7.5 The Equation of the Lorentz Force 41
In this way we can say that the Planck constant, which appears in Eq. 7.11, and
the spin are hidden parameters of the classical theory of the electron.
But one has to notice that the Dirac theory is reduced here to its dynamical
equation the equation D
I
. The part of the Dirac theory, equation D
II
which implies
the density ρ has not been taken into account.
Furthermore in all that precedes the spacetime curve C is to be considered in the
particular case where it is situated in a spacetime plane and where the space curve
of the electron is a straight line.
7.6 On the Passages of the Dirac Theory to the Classical
Theory of the Electron
In the particular case of a potential in the form of Eq. 7.10, the two points which
are implied in the classical theory of the electron, that is the absence of h and of the
probability density ρ, have been deduced exactly from the Dirac equation, limited
to the dynamic equation D
I
.
In the case of other forms of the potential A, such that the direction of the spin
plane varies sufficiently slowly in such a way that ω may be considered as a gradient,
and so is eliminated, one can deduce from D
I
as an approximation the classical
behaviour of the electron in presence of the field F = ∂ ∧ A.
And now, we can associated what precedes with the sentence of Einstein, “God
cannot play with dice", about the difference between a classical and a quantum
electron. On one side a particle which may be clearly situated in the space, which has
a trajectory, and on the other an object whose position seems to depend on hazard. In
agreement with the conviction of Einstein, we can say that, in what precedes, nothing
allows one to assert that hazard is to be associated with this object, which the Dirac
height scalars equations cannot specify exactly its position. Four of these equations
are relative to seven scalar variables corresponding to observable entities, and so the
determination of these entities only by means of these equations is impossible. But
and it is only a deficiency of our knowledge, not some caprice of Nature. One scalar
variable and four equations more, and the calculation of these entities is possible,
except the position which may be only approached in a statistical point of view.
However the case treated above of an electron both classical, because it has a
trajectory, and quantal, because it satisfies the Einstein formula of the photoeffect,
appears as a window open to the hitherto inaccessible reality.
References
1. R. Boudet, CR. Ac. Sc. (Paris). 272(767), 272A, 767 (1971)
2. R. Boudet, J. Math. Phys. 26, 718 (1985)
3. v. Halzen, D. Martin, Quarks and leptons. (J. Wiley and Sons, U.S.A., 1984)