3 The Dirac Electron and Basic Physical Concepts
71
alternatively, E y as ct and B z as x), with similar correspondences for the other two
components:
E y (B z ) ∼ x,
B z (E y ) ∼ ct;
E z (B y ) ∼ ±x,
B y (E z ) ∼ ±ct. (3.34)
Just as p 2
0 and x 2
0 (Eqs. (3.7) & (3.8)) are invariant under a change of frame, here
also there are two invariants [65]:
Y
2
= E
2
− B
2 ,
Z
2
= E · B.
(3.35)
As (dimensionwise) E ∼ B ∼ MLT −2 Q −1 ∼ L −2 Q −1 (in our system), these two
invariants are homologous to L −4 Q −2 . According to our previous remark on the
analogy between charge Q and curvature L −1 , one may then write: Y ∼ Z ∼ L −1 ,
homologous to p 0 , then to rest mass.
The electric and magnetic fields are different in that E is a polar vector (translational symmetry) while B in an axial vector (rotational symmetry). Now, in the
Dirac equation, space coordinates x are related to the external, linear momentum
p while the time coordinate x 0 = cτ 0 is related to the internal, spin momentum
p 0 = m 0 c. There is thus a correspondence between electric field, translation, linear momentum, and space, and a similar one between magnetic field, rotation, spin
momentum, and time.
While space homogeneity entails conservation of p, and time homogeneity conservation of p 4 ≡ E/c, space-time isotropy entails conservation of l + s, for l and
s cannot be measured independently [6]. A definition of isotropy consistent with
relativistic quantum mechanics then necessarily involves full space-time.
3.6 Conclusions
In this paper we have conforted our previous conjecture that the visible properties
of the electron, especially its rest mass, are determined by a subquantum massless
charge spinning at light speed within a Compton radius. In complement to the conclusions drawn in our previous paper [22], the following points can be stressed.
1. The rest mass energy m 0 c 2 of the electron is essentially a kinetic self-energy
related to its spin motion, with a contribution α-smaller of the potential selfenergy related to its charge content. For photons, p = ω/c (p being the external
linear momentum), whereas for electrons, 1/2/r C = m 0 c 2 /c (1/2 being the
spin angular momentum). This sets the Compton radius: r C = /2m 0 c, as the
range of the spin motion.
2. Spin itself being the ‘orbital momentum’ of Zitterbewegung, which in turn stems
from a wave beat between the electron and its mirror twin the positron, there
is no matter without antimatter: there is no need to search for antimatter for it
is around us and in us, as the two faces of a same coin or the two poles of a
magnetic moment.
The question now is, why ‘only one face of the coin’ shows up regarding
the electric charge, while magnetic poles always appear in couple. If charge is
71
alternatively, E y as ct and B z as x), with similar correspondences for the other two
components:
E y (B z ) ∼ x,
B z (E y ) ∼ ct;
E z (B y ) ∼ ±x,
B y (E z ) ∼ ±ct. (3.34)
Just as p 2
0 and x 2
0 (Eqs. (3.7) & (3.8)) are invariant under a change of frame, here
also there are two invariants [65]:
Y
2
= E
2
− B
2 ,
Z
2
= E · B.
(3.35)
As (dimensionwise) E ∼ B ∼ MLT −2 Q −1 ∼ L −2 Q −1 (in our system), these two
invariants are homologous to L −4 Q −2 . According to our previous remark on the
analogy between charge Q and curvature L −1 , one may then write: Y ∼ Z ∼ L −1 ,
homologous to p 0 , then to rest mass.
The electric and magnetic fields are different in that E is a polar vector (translational symmetry) while B in an axial vector (rotational symmetry). Now, in the
Dirac equation, space coordinates x are related to the external, linear momentum
p while the time coordinate x 0 = cτ 0 is related to the internal, spin momentum
p 0 = m 0 c. There is thus a correspondence between electric field, translation, linear momentum, and space, and a similar one between magnetic field, rotation, spin
momentum, and time.
While space homogeneity entails conservation of p, and time homogeneity conservation of p 4 ≡ E/c, space-time isotropy entails conservation of l + s, for l and
s cannot be measured independently [6]. A definition of isotropy consistent with
relativistic quantum mechanics then necessarily involves full space-time.
3.6 Conclusions
In this paper we have conforted our previous conjecture that the visible properties
of the electron, especially its rest mass, are determined by a subquantum massless
charge spinning at light speed within a Compton radius. In complement to the conclusions drawn in our previous paper [22], the following points can be stressed.
1. The rest mass energy m 0 c 2 of the electron is essentially a kinetic self-energy
related to its spin motion, with a contribution α-smaller of the potential selfenergy related to its charge content. For photons, p = ω/c (p being the external
linear momentum), whereas for electrons, 1/2/r C = m 0 c 2 /c (1/2 being the
spin angular momentum). This sets the Compton radius: r C = /2m 0 c, as the
range of the spin motion.
2. Spin itself being the ‘orbital momentum’ of Zitterbewegung, which in turn stems
from a wave beat between the electron and its mirror twin the positron, there
is no matter without antimatter: there is no need to search for antimatter for it
is around us and in us, as the two faces of a same coin or the two poles of a
magnetic moment.
The question now is, why ‘only one face of the coin’ shows up regarding
the electric charge, while magnetic poles always appear in couple. If charge is