CHAPTER 4
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 (see Chapter 2),
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.
CHAPTER 5
DISCUSSION AND CONCLUSION
This thesis began with a comparison of some similarities between mass and
charge. The particular similarity of
interest is the inertia of both mass and charge, their tendency to resist being
accelerated. The main concepts are
radiation reaction, electromagnetic mass, and renormalization, as applied to classical
models of the electron.
Radiation reaction can be seen as something of a
fundamental electrodynamic feedback process, and as such can introduce unstable
solutions to the electron’s equations of motion. As discussed in Section 4.2, the instabilities can be avoided by
an appropriate choice of radius for a spherical surface charge model of the
electron. Numerically, the minimum
value of the radius for this case is 2/3 the classical electron radius,
Although the model predicts
a stable electron, it does not seem to agree with current experimental
evidence. High energy scattering
experiments indicate the upper limit on the radius of the electron is on the
order of 10-18 meter.[xv] According to the extended charge delay-differential
equation discussed in Section 4.2, a radius this small means the
electromagnetic mass of the electron is greater than its experimental mass m, and therefore the model electron is
unstable with respect to runaway solutions.
Classical physics, however, is generally considered
inapplicable for length scales close to the electron’s Compton wavelength, on the order of 10-12 meter. Thus the classical electron is something of
a contradiction in terms. In the
quantum realm, however, if the Compton wavelength formally replaces the
electron radius, the extended charge model can be taken to the point limit
without runaway solutions or pre-acceleration. According to Sharp,[xvi]
“It is the fact that there is a new length scale in quantum theory which allows
this to happen.”
In classical field theory, the old dream of unifying
general relativity with electromagnetism is still being pursued. According to Cooperstock,[xvii]
in general relativity “the Kerr-Newman metric reveals a gyromagnetic ratio
which agrees with that of the electron,” which appears promising for
classically describing electron spin.
However, Cooperstock also points out that in a system of units that give
charge and mass in centimeters, “whenever gravity plays a significant role, the
charge-to-mass ratio e/m is [expected to be] of order unity or less,”
whereas the experimental value in these units is approximately 1021.
In both the quantum and classical realms, renormalization
raises the question of how much of the electron’s observed or experimental mass
is mechanical or bare and how much is electromagnetic. Classically, electrodynamics and relativity
provide a way to answer the question, although the answer still contains
ambiguities.[xviii]
Relativity says electromagnetic mass cannot be measured dynamically
as a separate entity from mechanical mass.
When this news first arrived in the physics community in 1905, it
dampened the hopes of determining whether the mass of the electron is entirely
electromagnetic. At the same time,
Poincaré’s solution to the 4/3 problem gave a partial answer to that very
question--although the entire mass of
the Abraham-Lorentz electron can be electromagnetic, it is not a stable charge
distribution.
Poincaré’s solution
leads to a question: Why is the
Abraham-Lorentz electromagnetic mass greater than the Einstein field-energy
electromagnetic mass? Feynman[xix]
points out that, because of the necessity of the Poincaré stresses--which must
be added to the Abraham-Lorentz model to make it agree with the field energy
model--it is “impossible to get all of the mass to be electromagnetic in the
way we hoped.”
Panofsky and Phillips[xx]
say that the force accounting for stability is due to mass: “The electromagnetic mass of the electron
does not account for the entire mass;
the electron must have nonelectromagnetic mass of unknown origin to
account for its stability.”
In common with other references consulted for this thesis,
Feynman and Panofsky and Phillips do not explicitly say that the “mass of
unknown origin” can be considered to be responsible for the binding energy of the classical
electron. [Note added later: P&P do mention a comparison to nuclear binding
energy.] The more massive
Abraham-Lorentz model is electrostatically unstable when compared to the
smaller electromagnetic mass of the field energy calculation. This is comparable to the mass deficit (or
mass defect) of a nucleus, where the missing mass of the nucleus as compared
with its separated nucleons is the binding energy of the nucleus.
Since the Abraham-Lorentz electromagnetic mass is 4/3 times
greater than the Einstein field-energy electromagnetic mass, the binding energy
term is 1/3 the field energy electromagnetic mass,
,
so that the spherical shell
charge has a binding energy of
This is the work required to
assemble the shell of charge.[xxi] The physical meaning of this amount of
energy in a realistic setting is unclear.
What is lacking at the present time is a theory relating
the mass and charge of the electron, proton, and other elementary
particles. The need for such a theory
was mentioned by Oppenheimer[xxii]
in 1930, when he found that “it is impossible on the present theory to
eliminate the interaction of a charge with its own field,” which led to
predicted infinite energy level shifts in atoms. Oppenheimer concluded: “It appears improbable that the difficulties
discussed in this work will be soluble without an adequate theory of the masses
of the electron and proton; nor is it certain that such a theory will be
possible [merely] on the basis of the special theory of relativity.” Although the problem of the infinite
electromagnetic mass of the electron was fixed by the renormalization procedure
of quantum electrodynamics, Oppenheimer and also Dirac[xxiii]
believed that some new theory was necessary.
Such a new theory has not yet been developed.
There are two ways to look at the problems posed by the
classical electron model. Why not just
say that an electron is given to us as a point charge and there is no sense in
asking what its structure is? That is
the working assumption of quantum electrodynamics. The answer to that question--the other way to look at the
classical model of the electron--is contained in the questions about the
electron’s (and proton’s) charge and mass posed in the Introduction. An improved classical relativistic theory of
the electron could lead to an improved quantum theory, and that could lead to a
theory that explains and relates the charges and masses of the elementary
particles.
APPENDIX
PLANCK’S DIPOLE RESONATOR
EQUATIONS
In the same year the electron was discovered, the first
theoretical expression for radiation damping was derived by Max Planck in his
research on the relation between the energy and entropy of the electromagnetic
field as it interacts with small “dipole resonators”.[xxiv] Using the assumption of incident
electromagnetic plane waves polarized parallel to the dipole axis, Planck found
an equation relating the incident electric field to the electric dipole moment
of the resonators:
where, in Gaussian units, E(x,t) is the incident electric field, K and
L are resonator-dependent constants,
and p(x,t) is the dipole
moment of the resonator. Planck showed that when the resonator’s conditions are
such that its energy changes slowly in comparison with variations in the
incident field--when damping is much slower than the frequency of the incident
electromagnetic wave--the third time derivative can be replaced with a first
time derivative. (This is simple
harmonic oscillation approximation: 
.) The result is
This equation is of the form
used in the classical model of the atom.
Planck was searching for an explanation via electrodynamics
for the increase in entropy of closed systems, the observed macroscopic
irreversibility in time not predicted by mechanical equations of motion, which
are time reversal invariant.
The appearance of the damping terms in the resonator
equations implies a partial breaking of the forward and backward symmetry in
time, since both 
and 
change sign when t (time) changes sign. Planck called radiation damping
“conservative damping” in order to distinguish it from the dissipative effects
of non-conservative damping forces such as friction.[xxv] Planck’s work, however, did not lead him to
an explanation of entropy based on electrodynamics. As Kuhn[xxvi]
explains:
Through most of the year 1897, Planck continued to believe that he could prove irreversibility directly, without the aid of any statistical or other special hypotheses. That proof had been his initial objective in taking up the black-body problem at all. But by the spring of 1898 he had recognized that that goal could not possibly be achieved, and the concepts deployed in his subsequent papers came more and more to resemble those developed by Boltzmann for gas theory.
Though Planck was not
successful at explaining entropy from fundamental electrodynamics, his research
resulted in his postulate that a charged oscillator must emit and absorb
radiation only in discrete amounts--the idea that gave birth to quantum
mechanics in 1900.
References