64
J. Maruani
have: p = (/2)/r C , yielding: E = c/2r C = m 0 c 2 according to Eq. (3.4). Writing
E = eV and making use of Eq. (3.4), one obtains:
V = c/2er C = k e α
−1 e/2r C .
(3.31)
What maintains the massless charge −e in a spinning orbit and provides it with the
energy accounting for the electron rest mass would then be the electric potential V
exerted by an opposite charge at a distance equal to twice the Compton radius, as if
the spheres of the electron and its mirror image were contiguous. This is indeed the
occurrence leading to Eq. (3.26). However, in this no man’s land (cf. Eq. (3.27)),
the inverse permittivity of the medium or, alternatively, the attracting virtual charge
would be multiplied by α −1 ∼ 137.
That an internal potential V acting on a massless charge −e be responsible for
its spin motion and rest mass is not contradictory with p 0 and x 0 (Eqs. (3.7) and
(3.8)) being invariant under an external field, while p 4 is modified by A 4 and p
by A (Eq. (3.13)). The spin momentum s = p 0 r C and rest mass m 0 = p 0 /c are
also insensitive to (A 4 , A), although these potentials can act on the associated magnetic moment and electric charge. The deep connection between spin and mass also
appears in the unitary irreducible representations of the group of isometries in relativistic space-time being indexed by spin and mass.
A corroboration of our conjecture can be found in more formal treatments. Barut
and Bracken [18–21] have derived the Dirac equation for a finite quantum system in
an arbitrary moving frame. There, spin appears as the orbital momentum associated
with the internal system, while the rest mass is the internal energy in the rest frame.
In another, stochastic electrodynamics approach [48], Haisch, Rueda and Puthoff
have shown that Zitterbewegung arises from the electromagnetic interaction of a
subelementary charged particle (parton) with the vacuum zero-point field (ZPF).
Inertia was interpreted as a resistance of ZPF to spectral distortion in an accelerated frame, and the van der Waals force generated by this oscillating motion was
identified with the Newtonian gravitational force. In addition, the inertial and gravitational masses thus derived were shown to be equivalent.
In Sect. 3.5, we shall see that charge is the only independent quantity that remains when one identifies time with length, and mass with length inverse. It is not
surprising then that the three related quantities: length, time, and mass, vary with
velocity, but not charge. If the electron rest mass essentially results from the spinning motion, over a sphere of radius r C , of a massless charge −e at light speed, then
the contribution of the electrostatic potential due to the charge distribution over this
sphere is [22]:
E 0 ∼ k e e
2 /2r C = α · m 0 c
2 .
(3.32)
The contribution of the electrostatic self-energy to the electron rest mass is less
than 1 % of the kinetic contribution due to the spin motion around the effective virtual charge α −1 e (Eq. (3.31)). This energy can be compared to the electron-nucleus
‘contact energy’ (Eq. (3.27)).
Using again the semi-classical picture of an electron ball, the hidden confined
motion of a massless charge at velocity c can be related to the visible free motion
J. Maruani
have: p = (/2)/r C , yielding: E = c/2r C = m 0 c 2 according to Eq. (3.4). Writing
E = eV and making use of Eq. (3.4), one obtains:
V = c/2er C = k e α
−1 e/2r C .
(3.31)
What maintains the massless charge −e in a spinning orbit and provides it with the
energy accounting for the electron rest mass would then be the electric potential V
exerted by an opposite charge at a distance equal to twice the Compton radius, as if
the spheres of the electron and its mirror image were contiguous. This is indeed the
occurrence leading to Eq. (3.26). However, in this no man’s land (cf. Eq. (3.27)),
the inverse permittivity of the medium or, alternatively, the attracting virtual charge
would be multiplied by α −1 ∼ 137.
That an internal potential V acting on a massless charge −e be responsible for
its spin motion and rest mass is not contradictory with p 0 and x 0 (Eqs. (3.7) and
(3.8)) being invariant under an external field, while p 4 is modified by A 4 and p
by A (Eq. (3.13)). The spin momentum s = p 0 r C and rest mass m 0 = p 0 /c are
also insensitive to (A 4 , A), although these potentials can act on the associated magnetic moment and electric charge. The deep connection between spin and mass also
appears in the unitary irreducible representations of the group of isometries in relativistic space-time being indexed by spin and mass.
A corroboration of our conjecture can be found in more formal treatments. Barut
and Bracken [18–21] have derived the Dirac equation for a finite quantum system in
an arbitrary moving frame. There, spin appears as the orbital momentum associated
with the internal system, while the rest mass is the internal energy in the rest frame.
In another, stochastic electrodynamics approach [48], Haisch, Rueda and Puthoff
have shown that Zitterbewegung arises from the electromagnetic interaction of a
subelementary charged particle (parton) with the vacuum zero-point field (ZPF).
Inertia was interpreted as a resistance of ZPF to spectral distortion in an accelerated frame, and the van der Waals force generated by this oscillating motion was
identified with the Newtonian gravitational force. In addition, the inertial and gravitational masses thus derived were shown to be equivalent.
In Sect. 3.5, we shall see that charge is the only independent quantity that remains when one identifies time with length, and mass with length inverse. It is not
surprising then that the three related quantities: length, time, and mass, vary with
velocity, but not charge. If the electron rest mass essentially results from the spinning motion, over a sphere of radius r C , of a massless charge −e at light speed, then
the contribution of the electrostatic potential due to the charge distribution over this
sphere is [22]:
E 0 ∼ k e e
2 /2r C = α · m 0 c
2 .
(3.32)
The contribution of the electrostatic self-energy to the electron rest mass is less
than 1 % of the kinetic contribution due to the spin motion around the effective virtual charge α −1 e (Eq. (3.31)). This energy can be compared to the electron-nucleus
‘contact energy’ (Eq. (3.27)).
Using again the semi-classical picture of an electron ball, the hidden confined
motion of a massless charge at velocity c can be related to the visible free motion