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we define the Gordon current:
e e e
Kµ = [ (pµÈ)È](0) = (ÈpµÈ)(0) = (pµÁ cos ² - qµÁ sin ²) . (6.49)
m m m
Or in vector form,
e
K = Á(p cos ² - q sin ²) . (6.50)
m
As anticipated in the last Section, from the last term in (6.48) we define the magnetization
e
M = ÁSei² . (6.51)
m
When (6.48) is inserted into (6.47), the pseudovector part must vanish, and vector part gives us the
so-called Gordon decomposition
J = K + · M. (6.52)
This is ostensibly a decomposition into a conduction current K and a magnetization current · M,
both of which are separately conserved. But how does this square with the physical interpretation
already ascribed to J? It suggests that there is a substructure to the charge flow described by J.
Evidently if we are to understand this substructure we must understand the role of the parameter
² so prominently displayed in (6.50) and (6.51). A curious fact is that ² does not contribute to
the definition (5.20) for the Dirac current in terms of the wave function; ² is related to J only
indirectly through the Gordon Relation (6.52). This suggests that ² characterizes some feature of
the substructure.
So far we have supplied a physical interpretation for all parameters in the wave function (5.16)
except duality parameter ². The physical interpretation of ² is more problematic than that of the
other parameters. Let us refer to this as the ²-problem. This problem has not been recognized in
conventional formulations of the Dirac theory, because the structure of the theory was not analyzed
in sufficient depth to identify it. The problem arose, however, in a different guise when it was
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noted that the Dirac equation admits negative energy solutions. The famous Klein paradox showed
that negative energy states could not be avoided in matching boundary conditions at a potential
barrier. This was interpreted as showing that electron-positron pairs are created at the barrier, and
it was concluded that second quantization of the Dirac wave function is necessary to deal with the
many particle aspects of such situations. However, recognition of the ²-problem provides a new
perspective which suggests that second quantization is unnecessary, though this is not to deny the
reality of pair creation. A resolution of the Klein Paradox from this perspective has been given by
Steven Gull.26
In the plane wave solutions of the Dirac equation (next Section), the parameter ² unequivocally
distinguishes electron and positron solutions. This suggests that ² parametrizes an admixture of
electron-positron states where cos ² is the relative probability of observing an electron. Then, while
Á = Á(x) represents the relative probability of observing a particle at x, Á cos ² is the probability
that the particle is an electron, while Á sin ² is the probability that it is an positron. On this
interpretation, the Gordon current shows a redistribution of the current flow as a function of ².
It leads also to a plausible interpretation for the ²-dependence of the magnetization in (6.51). In
accordance with (4.39), in the electron rest system determined by J, we can split M into
M = -P + iM, (6.53)
where, since v · s =0,
e
iM= SÁ cos ² (6.54)
m
is the magnetic moment density, while
e
P = - iSÁ sin ² (6.55)
m
is the electric dipole moment density. The dependence of P on sin ² makes sense, because pair
creation produces electric dipoles. On the other hand, cancelation of magnetic moments by created
pairs may account for the reduction of M by the cos ² factor in (6.54). It is tempting, also, to
interpret equation (6.4) as describing a creation of spin concomitant with pair creation.
Unfortunately, there are difficulties with this straight forward interpretation of ² as an antiparticle
mixing parameter. The standard Darwin solutions of the Dirac hydrogen atom exhibit a strange
² dependence which cannot reasonably be attributed to pair creation. However, the solutions also
attribute some apparently unphysical properties to the Dirac current; suggesting that they may be
superpositions of more basic physical solutions. Indeed, Heinz Krüger has recently found hydrogen
atom solutions with ² =0.27
It is easy to show that a superposition of solutions to the Dirac equation with ² = 0 can produce
a composite solution with ² = 0. It may be, therefore, that ² characterizes a more general class of
statistical superpositions than particle-antiparticle mixtures. At any rate, since the basic observables
v, S and p are completely characterized by the kinematical factor R in the wave function, it appears
that a statistical interpretation for ² as well as Á is appropriate.
7. ELECTRON TRAJECTORIES
In classical theory the concept of particle refers to an object of negligible size with a continuous
trajectory. It is often asserted that it is meaningless or impossible in quantum mechanics to regard
the electron as a particle in this sense. On the contrary, it asserted here that the particle concept
is not only essential for a complete and coherent interpretation of the Dirac theory, it is also of
practical value and opens up possibilities for new physics at a deeper level. Indeed, in this Section
it will be explained how particle trajectories can be computed in the Dirac theory and how this
articulates perfectly with the classical theory formulated in Section 3.
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David Bohm has long been the most articulate champion of the particle concept in quantum
mechanics.28 He argues that the difference between classical and quantum mechanics is not in the
concept of particle itself but in the equation for particles trajectories. From Schroedinger s equation
he derives an equation of motion for the electron which differs from the classical equation only
in a stochastic term called the Quantum Force. He is careful, however, not to commit himself
to any special hypothesis about the origins of the Quantum Force. He accepts the form of the
force dictated by Schroedinger s equation. However, he takes pains to show that all implications of
Schroedinger theory are compatible with a strict particle interpretation. The same general particle
interpretation of the Dirac theory is adopted here, and the generalization of Bohm s equation derived
below provides a new perspective on the Quantum Force.
We have already noted that each solution of the Dirac equation determines a family of nonin-
tersecting streamlines which can be interpreted as expected electron histories. Here we derive
equations of motion for observables of the electron along a single history x = x(Ä). By a space-time
split the history can always be projected into a particle trajectory x(Ä) =x(Ä)'"³0 in a given inertial
system. It will be convenient to use the terms history and trajectory almost interchangeably. The
representation of motion by trajectories is most helpful in interpreting experiments, but histories
are usually more convenient for theoretical purposes.
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