Modern Advances in Electron Theory

V. F. Weisskopf*)

The application of microwave technique in spectroscopy has significantly increased the accuracy of spectroscopic measurements. Modern experiments on the spectra of hydrogen and other elements have yielded results not fully consistent with the basic principles of the mechanics of electrons in the atom. In checking the values of the energy levels of the hydrogen atom, calculated by Sommerfeld’s formula, small deviations were found. The measured value of the magnetic moment of the electron differs by an amount of the order of one thousandth from the value obtained from the fundamental equation for the electron.

These experimental discoveries led to a revision of the theory and, in particular, of its weakest point: the interaction of the electron with radiation. This interaction was considered in the theory,

*) Reviews of Modern Physics 21, 305 (1949).

The translation of V. F. Weisskopf’s article “Modern Advances in Electron Theory” gives our reader the opportunity to become acquainted with those attempts that have been made in recent years by foreign physicists to overcome the well-known difficulties of electron theory. The discussion concerns the so-called method of renormalization of charge and mass. We speak of this method, and not of a theory, since the latter part of this teaching has not yet been developed. This method represents only a certain formal device for excluding infinite, physically meaningless quantities, to which both the classical and the quantum theory of the electron lead. However, one cannot fail to take into account that certain predictions of this method, which previously remained doubtful, are in fact receiving experimental confirmation (“zero-point” oscillations, polarization of the “vacuum”).

V. F. Weisskopf passes over in silence Soviet work in this same field of theoretical physics. References to these works the reader can find in Ya. A. Smorodinsky’s article “The Shift of Terms of Hydrogen-like Atoms and the Anomalous Magnetic Moment of the Electron,” UFN, vol. XXXIX, no. 3, November 1949. — Ed.

called “quantum electrodynamics,” which, from the time when its basic principles were formulated by Dirac in 1926, suffered from certain internal contradictions connected with the old problem of the internal structure of the electron. These contradictions made it impossible, within the framework of the theory indicated, to approach rigorously the solution of questions connected with radiation phenomena.

In recent years, however, theoretical works have appeared containing attempts to separate unresolved problems and contradictions within the framework of the theory and to increase the accuracy of its predictions, despite the fact that the structure of the electron and its role in physical phenomena are not understood. These works have yielded quite encouraging results. The predictions of the theory are in complete agreement with new experiments, which has to a considerable degree increased confidence in the basic principles of “quantum electrodynamics.”

In the present article an attempt is made to present these new works in a form accessible to physicists who are not specialists in the questions under discussion. In this connection it is possible to give only a qualitative and incomplete picture of the problems considered. It does not seem advisable to us to limit this article only to the most recent achievements; therefore it gives a brief survey of the development of ideas about electrons, beginning with the classical theory of Lorentz and ending with the theory of the positron. The significance of the modern problems cannot be assessed without analyzing the most important stages in the development of the theory.

1. CLASSICAL THEORY OF THE ELECTRON

It is difficult to find any other discovery that led to the understanding of so broad a range of phenomena as the discovery of electrons. Many phenomena that seemed unconnected with one another, such as, for example, optical, electrical, and chemical phenomena, were explained with the help of the electron theory by one and the same basic mechanism.

The chief credit for creating a consistent classical electron theory belongs to H. A. Lorentz. He expressed the following basic assumptions:

1) The electron is an elementary particle with charge $e$ and mass $m$.

2) The motion of the electron is determined by classical mechanics, and the force acting on the electron is given by the expression

\[ \mathbf{F}=e\mathbf{E}+\frac{e}{c}[\mathbf{v}\mathbf{H}], \]

where $e$ and $\mathbf{v}$ are the charge and velocity of the electron, while $\mathbf{E}$ and $\mathbf{H}$ are the intensities of the electric and magnetic fields. The electromagnetic field, however,

is described by the system of Maxwell equations

\[ \frac{1}{c}\frac{\partial \mathbf{E}}{\partial t}-\operatorname{rot}\mathbf{H}=4\pi\mathbf{j},\qquad \operatorname{div}\mathbf{E}=4\pi\rho, \]

\[ \frac{1}{c}\frac{\partial \mathbf{H}}{\partial t}+\operatorname{rot}\mathbf{E}=0,\qquad \operatorname{div}\mathbf{H}=0. \]

The sources of the field are the charge density \(\rho\) and the current density \(\mathbf{j}\), which are due to moving electrons.

In most cases the electron may be regarded as a point charge. In these cases it is easy to write the expression for the field produced by an electron. We shall give only one simple example: an electron at rest produces an electric field

\[ \mathcal{E}=\frac{e}{r^{2}}, \tag{1} \]

where \(r\) is the distance from the electron. The expression for the field surrounding an electron in motion is somewhat more complicated.

It was necessary to make several additional assumptions concerning the conditions of motion of the electron in matter. Lorentz assumed that there are several electrons in each atom, which are elastically bound to the point of equilibrium and therefore are capable of performing harmonic oscillations with a given frequency. In a conductor it was assumed that additional electrons exist which are capable of moving freely. With the aid of these basic premises it proved possible to explain a large number of phenomena, such as, for example: the absorption, scattering, and refraction of light in a medium, the Zeeman effect, the optical properties of a metal with respect to infrared radiation, and much else. However, in many cases the explanation was only qualitative. Some details of the phenomena were obscure. It was impossible to explain the presence of an elastic binding of the electron inside the atom, especially if one took the standpoint of the planetary model of the atom. With the aid of the theory it was impossible to understand the cause of the oscillations of the electron inside the atom, and also to determine their frequency.

Lorentz also studied another fundamental problem: within what limits can the electron be regarded as a point charge? He was compelled to make certain assumptions about the internal structure of the electron in order to apply the equations of electrodynamics inside it. We quote from his book The Theory of Electrons:

“When I speak of what occurs inside the electron as confidently as if I had the possibility of looking into this small particle, I fear that someone may get the idea that it would be better if I did not try to penetrate into all these details. My justification must be that it is hardly possible to refrain from such a detailed consideration when one wishes to have

completely definite system of equations. Moreover, as we shall see later, from our experiments something can be concluded about the dimensions of the electron. Secondly, it must be noted that in those cases where the internal structure of the electron may manifest itself, considerations of the kind just discussed are in any case of interest, regardless of whether they are correct or not. At the same time, they do not in the least harm so long as we may consider that the internal structure plays a small role.”

The basic problem concerning the structure of the electron can at present be formulated very simply, since the principle of equivalence of mass and energy*) has acquired universal significance. The total energy \(E_{\mathrm{st}}\) of the electrostatic field (1) of the electron is expressed as follows:

\[ E_{\mathrm{st}}=\frac{1}{8\pi}\int \mathcal{E}^{2}\,dv, \]

where the integration extends over all space. \(\mathcal{E}\) is given by equation (1) outside the electron, but, of course, its application “inside” the electron is unlawful. It is convenient to assume that the charge of the electron is concentrated on the surface of a sphere of radius \(a\). In this case \(\mathcal{E}\) must vanish inside the electron, and we obtain:

\[ E_{\mathrm{st}}=\frac{e^{2}}{2}\int_{a}^{\infty}\frac{dr}{r^{2}}=\frac{e^{2}}{2a}. \tag{2} \]

Any other assumptions concerning the charge distribution do not change the general character of (2): the energy of the electric field depends essentially on the radius of the electron. This energy must be part of the rest energy of the electron. Then, with the aid of Einstein’s relation, we obtain:

\[ m=m_{0}+\frac{E_{\mathrm{st}}}{c^{2}}=m_{0}+\frac{e^{2}}{2c^{2}a}, \]

where \(m_{0}\) is the “mechanical” mass of the electron, by which we mean the part of the mass that has a non-electromagnetic origin. Since the total mass \(m\) is known experimentally, one can find a lower bound for the radius of the electron if one assumes that the entire mass of the electron is of electrical origin (we exclude the rather artificial choice of negative values for \(m_{0}\)):

\[ a=\frac{e^{2}}{2mc^{2}}=r_{0}. \tag{3} \]

The radius of the electron is at least equal to \(r_{0}\). The quantity \(r_{0}\) is usually

*) The relation \(E=mc^{2}\) is meant. (Editor’s note.)

called the “classical radius” of the electron. Thus we are compelled to abandon the idea of a point charge.

There is no point in discussing in detail the conclusions that follow from the assumption of an extended classical electron. Subsequent investigations introduced so many new features into the theory that they completely changed the classical conceptions of the electron. It is necessary, however, to mention one further circumstance. At what energies should one, in the classical theory, expect significant discrepancies between the results of the theories of an extended and a point electron? This difference, as is easy to see, will show up in calculating the scattering cross section of electrons by electrons, or of electrons by particles with the same charge (for example, protons), if the energy of the particles is so great that they can approach one another to a distance smaller than \(a\). This will occur at energies exceeding \(2mc^2\), i.e., greater than 1 MeV. It may be noted that physicists of that time would have been very surprised if they had succeeded in carrying out such experiments. Instead of detecting the effect of the finite extension of the electron, they would have observed the creation of electron–positron pairs.

Below in this article the fundamental connection between pair creation and the problem of the structure of the electron will be discussed.

2. QUANTUM THEORY OF THE ELECTRON

The problems of the structure of the electron soon passed out of view owing to the successful development of the quantum theory of the electron. The discovery of the quantum of action, Bohr’s theory of quantized orbits in the atom, and the dualism of the wave and corpuscular properties of the electron ultimately led to the development of quantum mechanics. A new interpretation was introduced of mechanical concepts concerning momentum, energy, coordinate, and velocity, in order to describe consistently facts that at first sight seemed contradictory, such as, for example, the wave and corpuscular properties of the electron or the stability of planetary orbits in the atom against collisions. The new theory is best known in the mathematical formulation of Schrödinger’s wave equation. The success of quantum mechanics was enormous. Many unsolved problems of the classical electronic theory were solved. It now became possible to understand and calculate the resonance frequencies of atoms, the stability of electronic orbits, and many other facts that could not be explained from the point of view of classical mechanics. There is hardly any phenomenon in the physics of atoms and molecules which, at least in principle, cannot be explained with the aid of quantum mechanics. It is also worth noting the following: the quantum theory of the electron established that all forces between atoms, molecules, and electrons have a purely electromagnetic ...

origin. Quantum mechanics can answer all questions concerning the behavior of an electron (or other particles) in electromagnetic fields, if these fields are given as functions of time and coordinates. But in most problems of atomic physics what is required is precisely to obtain an answer to the question: “how does an electron move in an electric or magnetic field of a definite kind?” Difficulties arise, however, if one is interested in the question: “What fields are produced by charges in their motion?” For example, the existing theory cannot explain the fact that an atom in its ground state does not radiate, despite the fact that the charges are in rapid motion.

Nevertheless, it proved possible to formulate a certain number of definite rules for calculating the radiation of atomic systems. This was done with the aid of two principles. The first is the hypothesis of light quanta: light of frequency \(\nu\) can be emitted and absorbed only in quanta of energy \(h\nu\). Thus, the emission or absorption of a quantum must be accompanied by a transition from one quantum state to another, whose energies differ by \(h\nu\).

The second principle is the correspondence principle: a system in quantum states with a very large excitation energy exhibits mechanical properties that can be obtained from a classical treatment of the same problem. The radiation from these excited levels must be the same as that calculated classically. From these two principles it proved possible to derive rules by means of which one can successfully calculate the radiation, absorption, and scattering of light by atomic systems. If the wavelength of the light is large in comparison with the dimensions of the system, then the latter in the quantum state \(n\) is in many cases equivalent to an aggregate of classical electric oscillators with frequencies obtained from the formula:

\[ h\nu_{nk}=E_n-E_k, \]

where \(k\) is some other state of the system. The effective charge of these oscillators is determined from the formula:

\[ \varepsilon^2=e^2 f_{nk}, \]

where \(f_{nm}\) is the so-called oscillator strength:

\[ f_{nk}=\frac{2m}{h}\nu_{nk}\left|\int \psi_n^* r \psi_k\,dv\right|^2 , \tag{4} \]

where the integral is the matrix element of \(r\) for the transition from the state \(n\) to the state \(k\).

The problem of the structure of the electron does not enter into this theory. The theory allows the construction of an electronic wave packet with pro-

of arbitrarily small diameter, which may be made even smaller than \(a\), if wavelengths smaller than \(a\) are used to construct the packet. Since the problem of the generation of fields by quantum-mechanical systems had not yet been clearly formulated, no difficulties arose in considering the field produced by such a packet.

It should be noted, however, that in collisions of electrons with energy of the order of \(mc^2\) one can no longer regard the electron radius \(a = r_0\) as equal to the minimum distance between two colliding electrons. The wavelength corresponding to this energy is equal to \(\frac{h}{mc}\), a quantity considerably exceeding \(r_0\). Consequently, it is impossible to localize the electron more accurately than within \(\frac{h}{mc}\). During the collision the mean distance between the electrons will be \(\frac{h}{mc}\), and they will practically never approach one another to a distance of the order of the radius.

3. THE RELATIVISTIC WAVE EQUATION AND QUANTUM ELECTRODYNAMICS

The quantum theory of the electron required improvement in two directions: it had to be generalized to the case of high energies, in accordance with the theory of relativity, and it was also necessary to find a consistent treatment of the interaction of matter with radiation.

Dirac was the first to take steps in both directions. He succeeded in obtaining a wave equation for the electron which satisfied the requirements of the relativistic theory. He made use of the fact that the electron has an intrinsic angular momentum. The state of the electron can always be described as a superposition of states with spin parallel and antiparallel to a given direction of reference, just as is done for polarized light. Thus the electron wave was regarded as a “spinor” wave with two components corresponding to the two directions of spin. Dirac showed that for a relativistic wave equation it is necessary to introduce two further components, which in the case of small velocities are much smaller than the other two. The electron wave is completely described by specifying all four components.

Dirac’s relativistic wave equation determines the mechanical properties of such a four-component wave. For small kinetic energies (small in comparison with \(mc^2\)) two components become very small, while the other two in turn pass into the solutions of the nonrelativistic (Schrödinger)

wave equations, each of which corresponds to one of the two spin directions.

In the nonrelativistic theory it was necessary artificially to ascribe to the spins a certain magnetic moment $\mu$, whose value was taken from experiments. The Dirac relativistic equation implicitly contains the interaction of spin with a magnetic field. The resulting magnetic moment of the electron

\[ \mu=\frac{eh}{2mc} \]

is in almost exact agreement with experiment.

From the relativistic wave equation for the electron there follows a number of physical consequences with which it is difficult to agree. The equation admits solutions corresponding to states of a particle with negative rest mass. The kinetic energy of a particle in such states is negative, and this means that such a particle must move in a direction opposite to the motion of an ordinary particle. For example: such a particle with an electronic charge is repelled by the field of a proton. Such states, of course, are not realized in nature, if only because their energy is negative and, consequently, less than the energy of the very lowest state with positive rest mass. Consequently, transitions from ordinary states into states with negative rest energy, with the emission of light quanta, ought to take place. Thus there can be no stable ordinary states, since there is an infinite number of states with negative energy, and there is a probability of transition into these states with the emission of the corresponding light quantum. These states can be excluded from consideration only on the grounds that they do not exist in nature, since the ordinary states by themselves do not yet give a complete system of solutions. From the physical point of view this means the following: if, by means of a definite measurement, an electron is placed in some state, then it is very probable that it will be a combination of states, part of which will have negative rest mass. In the same case, when the electron is localized in a volume with linear dimensions smaller than the Compton “wavelength”

\[ \lambda_c=\frac{h}{mc}, \]

states with negative rest mass will be represented in large number.

Let us now turn to the Dirac treatment of radiation. In order to describe consistently the interaction between matter and radiation, it is necessary to “quantize” not only the motion of material particles, but also the electromagnetic field. By “quantization” we understand the consistent application of definite rules that make it possible to pass from classical mechanics to quantum mechanics.

It is relatively simple to apply these rules to the electromagnetic field in a vacuum. The field can be decomposed into “Fourier components” and regarded as a superposition of monochromatic waves. Each of these waves has dynamical properties very similar to those of a harmonic oscillator. Thus, the “quantization” of the electromagnetic field is equivalent to the quantization of a series of harmonic oscillators and, consequently, the energy of each monochromatic wave can change only by amounts that are multiples of \(h\nu\). Therefore, electromagnetic energy of frequency \(\nu\) must appear in portions of magnitude \(h\nu\). This is precisely the hypothesis of light quanta. The next important result is the presence of zero-point field fluctuations: a harmonic oscillator in the state of lowest energy still has some finite amplitude of oscillation. Applying this to the electromagnetic field, we conclude that even in the state of lowest energy there exist electromagnetic oscillations in space. The state of lowest energy is a state in which there are no light quanta, but in this state the mean squares of the field strengths do not vanish.

We shall now estimate the magnitude of the field fluctuations averaged over a volume \(V\) of linear dimensions \(a\) \((V=a^3)\). The amplitude \(B\) of the zero-point oscillations of an oscillator of frequency \(\nu\) is given by the expression:

\[ B \sim \left(\frac{h}{2m\nu}\right)^{\frac{1}{2}}; \]

it corresponds to an oscillation with energy \(\frac{h\nu}{2}\). The principal part of the field fluctuations in a volume \(a^3\) consists of oscillations with wavelength \(\lambda = \frac{c}{\nu} \sim a\). The amplitude corresponds to the energy \(\frac{h\nu}{2}\), i.e. to one half of a light quantum. The quantity \(\frac{1}{4\pi}\langle \mathcal{E}_{\mathrm{sr}}^2\rangle a^3\) is the energy of the field contained in \(a^3\); this must be equated to

\[ \frac{h\nu}{2}=\frac{hc}{2a}, \]

so that we obtain approximately:

\[ \mathcal{E}_{\mathrm{fluct}}^{2} \sim \frac{hc}{a^4}. \tag{5} \]

The field fluctuation is greater, the smaller the chosen volume.

The interaction between light and matter can now be described as the interaction between two quantized systems: the electromagnetic field on the one hand and the electron in the atom on the other. Such an interaction can be considered by means of the existing methods of quantum mechanics. The interaction energy is given by the classical expression:

\[ \int (\mathbf{j}\cdot \mathbf{A})\,dv, \]

where \(\mathbf{j}\) is the current density in the atom and \(\mathbf{A}\) is the vector potential of the field. The integral is taken over all space. The two variables \(\mathbf{j}\) and \(\mathbf{A}\) are physical quantities which are operated on according to the rules of quantum mechanics.

Dirac showed that by this method one can calculate the absorption, emission, and scattering of light, and that the result obtained is the same as when the correspondence principle is applied. For example, the emission of light in a transition from the state \(n\) to the state \(k\) is described in this theory in the following way. At a given moment, say \(t=0\), the emitting atom is in the excited state \(n\), and all electromagnetic oscillations are in their ground states. As a result of the interaction, the excitation energy \(E_n-E_k\) passes into one of the oscillations, whose frequency, of course, must satisfy the condition \(h\nu=E_n-E_k\). The probability \(P\) that, after the time \(t\), the excitation energy has passed into the field proves to have an exponential dependence on time: \(P=1-e^{-\Gamma t}\), where \(\Gamma\) is the probability of radiation per unit time.

The quantity \(\Gamma\) is given by the expression

\[ \Gamma_k=\frac{2e^2\nu_{nk}^{\,2}}{3mc^3}\,f_{nk}, \]

in agreement with the probability of radiation of an oscillator of strength \(f_{nk}\), determined according to (4). Dirac’s quantum electrodynamics gave a more consistent derivation of results obtained with the help of the correspondence principle, but it also introduced several new and serious difficulties.

The structure and size of the electron again appeared in the theory. The difficulties arose because of the interaction of the electron with the zero fluctuations of the field. Consider a free electron acted upon by the field of an oscillator of field strength \(\mathcal{E}=\mathcal{E}_0 e^{i\nu t}\); the electron performs forced oscillations of frequency \(\nu\) with displacement \(x_\nu\); the mean square \(\langle x_\nu^2\rangle_{\mathrm{av}}\) of this displacement and the mean square velocity \(\langle \dot{x}_\nu^2\rangle_{\mathrm{av}}\) of the free electron are given by the expressions:

\[ \left. \begin{aligned} \langle x_\nu^2\rangle_{\mathrm{av}}&=\frac{1}{2}\,\frac{e^2\mathcal{E}_0^{\,2}}{m^2\nu^4},\\[4pt] \langle \dot{x}_\nu^2\rangle_{\mathrm{av}}&=\frac{1}{2}\,\frac{e^2\mathcal{E}_0^{\,2}}{m^2\nu^2}. \end{aligned} \right\} \tag{6} \]

The kinetic energy of these oscillations of the electron is equal to

\[ E_\nu=\frac{1}{2}m\langle \dot{x}^{2}\rangle_{\mathrm{av}} =\frac{e^2\mathcal{E}_0^{\,2}\lambda^2}{4mc^2}, \tag{6a} \]

where \(\lambda\) is the wavelength corresponding to the frequency \(\nu\). Consequently,

zero oscillations of the field impart to the electron a certain amount of energy. Let us suppose for a moment that the electron is a sphere of radius \(a\). Then only waves with wavelength \(\lambda > a\) will act on the electron; waves with \(\lambda \gg a\) are not essential, so that in (6a) we may put \(\lambda = a\). If we now take the quantity (5) as the average value \(\langle \mathcal{E}_0^2\rangle_{\mathrm{av}}\) over the volume \(a^3\), we obtain for the energy \(E_{\mathrm{fluct}}\) acquired by the electron from the zero fluctuations of the field,

\[ E_{\mathrm{fluct}}\sim \frac{e^2 h}{4mca^2}. \tag{7} \]

In more exact calculations one must start from expression (6), which gives the action on the electron of the field strength of an oscillator with amplitude \(\mathcal{E}_0\) and frequency \(\nu\). To compute the quantity \(\mathcal{E}_0^2\) for zero oscillations, we enclose the electromagnetic field and the electron in a large volume \(\Omega\). The amplitude of the zero field \(\mathcal{E}_0 e^{i\nu t}\) of one corresponding oscillation can be computed by setting the total energy of the oscillation equal to

\[ \frac{1}{8\pi}\int \left(\mathcal{E}^2+H^2\right)\,dv = \frac{1}{8\pi}\mathcal{E}_0^2\Omega = \frac{h\nu}{2}; \qquad \mathcal{E}_0^2=\frac{4\pi h\nu}{\Omega}. \]

We use the well-known formula

\[ Z(\nu)\,d\nu=\Omega\left(\frac{\nu^2}{\pi^2 c^3}\right)d\nu, \]

which gives the expression for the number of normal oscillations in the frequency interval \(d\nu\). Since zero oscillations of different frequencies are statistically independent, their contributions to the value of the mean square displacement and velocity add, and we obtain expressions for these quantities:

\[ \langle x^2\rangle_{\mathrm{av}} = \int \langle x_\nu^2\rangle_{\mathrm{av}} Z(\nu)\,d\nu = \frac{2e^2h}{\pi m^2 c^3} \int_{\nu_0}^{\infty}\frac{d\nu}{\nu}, \tag{8} \]

\[ \langle \dot{x}^{\,2}\rangle_{\mathrm{av}} = \frac{2e^2h}{\pi m^2 c^3} \int \nu\,d\nu; \tag{9} \]

the integrals are taken from the lower limit \(\nu_0\) to \(\infty\). The frequency \(\nu_0\) depends on the character of the electron’s binding. \(h\nu_0\) is a quantity of the order of the binding energy. If the frequency of the field oscillations becomes smaller than the frequency \(\nu_0\), the electron can no longer be regarded as free, and expression (6) likewise becomes invalid. Taking account of the effect of binding is equivalent to discarding frequencies below \(\nu_0\).

Expressions (8) and (9) lead to infinities. This is especially unpleasant in the case of the square of the velocity, because it leads to an infinite kinetic energy \(E_{\text{fluct}}\) acquired by the electron owing to zero-point fluctuations,

\[ E_{\text{fluct}}=\frac{m}{2}\langle \dot{x}^{2}\rangle_{\text{av}} =\frac{e^{2}h}{\pi mc^{3}}\int_{0}^{\infty}\nu\,d\nu . \tag{10} \]

This expression contains a quadratically divergent integral. Since this energy is a necessary part of the total energy of the electron, it must constitute a part of its rest energy \(mc^{2}\). In order to keep the mass finite, it is therefore necessary to make certain assumptions about the structural properties of the electron which would exclude interaction with high frequencies of the field. We can do this by introducing an upper limit \(\nu_{\max}\) for the frequencies interacting with the electron. At this limit we cut off the integral in (10). The fluctuation energy takes the form:

\[ E_{\text{fluct}}=\frac{e^{2}h}{2\pi mc^{3}}\nu_{\max}^{2} \tag{7a} \]

and we can determine the upper value for \(\nu_{\max}\) by setting \(E_{\text{fluct}}\) equal to \(mc^{2}\):

\[ h\nu_{\max}\leqslant \left(2\pi\frac{hc}{e^{2}}\right)^{\frac{1}{2}}mc^{2} \simeq 15\ \text{Mev}. \tag{7b} \]

This restriction must eliminate the interaction of a quantum whose energy exceeds \(15\ \text{Mev}\) with a resting electron, which is very improbable. The introduction of \(\nu_{\max}\) corresponds to the assumption that the radius of the electron is \(a=c/\nu_{\max}\), whence it is clear that (7) and (7a) are equivalent.

From equation (7b) it follows that the quantity \(a\simeq\left(\dfrac{hc}{e^{2}}\right)^{-\frac{1}{2}}\). This quantity is larger than the classical limit (3); thus the radius of the electron obtained by considering the fluctuation energy turns out to be even larger than the electron radius that we obtain by considering the energy of the electric field. It should be noted, however, that in interactions with a light quantum whose energy is greater than \(2mc^{2}\), solutions with negative mass play an essential role. Thus these states will play an essential role in considering the problem of the self-energy of the electron.

The two Dirac generalizations of quantum mechanics—the relativistic wave equation and quantum electrodynamics—proved to be very significant for the solution of certain questions, namely: the explanation of the magnetic moment of the electron, the derivation of the Sommerfeld

fine-structure formulas, a consistent derivation of expressions for the absorption, emission, and scattering of light. But at the same time two fundamental difficulties arose:

1) The existence of electron states in which the mass is negative. This fact causes the instability of the normal bound state, owing to the possibility of emitting a quantum of high energy with a transition to a state with negative mass. Thus, the “normal” states of the electron have a very strong “resonant” interaction with a light quantum of high energy.

2) Quantization of the electromagnetic field introduces zero-point fluctuations of the field. In order that their contribution to the electron’s energy remain within the limits of the observed value of the mass, it is necessary radically to change the interaction of the electron with light quanta of energy \(h\nu > (137\,mc^2)^{\frac{1}{2}}\). In the following section it will be shown that the theory of the positron removed the first difficulty and completely changed the aspect of the second.

4. THEORY OF THE POSITRON

The phenomenon of the birth of a positron and an electron by a light quantum introduces a new aspect into the theory of the electron. This fundamental process may be described as follows: a light quantum whose energy is greater than \(2mc^2\) can be absorbed by the vacuum in the presence of strong electric fields; in this event there arises a pair consisting of a positive and a negative electron.

Two outstanding facts appear in this phenomenon: the existence of the positive electron and the fact that the vacuum has physical properties that allow it to absorb light and produce electrons. Consequently, the physical description of the vacuum must be more complex than hitherto, and must contain hidden electron pairs capable of being produced.

Dirac succeeded in turning a defect of the theory into its virtue by applying the inadmissible states with negative mass to the description of the vacuum. Dirac’s interpretation of these states gives an almost perfect description of the vacuum and explains the existence of positrons. States with negative mass correspond, in a certain sense, to states of a particle with the opposite charge, since they move in opposite directions in any electromagnetic field. However, they are still inadmissible, since they have negative kinetic energy. The interpretation that removes this difficulty may be formulated as follows: according to the Pauli exclusion principle, any state may be either occupied by one single electron or unoccupied.

Filling a state with energy \(E_i\) increases the total energy of the system by \(E_i\); the departure of an electron from this state decreases the total energy by \(E_i\). Dirac’s interpretation of states with negative mass consists in interchanging the words “filling” and “departure.” We decided to call a filled state with negative mass “empty,” and an empty state “occupied.” The transition from “empty” to “occupied” is then associated with a change of energy by \(-E_i\). Since \(E_i\) itself is negative, the energy in reality increases by \(+(E_i)\). The difficulty with negative energy is thus removed.

Now the vacuum can be formally described by assuming that all states with negative mass are occupied by electrons. “In reality” they are not occupied; this is only our interpretation, and therefore there is no need to worry about the infinite charge density that would result if all states with negative mass were in fact occupied. The wave functions that represent the absence of positrons are the very same functions that would represent the presence of electrons with negative mass. What is new here is that the “absence” of a particle is now described by a wave function. This is, of course, an expression of the fact that the vacuum has the physical properties described above; it is filled with “hidden” electrons. This interpretation immediately removes the difficulty introduced by states with negative mass. Since these states in the vacuum are occupied, an electron from normal states cannot jump into them. Thus the normal states are no longer unstable with respect to a transition into a state with negative mass. They are no longer in “resonant” interaction with an arbitrarily large light quantum.

The creation of pairs is now described as follows: a light quantum causes the transition of an electron from a filled state with negative mass into a state with positive mass. As a result, we obtain an electron in a state with positive mass and an unoccupied state with negative mass. The latter must be interpreted as an occupied state of a positron with positive mass. Consequently, the light quantum creates two particles—one positive and one negative—with positive masses.

Such a transition can occur only in the presence of external fields. Without these fields it is impossible to satisfy the laws of conservation of energy and momentum. The probability of such transitions can be calculated, and the results agree excellently with the experimental material. The opposite process is the annihilation of a positive and a negative electron, with the emission of either one quantum or two quanta in a free from

from the field space. This process may be described in our picture as the transition of an electron into an “unfilled” state, which is represented by a positron. This transition is accompanied by the emission of light quanta.

The new interpretation of the vacuum has had a powerful effect on the problem of the electron’s self-energy. The properties of the vacuum with respect to electrons are now, in a certain sense, analogous to its properties with respect to the electromagnetic field; namely, there exist zero-point fluctuations of the electric charge and of the electric current in the vacuum. These fluctuations are very small if they are averaged over a volume with linear dimensions greater than the Compton wavelength

\[ \lambda=\frac{h}{mc}; \]

they are virtual electron pairs which, under the action of light quanta, can become real.

Let us now consider the properties of the “vacuum” near a real electron. Then there will exist an interaction, mainly by virtue of Pauli’s exclusion principle, between this electron and the virtual charges. According to this principle, electrons cannot come close to one another. Two electrons of the same spin do not approach closer than a distance \(d\), determined by their relative momentum \(p\):

\[ d \sim \frac{h}{p}. \]

(They must not be in one and the same elementary volume of phase space.)

The presence of a real electron in the vacuum introduces certain changes into the “charge distribution” of the vacuum. In the absence of a perturbation this charge distribution must, on the average, be equal to 0. The wave functions which represent electrons of negative mass also change somewhat near a real electron. This change in the charge distribution, in comparison with the unperturbed vacuum, appears as a spreading out of the electron’s charge, since the vacuum electrons are repelled by the real electron. Calculations show that this spreading is sufficient to change the classical electrostatic self-energy to

\[ \frac{e^{2}}{h c} mc^{2}\ln\frac{\lambda_{c}}{a}. \]

Here \(a\) is the “radius” of the electron, or, more precisely, \(a\) is such a limiting wavelength that fields with \(\lambda<a\) do not interact with the electron. Still more radically changed, with the introduction of our new conception of the vacuum, is the influence of the oscillations of the zero-point field on the electron. This occurs because the field oscillations also interact with the virtual electron pairs of the vacuum. So long as their frequencies are much less than

\[ \frac{2mc^{2}}{h} \]

(the minimum quantum frequency at which a pair can be created), the influence of the “vacuum” is very small and the earlier calculations (6)

displacements \(\langle x_\nu^2\rangle_{\mathrm{av}}\) and their velocities \(\langle \dot{x}_\nu^2\rangle\) while the laws hold. For frequencies greater than \(\dfrac{2mc^2}{h}\), the oscillations of the field already strongly affect the virtual electron pairs, and the induced fluctuations of the charge and current of the vacuum interfere with the induced fluctuations of the electron itself. This interference somewhat reduces the magnitude of the induced displacement and velocity.

In order to estimate this reduction roughly, one must multiply expression (6) for \(h\nu>2mc^2\) by the factor \(\left(\dfrac{mc^2}{h\nu}\right)^2\).

\[ \left. \begin{aligned} \langle x_\nu^2\rangle_{\mathrm{av}}&=\frac{1}{2}\frac{e^2\mathcal{E}_0^2}{m\nu^4}\left(\frac{mc^2}{h\nu}\right)^2,\\ \langle \dot{x}_\nu^2\rangle_{\mathrm{av}}&=\frac{1}{2}\frac{e^2\mathcal{E}_0^2}{m\nu^2}\left(\frac{mc^2}{h\nu}\right)^2 \end{aligned} \right\} \quad \text{for } h\nu>2mc^2, \tag{6'} \]

Qualitatively, this effect is difficult to explain. It is connected with the Pauli exclusion principle, according to which electrons tend to move away from one another. Thus, the fluctuations of the charge and current of the vacuum near the electron tend to be in opposite phase with respect to the fluctuations of the electron itself and, therefore, the latter are diminished because of interference. Taking these effects into account constitutes a certain improvement of the theory. The mean displacement \(\langle x^2\rangle_{\mathrm{av}}\) no longer leads to infinities. The divergent integral in (8) now converges for frequencies greater than \(\dfrac{2mc^2}{h}\), to expressions (6′); then we obtain:

\[ \langle x^2\rangle_{\mathrm{av}}=\frac{2e^2h}{\pi m^2c^3}\ln\frac{fmc^2}{h\nu_0}, \tag{11} \]

where \(f\) is a factor of order 1, which can be determined if the contribution from higher frequencies is taken exactly into account. The mean square velocity (9) remains infinite, but the divergence is only logarithmic. From (6′) we obtain:

\[ \langle \dot{x}^2\rangle_{\mathrm{av}}\sim \frac{2e^2h}{\pi m^2c^3} \left[ \int_0^{\frac{2mc^2}{h}}\nu\,d\nu+ \left(\frac{mc^2}{h}\right)^2 \int_{\frac{2mc^2}{h}}^\infty \frac{d\nu}{\nu} \right]. \]

The fluctuation energy

\[ E_{\mathrm{fluct}}=\frac{m}{2}\langle \dot{x}^2\rangle_{\mathrm{av}} \]

takes the form

\[ E_{\mathrm{fluct}}=\frac{e^2}{\pi hc}mc^2\ln\left(\frac{fh\nu_{\max}}{mc^2}\right), \]

where \(f\) is a numerical factor and \(\nu_{\max}\) is the frequency at which the cutoff occurs. To keep this energy smaller than the total energy \(mc^2\) of the electron, it is now sufficient to take

\[ a=\frac{c}{\nu_{\max}} \]

larger than

\[ \frac{h}{mc}\exp\left(-\frac{he}{e^2}\right). \]

This lower limit is much smaller than any length considered up to now. There is now no need to interfere arbitrarily with the interaction of the electron with light quanta of energies of several \(M_{\beta\theta}\).

Of course, it is still unsatisfactory that one cannot choose a limit equal to infinity without thereby obtaining infinite proper energies; thus, the internal structure of the electron will appear somewhere in the theory. However, some changes in the interaction between light and matter must certainly occur at very high energies, when we can quite expect the appearance of new phenomena (of the nuclear or meson type). Up to now we have discussed the influence of the “real” electron on the vacuum, which takes place as a consequence of the Pauli principle. There is still another influence, although weaker, caused by the electrical interaction and manifested in the form of a displacement of the vacuum electrons. This effect is easier to describe not for a “real” electron, but for a proton placed in the vacuum. The wave functions of all states with negative masses are deformed in the presence of the proton. Since the vacuum is described by undeformed states, the difference between the deformed and undeformed states must give the ordinary charge density. This is called the polarization of the vacuum by an external charge (the proton).

The proton induces in the vacuum a charge density \(\rho_i\). Calculation shows that \(\rho_i(\mathbf r)\), as a function of position \(\mathbf r\), has the following form:

\[ \rho_i(\mathbf r)=A\rho_0(\mathbf r)+\int G(\mathbf r-\mathbf r')\rho_0(\mathbf r')\,d\mathbf r'. \tag{12} \]

Here \(\rho_0(\mathbf r)\) is the density of the external charge; in our case \(\rho_0\) is the charge density of the proton, \(A\) is a constant, and \(G(\mathbf r-\mathbf r')\) is a function of the distance between the points \(\mathbf r\) and \(\mathbf r'\). The integral is extended over all points \(\mathbf r'\). The expression for the induced charge consists of two parts. The first term is directly proportional to the density of the inducing charge \(\rho_0\); the second part is an action at a distance.

According to this term, a point charge at \(r=0\) (for example, a proton) must give a charge distribution \(G(r)\). \(G(r)\) is different from zero only at distances smaller than the Compton wavelength \(\lambda_c\). The effect is just the same as if the dielectric constant of the vacuum were different from unity by approximately \(1/137\).

within a region of the order of \(\lambda_c\). It is important to note that the first part is in principle unobservable*). It cannot be separated from the initial charge density \(\rho_0\), since it is always induced by it. What is usually measured in practice as the charge of the proton is not \(e\), but \((1+A)e\). Thus, this is nothing other than a renormalization of the charge. Only the second term has physical meaning. But with such an interpretation we encounter a serious difficulty—the factor \(A\) diverges logarithmically

\[ \left( A \sim \frac{e^2}{hc}\ln \frac{\lambda_c}{a} \right), \]

if the “cutoff” radius is set equal to zero. This would mean that the external charge of the proton \(\rho_0\) induces in the vacuum, at the place where it is located, a charge that changes the initial value by an infinite amount. True, this change is not observable by itself, since in practice one always observes the total charge—the external plus the induced one. However, the fact that the induced charge is infinite for \(a=0\) presents a serious difficulty for the theory.

The vacuum is polarized not only by the proton, but also by the electron. In this case the situation is complicated by exchange phenomena between the external electron and the electrons of the vacuum. However, the fact remains valid that the electron, regarded as a point \((a=0)\), also induces in the vacuum a charge that is an infinite addition to its initial charge. Thus, the internal structure of the electron is related not only to its mass, but also to its charge.

One can make these infinite additions finite, without changing the second term in (12), if one arbitrarily discards the interaction with electrons of the field having a wavelength smaller than \(a\). Here, just as in the case of the self-energy, the infinity is obtained from interaction at very high energies. One may hope that a future theory will modify this interaction in such a way that the constant \(A\) remains finite and small. In spite of these difficulties, the theory of the positron may be regarded as a great step forward in our understanding of the electron. Indeed, with the aid of Dirac’s new interpretation of states with negative masses, it became possible to explain the new phenomena of pair creation and annihilation and to remove several fundamental difficulties of the Dirac equation:

1) Radiative transitions from ordinary states to states with negative masses are excluded.

*) As the reader has noted, Weisskopf sometimes formulates objective physical regularities in the language of subjective perception, characteristic of the positivists. In fact, in the present context the point is not the fundamental unobservability of the effect, but that its first part is inseparable from the very concept of the magnitude of charge. (Ed.)

2) The energy of the fluctuations is much less sensitive to the structure of the electron, owing to its logarithmic dependence on the radius of the electron.

3) The mean-square displacement of the electron under the action of field fluctuations is finite and does not depend on the radius or structure of the electron.

5. EXPERIMENTAL TEST OF QUANTUM ELECTRODYNAMICS

The quantization of the electromagnetic field has so far not been especially successful. True, it has made it possible to derive expressions for the absorption, emission, and scattering of light, which formerly were based only on recipes obtained with the aid of the correspondence principle. On the other hand, new difficulties have appeared, connected with the zero-point oscillations of the electromagnetic field and their influence on the electron’s self-energy. Quantum electrodynamics has not yet shown its superiority over the correspondence principle. On the contrary, its modern expressions for electromagnetic phenomena become meaningless, since, consistently adhering to the theory, we have to take the mass of the electron to be infinite in all cases where it occurs.

Encouraged by some new experiments, which will be discussed later, scientists have recently attempted to find observable effects directly connected with the new features introduced by quantum electrodynamics. The principal theoretical difficulty lay in how to separate the infinities of masses and charges from the rest of the theory in order to obtain results having physical meaning. This was done by separating the expressions for the infinite mass and charge within the theory, in the hope that the mass and charge become finite upon further improvement of the theory. This procedure is possible because the terms of the self-energy and the infinite charge arise chiefly from interaction with quanta of light of very high energies and therefore, over wide limits, do not depend on the interaction of the electron with the fields usually encountered in nature. Therefore they can be isolated as an additional mass and charge of the electron. For the charge this was already shown in the preceding paragraph in the discussion of expression (12). The separation of the term representing the mass is mathematically much more complicated, but can be done in an analogous way. It turns out that the transformation relativistic properties of the terms occurring in the calculation are very important for finding a consistent rule indicating what part of the expression for the self-energy is to be regarded as the term,

representing mass. It was necessary to formulate quantum electrodynamics anew so that the relativistic invariance of the theory would leave no doubt and would become clearer. This very difficult task was carried out by Schwinger and, independently of him, by S. Tomonaga.

There is, however, a small part of the self-energy which is not contained in the mass and which arises from interaction with low-frequency oscillations. It depends on external conditions and can lead to a small shift of the energy levels, depending on the binding conditions, and also to a small change in some of the fundamental properties of the electron. This is due mainly to the effect of the displacement of the electron under the action of zero-point oscillations, whose mean square \(\langle x_\nu^2\rangle_{\mathrm{av}}\) turns out to be finite and is entirely determined by interaction with low frequencies. In the case corresponding to the real experiment, namely in the case of the displacement of the levels of hydrogen-like atoms, this can be shown by quite simple calculations.

Let us consider a stationary state \(n\) of an electron in a Coulomb field, whose wave function is \(\psi_n\). The Coulomb field is described by the potential energy \(V(r)=Ze^2/r\), where \(r\) is the distance from the nucleus. The mean potential energy \(\langle V\rangle_{\mathrm{av}}\) in the state \(n\) can be expressed in the form:

\[ \langle V\rangle_{\mathrm{av}}=\int V(r)|\psi_n(\mathbf r)|^2\,dv, \tag{13} \]

where \(|\psi_n(\mathbf r)|^2\) is the well-known probability of finding the electron at the point \(\mathbf r\); the integral is taken over volume. This expression must be changed, taking into account the existence of zero-point oscillations. It is assumed in advance that the effect of these oscillations on the electromagnetic mass is contained in the observed mass \(m\) of the electron. However, there is also an influence on the potential energy, since the electron is forced to oscillate about the position \(\mathbf r\). It will be shown below that this oscillation slightly changes the mean value of the potential energy. This change is what causes the shift of the energy levels.

To calculate this change, in equation (13) we replace \(V(\mathbf r)\) by \(V(\mathbf r+\mathbf x)\), where \(\mathbf x\) is the zero-point oscillation of the electron. We apply the Taylor expansion, owing to the smallness of the quantity \(\mathbf x\)*:

\[ V(\mathbf r+\mathbf x)=V(\mathbf r)+\operatorname{grad}V\cdot\mathbf x+\frac12\,\Delta V\,\frac{\mathbf x^2}{3}, \tag{14} \]

\[ \text{*) The simplest form of the third term in (14) is obtained because, on the average, } \langle x_x x_y\rangle_{\mathrm{av}}=0,\quad x_x^2=x_y^2=x_z^2=\frac{\mathbf x^2}{3}. \]

where \(\Delta V\) is the Laplace operator acting on \(V\):

\[ \Delta V=\left[\frac{\partial^2}{\partial x^2}+\frac{\partial^2}{\partial y^2}+\frac{\partial^2}{\partial z^2}\right]V . \]

The second term is on the average equal to zero, since \(X\) is a random quantity. Thus, the addition \(\delta E_n\) to the mean potential energy of state \(n\) has the form:

\[ \delta E_n=\frac{1}{6}\int \Delta V\cdot \langle x^2\rangle_{\mathrm{cp}}|\psi_n(\mathbf r)|^2\,dv . \]

The Laplacian of the Coulomb potential is proportional to the density of the charge \(\rho_0\) that creates it: \(\Delta V=4\pi e\rho_0\); \(\rho_0\) is the charge density of the nucleus, which we approximately represent by a \(\delta\)-function\(^*\) \(\rho_0=Ze\delta(r)\), where \(Ze\) is the charge of the nucleus. Thus, for \(\delta E_n\) we obtain:

\[ \delta E_n=\frac{2\pi}{3}Ze^2|\psi_n(0)|^2\langle x^2\rangle_{\mathrm{cp}}, \tag{15} \]

where \(|\psi_n(0)|^2\) is the intensity of the wave function in the nucleus; to calculate the displacement of the level, we substitute the expression (11), found for \(\langle x^2\rangle_{\mathrm{cp}}\), into equation (15).

The frequency \(\nu_0\) appearing in equation (11) depends on the electron binding and is of the order of the Rydberg frequency \(\nu_R\) for an electron in a hydrogen-like atom. Since \(\nu_0\) appears only under the logarithm sign, it is not important to know its exact value.

Bethe showed that for the quantum state \(n\), \(\nu_0\) is expressed by the formula

\[ \ln h\nu_0= \frac{\sum_m |P_{nm}|^2(E_m-E_n)\ln|E_m-E_n|} {\sum_m |P_{nm}|^2(E_m-E_n)} , \]

where \(E_n\) is the energy of state \(n\), and the sums are extended over all other quantum states \(m\). \(P_{nm}\) is the matrix element of the momentum for the transition from state \(n\) to state \(m\).

It can be seen that \(|\psi(0)|^2\) vanishes for all states except \(S\)-states (states with orbital angular momentum equal to zero), for which the simple relation holds:

\[ |\psi_n(0)|^2=\frac{Z^3}{\pi l^3 n^3}, \tag{16} \]

where \(l=\dfrac{h^2}{me^2}\) is the Bohr radius. Thus, the displacement of levels vanishes for states with orbital angular momentum different from zero. For \(S\)-states, the level displacement is obtained finally from

\[ \text{* } \delta\text{-function } \delta(r) \text{ is equal to zero everywhere except } r=0. \text{ It is normalized so that the volume integral } \int \delta(r)\,dv \text{ is equal to } 1. \]

equations (15), (11), (16). In practice it is convenient to express it in the form of a relative shift, by dividing \(\delta E_n\) by the energy \(E_n\) of the level, which is given by the Balmer relation \(E_n=\dfrac{Z^2me^4}{2\hbar^2n^2}\):

\[ \frac{\delta E_n}{E_n} = \frac{8}{3\pi} \left(\frac{e^2}{\hbar c}\right)^3 \frac{Z^3}{n} \ln\frac{fmc^2}{h\nu_0}. \tag{17} \]

An exact calculation for the term \(2S_{1/2}\) gives the value: \(\nu_0=18\nu_r\), \(f=1.3\). Thus the \(S\)-levels of hydrogen-like atoms must be shifted upward (\(\delta E_n\) positive) by small amounts relative to the values given by Sommerfeld’s formula. This is the direct action of zero-point oscillations, and its experimental verification gives strong support to quantum electrodynamics.

The polarization of the vacuum by the proton also gives a shift which must be added to (17).

As follows from the arguments in the last paragraph, the only observable effect is the small polarization around the proton over a distance \(\lambda_c\). Calculation shows that it causes a shift of the line \(\delta E'_n\):

\[ \frac{\delta E'_n}{E_n} = -\frac{8}{15\pi} \left(\frac{e^2}{\hbar c}\right)^3 \frac{Z^3}{n}. \tag{18} \]

Its magnitude is only about \(1/40\) of the shift \(\delta E_n\). The most reliable experiment on the determination of the line shift was carried out by Lamb and Retherford on hydrogen. According to Sommerfeld’s formula, the levels \(2S_{1/2}\) and \(2P_{1/2}\) of the hydrogen atom coincide in energy, while the level \(2P_{3/2}\) should lie 10,000 megacycles higher. Lamb and Retherford measured the level \(2S_{1/2}\) relative to two other levels and found that the level \(2S_{1/2}\) is shifted upward by approximately \(1060\) megacycles, which agrees well with the theoretical formula (17). Similar shifts were found by Mack and Kusch and by Paul in helium. The measurements carried out are not accurate enough to reveal such small shifts as those caused by vacuum polarization (see (18)). Subsequent experiments will show whether this additional effect may be regarded as actually existing.

Another important result obtained by these methods is the correction to the \(g\)-factor for the electron. According to the Dirac equation, the magnetic moment of the electron \(\mu_e\) is equal to \(\dfrac{h e}{2mc}\). The ratio of this quantity to the mechanical moment of the electron \(\dfrac{h}{2}\) is equal to \(g\left(\dfrac{e}{2mc}\right)\), where \(g=2\), in contrast to the value of this ratio for orbital motions, for which \(g=1\). If the interaction of the electron with the radiation field is correctly taken into account, it is found that \(g\) is not exactly equal to 2, but

\[ g=2+\frac{e^2}{\pi\hbar c}. \]

Unfortunately, it is impossible to give a qualitative description of this effect in the same way as the shift of levels was explained.

The spin of the electron in itself is a phenomenon that cannot be represented visually. A path to understanding this effect may be found if one recalls that the magnetic moment of the Dirac electron is a consequence of circular currents with radius \(\frac{h}{mc}\). The zero-point oscillations of the electromagnetic field affect these currents to some extent, and this creates fluctuations of the current induced in the “vacuum.” These interactions cause a small change in the magnetic moment. The quantitative result is in excellent agreement with the latest experimental measurements\(^2\). The magnetic moment of the electron was determined with great accuracy from the Zeeman effect on several doublets of fine structures. Although the correction to the \(g\)-factor cannot be interpreted simply, it is a most important result of quantum electrodynamics, since it concerns one of the fundamental properties of the free electron—its magnetic moment.

The great successes of quantum-electrodynamical ideas in these two separate cases prove that there is a large element of truth in the basic ideas. The main achievement of recent investigations consists in the fact that a consistent and relativistically invariant way was found to separate those effects of interaction between light and the electron which can be interpreted as an additional mass and charge from the other effects giving the observed phenomena. The additional mass and charge are contained in the observed values \(m\) and \(e\) and can never be observed by themselves. It should not be forgotten, however, that these quantities are still infinite in the present theory. This means that the interaction of the electron with light quanta of very high energy is still not understood. At some very high energies the internal structure of the electron, in some manner still unknown, will have to play an essential role in the future theory. At present this structure manifests itself in the form of an arbitrary length \(a\), which we have introduced as the radius of the electron in order to make its mass and charge finite quantities. The importance of the latest achievements lies in the recognition of the following fact: in problems connected with the structure of the atom, only the mass and charge of the electron are “structure-dependent” (i.e. depend on the magnitude \(a\) and tend to infinity when \(a\) is equal to zero), whereas all other effects, such as scattering cross sections, energy levels, magnetic moments, etc., can be calculated without any assumptions whatever concerning the structure of the electron.

It may be of some significance that the theory of the electron in the region of high energies cannot be given in a completely satisfactory form without introducing something new. It cannot be a pure accident that the charge of the proton and of the meson

is equal to the charge of the electron, or that the classical radius of the electron, $r_0$, is almost equal to the range of action of the nuclear forces. There must exist a connection between quantum electrodynamics and the future theory of mesons and nuclear forces, which at present exists only in embryonic form. The connection between these theories should be significant for the electron only at energies of the order of the meson rest energy or higher. This value is sufficiently large (more than $100$ MeV), and therefore the results of the theory in the region of atomic energies remain unchanged. One may hope that an understanding of this connection will lead to the solution of the problem of the electromagnetic mass and the induced charge of the electron. We noted, in considering the classical electron theory, that the scattering experiment by means of which the limits of applicability of the classical theory are established reveals the existence of positrons. This fact is of fundamental importance for the further development of the theory.

The experiment by means of which attempts are made to test the applicability of the modern theory in the region of high energies ($100$ MeV and above) will probably lead to the production of mesons. It is possible that this will indicate the large role of mesons in the future theory of the electron. With the aid of the new accelerators now being constructed, it will be possible to reach these critical energy values. One may hope that the phenomena discovered by means of these new instruments will shed new light on the fundamental problems of the connection between elementary particles.

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