(This is my master's thesis in physics, Southwest Texas State University, 1998. SWT is now called Texas State University-San Marcos. My thesis, which is called The Binding Energy of the Classical Electron, is more history and its interpretation than actual physics. This blog version is being edited by me as I have time. At the moment it doesn't have the equations that are supposed to go with it. I hope to include them after I figure out how to do that!)
INTRODUCTION
In the 100 years since the discovery
of the electron, no one has produced a satisfactory answer to the most
elementary questions about this first inhabitant of what has been called[i]
the subatomic particle zoo. Although it
has been defined by precise measurements of its charge, charge-to-mass ratio,
and magnetic moment, and even though the experimental value of magnetic moment
fits very snugly into the calculational confines of quantum electrodynamics,
the electron is still a mystery. Why
does it have the charge and mass it has?
Why is the charge precisely the same as the proton’s charge (but
opposite in sign), and why do the electron and proton have the charge-to-mass
ratios they have? Until these conceptual
questions are answered and the mass and charge of the electron are predicted by
a theory, rather than inserted as parameters in quantum electrodynamics as they
are now, we cannot say we understand what an electron is.
The discovery of the
electron in 1897 was accomplished by J. J. Thomson’s experimental determination
of the charge-to-mass ratio for the formerly unexplained cathode rays, showing
that the “rays” are actually streams of charged particles. Several kinematical or dynamical similarities
between charge and mass indicate the charge-to-mass ratio is of fundamental
importance in the development of a theory of the electron: Point charges and point masses produce
inverse square fields, charge and mass are both relativistic invariants
(scalars)--although mass is often improperly described as varying with
velocity,[ii]
and charge and mass both resist being accelerated. Mass resists acceleration because of its
mysterious property called inertia.
Charge resists acceleration because it creates a time-dependent
electromagnetic field when accelerated and, in conserving momentum, the field
reacts back on the charge with a force called the radiation reaction
force.
The radiation reaction phenomenon has been chosen as the
subject of this thesis because the difficulties which arose in classical models
of the electron’s radiation reaction have not been solved by quantum
electrodynamics. The phenomenon of radiation reaction also represents a partial
unification of the ideas of charge and mass, because the radiation reaction force
causes an electron to act as if it has an electromagnetic mass in addition to
its bare or mechanical mass. This
apparent combination of two kinds of inertial mass--and their apparent
inseparability--is a problem “we still don’t understand,” as Pais says.[iii]
In classical
electrodynamics, the Maxwell equations are used to calculate fields when
charges and currents are specified, or fields can be given and the
distributions of charge and current calculated.
Forces on charge and current distributions can also be calculated from
the Lorentz force equation. As Jackson[iv]
notes, however, both regimes usually ignore the radiation reaction effect,
partly because there is “negligible error” when acceleration of charges is not
extremely rapid, and partly because “a completely satisfactory treatment of the
reactive effects of radiation does not exist.”
Although quantum electrodynamics has been successful in
giving precise calculations of radiative effects, it fails to put such
calculations on a sound theoretical foundation.
As Jackson[v]
says:
The
difficulties presented by this problem touch one of the most fundamental
aspects of physics, the nature of an elementary particle....One might hope that
the transition from classical to quantum-mechanical treatments would remove the
difficulties....It is one of the triumphs of comparatively recent years
(1948-1950) that the concepts of Lorentz covariance and gauge invariance were
exploited sufficiently cleverly to circumvent these difficulties in quantum
electrodynamics and so allow the calculation of very small radiative effects to
extremely high precision, in full agreement with experiment. From a fundamental point of view, however,
the difficulties still remain.
One very significant domain where accelerations of a charge
are rapid enough that the radiation reaction force cannot be neglected is in
classical and quantum models of light emission and absorption by atoms. Because of the oscillatory nature of the
electron’s motion in this case, radiation reaction is called radiation
damping. In the classical model of light
emission and absorption, the electron is bound to the nucleus by a linear
(Hooke’s law) restoring force, and there is a velocity-dependent radiation
damping term in the equation of motion of the electron. However, this velocity-dependent term is just
the simple harmonic motion approximation to the more general case where
radiation damping is proportional to
, the rate of change of acceleration.
The need for a rate of change of acceleration in a
single-particle force law is the distinguishing characteristic of radiation
reaction. The presence of
in the electron’s
equation of motion--the classical Abraham-Lorentz equation--leads to physically
senseless solutions. In one case,
exponentially increasing “runaway” solutions are predicted. Choosing reasonable asymptotic conditions on
the electron’s acceleration and avoiding the runaway solutions leads to
“pre-acceleration,” the electron accelerating before an external force is applied. These problems are discussed
in Chapter 2 of this thesis.
Another difficulty of both classical and quantum
electrodynamic models of radiation reaction is that the electromagnetic mass of
a point electron is infinite. This
subject is discussed in Chapter 3, along with the so-called “4/3 problem”--a
disagreement between relativistic and purely electrodynamic predictions of the
electromagnetic mass of a spherical shell of charge. In Chapter 4, classical models of the point
charge electron and the extended spherical shell electron are discussed, and
the spherical shell model is shown to be free of the difficulties of runaway
solutions, pre-acceleration, and infinite electromagnetic mass.
In Chapter 5, Discussion and Conclusion, the 4/3 problem
is considered as logical consequence of
the classical electron requiring a relativistic binding energy, just as a
nucleus has a binding energy holding together its positively charged protons. The problem of predicting the observed
stability of the electron is also discussed.
This thesis seeks to put into perspective the classical
ideas of radiation reaction and electromagnetic mass, and to discuss how the
classical ideas might influence quantum electrodynamics and thereby improve our
understanding of the electron as an elementary particle.
[i]
Cindy Schwarz, A Tour of the Subatomic
Zoo: A Guide to Particle Physics ,
2nd ed. (American Institute of Physics, New York, 1997).
[ii]
Lev B. Okun, “The Concept of Mass,”
Physics Today, June 1989; Carl G. Adler, “Does mass really depend on
velocity, Dad?” Am. J. Phys. 55,
739-743 (1987).
[iii]
Abraham Pais, ‘Subtle is the Lord...’:
The Science and the Life of Albert Einstein
(Oxford University Press, New York, 1982), p.155.
[iv]
J. D. Jackson, Classical Electrodynamics,
2nd ed. (Wiley, New York, 1975), p. 781.
[v]
Ibid.
