DISCUSSION AND CONCLUSION
This thesis began with a comparison
of some similarities between mass and
electric 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.[i] 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,[ii]
“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,[iii]
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.[iv]
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[v]
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[vi]
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. 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.[vii] 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[viii]
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 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[ix]
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.
[i]
F. I. Cooperstock, “Non-linear gauge invariant field theories of the electron
and other elementary particles,” in The
Electron: New Theory and Experiment, ed. D. Hestenes and A. Weingartshofer
(Kluwer, Dortrecht, 1991), pp. 171-181.
[ii]
D. H. Sharp, p. 130.
[iii]
Cooperstock, p. 176.
[iv]
F. Rohrlich, “The dynamics of a charged sphere and the electron,” Am. J. Phys. 65, 1051-1056 (1997). The discussion of such issues is in the
appendix of this paper, although the
electron is not what is being discussed.
[v]
Feynman, et al, p. 28-4.
[vi]
Wolfgang Panofsky and Melba Phillips, Classical
Electricity and Magnetism (Addison-Wesley, Reading MA, 1955), p. 320.
[vii]
T. H. Boyer, p. 3250. Instead of using
the term “work”, Boyer says this extra energy is needed “to maintain the
validity of the force-momentum balance.”
[viii]
J.R. Oppenheimer, p. 477.
[ix]
P.A.M. Dirac, Directions in Physics,
ed. H. Hora and J.R. Shepanski (Wiley, New York, 1978) p. 37.
