**Mass-charge
relation for the electron**

**(on electromagnetism,
gauge invariance and the electric charge)**

**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}*

where a four-vector *A _{m}*
must be introduced since

*e D _{m }*ln(

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

*A' _{m}*
=

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}*
=

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}*
=

*y* = *y*_{0}
*e ^{i e }*

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, *e*_{0}
-> *e, V *=* lnr*
= *ln *(*e*_{0}/*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* = *y*_{0}
*exp *[*i* (*e*^{2}/* h c*)

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

*e*^{2} = 4*p* *a
h*

Eq. (20) becomes,* *

*y* = *y*_{0}*
e ^{i }*

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*

We recognize in -*i h*

~~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*
.

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 }*

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

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}*) =

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 *a*_{0}^{-1}
from the high energy U(1) and SU(2) couplings:

*a*_{0}^{-1 }= (3/8)
*a*_{2}^{-1} + (5/8) *a*_{1}^{-1}_{
}. (36)

This running coupling is such that *a*_{0}
= *a*_{1 }= *a*_{2}
at unification scale and is related to the fine structure constant at *Z*
scale by the relation *a*_{ }=
3*a*_{0}/8. It is *a*_{0}
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:

(8*a*_{e}*/
*3)_{ }ln(*m** _{P}*/

where *a*_{e}
is the fine structure cosntant, *m*_{e}
is the electron mass and *m*_{P} = (* hc
*/

**Figure 1**. Variation in function of length-scale
*r *of the running "electromagnetic" inverse coupling *a*_{2}^{-1}
+ (5/3) *a*_{1}^{-1}
, from the Planck scale (*P*) to the electron scale (e), and of 8*C*(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 8*C*(*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*
/

*a*(*r*) = *a*_{e
}{1 + (2*a*_{e}/3*p*)
[ 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 + (3*a*_{e}/2*p*)
[ 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 8*C*_{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}*/

ln (*m _{P}*/

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 Genera*l",
(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).