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Mass-charge relation for the electron  

 (http://www.daec.obspm.fr/users/nottale/ukmenure.htm)

   (on electromagnetism, gauge invariance and the electric
                          charge)



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

  Nottale, L., 1994, invited conference in "Relativity in General",
(Spanish Relativity Meeting'93), Salas, Eds. J. Diaz Alonso and M.
Lorente Paramo (Fronti�res), p.121-132. 

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

  Nottale, L., 1996, Chaos, Solitons and Fractals, 7, 877-938 

"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 = - Am dx m , (16)

where a four-vector Am 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 Dm ln(l/e) = e dm ln(l/e) + Am . (17)

However, if one wants the "field" Am 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 dxm, so that we obtain:

A'm = Am + e  dm 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 Fmn = dm An - dn Am . This field
is,
contrarily to Am, independent of the scale factor. One recognizes in
Fmn the
analog of an electromagnetic field, in Am 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 = Am - dm 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 Am = A0m - dm c, the wave function of an
electron of charge e becomes:

y = y0 ei 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 ei 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' = ei 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/LP), where LP 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 / LP) =
ln(mP/m),
while a = a(r) is also running. (Here mP 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:

(8ae/ 3)  ln(mP/me)  =  1,   (37)

where ae is the fine structure cosntant, me is the electron mass and
mP = (hc
/G)1/2 is the Planck mass. Their experimental values yield
respectivelyCe =
ln(mP/me) = 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/LP) 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.


Correction for threshold effects.

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 << le, where le= h /
me c
is the Compton scale of the electron) [36], [3, chapt. 6.2]:

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

where ae  is the low energy fine structure constant, such that (ae)-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) = me {1 + (3ae/2p) [ ln (le/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 8Ce / 3 is
equal, within a relative error of 10-4, to aas-1 at this scale
(Eq.40).



We now identify the scale relativity constant C= ln (mP/me) 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 (mP/me) = (3/8) ae-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 mP.

Equation (40) can be used to theoretically predict the electron mass
from the
experimental value of the fine structure constant. One finds me(th) =
1.007
me(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 Fronti�res, 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).