 We have been calculating the self-force of a simple dumbbell model of the electron, consisting of two-point charges Q separated by a distance d equal to 2r. For uniform velocity, we found the self-force vanished as required by special relativity. For uniform acceleration, we found a self-force minus Q squared over r times acceleration. If we let r go to zero so our model shrinks to a single point charge, this force becomes infinite and it would be impossible to accelerate our electron. This seems to imply that the electron is not a point charge since electrons most definitely can be accelerated. We now continue our analysis of the self-force. Let's add another term to z of t, 1 over 6 times a dot t cubed. The derivative of position is the velocity at plus one half a dot t squared and the derivative of velocity is the acceleration a plus a dot t, a linearly varying acceleration with the parameter a dot giving the accelerations rate of change. Our previous tau expressions are modified by the addition of a plus 1 over 6 a dot tau 1 cubed term in the first case and a minus 1 over 6 a dot tau 2 cubed term in the second case. And our electric field expressions have additional a dot tau 1 or 2 squared over 2 terms in the velocities. Again, we use maxima for the calculations and we write a sub d for a dot. The Taylor series for the force expression to the zero power in d is our previous expression minus 2 a q squared over d plus 4 over 3 a dot q squared. So the self-force is minus q squared over r times a plus 4 over 3 a dot q squared. We still have 1 over r dependence in the acceleration term, but the a dot term is independent of r. Now let's see how we can deal with the 1 over r factor in the self-force expression. We will be led to perform a so-called renormalization of the electrons mass. This will be for the classical case. Later, we will consider the analogous process required in the quantum case. For a spherical shell electron model, the type of analysis we have performed is much more complicated but can be carried out. The result for the self-force is minus 2 over 3 e squared over r times a plus 2 over 3 e squared times a dot plus terms with positive powers of r. We assume r is small enough that these last terms can be neglected. Here minus e is the electron charge and we are using units where the speed of light is 1. a is acceleration and a dot is quite appropriately called jerk. Thinking of force equals mass times acceleration, a constant force results in a constant acceleration. A time varying force results in a time varying acceleration, an a dot term. To jerk something around is to push and pull it, to subject it to a time varying force. When, around 1900, researchers such as Max Abraham and Hendrick Lawrence derived these results, it was very satisfying because the jerk term precisely describes the phenomenon of radiation reaction which is required if energy is to be conserved by a radiating charged body. Here is the electric field of a point charge rendered in false color with black the weakest field strength up through blue, green, red and with white the strongest. If we jerk the charge up and down, electromagnetic waves are created, which peel off and propagate to infinity. Here is a variation where we jerk the charge around a circular path. In any case, electromagnetic radiation carries energy away from the charge. Where does this energy come from? Let's look at the circular path case. If the circle has radius r and the charge moves with angular velocity omega, the charge has x, y coordinates, r cosine omega t and r sine omega t. The velocity is the time rate of change of position. The acceleration is the time rate of change of velocity and the jerk is the time rate of change of acceleration. Acceleration is minus omega squared times position and points from the charge toward the center of the circle. To keep the charge moving along the circle requires a force pointing in this direction. This could be produced by an opposite charge at the circle's center, as in the planetary or Rutherford model of the hydrogen atom. Or it could be produced by a magnetic field perpendicular to the screen, as in a cyclotron particle accelerator. Jerk is minus omega squared times velocity. This is anti-parallel to the direction of motion, and this force component will reduce the charge's velocity and kinetic energy. The loss of kinetic energy precisely equals the energy carried away by the radiation. For the charge to maintain a constant speed, we would have to apply a force parallel to the velocity to cancel this component of the self-force. The supplied force would do work as the charge follows its circular path, and that energy would be transformed into the energy of the radiated field. So the jerk term of the self-force makes complete sense physically, and notice that it does not depend on the radius r. In fact, it does not depend on the details of the charge distribution, uniform sphere, spherical shell, etc., only on the total charge. Now let's look at the acceleration term. Since force equals mass times acceleration, the factor in parentheses can be interpreted as a mass, and we call this the electromagnetic mass. Writing Newton's second law of motion, mass times acceleration equals force, we have M0A equals minus the electromagnetic mass times A, plus the radiation reaction force, plus any additional external forces applied to the electron. We call M0 the bare mass. It's the intrinsic mechanical mass of the electron. The logical thing to do at this point is to combine the two acceleration terms, so the equation reads bare mass plus electromagnetic mass times acceleration equals radiation reaction plus external force. Now the coefficient of acceleration in the force law must be, by definition, the observed mass of the electron. So, apparently the mass of a charged particle has two contributions, the mechanical mass, whatever that is, and the electromagnetic mass, due to the corresponding term in the self-force expression. Now, note that due to the 1 over r dependence of the electromagnetic mass, by appropriate choice of the radius, the electromagnetic mass expression can take on any positive value from 0 to infinity, and this was, at least initially, another very satisfying result. Because it suggested that the radius might be such that the observed mass is nothing but the electromagnetic mass, that is, there is no additional bare mass component. The value that gives this result is about 2.8 femtometers, called the classical electron radius. This was, at least initially, an amazing result. The mysterious inertial property of all matter that we call mass might simply be the electromagnetic self-force effect. And, since mass is the source of the gravitational field, maybe gravity is an electromagnetic effect. Maybe electromagnetism explains all physical phenomena, but there are some problems with this idea. As we've already mentioned, if an electron is a distribution of negative charge, what holds this charge together and keeps the electron from flying apart due to the electrical repulsion between like charges? We have to assume some non-electromagnetic attractive force that counteracts the electrical repulsion, and that ruins the beauty of the everything-is-electromagnetism idea. Moreover, experiments have set an upper bound for the electron's radius, that is, orders of magnitude smaller than the classical electron radius. This would give an electromagnetic mass orders of magnitude larger than the observed mass. So, we might as well treat the electron as an elementary particle, with no substructure, a point particle. But if we let the electron shrink to a point, the electromagnetic mass goes to infinity, then how could the electron have a finite observed mass? Well, we know it does, so let's assume that m equals the bare mass plus the electromagnetic mass remains finite and equal to the observed mass. But wait, if the electromagnetic mass goes to infinity, doesn't that mean that the bare mass must go to minus infinity so that their sum remains finite? What does that even mean? Okay, how about this? We replace the coefficient of acceleration with a black box. Any mass factors from whatever source get thrown into this black box and then forgotten. At the end, we call the total contents of the black box the observed mass of the electron. That is the basic idea of mass renormalization, and it works. It allows us to treat the electron as a point charge with finite mass, while keeping the necessary radiation reaction term. If you think that there has to be a better way to resolve this problem, let me point out that people have spent about a century trying to fix it, yet it remains with us to this day. Here are some excerpts discussing this issue in the Feynman Lectures on Physics. Feynman notes that many physicists hope that it might be possible to modify Maxwell's theory of electrodynamics so that the idea of an electron as a simple point charge could be maintained. However, many attempts have been made and some of the theories were even able to arrange things so that all of the electron mass was electromagnetic, but all of these theories have died. In spite of this, he was hoped that it could turn out that in quantum electrodynamics, the difficulties will disappear or may be resolved in some other fashion, but the difficulties do not disappear in quantum electrodynamics. That is one of the reasons that people have spent so much effort trying to straighten out the classical difficulties, hoping that if they could straighten out the classical difficulty and then make the quantum modifications, everything would be straightened out. It turns out, however, that nobody has ever succeeded in making a self-consistent quantum theory out of any of the modified theories. So, after more than 100 years, the electron radius and mass problem is still with us. If we want to make progress in quantum electrodynamics, it looks like we are going to have to accept the mass renormalization workaround and move onward. If you find this process a bit unsettling, sweeping infinities into a black box and calling the net result finite, you are in good company. Here is Durak's opinion. I am very dissatisfied with the situation because this so-called good theory does involve neglecting infinities which appear in its equations. Sensible mathematics involves neglecting a quantity when it turns out to be small, not neglecting it just because it is infinitely great and you do not want it. In Feynman's opinion, the shell game that we play is technically called renormalization. It is what I would call a dippy process. Having to resort to such hocus pocus has prevented us from proving that the theory of quantum electrodynamics is mathematically self-consistent. Such a bunch of words is not good mathematics. On the other hand, renormalization has allowed quantum electrodynamics to be formulated in a manner that enables accurate predictions to be made which have been experimentally confirmed to high levels of precision. A more generous, arguably more modern view of renormalization was stated by Steven Weinberg. Taking account of the difference between the bare charge and mass of the electron and their measured values is not merely a trick that is invented to get rid of infinities. It is simply a matter of correctly identifying what we are actually measuring in laboratory measurements of the electron's mass and charge.