 This is Quantum Field Theory 5. Welcome. In this video, we will consider some results from classical electrodynamics that play a direct role in, or have counterparts in, quantum electrodynamics. First up is the phenomenon of the electron's self-force, which is intimately connected with the structure of the electron. At the end of his seminal 1932 paper, Fermi wrote, in conclusion we may therefore say that practically all the problems in radiation theory, which do not involve the structure of the electron, have their satisfactory explanation. While the problems connected with the internal properties of the electron are still very far from their solution, these problems took the form of troubling infinities that kept showing up in calculations and seemed to threaten the very foundations of quantum field theory for years to come. Let's start by asking, what does an electron look like? We know it has a certain mass and carries a certain amount of electric charge. By symmetry, we might guess it's a sphere of some radius R with the charge uniformly distributed throughout its volume. Or, since like charges repel, maybe the charge is concentrated in the spherical shell of radius R, such as happens when a conducting sphere is charged. In any case, if the electron is a distribution of electric charge, the pieces should exert forces on each other. Consider any two patches on the surface of the shell model. These charges will exert equal magnitude but oppositely directed forces on each other. While the net force on the entire electron is zero, each patch of the surface will feel a radial force trying to push it off to infinity. This immediately raises the question, what holds the pieces together? And, if the pieces are held together by forces in equilibrium, then can they vibrate? That is, can the electron be excited into various oscillation modes? Along the same lines, can we add or subtract pieces? Can we form a half electron? The modern response to these types of questions seems to have been first put forward by Frankel in 1925. The electron is elementary and has no substructure. Now, what geometric object has no substructure? A point. So the electron is a point particle, a point charge. We will discuss some of the reasons for this conclusion later. For now, let's continue thinking about the possible substructure of the electron. Analyzing a spherical distribution of charge leads to difficult calculations. We will instead look at a toy problem. A dumbbell electron, consisting of two parts, separated by distance d along the z-axis. Each part is a half electron, a point mass with charge q, half of the total electron charge. Part one creates an electric field E1. The electron charge is negative, but we will treat it as positive since then the electric field points in the same direction as the resulting force. Our final result will depend only on the square of the charge, so it won't matter whether it's positive or negative. Part two creates an electric field E2. At the positions of the other parts, these fields have equal magnitude but opposite direction. Therefore, while each part will feel a force of repulsion trying to tear the electron apart, there is no net force on the electron as a whole. If we move to the right of part one, we are moving closer to part two, so field E2 is stronger. If we move to the right of part two, we are moving farther from part one, so field E1 is weaker. Now imagine the fields are produced by the charges at their original positions, Z1r and Z2r. Then, without changing the fields, the charges are moved rightward to positions Z1 and Z2. Assume each part only sees the field produced by the other part. The result would be that part one would feel a stronger force toward the left and part two would feel toward the right. And there would be a net leftward force on the entire electron. Could such a process occur? Well, imagine the distance D is one light second, roughly the distance between Earth and Moon. The electron has been at rest for a long time with the pieces at positions Z1r and Z2r, what we call the retarded positions. Then the electron starts moving to the right. As the charges move, the fields they produce will change. However, these changes do not occur instantaneously throughout space, but propagate at the speed of light. So for about a second, the field E1 seen by part two and the field E2 seen by part one will continue to be those produced at the retarded positions. And this net force process will indeed occur. This will be true for any finite value of D, although the time scale will scale accordingly. This is the same phenomenon that leads us to say that the stars we see in the sky appear not as they are now, but as they were in the past, when the light entering our eyes was emitted. So each piece of an electron sees the other pieces, not as they are now, but as they were some very short time in the past. This is the source of the electron self-force. If the electric field magnitude is charge over distance squared, and if each part has charge Q, E1 is Q over Z2 minus Z1r squared, and E2 is minus Q over Z1 minus Z2r squared. The minus sign indicates the field points to the left in the minus z direction. According to electromagnetic theory, if the charges were in motion when the fields were produced, the field magnitudes are modified by the velocity factors shown here, where V1r and V2r are the velocities of the parts when they were at the retarded positions Z1r and Z2r. Notice that if the electron had been moving to the right, so both velocities are positive, the V1r factor will increase the magnitude of V1, and the V2r factor will decrease the magnitude of V2. If we use units in which the speed of light is 1, then a distance such as Z2 minus Z1r equals the time it takes light to travel that distance, call it tau 1. So, we replace the distances squared in the denominators with time squared, tau 1 squared, and tau 2 squared. Let's call z of t the position of the center of the electron, with z at time 0 equal to 0. Then part 1 is located at z1 equals z of t minus r, where the radius r is half of the diameter d, and part 2 is located at z2 equals z of t plus r. The trajectories of the two parts are identical curves offset vertically by d. At time t, part 2 will see the field produced by part 1 at a time tau 1 in the past. The corresponding distance, z2 of t minus z1 of t minus tau 1 equals the time delay tau 1. We can understand this graphically using our trajectory curves. In our units, the speed of light is 1. So starting at a point on the z2 curve, we travel back and down at a slope of 1, an angle of 45 degrees, until we hit the z1 curve. The horizontal offset is tau 1. The vertical offset is z2 of t minus z1 of t minus tau 1. Substituting the z1 and z2 expressions, we have tau 1 equals d plus z of t minus z of t minus tau 1. At a time t, part 1 will see the field produced by part 2 at a time tau 2 in the past. The corresponding distance, z2 of t minus tau 2 minus z1 of t equals the time delay tau 2. Starting at a point on the z1 curve, we travel back and up at 45 degrees until we hit the z2 curve. The horizontal offset is tau 2. The vertical offset is z2 of t minus tau 2 minus z1 of t. And we find tau 2 equals d plus z of t minus tau 2 minus z of t. For simplicity, let's take t equals 0. Then tau 1 equals d minus z of minus tau 1 and tau 2 equals d plus z of minus tau 2. Let's evaluate the self-force for the case of uniform motion z of t equals vt. Our tau expression reads tau 1 equals d plus v tau 1 and tau 2 equals d minus v tau 2. We can solve these for tau 1 equals d over 1 minus v and tau 2 equals d over 1 plus v. Substituting this into our expression for e1 and canceling a common factor of 1 minus v, we get q over d squared times 1 minus v squared. For e2, we get minus q over d squared 1 minus v squared. Since e1 plus e2 equals 0, there is no self-force in this case. This is great news because it's required by special relativity, according to which there is no physical way to establish an absolute state of motion. If there was an electron self-force which varied with velocity relative to some reference frame, then that frame could be taken to represent absolute rest in violation of special relativity. Constant velocity produces no self-force, but let's look at constant acceleration. z of t equals 1 half a t squared. The time derivative of this is the instantaneous velocity v of t equals a t. Our tau expressions now read tau 1 equals d minus 1 half a tau 1 squared and tau 2 equals d plus 1 half a tau 2 squared. And our electric field expressions are e1 equals q over tau 1 squared 1 plus minus a tau 1 over 1 minus minus a tau 1. e2 equals minus q over tau 2 squared 1 minus minus a tau 2 over 1 plus minus a tau 2. We will do the calculations using the open source computer algebra system Maxima. We solve the tau equations iteratively to sufficient accuracy for our purposes. First set tau 1 equal to d, then substitute this value into the equation tau 1 equals d minus 1 half a tau 1 squared. Do the same for tau 2. Then calculate the fields e1 and e2 and finally compute the net force f equals qe1 plus qe2. This is a messy expression, but we can have the computer expand the expression for f in a so-called Taylor series in the variable d about the value d equals 0 and up to the minus 1 power of d. The result is minus 2 aq squared over d plus terms and higher powers of d. Writing 1 over r in place of 2 over d, the self-force for an accelerating dumbbell electron is minus q squared over r times a. Due to the r in the denominator, if we let r go to 0, corresponding to the dumbbell electron collapsing to a single point charge, the self-force becomes minus infinity times a, an infinite force that opposes acceleration, so no point charge could ever accelerate. Obviously, charged particles can accelerate, so apparently we can rule out the existence of point charges.