3 The Dirac Electron and Basic Physical Concepts
59
One may then see Eq. (3.8) as involving formally three space dimensions and two
time dimensions.
In the simple classical picture of a particle endowed with charge e and mass m 0
moving at velocity c around a loop of radius r C , the intrinsic angular momentum
would be: s = m 0 c · r C = r C · 2π/λ C , from the definition of λ C in Eq. (3.2). As in
the Bohr model for the orbital motion of an electron around a nucleus, the quantum
number s/ = 2πr C /λ C takes on a (half) integer value if the circumference 2πr C
involves a (half) integer number of wave-lengths λ C (the half stemming from the
Zitterbewegung frequency being that of a wave beat between the positive and negative energy states). This loop could then be seen as a kind of ‘intrinsic orbit’ with
range 2r C .
If then the electron (m 0 ∼ 0.5 MeV) is viewed as the ‘lowest (stable) state’ of
a kind of ‘hidden structure’ similar to the Bohr atom, then the related muon and
tau particles (m μ ∼ 106 MeV, m t ∼ 1800 MeV) could be seen as ‘excited (unstable) states’ of this quasi-Bohr substructure. In a classical (spinless) extensible
model of the electron as a charged conducting surface [32], Dirac showed that its
first excited state with spherical symmetry has a rest mass about four times smaller
(m ∗
1s ∼ 27 MeV) than the muon’s. A large part of the rest mass should then arise
from the spin motion.
The muon and tau particles belonging to the same (lepton) family as the electron, their Compton wavelength λ C , Eq. (3.2), if it was measurable in spite of their
very short lifetime (τ μ ∼ 2.10 −6 , τ t ∼ 3.10 −13 ), would be much smaller than the
electron’s. In hydrogenoid atoms, the smaller the ‘Bohr’ (average) radius r n of a
given (spherically symmetric) ns orbital, the larger the ionization energy I n from
this state. In our quasi-Bohr lepton substructure, the smaller the ‘Compton radius’
r C , the larger the rest mass energy m 0 c 2 .
In the hydrogen atom, there is an infinite sequence of excited states, with higher
and higher discrete energies bounded by the Rydberg energy R H (∼13.6 eV) and
ending in the continuum. Similarly, one may conjecture that in the electron family,
there is an infinite sequence of excited states, with larger and larger discrete energies
bounded by the Planck energy E P (∼1.96 × 10 9 J ∼ 1.22 × 10 22 MeV), corresponding to a Planck spinning range 2r P (∼1.62 × 10 −35 m) and a Planck time scale τ P
(∼5.39 × 10 −44 s).
We have tried to find some regularity in the sequence of known members of
the electron family, so as to estimate what could be the next member in the series. Results of a quadratic fit of the logarithms of reduced energies, including the
Planck limit, are shown in Table 3.1. The next member would have a rest mass
m ν ∼ 122 GeV, very close to that (125 GeV) of the particle identified in 2012 at
CERN as the celebrated ‘Higgs boson’: it could be detected in the same energy
range. Figure 3.1 displays a fit of these masses.
The proton and the neutron belong to a different (baryon) family: they are composite particles (both are made up of three quarks) and are sensitive to the strong
nuclear force. Their charge radii r N (N for nucleon), measured by electron scattering, differ from their Compton radii r C by one to two orders of magnitude, and
their magnetic moments μ N from the nuclear magneton μ P = ecr P (r P being the
Compton radius of the proton) by factors 2.79285 and −1.91315, respectively.
59
One may then see Eq. (3.8) as involving formally three space dimensions and two
time dimensions.
In the simple classical picture of a particle endowed with charge e and mass m 0
moving at velocity c around a loop of radius r C , the intrinsic angular momentum
would be: s = m 0 c · r C = r C · 2π/λ C , from the definition of λ C in Eq. (3.2). As in
the Bohr model for the orbital motion of an electron around a nucleus, the quantum
number s/ = 2πr C /λ C takes on a (half) integer value if the circumference 2πr C
involves a (half) integer number of wave-lengths λ C (the half stemming from the
Zitterbewegung frequency being that of a wave beat between the positive and negative energy states). This loop could then be seen as a kind of ‘intrinsic orbit’ with
range 2r C .
If then the electron (m 0 ∼ 0.5 MeV) is viewed as the ‘lowest (stable) state’ of
a kind of ‘hidden structure’ similar to the Bohr atom, then the related muon and
tau particles (m μ ∼ 106 MeV, m t ∼ 1800 MeV) could be seen as ‘excited (unstable) states’ of this quasi-Bohr substructure. In a classical (spinless) extensible
model of the electron as a charged conducting surface [32], Dirac showed that its
first excited state with spherical symmetry has a rest mass about four times smaller
(m ∗
1s ∼ 27 MeV) than the muon’s. A large part of the rest mass should then arise
from the spin motion.
The muon and tau particles belonging to the same (lepton) family as the electron, their Compton wavelength λ C , Eq. (3.2), if it was measurable in spite of their
very short lifetime (τ μ ∼ 2.10 −6 , τ t ∼ 3.10 −13 ), would be much smaller than the
electron’s. In hydrogenoid atoms, the smaller the ‘Bohr’ (average) radius r n of a
given (spherically symmetric) ns orbital, the larger the ionization energy I n from
this state. In our quasi-Bohr lepton substructure, the smaller the ‘Compton radius’
r C , the larger the rest mass energy m 0 c 2 .
In the hydrogen atom, there is an infinite sequence of excited states, with higher
and higher discrete energies bounded by the Rydberg energy R H (∼13.6 eV) and
ending in the continuum. Similarly, one may conjecture that in the electron family,
there is an infinite sequence of excited states, with larger and larger discrete energies
bounded by the Planck energy E P (∼1.96 × 10 9 J ∼ 1.22 × 10 22 MeV), corresponding to a Planck spinning range 2r P (∼1.62 × 10 −35 m) and a Planck time scale τ P
(∼5.39 × 10 −44 s).
We have tried to find some regularity in the sequence of known members of
the electron family, so as to estimate what could be the next member in the series. Results of a quadratic fit of the logarithms of reduced energies, including the
Planck limit, are shown in Table 3.1. The next member would have a rest mass
m ν ∼ 122 GeV, very close to that (125 GeV) of the particle identified in 2012 at
CERN as the celebrated ‘Higgs boson’: it could be detected in the same energy
range. Figure 3.1 displays a fit of these masses.
The proton and the neutron belong to a different (baryon) family: they are composite particles (both are made up of three quarks) and are sensitive to the strong
nuclear force. Their charge radii r N (N for nucleon), measured by electron scattering, differ from their Compton radii r C by one to two orders of magnitude, and
their magnetic moments μ N from the nuclear magneton μ P = ecr P (r P being the
Compton radius of the proton) by factors 2.79285 and −1.91315, respectively.