POINT CHARGE AND EXTENDED CHARGE
MODELS
4.1 RELATIVISTIC POINT CHARGE MODEL
The
infinite electromagnetic mass of a point electron can be avoided by some method
of renormalization that makes the electromagnetic mass zero. This is the antithesis of Lorentz’s idea that
all the mass of the electron should be electromagnetic, yet it is precisely the
solution proposed by Dirac in 1938:[i]
One of
the most attractive ideas in the Lorentz model of the electron, the idea that
all mass is of electromagnetic origin, appears at the present time to be wrong,
for two separate reasons. First the
discovery of the neutron has provided us with a form of mass which it is very
hard to believe could be of electromagnetic nature. Secondly, we have the theory of the
positron--a theory in agreement with experiment so far as is known--in which
positive and negative values for the mass of an electron play symmetrical
roles. This cannot be fitted in with the
electromagnetic idea of mass, which insists on all mass being positive, even in
abstract theory.
Dirac
accomplished the elimination of electromagnetic mass in the point charge model
of the electron by keeping both the retarded and advanced field solutions to
Maxwell’s equations. An explanation of
Dirac’s theory requires some more discussion of how the Abraham-Lorentz model
fits into Maxwell’s theory.
In fact, the theory of Abraham and
Lorentz is only based on the Maxwell equations insofar as it uses the retarded
vector potentials of Liénard and Wiechert.
Thus, Erber[ii]
says, the Abraham-Lorentz equation and its early relativistic generalizations,
are “largely phenomenological.” Dirac, in contrast, applied the Maxwell
equations and the relativistic Lorentz force equation to the self-interaction
of a charged particle, then adopted “boundary conditions different from the
‘logical’ one in which only retarded fields are allowed.”[iii]
The retarded fields are responsible
for the retarded self-force computed by Lorentz and Abraham (from ),
and the
advanced fields--at least from a mathematical point of view--cause an advanced
self-force obtained by replacing t with -t ,
giving[iv]
Dirac’s elimination of the
electromagnetic mass term is then accomplished by taking one-half the
difference of the advanced and retarded self-interactions,
The
electromagnetic mass term in the retarded force cancels the one in the advanced
force equation. Higher order terms that
don’t cancel in this subtraction are all proportional to positive powers of the
model electron’s radius, which causes them to become identically zero when the
point charge limit is taken.
Misner, Thorne, and Wheeler[v]
give the fully relativistic result of Dirac’s calculation (in four-vector
notation and Gaussian units) as
where t is
proper time and Fmn is the
electromagnetic field strength tensor.
These authors then comment:
“Every acceptable line of reasoning has always led to [this]
expression. It also represents the field
required to reproduce the long-known and thoroughly tested law of radiation
damping.”
However, the long-known theory of
radiation damping--that is, radiation reaction in the case of general
oscillatory motion--has no runaway solution.
As discussed in Section 2.4, the general solution for the acceleration in
the radiation reaction equation shows an exponential increase with time. The relativistic version of the equation is
no different in that respect.[vi]
For the solution in terms of the
acceleration given in Section 2.4,
Dirac’s
resolution of the runaway nature of this equation was to propose the asymptotic
initial condition
which gives
the general solution[vii]
This
equation says that an acceleration at time t
is caused by a force acting at time later than t , which violates the common notion of causality.
The reasoning behind Dirac’s
subtraction renormalization and his asymptotic initial condition is purely
mathematical rather than physical.
Wheeler and Feynman[viii]
used retarded and advanced fields to develop a more physical theory based on
the assumption that an electron, considered to be a point charge, does not
interact with itself. The interaction
with other charges occurs by a sum of half the advanced field plus half the retarded
field. When another charge a distance d away absorbs an electromagnetic wave from an
accelerating electron at the retarded time t’
= t + d/c, the charge also emits an advanced wave that reaches the electron
at time t’ - d/c = t + d/c - d/c = t. Thus, the advanced wave acts on the electron
just as it begins to accelerate, giving the radiation reaction effect without
electron self-interaction.
As far as Dirac’s point charge model
is concerned, his reasoning against Lorentz’s idea of electromagnetic mass is
now outdated, since the neutron does have electromagnetic energy as a
consequence of its magnetic moment, and the mass of the positron is now
considered to be positive. Also,
presuming the existence of advanced fields, as in the Dirac and Wheeler-Feynman
theories, is physically counter-intuitive, since the retarded fields “are the
ones measured in a typical experiment.”[ix] As Feynman[x]
himself says, “You can see what tight knots people have gotten into trying to
get a theory of the electron!”
4.2 NONRELATIVISTIC EXTENDED CHARGE MODEL
A point
charge theory such a Dirac’s or the Wheeler-Feynman absorber theory is
desirable from the point of view of relativity because a perfectly rigid sphere
is assumed to transmit mechanical waves instantaneously through its interior,
thus violating the speed-of-light limitation of special relativity. Also, an extended charge model that is not
perfectly rigid would presumably have observable modes of oscillation, and so
far the electron has not revealed such observable oscillations. For these reasons, the extended charge model
of the electron has remained in the backwaters of theoretical physics.
In spite of its difficulties, the
extended charge model is a successful nonrelativistic model which does not have
infinite electromagnetic mass or runaway solutions or pre-acceleration
problems. In addition, certain modes of
oscillation of the extended charge are predicted to be radiationless, and thus
would be undetectable by radiation detectors.
The most significant accomplishment of
certain extended charge models is the replacement of Lorentz’s infinite series
expansion with a delay-differential equation that has no third or higher order
time derivatives of position.
For the spherical shell of charge of
radius a, the expression for the
charge distribution can be written in terms of the three dimensional Dirac
delta function as
where r = |r|. The Lorentz series
expansion terms can then be summed to give the delay-differential equation[xi]
or in terms
of the acceleration of the electron’s center of mass R,
where m is the experimental mass of the electron,
where
(Section 2.2)
Then the
experimental mass of the electron, the speed of light, and the model electron’s
radius can all be included in one factor,
which can be put in the
electron center of mass self-acceleration equation above with terms rearranged
to give
This equation for the
spherical shell charge model was derived by Bohm and Weinstein[xii]
and others. It implicitly shows that
there are runaway solutions only when the bare mass of the electron is
negative. That condition occurs when ct > a or
when the electromagnetic
mass of the spherical shell charge model is greater than the observed mass of
the electron. This condition is derived
in a more explicit manner by Levine, Moniz, and Sharp.[xiii]
One appealing aspect of the Bohm-Weinstein derivation when
it was published in 1948 was their demonstration that the model electron’s
self-oscillations (harmonic motions about the center of mass) could be
quantized and the energy of the first excited state was approximately equal to
the rest energy of a p meson or, in modern
language, a pion. In modern theory,
however, the pion is a hadron rather than a lepton, and is thus composed of
quarks.
Overall, the extended charge model, although
nonrelativistic, avoids the problems of infinite electromagnetic mass, runaways,
and pre-acceleration. As described by
Milonni:[xiv]
For most of the twentieth century the
classical electron theory, based on the presumption of a point electron, has suffered from the runaway and preacceleration
maladies, as well as the divergent electromagnetic mass. It is seldom acknowledged that the classical
theory is free of runaways if the radius of an extended charged particle is larger than the radius for which its
observed mass would be entirely electromagnetic.
[i]
Ibid., p. 148.
[ii]
Erber, p. 350.
[iii]
Milonni, p. 161.
[iv]
Ibid.
[v]
C.W. Misner, K.S. Thorne, and J.A. Wheeler, Gravitation,
(W.H. Freeman, San Francisco, 1973), p. 474.
[vi]
Rohrlich, p.22.
[vii]
Milonni, p. 157.
[viii]
J.A. Wheeler and R.P. Feynman, “Interaction with the Absorber as the Mechanism
of Radiation,” Rev. Mod. Phys. 17,
157- (1945).
[ix]
Rohrlich, p. 22.
[x]
Feynman, et al., Chapter 28.
[xi]
Milonni, p. 166.
[xii]
D. Bohm and M. Weinstein, “The Self-Oscillations of a Charged Particle,” Phys.
Rev. 74, 1789-1798 (1948).
[xiii]
H. Levine, E.J. Moniz, and D.H. Sharp,
“Motion of extended charges in classical electrodynamics,” Am. J. Phys. 45, 75-78 (1977).
[xiv]
Milonni, p.168.
