Attempts to modify the Maxwell theory
We would like now to discuss how 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. Many attempts have been made, and some of the theories were even able to arrange things so that all the electron mass was electromagnetic. But all of these theories have died. It is still interesting to discuss some of the possibilities that have been suggested—to see the struggles of the human mind. We started out our theory of electricity by talking about the interaction of one charge with another. Then we made up a theory of these interacting charges and ended up with a field theory. We believe it so much that we allow it to tell us about the force of one part of an electron on another. Perhaps the entire difficulty is that electrons do not act on themselves; perhaps we are making too great an extrapolation from the interaction of separate electrons to the idea that an electron interacts with itself. Therefore some theories have been proposed in which the possibility that an electron acts on itself is ruled out. Then there is no longer the infinity due to the self-action. Also, there is no longer any electromagnetic mass associated with the particle; all the mass is back to being mechanical, but there are new difficulties in the theory. We must say immediately that such theories require a modification of the idea of the electromagnetic field. You remember we said at the start that the force on a particle at any point was determined by just two quantities—$\FLPE$ and $\FLPB$. If we abandon the “self-force” this can no longer be true, because if there is an electron in a certain place, the force isn’t given by the total $\FLPE$ and $\FLPB$, but by only those parts due to other charges. So we have to keep track always of how much of $\FLPE$ and $\FLPB$ is due to the charge on which you are calculating the force and how much is due to the other charges. This makes the theory much more elaborate, but it gets rid of the difficulty of the infinity. So we can, if we want to, say that there is no such thing as the electron acting upon itself, and throw away the whole set of forces in Eq. (28.9). However, we have then thrown away the baby with the bath!  Because the second term in Eq. (28.9), the term in $\dddot{x}$, is needed. That force does something very definite. If you throw it away, you’re in trouble again. When we accelerate a charge, it radiates electromagnetic waves, so it loses energy. Therefore, to accelerate a charge, we must require more force than is required to accelerate a neutral object of the same mass; otherwise energy wouldn’t be conserved. The rate at which we do work on an accelerating charge must be equal to the rate of loss of energy by radiation. We have talked about this effect before—it is called the radiation resistance. We still have to answer the question: Where does the extra force, against which we must do this work, come from? When a big antenna is radiating, the forces come from the influence of one part of the antenna current on another. For a single accelerating electron radiating into otherwise empty space, there would seem to be only one place the force could come from—the action of one part of the electron on another part. We found back in Chapter 32 of Vol. I that an oscillating charge radiates energy at the rate \begin{equation} \label{Eq:II:28:10} \ddt{W}{t}=\frac{2}{3}\,\frac{e^2(\ddot{x})^2}{c^3}. \end{equation}  Let’s see what we get for the rate of doing work on an electron against the bootstrap force of Eq. (28.9). The rate of work is the force times the velocity, or $F\dot{x}$: \begin{equation} \label{Eq:II:28:11} \ddt{W}{t}=\alpha\,\frac{e^2}{ac^2}\,\ddot{x}\dot{x}- \frac{2}{3}\,\frac{e^2}{c^3}\,\dddot{x}\dot{x}+\dotsb \end{equation}  The first term is proportional to $d\dot{x}^2/dt$, and therefore just corresponds to the rate of change of the kinetic energy $\tfrac{1}{2}mv^2$ associated with the electromagnetic mass. The second term should correspond to the radiated power in Eq. (28.10). But it is different. The discrepancy comes from the fact that the term in Eq. (28.11) is generally true, whereas Eq. (28.10) is right only for an oscillating charge. We can show that the two are equivalent if the motion of the charge is periodic. To do that, we rewrite the second term of Eq. (28.11) as \begin{equation*} -\frac{2}{3}\,\frac{e^2}{c^3}\,\ddt{}{t}(\dot{x}\ddot{x})+ \frac{2}{3}\,\frac{e^2}{c^3}(\ddot{x})^2, \end{equation*}  which is just an algebraic transformation. If the motion of the electron is periodic, the quantity $\dot{x}\ddot{x}$ returns periodically to the same value, so that if we take the average of its time derivative, we get zero. The second term, however, is always positive (it’s a square), so its average is also positive. This term gives the net work done and is just equal to Eq. (28.10). The term in $\dddot{x}$ of the bootstrap force is required in order to have energy conservation in radiating systems, and we can’t throw it away. It was, in fact, one of the triumphs of Lorentz to show that there is such a force and that it comes from the action of the electron on itself. We must believe in the idea of the action of the electron on itself, and we need the term in $\dddot{x}$. The problem is how we can get that term without getting the first term in Eq. (28.9), which gives all the trouble. We don’t know how. You see that the classical electron theory has pushed itself into a tight corner. There have been several other attempts to modify the laws in order to straighten the thing out. One way, proposed by Born and Infeld, is to change the Maxwell equations in a complicated way so that they are no longer linear. Then the electromagnetic energy and momentum can be made to come out finite. But the laws they suggest predict phenomena which have never been observed. Their theory also suffers from another difficulty we will come to later, which is common to all the attempts to avoid the troubles we have described. The following peculiar possibility was suggested by Dirac. He said: Let’s admit that an electron acts on itself through the second term in Eq. (28.9) but not through the first. He then had an ingenious idea for getting rid of one but not the other. Look, he said, we made a special assumption when we took only the retarded wave solutions of Maxwell’s equations; if we were to take the advanced waves instead, we would get something different. The formula for the self-force would be \begin{equation} \label{Eq:II:28:12} F=\alpha\,\frac{e^2}{ac^2}\,\ddot{x}+ \frac{2}{3}\,\frac{e^2}{c^3}\,\dddot{x}+ \gamma\,\frac{e^2a}{c^4}\,\ddddot{x}+\dotsb \end{equation}  This equation is just like Eq. (28.9) except for the sign of the second term—and some higher terms—of the series. [Changing from retarded to advanced waves is just changing the sign of the delay which, it is not hard to see, is equivalent to changing the sign of $t$ everywhere. The only effect on Eq. (28.9) is to change the sign of all the odd time derivatives.] So, Dirac said, let’s make the new rule that an electron acts on itself by one-half the difference of the retarded and advanced fields which it produces. The difference of Eqs. (28.9) and (28.12), divided by two, is then \begin{equation*} F=-\frac{2}{3}\,\frac{e^2}{c^3}\,\dddot{x}+ \text{higher terms}. \end{equation*}  In all the higher terms, the radius $a$ appears to some positive power in the numerator. Therefore, when we go to the limit of a point charge, we get only the one term—just what is needed. In this way, Dirac got the radiation resistance force and none of the inertial forces. There is no electromagnetic mass, and the classical theory is saved—but at the expense of an arbitrary assumption about the self-force. The arbitrariness of the extra assumption of Dirac was removed, to some extent at least, by Wheeler and Feynman, who proposed a still stranger theory. They suggest that point charges interact only with other charges, but that the interaction is half through the advanced and half through the retarded waves. It turns out, most surprisingly, that in most situations you won’t see any effects of the advanced waves, but they do have the effect of producing just the radiation reaction force. The radiation resistance is not due to the electron acting on itself, but from the following peculiar effect. When an electron is accelerated at the time $t$, it shakes all the other charges in the world at a later time $t'=t+r/c$ (where $r$ is the distance to the other charge), because of the retarded waves. But then these other charges react back on the original electron through their advanced waves, which will arrive at the time $t''$, equal to $t'$ minus $r/c$, which is, of course, just $t$. (They also react back with their retarded waves too, but that just corresponds to the normal “reflected” waves.) The combination of the advanced and retarded waves means that at the instant it is accelerated an oscillating charge feels a force from all the charges that are “going to” absorb its radiated waves. You see what tight knots people have gotten into in trying to get a theory of the electron! We’ll describe now still another kind of theory, to show the kind of things that people think of when they are stuck. This is another modification of the laws of electrodynamics, proposed by Bopp. You realize that once you decide to change the equations of electromagnetism you can start anywhere you want. You can change the force law for an electron, or you can change the Maxwell equations (as we saw in the examples we have described), or you can make a change somewhere else. One possibility is to change the formulas that give the potentials in terms of the charges and currents. One of our formulas has been that the potentials at some point are given by the current density (or charge) at each other point at an earlier time. Using our four-vector notation for the potentials, we write \begin{equation} \label{Eq:II:28:13} A_\mu(1,t)=\frac{1}{4\pi\epsO c^2} \int\frac{j_\mu(2,t-r_{12}/c)}{r_{12}}\,dV_2. \end{equation}  Bopp’s beautifully simple idea is that: Maybe the trouble is in the $1/r$ factor in the integral. Suppose we were to start out by assuming only that the potential at one point depends on the charge density at any other point as some function of the distance between the points, say as $f(r_{12})$. The total potential at point $(1)$ will then be given by the integral of $j_\mu$ times this function over all space: \begin{equation*} A_\mu(1,t)=\int j_\mu(2,t-r_{12}/c)f(r_{12})\,dV_2. \end{equation*}  That’s all. No differential equation, nothing else. Well, one more thing. We also ask that the result should be relativistically invariant. So by “distance” we should take the invariant “distance” between two points in space-time. This distance squared (within a sign which doesn’t matter) is  \begin{align} s_{12}^2&=c^2(t_1-t_2)^2-r_{12}^2\notag\\[3pt] \label{Eq:II:28:14} &=c^2(t_1-t_2)^2-(x_1-x_2)^2-(y_1-y_2)^2-(z_1-z_2)^2. \end{align} 
 \begin{align} s_{12}^2=c^2&(t_1-t_2)^2-r_{12}^2\notag\\[4pt] =c^2&(t_1-t_2)^2-(x_1-x_2)^2\notag\\ \label{Eq:II:28:14} &-\;(y_1-y_2)^2-(z_1-z_2)^2. \end{align}  So, for a relativistically invariant theory, we should take some function of the magnitude of $s_{12}$, or what is the same thing, some function of $s_{12}^2$. So Bopp’s theory is that \begin{equation} \label{Eq:II:28:15} A_\mu(1,t_1)=\int j_\mu(2,t_2)F(s_{12}^2)\,dV_2\,cdt_2. \end{equation}  (The integral must, of course, be over the four-dimensional volume $cdt_2\,dx_2\,dy_2\,dz_2$.) All that remains is to choose a suitable function for $F$. We assume only one thing about $F$—that it is very small except when its argument is near zero—so that a graph of $F$ would be a curve like the one in Fig. 28–4. It is a narrow spike with a finite area centered at $s^2=0$, and with a width which we can say is roughly $a^2$. We can say, crudely, that when we calculate the potential at point $(1)$, only those points $(2)$ produce any appreciable effect if $s_{12}^2=c^2(t_1-t_2)^2-r_{12}^2$ is within $\pm a^2$ of zero. We can indicate this by saying that $F$ is important only for \begin{equation} \label{Eq:II:28:16} s_{12}^2=c^2(t_1-t_2)^2-r_{12}^2\approx\pm a^2. \end{equation}  You can make it more mathematical if you want to, but that’s the idea. Now suppose that $a$ is very small in comparison with the size of ordinary objects like motors, generators, and the like so that for normal problems $r_{12}\gg a$. Then Eq. (28.16) says that charges contribute to the integral of Eq. (28.15) only when $t_1-t_2$ is in the small range \begin{equation*} c(t_1-t_2)\approx\sqrt{r_{12}^2\pm a^2}=r_{12} \sqrt{1\pm\frac{a^2}{r_{12}^2}}. \end{equation*}  Since $a^2/r_{12}^2\ll1$, the square root can be approximated by $1\pm a^2/2r_{12}^2$, so \begin{equation*} t_1-t_2=\frac{r_{12}}{c}\biggl(1\pm\frac{a^2}{2r_{12}^2}\biggr) =\frac{r_{12}}{c}\pm\frac{a^2}{2r_{12}c}. \end{equation*}   What is the significance? This result says that the only times $t_2$ that are important in the integral of $A_\mu$ are those which differ from the time $t_1$, at which we want the potential, by the delay $r_{12}/c$—with a negligible correction so long as $r_{12}\gg a$. In other words, this theory of Bopp approaches the Maxwell theory—so long as we are far away from any particular charge—in the sense that it gives the retarded wave effects. We can, in fact, see approximately what the integral of Eq. (28.15) is going to give. If we integrate first over $t_2$ from $-\infty$ to $+\infty$—keeping $r_{12}$ fixed—then $s_{12}^2$ is also going to go from $-\infty$ to $+\infty$. The integral will all come from $t_2$’s in a small interval of width $\Delta t_2=2\times a^2/2r_{12}c$, centered at $t_1-r_{12}/c$. Say that the function $F(s^2)$ has the value $K$ at $s^2=0$; then the integral over $t_2$ gives approximately $Kj_\mu\Delta t_2$, or \begin{equation*} \frac{Ka^2}{c}\,\frac{j_\mu}{r_{12}}. \end{equation*}  We should, of course, take the value of $j_\mu$ at $t_2=t_1-r_{12}/c$, so that Eq. (28.15) becomes \begin{equation*} A_\mu(1,t_1)=\frac{Ka^2}{c} \int\frac{j_\mu(2,t_1-r_{12}/c)}{r_{12}}\,dV_2. \end{equation*}  If we pick $K=1/4\pi\epsO ca^2$, we are right back to the retarded potential solution of Maxwell’s equations—including automatically the $1/r$ dependence! And it all came out of the simple proposition that the potential at one point in space-time depends on the current density at all other points in space-time, but with a weighting factor that is some narrow function of the four-dimensional distance between the two points. This theory again predicts a finite electromagnetic mass for the electron, and the energy and mass have the right relation for the relativity theory. They must, because the theory is relativistically invariant from the start, and everything seems to be all right. There is, however, one fundamental objection to this theory and to all the other theories we have described. All particles we know obey the laws of quantum mechanics, so a quantum-mechanical modification of electrodynamics has to be made. Light behaves like photons. It isn’t $100$ percent like the Maxwell theory. So the electrodynamic theory has to be changed. We have already mentioned that it might be a waste of time to work so hard to straighten out the classical theory, because 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. The Maxwell theory still has the difficulties after the quantum mechanics modifications are made.  The quantum effects do make some changes—the formula for the mass is modified, and Planck’s constant $\hbar$ appears—but the answer still comes out infinite unless you cut off an integration somehow—just as we had to stop the classical integrals at $r=a$. And the answers depend on how you stop the integrals. We cannot, unfortunately, demonstrate for you here that the difficulties are really basically the same, because we have developed so little of the theory of quantum mechanics and even less of quantum electrodynamics. So you must just take our word that the quantized theory of Maxwell’s electrodynamics gives an infinite mass for a point electron.  It turns out, however, that nobody has ever succeeded in making a self-consistent quantum theory out of any of the modified theories. Born and Infeld’s ideas have never been satisfactorily made into a quantum theory. The theories with the advanced and retarded waves of Dirac, or of Wheeler and Feynman, have never been made into a satisfactory quantum theory. The theory of Bopp has never been made into a satisfactory quantum theory. So today, there is no known solution to this problem. We do not know how to make a consistent theory—including the quantum mechanics—which does not produce an infinity for the self-energy of an electron, or any point charge. And at the same time, there is no satisfactory theory that describes a non-point charge. It’s an unsolved problem. In case you are deciding to rush off to make a theory in which the action of an electron on itself is completely removed, so that electromagnetic mass is no longer meaningful, and then to make a quantum theory of it, you should be warned that you are certain to be in trouble. There is definite experimental evidence of the existence of electromagnetic inertia—there is evidence that some of the mass of charged particles is electromagnetic in origin. It used to be said in the older books that since Nature will obviously not present us with two particles—one neutral and the other charged, but otherwise the same—we will never be able to tell how much of the mass is electromagnetic and how much is mechanical. But it turns out that Nature has been kind enough to present us with just such objects, so that by comparing the observed mass of the charged one with the observed mass of the neutral one, we can tell whether there is any electromagnetic mass. For example, there are the neutrons and protons. They interact with tremendous forces—the nuclear forces—whose origin is unknown. However, as we have already described, the nuclear forces have one remarkable property. So far as they are concerned, the neutron and proton are exactly the same. The nuclear forces between neutron and neutron, neutron and proton, and proton and proton are all identical as far as we can tell. Only the little electromagnetic forces are different; electrically the proton and neutron are as different as night and day. This is just what we wanted. There are two particles, identical from the point of view of the strong interactions, but different electrically. And they have a small difference in mass. The mass difference between the proton and the neutron—expressed as the difference in the rest-energy $mc^2$ in units of MeV—is about $1.3$ MeV, which is about $2.6$ times the electron mass. The classical theory would then predict a radius of about $\tfrac{1}{3}$ to $\tfrac{1}{2}$ the classical electron radius, or about $10^{-13}$ cm. Of course, one should really use the quantum theory, but by some strange accident, all the constants—$2\pi$’s and $\hbar$’s, etc.—come out so that the quantum theory gives roughly the same radius as the classical theory. The only trouble is that the sign is wrong! The neutron is heavier than the proton. Nature has also given us several other pairs—or triplets—of particles which appear to be exactly the same except for their electrical charge. They interact with protons and neutrons, through the so-called “strong” interactions of the nuclear forces. In such interactions, the particles of a given kind—say the $\pi$-mesons—behave in every way like one object except for their electrical charge. In Table 28–1 we give a list of such particles, together with their measured masses. The charged $\pi$-mesons—positive or negative—have a mass of $139.6$ MeV, but the neutral $\pi$-meson is $4.6$ MeV lighter. We believe that this mass difference is electromagnetic; it would correspond to a particle radius of $3$ to $4\times10^{-14}$ cm. You will see from the table that the mass differences of the other particles are usually of the same general size. Now the size of these particles can be determined by other methods, for instance by the diameters they appear to have in high-energy collisions. So the electromagnetic mass seems to be in general agreement with electromagnetic theory, if we stop our integrals of the field energy at the same radius obtained by these other methods. That’s why we believe that the differences do represent electromagnetic mass. You are no doubt worried about the different signs of the mass differences in the table. It is easy to see why the charged ones should be heavier than the neutral ones. But what about those pairs like the proton and the neutron, where the measured mass comes out the other way? Well, it turns out that these particles are complicated, and the computation of the electromagnetic mass must be more elaborate for them. For instance, although the neutron has no net charge, it does have a charge distribution inside it—it is only the net charge that is zero. In fact, we believe that the neutron looks—at least sometimes—like a proton with a negative $\pi$-meson in a “cloud” around it, as shown in Fig. 28–5. Although the neutron is “neutral,” because its total charge is zero, there are still electromagnetic energies (for example, it has a magnetic moment), so it’s not easy to tell the sign of the electromagnetic mass difference without a detailed theory of the internal structure.  We only wish to emphasize here the following points: (1) the electromagnetic theory predicts the existence of an electromagnetic mass, but it also falls on its face in doing so, because it does not produce a consistent theory—and the same is true with the quantum modifications; (2) there is experimental evidence for the existence of electromagnetic mass; and (3) all these masses are roughly the same as the mass of an electron. So we come back again to the original idea of Lorentz—maybe all the mass of an electron is purely electromagnetic, maybe the whole $0.511$ MeV is due to electrodynamics. Is it or isn’t it? We haven’t got a theory, so we cannot say. We must mention one more piece of information, which is the most annoying. There is another particle in the world called a muon which, so far as we can tell, differs in no way whatsoever from an electron except for its mass. It acts in every way like an electron: it interacts with neutrinos and with the electromagnetic field, and it has no nuclear forces. It does nothing different from what an electron does—at least, nothing which cannot be understood as merely a consequence of its higher mass ($206.77$ times the electron mass). Therefore, whenever someone finally gets the explanation of the mass of an electron, he will then have the puzzle of where a muon gets its mass. Why? Because whatever the electron does, the muon does the same—so the mass ought to come out the same. There are those who believe faithfully in the idea that the muon and the electron are the same particle and that, in the final theory of the mass, the formula for the mass will be a quadratic equation with two roots—one for each particle. There are also those who propose it will be a transcendental equation with an infinite number of roots, and who are engaged in guessing what the masses of the other particles in the series must be, and why these particles haven’t been discovered yet.