e^-1 + (1/4p) (g + 13/12 ),   (40)

One can interpret this relation as meaning that the electron mass is mainly of electromagnetic origin. Indeed, the two members of this relation [ 51.528(1) vs 51.521 ] agree experimentally within 1.5 10^-4 (relative difference). The main uncertainty comes from the gravitation constant G, that interveves in the calculation of m_P.

Equation (40) can be used to theoretically predict the electron mass from the experimental value of the fine structure constant. One finds m_e(th) = 1.007 m_e(exp). This means that the electron mass (self-energy) may not be fully accounted for in this way, since it is experimentally known with an uncertainty of 0.3 ppm. This difference may be the signature, either of an incomplete treatment of the threshold effects, or of the existence of another contribution, e.g., of weak origin, to the electron mass.

References

[3] Nottale L., Fractal Space-Time and Microphysics: Towards a Theory of Scale Relativity (World Scientific, London, 1993).

[6] Nottale L., Chaos, Solitons and Fractals, 7 (1996) 877.

[7] Nottale L., Int. J. Mod. Phys. A7 (1992) 4899.

[8] Nottale L., in "Relativity in General", (1993 Spanish Relativity Meeting, Salas, Spain), Ed. J. Diaz Alonso, (Editions Frontières, 1994), pp.121-132.

[11] Nottale L., Astron. Astrophys. 327(1997) 867.

[22] Weyl H., in "The Principle of Relativity", (Dover publications, 1918), p. 201.

[23] Dirac P.A.M., Proc. Roy. Soc. Lond. A 333 (1973) 403.

[36] Landau, L., & Lifchitz, E., Relativistic Quantum Theory (Mir, Moscow, 1972).

[37] Itzykson, C., & Zuber, J.B., Quantum Field Theory (McGraw-Hill, 1980).