The Theory of the Electron 
                       __________________________
             and the Relation Between Field and Particle
             ___________________________________________
                     in Classical Electrodynamics
                     ____________________________

                          Daniel Stubbs

                  University of Western Ontario 
                     London, Ontario, Canada

                     ds@look1.apmaths.uwo.ca

                    (received June 10, 1997)


Historical Background
_____________________
 
		Before Maxwell people thought of physical reality 
	--- insofar as it represented events in nature --- as 
	material points, whose changes consist only in motions which 
	are subject to total [i.e. ordinary] differential equations. 
	After Maxwell, they thought of physical reality as represented 
	by continuous fields, not mechanically explicable, which are 
	subject to partial differential equations

			--- Albert Einstein 
 
	The basic nature of Newtonian mechanics is a universe of particles, 
each of  which interacts with the others either through direct collision 
(as was the  view in mechanistic philosophies such as Descartes'), or by 
means of various  forces acting at a distance, of which the most famous 
example is of course  gravitation. Mathematically, this is made most 
evident in Newton's second law  of motion, which is an *ordinary* 
differential equation: 
                         ..        .     
                       m x  =  F ( x(t), x(t),t)
 
Newton remained all his life a passionate defender of the corpuscular, or 
particle-like, conception of light. Throughout the eighteenth century, the 
great prestige of Newton, as well as the observational evidence at hand, 
helped to ensure the dominance of this particle-like conception of physical 
reality --- the universe was a giant pool table, on which a finite collection 
of massive particles interacted, the whole system smoothly evolving through 
time.
 
	Beginning in the nineteenth century, though, the use of smooth quantites, 
with an uncountably infinite number of spatial degrees of freedom, became more 
prevalent, first as a calculational aid in such subjects as heat flow and 
fluid mechanics: no one really doubted the existence of many millions of tiny 
particles, but it was simply more convenient to work with *partial* 
differential equations for bulk macroscopic variables such as velocity or 
pressure than to try and formulate and solve many millions of ordinary 
differential equations. But with Faraday's proposed magnetic field, another 
occult force (albeit one that continued to obey an inverse square law, just 
like Newton's law of universal gravitation) seemed to have been introduced, a 
force for which, like gravitation, no corpuscular interpretation seemed ready 
at hand. Young's famous experiment exhibiting the interefence of light seemed 
to cast further doubt on the wisdom of Newton's corpuscular views, and by the 
late nineteenth century a minor revolution was underway in physics, as Maxwell 
proposed the world's first complete *field* theory, which sought to unite the 
magnetic and electric fields into one electromagnetic field, which existed 
even in the absence of sources of any kind, and seemed indeed to knock the 
corpuscular view of physics off its throne, to be replaced with a 
field-theoretic conception of the natural universe: space was inhabited by 
various fields, interacting with each other, and the "particles" we observe 
are really nothing other than local excitations of these fields.
 
	Then, of course, came Thomson's discovery of the electron in 1897; the 
old Newtonian view had found a new lease on life, but how could this "particle" 
be reconciled with Maxwell's field conception of electromagnetism? What was the 
appropriate equation for the motion of this new particle in the presence of an 
electromagnetic field? How did the field, as carrier of charge, interact with 
a charged *particle*, rather than a smoothly varying charge density or current 
vector, particularly if the particle was undergoing accelerated (i.e. 
noninertial) motion? Over the next fifteen years, these questions were to 
result in much controversy and discussion within theoretical physics, 
involving some of the most well-known names of that period: Lorentz, Einstein, 
Poincare and Dirac, to name only some of the most prominent.
 
The Lorentz Force and Equation of Motion
________________________________________
 
	The first main requirement for obtaining an equation of motion for a 
charged particle in the presence of an electromagnetic field is a force law, 
to whose value we may equate the product of the mass and the acceleration of 
the point charge. Such a force law was first proposed by Lorentz, and can be 
expressed as (in units where c = e = 1, which we employ throughout this section)
 
                        F = q [ E + v * B]
 
where `E' is the electric field vector, `B' that of the magnetic field, `v' 
and `q' are the velocity and charge of the particle. Lorentz had hoped to give 
a purely electromagnetic model of the electron (see [1], Note 18, p. 252), but 
as Poincare later pointed out, such a purely electromagnetic charged particle 
was impossible because classically, a charge distribution by itself is 
unstable. The Newtonian equation can then be applied to give us the equation 
of motion
                        ..          .
                      m x = q [ E + x * B]
 
	But in writing down this equation we have neglected a crucial new 
development that emerges from the duality of our conceptions of field and 
particle: the particle itself carries with it an electromagnetic field of its 
own, which will contribute to the expressions for `E'  and `B', and the 
effects of these new electromagnetic fields, both external ones and the 
"self-field" of the particle, travel at a *finite~ velocity `c', rather than 
the infinitely quick action-at-a-distance theory of Newtonian gravitation. 
Among other things, we know that this maximum velocity forces us to include 
relativistic effects in our equations, specifically with the mass term, which 
is transformed from `m' to  `m_0 gamma(v)', where gamma(v) is the relativistic 
Lorentz factor.
 
	Because of this, our naive attempt at writing down an equation of 
motion cannot be complete; to take account of this duality between field and 
particle, we will have to couple this Lorentz force equation to the Maxwell 
equations describing the evolution of the electromagnetic field itself. The 
resulting system has the form
 

		d E
		___  	=  del * B - 4 pi (J_ext + J_self)
		d t

		d B
		___ 	=  - del * E
		d t

		del . E	= 4 pi (rho_ext + rho_self)

		del . B = 0

		d         .  .       q        .
		-- (gamma(z) z )  = --- ( E + z * B )      .
		d t                  m


Here, `rho_ext' and `J_ext' are the charge density and current vector arising 
from the external fields, while the charge density and current vector for the 
self-field take the form of scalar-valued and vector-valued distributions.
 
	Given expressions for `rho_ext(x)', `J_ext(x)'  and appropriate 
initial and boundary data, we should then be in a position to solve the 
coupled Lorentz-Maxwell equations. The direct solution of these equations, 
though, is extremely difficult, so we will exploit a knowledge of the retarded 
field of a point charge to find what we hope will be the correct equation of 
motion of a point charge in classical electrodynamics. This retarded field, 
the Lienard-Wiechert field, can be written as (in a frame in which the 
particle is at rest at the origin at the retarded time)
 
                  n       w                          w * n
	E_ret = q ___ - q ___   and       B_ret = q _________
                 r^2      r                            r
 
where `n' is a unit vector in the direction of the field point, `w' is the 
component of the acceleration vector orthogonal to `n' (and `v') and `r' is 
the distance from the origin to the field point. The Poynting vector  (Jackson 
[2] equ 6.109)  for this field can be easily calculated to be just
 
			       q^2
			P = _________  | w |^2 n   +  O(1/r^3)
			    4 pi r^2 

giving a total energy outflow through a retarded sphere of
			                      .
			E_out  = - (2/3) q^2 |v|^2
  
the Larmor radiation law. Now that we understand how much energy is being 
radiated by the electron as it moves, we can (hopefully) correct our original 
Lorentz equation to take account of this energy loss.
 
	So, starting with 
                          .
			m v = F_ext + F_rad
 
and following the presentation in Jackson [2], pp. 783-4, we can show that the 
equation of motion of the electron assumes the unusual form
			    .           ..
			m ( v - ___ q^2 v )  =  q F_ext  .	(equ 1)
 
	This purely classical equation is called the Abraham-Lorentz equation, 
after the two men who were largely responsible for its derivation. The first, 
striking, fact about this equation is that it is *third* order, and so 
requires the specification of not only the initial position and velocity of 
our hypothetical point particle, but also of its initial acceleration. There 
are other problems: let us suppose that F_ext = 0, then there exist two 
solutions to this equation, the obvious one
                .                 .           3 m t
 		v = 0     and     v = a  exp( _____ )
		                              2 q^2
 
where `a' is the initial acceleration, a solution which diverges as 
t->infinity ! We can avoid these unwelcome "runaway" solutions by replacing 
this equation with an integro-differential equation by means of an integrating 
factor: let epsilon = 2q^2/(3m) and multiply both sides of (equ 1) by 
exp(-t/epsilon)/(epsilon m), yielding

		d    .                       3
		___ (v exp(-t/epsilon)) = - ___ F_ext      .
		d t                         2 q
  
Integrating both sides gives us
                              infinity 
          3                   /
  x(t) = ___ exp(-t/epsilon)  |  F_ext ( x(s) ) exp(-s/epsilon) ds (equ 2)
         2 q                  /
                             t

which is equivalent to the original Abraham-Lorentz equation under the 
condition that           .
	lim(t->infinity) v(t) exp(-t/epsilon) = 0   .
 
We note here that nothing in this reduction of (equ 1) to (equ 2) 
assures us that this limit will be well-behaved, though we assumed 
this to vanish in order to derive (equ 1) in the first place.
Another problem with our integro-differential equation is that 
*pre-acceleration*: the acceleration of the particle at time `t' in (equ(2) 
seems to depend on the value of the external force at times  s > t , i.e. at 
*later* times --- the particle can begin to accelerate before the external 
field is applied! An *ad hoc* dismissal of this problem can be made by arguing 
that these violations of causality should occur over time intervals of 
approximate duration `epsilon', due to the damping factor e^(-s/epsilon) in 
the integrand, and for such obvious examples of point particles as the 
electron, `epsilon' is of the order of magnitude of 10^(-23) seconds, ruling 
out any hope of experimental observation. Moreover, when dealing with external 
forces turned on over an interval of length `epsilon', there will be an energy 
uncertainty of `Delta-E ~ h-bar / epsilon' according to the Heisenberg 
Uncertainty Principle; if this energy uncertainty is of the order of the 
particle's rest mass `m c^2', then the motion of the particle will in fact be 
far from the domain of validity of classical mechanics. Thus many authors 
argue that the whole issue of the electron's motion can only be properly dealt 
with in quantum electrodynamics (QED), but the logical and philosophical 
foundations of QED are, if anything, even murkier and less well understood 
than those of classical electrodynamics.
 
	It might be objected that had we done this calculation in a fully 
relativistic manner, the equation thus obtained would not suffer from the 
defects of the Abraham-Lorentz equation. In fact, such is not the case as the 
relativistic calculation (see [3], pp. 136-43) adds only one additional term 
and leads to the Lorentz-Dirac equation,
	                           infinity   
	                              /       3
	a_mu(tau) = exp(tau/epsilon)  |   [  ___ F_mu^ext( u(s) ) 
	                              /      2 q
                                     tau      

		+ a_nu(s) a^nu(s) u_mu(s) exp(-s/epsilon) ] ds
 
where `tau' is the proper time for the particle, `u_mu' is its four-velocity, 
`a_mu' the four-acceleration and indices are raised and lowered using the 
standard Minkowski metric. This equation too suffers from the violation of 
micro-causality, and despite having been the object of much study over the 
past fifty-nine years since Dirac derived it (see [4]), very little of a 
definitive nature is known about its solutions. All that is known for certain 
is that some solutions can behave in ways that seem very counter-intuitive: 
for instance, let us suppose we have two particles in symmetric straight-line 
motion, that is
 		u_1^in = gamma(v) (1,v)  and   u_2^in = gamma(v) (1,-v)  
 
with the only fields present being those of the retarded fields of the two 
particles themselves. If the two particles have equal but opposite charges and 
`v_in = 0', it can be proved that they never collide and that they end up 
travelling in opposite directions with velocities asymptotic to that of light 
(see [5]).
 
	It can also be shown that this same problem of point electrodynamics 
also leads to strange behaviour in the context of curved spacetime, where the 
usual Lorentz-Dirac equation seems to lead to violations of the strong 
equivalence principle. The equation of motion for free fall of a charged test 
particle in a curved spacetime assumes the form
 
	m a^mu  = Gamma^mu  - (1/3) v^alpha  (  R_alpha^mu  + 

			(1/c^2) R_alpha_beta  v^beta v^mu  )  + phi^mu
	 
	This equation clearly depends on the mass of the particle, and so the 
strong principle of equivalence (footnote 1) is *not* valid for charged 
particles in true gravitational fields (by true, we mean gravitational fields 
caused by genuine spacetime curvature, so that `phi^mu not= 0' and 
`R__mu_nu not= 0'. It is, however, true for charged particles in apparent 
gravitational fields since then the only solution of the equation consistent 
with the boundary condition at infinity is $\ddot z^\mu =0$. Rohrlich observes:
 
		One can speculate about the consequence of this 
	result. [...] Now we are faced with the fact that charged 
	particles satisfy Einstein's principle (i.e. the weak equivalence 
	principle) but not the principle (E) (i.e. the strong principle). 
	Does this imply that a geometrization of electromagnetic interactions 
	is not possible? This conclusion is certainly suggested. The 
	consistent lack of success of the many attempts at a fully unified 
	field theory seems to point in the same direction. It may well 
	be that electromagnetic interactions have much more in common with 
	strong and weak interactions than with gravitational ones and will 
	correspondingly find their next level of physical theory aligned 
	with those interactions.[6] 
 
	Once again, we see the rather mysterious problems that arise out of 
the field/particle duality that occurs not only in classical electrodynamics 
but also in the other great field theory of classical physics, the general 
theory of relativity. It has long been known that Huygens' Principle can fail 
to hold in general Riemannian spaces, a failure which manifests itself in the 
"tail-term" `phi^mu' for the equation of motion, and it would seem that a 
charged particle also seems to contradict the strong equivalence principle, 
though once again no laboratory measurements have ever been able to measure 
such violations due to the weakness of gravitational fields on Earth, 
particularly with respect to electrons.
 
Conclusions
___________
 
	From the examples given in the preceeding section, one can see clearly 
that the relationship between fields, the archetype of twentieth century 
physical thought, and particles, has always been a little mysterious and 
controversial, and often inexplicable. A great deal remains unknown, a hundred 
years after Thomson first discovered the electron, the particle whose 
existence first begged the question as to how a charged point particle 
interacts with a continuous field, either an electromagnetic one, or even the 
"geometric" fields used by Einstein in constructing general relativity. Even 
today, when the electron (along with the other leptons, the muon and tauon) 
continues to exhibit no internal structure down to 10^(-15) m, reinforcing our 
conviction in its essentially point-like nature, no general existence and 
uniqueness proof exists for the Lorentz-Dirac integrodifferential equation, 
much less one assuring us of continuous dependence on initial data and other 
features we would expect in a full and complete physical theory. It was 
Dirac's belief that until we had a consistent and coherent classical theory of 
point charges, any attempt at quantization was doomed to merely reproduce, or 
even magnify, the flaws of the original classical theory, though others have 
argued that the entire notion of a point particle is, in the context of 
twentieth century physics with its uncertainty principles between position and 
momentum and energy and time, untenable and ought to be banished from physics. 
Yet the electron persists in its failure to exhibit any internal structure, 
and there remains no coherent theory of the electron as an extended entity. At 
present, the tension between particle and field in modern physics continues to 
push us towards a better understanding of the nature of matter and energy and 
which, if either, representation of physical reality is the more fundamental.
 
Footnotes
_________
 
1. The equations of motion of a non-rotating test body in free fall in a 
gravitational field are independent of the energy content of that body (here, 
energy content simply means *inertial mass*, since this is what is 
experimentally measured).
 
References
___________
 
[1] Lorentz, H. A. "The Theory of Electrons", 2nd edition. New York: Dover, 
1952
 
[2] Jackson, J. "Classical Electrodynamics", 2nd edition. New York: John Wiley 
and Sons, 1975
 
[3] Parrott, S. "Relativistic Electrodynamics and Differential Geometry". New 
York: Springer-Verlag, 1987
 
[4] Dirac, P. , Proc. Roy. Soc. London, A167, p. 148 (1938)
 
[5] Eliezer, C., Proc. Camb. Phil. Soc., 39, p. 173 (1943)
 
[6] Rohrlich, F. "Classical Charged Particles", p. 223, Reading, MA,  
Addison-Wesley.