Dirac interpreted this as meaning that the electron has a spin kinetic momentum:
s = ðℏ ̸ 2Þ σ, which is to be added to the orbital kinetic momentum l to get a constant
of the motion. The directions of s and μ s being defined by the same matrix vector σ,
one has:
μ s = − ðe ̸ m 0 Þs.
ð16Þ
The spin magnetic moment μ s differs from the orbital magnetic moment μ l by a
factor 2:
μ l = − ðe ̸ 2m 0 Þl,
ð17Þ
as if the ‘loop’ described by the electron in its spin motion had half the length of its
orbital ‘loop’. The total magnetic moment can then be written:
μ t = − ðμ B ̸ ℏÞ ðl + g e sÞ,
ð18Þ
where we introduce the dimensionless factor g e , whose deviations from 2 (the value
given in Dirac’s theory) will be discussed in this paper. This factor distinguishes the
spin magnetic moment from the orbital magnetic moment derived classically. Being
the ratio of μ s (in units of μ B ) to s (in units of ℏ), it is called gyromagnetic factor.
In another computation, Dirac used a field-free Hamiltonian to determine at
which velocity the electron ‘spins’ to acquire kinetic and magnetic momenta [10]:
H = cðα 0 p 0 + α 1 p 1 + α 2 p 2 + α 3 p 3 Þ.
ð19Þ
Making use of the properties of the α k ’s he obtained, for any component v k of the
electron velocity:
iℏ ∂ x k ̸ ∂t = ½x k , H = iℏ c α k → v k = ∂ x k ̸ ∂t = ±c.
ð20Þ
The paradox of an electron moving at light velocity was elucidated by Schrödinger
[14] while investigating the Dirac velocity operators v k = cα k . He showed that:
iℏ ∂
2
α k ̸ ∂t
2 = 2 ð∂ α k ̸ ∂tÞ H.
ð21Þ
This differential equation can be integrated twice, yielding the explicit time
dependence of the velocity and then of the position. One first obtains:
v k = c α k = c
2 p k H
− 1 + ðiℏc ̸ 2Þγ
0
k e
− iωt H
− 1 ,
ð22Þ
where ω = 2H ̸ ℏ and γ
0
k = ∂ α k ̸ ∂t at t = 0. As H = mc
2 , the first term is a constant
of the order of p k /m, the classical relation between momentum and velocity. But
here also there is an extra term, oscillating at the ‘Zitterbewegung’ frequency [14]:
The Dirac Electron and Elementary Interactions …
365
s = ðℏ ̸ 2Þ σ, which is to be added to the orbital kinetic momentum l to get a constant
of the motion. The directions of s and μ s being defined by the same matrix vector σ,
one has:
μ s = − ðe ̸ m 0 Þs.
ð16Þ
The spin magnetic moment μ s differs from the orbital magnetic moment μ l by a
factor 2:
μ l = − ðe ̸ 2m 0 Þl,
ð17Þ
as if the ‘loop’ described by the electron in its spin motion had half the length of its
orbital ‘loop’. The total magnetic moment can then be written:
μ t = − ðμ B ̸ ℏÞ ðl + g e sÞ,
ð18Þ
where we introduce the dimensionless factor g e , whose deviations from 2 (the value
given in Dirac’s theory) will be discussed in this paper. This factor distinguishes the
spin magnetic moment from the orbital magnetic moment derived classically. Being
the ratio of μ s (in units of μ B ) to s (in units of ℏ), it is called gyromagnetic factor.
In another computation, Dirac used a field-free Hamiltonian to determine at
which velocity the electron ‘spins’ to acquire kinetic and magnetic momenta [10]:
H = cðα 0 p 0 + α 1 p 1 + α 2 p 2 + α 3 p 3 Þ.
ð19Þ
Making use of the properties of the α k ’s he obtained, for any component v k of the
electron velocity:
iℏ ∂ x k ̸ ∂t = ½x k , H = iℏ c α k → v k = ∂ x k ̸ ∂t = ±c.
ð20Þ
The paradox of an electron moving at light velocity was elucidated by Schrödinger
[14] while investigating the Dirac velocity operators v k = cα k . He showed that:
iℏ ∂
2
α k ̸ ∂t
2 = 2 ð∂ α k ̸ ∂tÞ H.
ð21Þ
This differential equation can be integrated twice, yielding the explicit time
dependence of the velocity and then of the position. One first obtains:
v k = c α k = c
2 p k H
− 1 + ðiℏc ̸ 2Þγ
0
k e
− iωt H
− 1 ,
ð22Þ
where ω = 2H ̸ ℏ and γ
0
k = ∂ α k ̸ ∂t at t = 0. As H = mc
2 , the first term is a constant
of the order of p k /m, the classical relation between momentum and velocity. But
here also there is an extra term, oscillating at the ‘Zitterbewegung’ frequency [14]:
The Dirac Electron and Elementary Interactions …
365