By F. Rohrlich
State University of Iowa
[Read before the Society May 11, 1962]
THIS year, 1962, marks the 70th anniversary of the birth of the theory of the electron. In 1892, H. A. Lorentz published his first account1 of this theory which was to contain the leading ideas of many future investigations for more than a generation to come. It should be noted that this event was prior to the experimental clarification of the basic properties of the electron; those were not measured until 1897 and 1898 when J. J. Thomson succeeded in a determination of the charge to mass ratio and in a first measurement of the elementary charge e.
As a theorist, and especially as one concerned with fundamental interactions, I consider the physical world and all its phenomena in principle reducible to the problem of the interaction of the so-called elementary particles (over thirty known at the present time) by means of the four types of fundamental forces; these are conveniently labelled electromagnetic, gravitational, strong, and weak. The basic problem here is the construction of a mathematical framework in which the static and the dynamical properties of these particles and their interactions can be reduced to a minimum amount of experimental information. If we know the mass of one particle, how can we compute the masses of all the other particles? Is there one interaction of which the electromagnetic and the gravitational interaction are special eases? The answer to questions of this type is the ultimate goal of physicists working in fundamental theory.
So far, we have barely scratched the surface concerning these matters. Of the four interactions the electromagnetic one is certainly most thoroughly investigated and best understood. But it is not at all completely understood. Of the electrically charged particles we know most about the electron. The theory of the electron has throughout its history been very closely linked to the theory of fundamental electromagnetic interactions. It has been the model and guide for the understanding of the other interactions. Looking at the electron you will see elementary particle theory at its best. You will also see how poor this "best" really is, and how much still remains to be done.
I want to show you some of the salient features of electron theory both before and after the advent of quantum mechanics just about half-way in its seventy-year history. I also want to exhibit for you some of its difficulties and how they were resolved, as well as some difficulties whose solution is so far not known.
The mathematical formulation of macroscopic electromagnetic phenomena was beautifully accomplished by J. C. Maxwell about thirty years prior to Lorentz's theory. Lorentz constructed a microscopic theory by using Maxwell's equations and adding to it an expression for the force which a charged particle experiences in the presence of electric and magnetic fields. This microscopic theory is a description of matter in terms of its charged atomic fragments, ions and electrons. The success of this microscopic theory lay in the proof first provided by Lorentz that the macroscopic Maxwell theory can be deduced from this microscopic theory by a suitable averaging process over the motion of the individual ions and electrons. Thus, Lorentz's theory became the primary theory and Maxwell's theory can be reduced to it.
However, Lorentz went beyond this: having successfully described the electromagnetic force acting on a charged particle due to externally present fields, he attempted to describe the structure of an individual electron. His aim was to show that the electron is a completely electromagnetic object. In particular, its mass was to be the mass equivalent of its electromagnetic energy contents; its inertia, i.e., the inertial term in Newton's equations of motion, was to be entirely due to its own electromagnetic field. Accelerating the electron means changing or distorting the field produced by the electron; this requires work. Therefore, the electron exhibits a certain inertia in following the force acting on it.
These ideas were contained, partly implicitly, in Lorentz's work. They were later clarified and extended to fast moving electrons by Abraham and others especially after Einstein's special theory of relativity became accepted. However, relativity was not applied to the theory in a consistent way and, consequently, the difficulty that the union of relativity and electron theory could have removed, remained there for many years. Let us therefore look at the non-relativistic theory.
As I mentioned previously, the starting point of the theory is the Lorentz force and the microscopic equations which, when averaged, produce Maxwell's equations. When the Lorentz force is used to describe the action which the electron's own electromagnetic field exerts on its source, the electron, the following equation of motion is obtained:
In this equation the acceleration a and the force F are three-vectors and the dot indicates a time derivative. m and e are mass and charge of the electron and c is the velocity of light. The famous Lorentz force F is due to the external electric field E and magnetic flux density B, and is a sum over the charge density r,
The electron's structure is therefore characterized by a charge density distribution. This quantity has to be assumed, since there is nothing in the theory which would determine it. Only the total charge e is known from experiment.
The mass m is completely electromagnetic and is therefore given by the energy W of the electric field when the particle is at rest,
In this equation the radius r0 of the electron enters.
Equation (1) now exhibits three difficulties, associated, respectively, with each of the three terms appearing on the left hand side:
(I) The inertial term differs by a factor 4/3 from Newton's classical "mass times acceleration." This is a kinematic problem which implies that the relationship between momentum and velocity for a particle in Newtonian mechanics differs from that for the completely electromagnetic electron. This would have dire consequences and is an intolerable defect of the theory. Luckily, it can be corrected rather easily. A conscientious merger of this theory with special relativity assures that this factor disappears, since it is incompatible with the relativistic transformation properties. For a finite electron this was first pointed out by Fermi2 in 1922. It is closely related to the definition of rigidity in special relativity where the difference in simultaneity of relatively moving observers plays an essential role. Unfortunately, Fermi's paper was either never understood or soon forgotten. In any case, the factor 4/3 can still be found in some of today's texts. For point electrons the removal of this factor was later rediscovered several times.3 Let me summarize then by saying that a relativistic generalization of (1) will not show this factor; if the non-relativistic theory is derived as the limit of the relativistic one, this factor will disappear from (1). A study of its origin reveals an unjustified and incorrect assumption about the relationship between the Poynting vector and the momentum of non-radiative electromagnetic fields.
(II) The second difficulty lies in the terms marked "structure terms." These terms depend on the charge distribution and radius of the electron. This dependence is a difficulty here while it is not a difficulty in the first term. In the latter, the mass must be determined experimentally anyway, while here the whole dynamics becomes explicitly dependent on the electron structure. This difficulty can be removed only by eliminating the electron structure altogether. If we assume the particle to be a point particle it will obviously have no structure. Furthermore, if, for this purpose, we let the electron radius r0 shrink to zero the "structure terms" all vanish. Since no experimental evidence for structure exists (the Coulomb potential is correct down to smallest distances measured) this is a satisfactory procedure indeed. The removal of the structure terms, however, produces a new difficulty, viz., that when the radius shrinks to zero, the electron mass becomes infinitely large. According to (3),
r0 ® 0 implies m ® ¥. (4)
This is the famous problem of the electron self-energy. It exists in the classical theory of the electron as well as in the quantum theory. A satisfactory solution of it is not known. An at best temporary solution is provided by the renormalization procedure and will be discussed later. In any case, this procedure no longer allows for a completely electromagnetic mass.
It is therefore not surprising that many attempts were made to keep the electron of finite extension and carry along throughout the development of the theory an explicit electron structure. The debate pro and con the point electron is in fact as old as electron theory. Both sides agree, however, that since the advent of quantum mechanics, the problem of the structure of the electron has become a quantum mechanical one. Surely, it does not belong into the field of competence of classical electrodynamics. Consequently, it is to some extent a matter of taste, exactly how the classical theory describes the electron outside its limits of validity. I believe, however, that, just because the classical theory is not competent to discuss structure, no structure should be introduced if at all possible. In fact, I think most physicists would agree that a classical point electron theory is preferable if it can be constructed without divergences. This cannot be done on the Lorentz level of development, but, as we shall see, it can be done at a later stage.
(III) The third difficulty evident in the Lorentz equation (1) is the following. Even if the structure terms were absent, there is still a term in the equation which involves a time derivative of the acceleration. Now Newtonian mechanics is based on equations of motion which involve no such time derivatives. Consequently, they are second order differential equations in position, and their solutions are determined by two initial conditions: if the position and the velocity of a particle is given at the starting time, its whole trajectory is completely determined. Not so for the Lorentz equation: there we have at least a third-order equation (viz. when there are no structure terms) and, therefore, the initial acceleration would have to be known too. This completely contradicts the Newtonian concepts of mechanics. The term also has a rather nebulous physical interpretation. It corresponds to a reversible change of energy, somewhat like the kinetic energy. In fact, this energy was named "acceleration energy" in contradistinction to the "velocity energy" i.e., the kinetic energy. After Schott, it is sometimes called the Schott energy. While the second term in (1) obviously arises from the self-field of the electron and is therefore often referred to as "radiation reaction." I shall not enter upon any further discussion of it, because this term will eventually disappear from the equation.
There is one more difficulty in the Lorentz-Abraham theory which is not apparent from the equation of motion. This difficulty relates to the stability of the electron. Obviously, an accumulation of negative charge is not a stable configuration: Any finite charge distribution of only one sign would explode. The completely electromagnetic theory of the electron there cannot even account for the very existence of this particle. This is a very old problem. A solution to it was proposed by Poincaré who showed that an attractive and consequently non-electromagnetic force can always be added to the theory so as to just balance the stresses and establish stability. It is a very ad hoc solution and not at all fitting a "fundamental" theory. On the other hand, in the point electron theory the electromagnetic self-stress is infinite and so must be the Poincarés cohesive force. This divergence is closely related to the self-energy divergence of the point electron. In fact, one can show that in a relativistic theory in which the electron self-energy (i.e., its mass) is not infinite, the self-stress will actually vanish and the particle will be stable. Thus, this problem need not be further discussed since it will be resolved simultaneously with the self-energy problem in any relativistic theory. Physically, this is quite reasonable, because, if the electron is the elementary charge, a satisfactory theory must be so as to admit no Coulomb-type self-interactions, since such interactions exist only between charges. Therefore, in such a theory it is no longer possible that part of the elementary charge acts on another part of the same elementary charge. Consequently, the electron will no longer explode but will be stable.
The next major development occurred in 1938 when Dirac4 published a most important paper. He started with the microscopic Maxwell-Lorentz equations. By a judicious use of the conservation laws and a covariant formalism throughout, he derived an equation of motion for a point electron which in the non-relativistic limit is exactly the Lorentz equation (1), but without the factor 4/3 and without the structure terms,
Here, a, v, and F are four-vectors of acceleration, velocity and force. The new term on the right involves the radiation rate, Â, i.e., the energy per second of electromagnetic radiation emitted at a given instant by the electron. This term vanishes in the non-relativistic limit.
Since this equation is almost exactly the Lorentz equation in the point particle limit, and in order to avoid confusion with Dirac's equation for the electron in relativistic quantum mechanics, this equation is referred to as the "Lorentz-Dirac Equation." In fact, the radiation term was also known long before Dirac. The difference between (5) and Newton's ma = F first obtained by Abraham as the relativistic generalization of the Lorentz radiation reaction term. These two terms are known as the "Abraham Radiation Reaction Four-vector." At this point one must note that, while the term has a rather questionable physical meaning, the Âv term has a very well defined meaning. It is exactly the rate at which energy and momentum in the form of radiation leaves the electron.
One more comment is essential concerning the derivation of the Lorentz-Dirac equation. In addition to the Maxwell-Lorentz equations one new assumption and one new procedure enter this derivation. The assumption is that the equation be of third order at most and not higher. As Dirac put it in his 1938 paper concerning higher order equations: " they are all much more complicated, so that one would hardly expect them to apply to a simple thing like an electron."
The new procedure used is that of "mass renormalization." This means that the mass equivalent of the electromagnetic self-interaction is added to whatever other mass the electron has ("bare mass") and the sum is identified with the observed mass. This originally classical procedure was not sufficiently appreciated at the time and was not used in the quantum domain until almost ten years later, yielding essential advances in quantum field theory; it became the basis for the successful computations of radiative corrections.
The elimination of the kinematic difficulties in Dirac's approach is not surprising. As mentioned earlier, it is assured by the consistent relativistic treatment used. Neither is it surprising that this point-electron equation contains no structure terms: there is of course an infinite electromagnetic mass term hidden in the renormalization procedure. For this reason the point electron limit can be carried out after renormalization. What is remarkable is that here Dirac "almost" succeeded in deriving the equation of motion of the electron from the Maxwell-Lorentz equation. It is well-known that the Lorentz force is a separate assumption, independent of Maxwell's equations. But here it is almost reduced to them. If this were actually accomplished, one would have a situation very similar to general relativity. The field equations there imply the equations of motion. That means that the equations which permit the computation of the gravitational field from the motion of the masses also determine the motion of these masses under the influence of the gravitational fields. The equations of motion are part of a self-consistency requirement for the field equations.
The main difficulty that still remains in the Lorentz-Dirac equation (ignoring for the time being the infinite mass renormalization) is the appearance of an term: the equation is of third order rather than of second order. This difficulty already exists in the Lorentz equation, but because of the structure terms it was perhaps less acute. The term requires the initial acceleration for a complete determination of the solution.
In order to appreciate the import of this situation, it is necessary to mention a difficulty which is not evident from the equation and which is actually closely related to the term. If the initial acceleration is not specified, an infinity of solutions results, all whichwith only one single exceptionare physically meaningless. They lead to trajectories according to which a particle would accelerate and approach the velocity of light in the distant future, no matter what the applied force is. In fact, even when no force at all is acting on the particle, such solutions exist. These solutions are therefore called "self-accelerating" or "run-away."
It follows that the initial acceleration is not only necessary to determine the solution uniquely, but the "right" initial acceleration must be specified in order to assure the physical solution. The smallest error in the initial acceleration yields a run-away solution.
How can these non-physical solutions be avoided? What principle or condition must be invoked in order to permit only the physical solution? Dirac mentioned that if one required that the acceleration should vanish in the distant future, one will have no run-away solutions. But this is almost a tautology. Furthermore, it is an ad hoc condition, since no physical principle seems to be involved.
However, a much better approach is possible. If one invokes the fact that only a finite amount of energy is available for a given system, then it follows from energy conservation that a physical system can radiate only a finite amount of energy, even if it is radiating for an infinite time. Thus, we require
Â ® 0 for t ® ¥. (6)
This very physical and reasonable requirement indeed implies a®0 for t ® ¥ as suggested by Dirac. The essential point concerning this condition is, however, that it must be an integral part of the equation of motion (5). Otherwise, the selection of the physical solution is not assured by the equation itself. The condition (6) is not an asymptotic condition imposed to specify a particular problem, as is done, e.g., in scattering theory, but it is a necessary condition which must be fulfilled no matter what specific system is under consideration.
By a few simple mathematical steps one can now prove the following.5 The Lorentz-Dirac equation (5) and the asymptotic condition (6) are together equivalent to the one equation
where K stands for the right hand side of (5). From what has been said, it follows that this equation should be taken as the equation of motion for the electron instead of the Lorentz-Dirac equation, since the latter is not meaningful without the condition (6).
The new equation of motion has many remarkable properties:
(a) It has no run-away solutions, since it was constructed that way.
(b) It is an integro-differential equation of second order, so that its solutions are determined in the Newtonian spirit of mechanics by initial position and velocity only. The transition from (5) to (7) eliminated the term and the run-away solutions at the same time.
(c) The absence of the avoids the question of that term's exact physical meaning.
(d) Only the term Âv occurs as radiation reaction. This term vanishes if and only if there is no radiation (Â = 0) which was not the case previously. This has led to a great deal of confusion in connection with uniformly accelerated charges which seem to radiate with zero radiation reaction.
(e) The solutions of this equation are exactly consistent with an extension of the principle of equivalence to electromagnetic systems. In particular, a neutral and a charged particle in a static homogeneous gravitational field fall equally fast according to (7). Nevertheless, an observer supported in such a field will see the charged particle radiate. This leads to an apparent contradiction with energy conservation which is resolved by Eq. (7) as follows: When this equation is written
where x is of order one, this form of the equation of motion says that the net force acting on the particle is the external force reduced by the radiation reaction. This net force at the time t + xt0 is equal to mass times acceleration at the time t. Consequently, it differs from Newton's second law by a time interval between the net force and the inertial term. This time interval is of order t0. Energy conservation says, accordingly, that the work done by the external force at time t + xt0 accounts exactly for the energy radiated at that time and the increase in kinetic energy at the slightly earlier time t.
Here an essentially new feature emerges, a feature which was not expected and which does not fit into the concepts of classical physics of which this theory is part: the new equation of motion has a non-local behavior in time, a certain lack of instantaneity which brings with itself a lack of causality over time intervals of the order of t0. In particular, energy conservation is no longer satisfied at every instant of time, but is spread out over a time interval of about t0. This is a very serious matter and one might be inclined to discard this theory just for that reason alone.
At this point, however, the size of t0 should be considered. This constant is given in terms of the electron's mass and charge as
Clearly, such time intervals are entirely outside the domain of competence of classical physics. It follows that the acausality here arises only outside the validity limits of the theory.
The situation can now be summarized as follows. By means of a derivation which, because of infinite mass renormalization, etc., must be considered as heuristic, a new equation of motion was obtained. This equation together with the Maxwell-Lorentz field equations define the classical theory of the point-electron. This theory cannot be derived from a Lagrangian or Hamiltonian approach, but is based on these equations as the starting point. Nor does it permit questions about the self-energy or self-stress of the electron. However, it does describe a point-particle and the complete dynamics associated with it. The mass and charge are phenomenological constants which enter the theory. The theory is classical and has corresponding validity limits. The theory agrees in detail with the requirements of the principle of equivalence. But it shows certain strange non-causal features outside its physical domain of validity which are associated with critical tunes of the order of t0 = 6x10-24 sec. The theory is mathematically well defined. It agrees with experiment in all cases which have been tested.
Phrased in these terms, a comparison of this theory with the Lorentz-Abraham theory exhibits an interesting instance in the study of scientific theory construction.
A typical physical theory (Newtonian mechanics, geometrical optics) has a certain domain of validity outside of which it no longer agrees with experiments. These limits are usually expressed by inequalities (v >> c, l >> a) characterizing the accuracy beyond which disagreement will be found. The mathematical structure of the theory does not break down at these limits. In the case of the Lorentz-Abraham theory there are of course also physical validity limits (classical nature of the theory, electron structure not meaningfully described by it). But in the point electron limit, the theory is mathematically not well defined. Based on the new equation of motion we have a mathematically well-defined theory which exhibits strange physical features outside its validity limits. This latter fact is mostly irrelevant, but the unambiguous mathematical nature of the theory is essential. Thus, the old theory was completed mathematically and thereby certain new physical features were introduced. But these lie outside experimental verification, while the experimentally accessible features of the Lorentz theory have not been changed. This situation, I believe, is new in the history of theory construction in natural science.
The discovery of quantum mechanics in 1925 and 1926 called for an extension of the classical theory of the electron to the atomic and sub-atomic domain. At first, however, the electron differed in no way from any other particle which is treated quantum mechanically. As required by experiment, this formalism permits one to explain wave-like as well as particle-like characteristics of the electron.
A most important and unexpected property of electrons was found almost simultaneously with the establishment of this new mechanics; it was discovered that the electron has an intrinsic angular momentum, a spin, and associated with it a magnetic moment. Based on a study of spectra emitted by atoms in a magnetic field, Pauli established a mathematical formalism to account for the apparently dichotomous behavior of electrons (1925). Uhlenbeck and Goudsmit6 thereupon provided the physical picture of a spin angular momentum characterized by ½, i.e. 1/(4p) times Planck's constant, and two possible orientations.
Essential progress towards a relativistic quantum mechanical description of electrons was made in 1928 when Dirac7 proposed a new equation which implied a characteristic symmetry property in the sign of the electric charge. His "hole theory" lead him to the prediction of the existence of positive electrons (first incorrectly identified with protons). Four years later Anderson found the positron in cosmic rays. This symmetry between particles (e.g. "negatons" or negative electrons) and antiparticles (e.g. "positons" or positive electrons) is today one of the most fundamental features of elementary particle theory
The need for a mathematical formalism that permits not only a quantum mechanical description, but also a mechanism for the creation and annihilation of photons and other particles led in short time to field quantization. The fundamental papers by Heisenberg and Pauli and by Wigner and Jordan provided the basis for the quantum field theory of particles of spin zero and one-half. While at first charged particles and their electromagnetic interaction were in the center of interest (quantum electrodynamics), quantum field theory was successfully extended to nuclear interactions by Yukawa (1935-36). This extension which was actually an analogue of quantum electrodynamics led to the prediction of the existence of mesons, i.e. particles which play the same role for nuclear interactions that photons play for electromagnetic interactions. The success of this "meson theory," however, lay mainly in that prediction because, despite great efforts, at best partial and qualitative agreements could be obtained.
The success of quantum field theory was of a very different nature when restricted to quantum electrodynamics. Here excellent experimental agreement resulted from the computations to first approximation of Compton scattering, pair production, bremsstrahlung and other processes. The work of Bethe and Heitler is especially well known.
The attempts to overcome the difficulties of computing effects beyond the first approximation were at first unsuccessful and later interrupted by the second world war. After that, the experiments by Lamb and Retherford ("Lamb shift") signalled the beginnings of new progress (1947). These results posed a great challenge to the theorists: the next approximation was measured for the first time. The break-through resulted from a combination of two factors. One was the reformulation of quantum field theory in covariant form, which was done in-dependently by Tomonaga and by Schwinger. The other was the appreciation of lessons learned in classical electron theory: Kramers taught mass renormalization to the young American theorists Feynman and Schwinger. The first results8 a were the anomalous magnetic moment of the electron (Schwinger, 1948) and an approximate theory for the "Lamb shift" (Bethe, 1947). The latter was then improved by others so as to give the same accuracy as the experiments. The general formulation, permitting consistent computations to arbitrarily high order was then established by Dyson on the basis of Feynman's diagrammatic methods.9 Renormalized quantum electrodynamics (within the framework of perturbation theory) was thus established in a few years, although somewhat later contributions proved essential for certain details (Ward and Salam).
The resultant predictive power of the theory is truly remarkable. The energy level difference between the 2s and 2p½ level of Hydrogen was measured to 0.1 Mc/sec accuracy. This means that the corresponding computation relative to the ground state must have an accuracy of about one part in 1010. The theory is completely capable of providing the figures to that accuracy. Similar feats relate to the hyper-fine structure and the electron's magnetic moment.
Thus, present quantum electrodynamics is one of the strangest achievements of the human mind. No theory has been confirmed by experiment to higher precision; and no theory has been plagued by greater mathematical difficulties which have withstood repeated attempts at their elimination. There can be no doubt that the present agreement with experiments is not fortuitous. Nevertheless, the renormalization procedure can only be regarded as a temporary crutch which holds up the present framework. It should be noted that, even if the renormalization constants were not infinite, the theory would still be unsatisfactory, as long as the unphysical concept of "bare particle" plays a dominant role.
If one considers quantum electrodynamics as a phenomenological theory with respect to the mass and charge of the interacting particles, and if one consequently condones the necessity of infinite mass and charge renormalizations, one is tempted to consider quantum electrodynamics as a pretty satisfactory theory. I would like to point out that this can be true only for charged particles of spin one-half, i.e. electrons and mu-mesons. For the case of charged spinless particles (p- or K-mesons) an essentially new situation arises: the electromagnetic interaction of these mesons diverges, and can be made consistent even within the framework of renormalization theory only by postulating a direct interaction of unknown strength to be determined by experiment. Such an interaction is entirely foreign to Maxwell's equations and can in fact not reasonably be regarded as electromagnetic. It occurs only in strongly interacting particles and is consequently masked by nuclear interactions. For this reason its determination has so far escaped detection. In fact, it seems not at all possible to separate this direct interaction from a nuclear interaction until a complete theory of nuclear forces is established. I believe, that such a separation will actually not be possible, because this interaction must be expected to be of nuclear type. Thus, a satisfactory quantum electrodynamics for spinless particles cannot be constructed unless it is concomitant with a theory of strong interactions. Conversely, I believe that the theory of nuclear forces can be constructed in a mathematically satisfactory way only when it embraces electrodynamics. When I discovered the need for a direct interaction for spinless electrodynamics, in 1950, I did not fully realize the meaning of this result. I have no doubt now that it is an important clue for elementary particle theory.
Restricting ourselves now to the theory of electrons and photons, what are the basic difficulties and how do they compare with those of the classical theory?
First and foremost, there is the self-energy problem. In the classical theory this problem involved divergent integrals associated with the electromagnetic mass of the point electron, or equivalently, it involved the structure dependence of the whole theory in the case of finite size electrons. As we have seen, this difficulty can be circumvented by starting the theory not in the Lagrangian or Hamiltonian way, but by means of an equation of motion which contains only the experimental (i.e. renormalized) mass. In quantum electrodynamics the situation is much worse. The self-energy difficulties are not restricted to the electron, but occur also for the photon. Furthermore, they involve not only divergent integrals associated with the mass, but in fact all integrals (in perturbation theory) which describe dynamical self-interactions, i.e. the emission and reabsorption of photons, are divergent. These, however contain all observable effects. It is indeed amazing to see an unambiguous procedure by which these meaningless integrals can be separated into another meaningless integral and a convergent one, so that the latter agrees with some of the most accurate experiments ever conducted.
It is possible also in quantum electrodynamics to start with renormalized equations. A formulation of this nature involving the renormalization constants explicitly was given by G. Källen; however, this formulation seems to be inconsistent since it leads to divergences, as was shown by its author. A formulation which does not involve reference to renormalization was given by Lehmann, Symanzik, and Zimmermann, but it has so far not been carried far enough to serve as a replacement for the present unsatisfactory form of quantum electrodynamics. Quantum field theory is therefore obviously not in satisfactory mathematical form. It is therefore presently not in a state of completion analogous to the classical theory of the electron.
There are of course many unsolved problems in quantum electrodynamics, but none is comparable to the one just mentioned or should pose a fundamental obstacle. Of these I want to mention only one, viz. the problem of obtaining classical electrodynamics, and in particular the integro-differential equation of motion discussed above, as a limit of quantum electrodynamics. That the classical theory must he derivable from the quantum theory is necessitated by the logical structure of physics. It is furthermore implied by every practicing physicist who uses the classical theory for the construction of accelerators (orbit calculations) and the quantum theory for the prediction of the experiments he plans to conduct with these machines. This problem is a very essential logical link in the theory of the electron.
The basic remaining problem, however, is that of a reformulation, so as to clad the completely satisfactory experimental predictions into a clean mathematical structure. It is my belief that considerable progress in that direction can be made by trying to bring the formalism closer to the physical situation. This idea has in fact been fruitful recently when it was possible to eliminate certain mathematical complexities by means of it. The case in point is the following.
The usual formulation of quantum electrodynamics is based on the potential four-vector as the fundamental quantity which describes the electromagnetic field. This has partly historical reasons: In classical electrodynamics it was found that the Maxwell equations for the field strengths can be greatly simplified if one introduces the potentials (scalar and vector potential) from which the field strengths can be derived. It was realized that these potentials have no direct physical significance and only field strengths can be measured; but it was a small price to pay for the great mathematical simplification. When these potentials are introduced in classical electrodynamics, they bring with themselves superfluous degrees of freedom. However, these are easy to eliminate by means of the Lorentz condition and the requirement of gauge-invariance.
In quantum electrodynamics this same introduction of potentials leads to various complications. Consistency with the quantization conditions (commutation relations) requires modifications of the Lorentz condition. Further difficulties arise and further mathematical complications (indefinite metric) are required. Thus, by the time a consistent formulation is achieved, the simplification for which the potentials were originally intended has been completely off-set by a rather formidable mathematical apparatus. At the same time, all these complications refer to the unobservable potentials and are therefore of no physical relevance.
Recently, it has been possible to reformulate quantum electrodynamics completely in terms of the field strengths of the electromagnetic radiation.10 The static and quasi-static interactions between moving charges are retarded and are of a classical nature: they are not associated with photons. Only the radiation fields are quantized. They lead to transverse photons only and no other photons occur in the theory. In this form the theory is free of questions of gauge-invariance and no supplementary condition need be satisfied. These are tremendous simplifications. At the same time, many unphysical features are eliminated.
At the present time no one can tell what a future mathematically satisfactory quantum electrodynamics will be like. But there are many guesses which have their origin in the relative importance various physicists ascribe to those principles and prejudices on which the present theory is based. Since a solution of this problem may not come within the immediate future, it seems safe to add my own ideas on this matter.
As I indicated before, the theory of the electron does not seem to be "coupled" to the strong interactions in contradistinction to the theory of charged boson, especially of the p- and K-mesons. Furthermore, since the reduction of classical to quantum electrodynamics must surely exist and the latter is presently in relatively satisfactory shape, various characteristic features can be gleaned from it. First, there is a strong hint that Lagrangian and Hamiltonian formulations are not essential and in fact not desirable: the classical Lorentz-Dirac equation cannot be derived from a Lagrangian.
Secondly, the asymptotic properties of the system (both in time and in space) play a much more fundamental role than has hitherto been realized. In general relativity a similar emphasis pervades recent work. The asymptotic formulation of field theory, begun by Lehmann, Symanzik, and Zimmermann seems therefore very much in the right direction.
Thirdly, classical electrodynamics teaches us that the theory in the small will most likely be of a non-local nature. This is confirmed by the formulation of quantum electrodynamics in terms of field strengths: Some of the important quantities of the theory are given by integrals over all space and are not point functions. Further confirmation of this conjecture is obtained from a renormalization of the theory in this formulation: the renormalized commutation relations are non-local;11 in particular, they no longer vanish outside the light cone (micro-causality). Only asymptotically do these commuters vanish (asymptotic or macro-causality). This feature is again closer to the experimental situation than the local field theory: quantum mechanical measurements are necessarily restricted to averages over finite (though small) space-time regions and strict micro-causality is not observable in principle.
Fourthly, associated with the non-local nature, as it emerges here, is a lack of local Lorentz invariance. Various quantities depend on space-like planes and are no longer scalars or tensors as in the local theory. Only in the distant past and future (incident and outgoing particles) are the transformation properties restored. Thus, we have asymptotic Lorentz invariance. It is quite possible that other invariance properties of the local theory will also become valid only asymptotically.
These are the features which seem to emerge and point toward the possible characteristics of the future quantum electrodynamics and perhaps of quantum field theory in general. They can be generally described as a fuzziness and indeterminacy for small space-times, with a consequent blurring of detail in the microscopic description, while leaving all known features unchanged asymptotically. The macroscopic theory is unchanged.
It is possible and in fact likely that some basic new principles and ideas will have to be added to what will be left of standard quantum field theory in order to provide for a complete and consistent description of electromagnetic interaction, and, in particular, of the electron. The future will tell which of these various conjectures are valid. If, nineteen years from now, the fiftieth Joseph Henry Lecturer should again discuss the theory of the electron, the chances are good that he will know the answer to many of these questions.
Progress seems faster than anticipated. One of my students, Robert E. Pugh, succeeded in his doctoral dissertation to develop further the asymptotic formulation of quantum field theory of Lehmann, Symanzik, and Zimmermann. He obtained a set of equations which can be solved completely in perturbation expansion and which reproduces exactly the experimental predictions of the usual, unsatisfactory, formulation. As conjectured, the commutation relations no longer play the central role they have in the usual theory and occur only asymptotically. The Lagrangian and Hamiltonian formalism is not used. This new formulation of quantum field theory is completely free of divergencies and involves no renormalization; it is "already renormalized." The outstanding problem now is to find solutions of the basic equations which are not expanded in a perturbation series.
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