PROBLEMS OF THE NEW QUANTUM THEORY OF THE ELECTRON*

V. Weisskopf, Zurich

1. Introduction and general survey. 2. Schrödinger’s wave mechanics of the electron. 3. Relativistic Schrödinger equation. 4. Electron spin. 5. Dirac’s electron theory. 6. Positron theory. 7. Interaction of matter and the electromagnetic field.

1. Introduction and General Survey

Planck’s discovery of the quantum of action brought about a revolution in atomic physics; after the construction of quantum, or wave, mechanics, the transformation of atomic physics attained a certain completeness.

The foundations of the new atomic physics showed what concepts and what mode of thought are necessary for ordering and for a principled understanding of the whole mass of newly discovered and, it seemed, contradictory facts. In the present article the limits of the new atomic physics will be considered, as well as the difficulties that arose in the course of its development. The discussion will not concern such newly discovered phenomena which cannot be described, or can be described only with difficulty, by the quantum mechanics of the atom, such as, for example, the results of nuclear physics; within the theory itself difficulties and contradictions have been found which have shown that the new atomic mechanics can represent only a first approximation to a correct theory, from whose true understanding we are, unfortunately, still very far.

Atomic mechanics is essentially the mechanics of electrons or other very light particles under the influence of electromagnetic fields.

Classical mechanics and electrodynamics are applicable to such small particles only with great restriction. This restriction is connected with Heisenberg’s uncertainty relation: thus, for example, it has no physical meaning to specify simultaneously the position and momentum of an electron with arbitrary precision. The permissible precision is so great that this restriction has no significance in the macroscopic picture; however, for the electron this

* Naturwissenschaften 23, 631, 1935, translated by S. A. Kameneshkov.

the limitation is of fundamental importance, preventing an exact space-time determination of the state of motion. This circumstance gives the key to understanding the apparent contradiction in the behavior of the electron, which exhibits now the properties of a wave, now the properties of a particle; this contradiction may be regarded simply as a difficulty of mental representation, since an unambiguous experimental resolution of the question of the correctness of the wave or corpuscular picture—so that one could consider one of them true and the other false—is impossible: for this it would be necessary to make measurements with greater accuracy than is permitted by the Heisenberg limitation.

Every state of motion of the electron can in quantum mechanics be described by a certain wave function. Experiments with the refraction and interference of electron beams have shown that essential features of the observed phenomena of motion can be represented in the form of wave propagation, and the waves may be superposed and annihilate one another, as is well known for other wave fields (the principle of superposition of matter waves). However, at the very foundation of wave mechanics lies the circumstance that, for the description of all the properties of the electron, neither a purely corpuscular nor a purely wave picture is suitable.

Therefore the wave function corresponding to a certain state of motion is not a measurable wave propagating in space and in time, although it does possess the corresponding properties in certain experiments. Its physical meaning is less concrete; rather, it has a symbolic character; but precisely because of this the significance of the wave function is broader: by means of this same function one can compute the results also of those experiments in which the electron manifests itself as a corpuscle.

The wave functions of the electron are determined by the wave equation, which was first obtained by Schrödinger. In the present article we shall be concerned specifically with developing the theory of electrons beyond the limits of Schrödinger wave mechanics.

Quantum mechanics, constructed on the basis of Schrödinger’s wave equation, is a complete and internally consistent theory which has significant achievements. It made it possible to explain the apparent paradoxes that are unavoidable under a visual space-time interpretation of atomic phenomena, and was able to interpret qualitatively and quantitatively a large number of important experimental facts of atomic physics.

These include: the structure and values of the spectral terms of atoms and molecules, if one disregards the multiplet structure of the terms, the intensities and selection rules of spectral lines, scattering phenomena and the Raman effect, the homeopolar bond, and much else that we shall not enumerate separately here.

However, Schrödinger wave mechanics must be regarded only as a first approximation to reality for the following reasons:

a) it does not satisfy the requirements of the theory of relativity, and therefore can be applicable only to particles whose velocity is small in comparison with the velocity of light;

b) it takes no account at all of the spin of the electron, which manifests itself above all in the phenomena of the multiplet structure of spectral terms and in the anomalous Zeeman effect;

c) it does not consider the process of the creation or disappearance of positive and negative electrons in the absorption or emission of light (the creation of electron pairs);

d) it does not resolve the difficulties already present in the classical theory of electrons, connected with the structure of the electron and leading to an infinite energy of its own electromagnetic field.

The problems listed here require an extension of wave mechanics, which, however, could not be carried out in such a way that the original completeness and consistency of the theory would not suffer. All attempts made in this direction have led, alongside remarkably good quantitative results, to new problems and contradictions, so that we are still very far from a complete and finished theory of the electron.

The possibility of the transformation of light into matter and conversely has at present already been proved by numerous experiments. Therefore we shall only briefly indicate what can be said about this process without resorting to any further hypotheses concerning the very mechanism of the phenomenon. The results of the experiments are limited only by the fact that the absorption of light can produce one negative and one positive electron. Since, on the basis of the theory of relativity, every mass \(m\) is equivalent to an amount of energy \(mc^2\) (where \(c\) is the velocity of light in vacuum), the amount of absorbed light energy must be at least equal to \(2mc^2\), if \(m\) is taken to be the mass of the electrons, the same for both kinds of electrons. The excess of absorbed energy will pass into the kinetic energy of both newly formed electrons. Thus, if the process under consideration takes place as a result of the absorption of only one single light quantum, then the frequency of the light must be at least equal to \(\frac{2mc^2}{h}\), and, consequently, the wavelength less than \(25\ X\). Thus, what is involved here can only be very hard \(\gamma\)-radiation.

Further, from the law of conservation of total momentum one can conclude that the creation of a pair of electrons upon absorption of a single light quantum cannot occur in space free of any field, but only in the vicinity of strongly charged nuclei, which can take up the excess momentum of the light quantum thanks to the presence of their electric field. If, for example, we suppose that the light quantum had an energy \(h\nu\), only slightly exceeding \(2mc^2\), then it must have possessed a momentum

\(h\nu = 2mc^2\), which is available only to electrons moving with velocities close to the speed of light. But this considerable impulse cannot be absorbed by the emerging pair of electrons, since almost all the energy of the light quantum will be spent on the formation of the masses of the electrons themselves, so that this pair of electrons will receive an entirely insignificant kinetic energy. But two light quanta, the sum of whose energies is at least \(2mc^2\), can give rise to a pair of electrons also in space free of fields.

Analogous reasoning may also be applied to the inverse process—the annihilation of one positive and one negative electron with the emission of one or several light quanta; they are always justified in principle, since in general, with modern experimental possibilities, a quantitative check can be carried out.

In the next section we shall first consider Schrödinger’s wave mechanics, in order to establish a connection with the subsequent arguments. The third section is devoted to the relativistic generalization of wave mechanics, carried out by Schrödinger and Gordon, which led to the so-called scalar relativistic wave equation. This attempt soon had to be abandoned, since, on the one hand, it could not take account of the electron spin and, on the other hand, it led to positive and negative electron charges. This latter circumstance, before the discovery of positive electrons—which occurred considerably later than the establishment of the scalar relativistic wave equation—constituted a shortcoming; at the present time this generalization is of great interest and, perhaps, deserves more detailed consideration. The fourth section examines the influence of spin on wave mechanics: the electron wave must be regarded as having several components. However, the properties of spin require a peculiar relativistic generalization, which led to the establishment of Dirac’s wave equation, as we shall see in the fifth section. Thanks to this, it proved possible to encompass simultaneously the properties of spin and the relativistic properties of the electron, and in applications to many atomic problems exceptionally good quantitative results were obtained. Unfortunately, this wave equation contains remarkable states of the electron in which it possesses negative mass and which cannot be excluded in any way. Dirac, by his bold idea of the theory of “holes,” wished to turn this defect into an advantage and at the same time to explain the existence and origin of positive electrons. This attempt and its problems will be discussed below, in the sixth section. Finally, in the last section the difficulties arising when the interaction of the electron and the electromagnetic field is taken into account are considered; these difficulties, apparently, must lead to a revision of our conceptions concerning the connection between matter and field.

2. Schrödinger’s Wave Mechanics of the Electron

In constructing the wave mechanics of the electron, it is best to start from rectilinear motions in this space, free from the action of forces. The wave functions corresponding to these motions are easily determined directly from experiment. Namely, experiments on the diffraction and interference of electron beams have shown that they should be represented in the form of plane waves, whose wavelength \(\lambda\) is related to the momentum \(p\) of the electron by de Broglie’s relation \(\lambda = \frac{h}{p}\). This relation must be placed at the foundation of any wave theory of electrons. However, whereas the wavelength can be determined directly from experiment, the frequency of the wave is given only by the relation, known in the quantum theory of light, between the frequency and the energy \(E\) of the electron: \(h\nu = E\). This relation cannot be verified experimentally, since the frequency of a matter wave is not a directly measurable physical quantity (on this see below). Nevertheless, specifying the wavelength and frequency completely determines a plane wave; thus, if both these determining data are known, one can immediately write down the wave equation of an electron moving in space free from the action of forces.

Schrödinger’s wave mechanics is based on nonrelativistic mechanics and the electrodynamics of the electron. Therefore the kinetic energy of a free electron here has the form:

\[ E = \frac{m}{2} v^2 = \frac{1}{2m} p^2, \tag{1} \]

whence there follows directly the relation between the frequency \(\nu = \frac{E}{h}\) and the wavelength \(\lambda = \frac{h}{p}\) for the wave associated with an electron in space free from the action of forces:

\[ \nu = \frac{h}{2m}\frac{1}{\lambda^2}. \tag{2} \]

Thus we have obtained Schrödinger’s wave equation for empty space. In fact, the only differential equation that admits these and only these wave functions is

\[ \Delta \psi = + \frac{4\pi i m}{h}\frac{\partial \psi}{\partial t} \quad (\Delta \text{ is the Laplace operator}). \tag{3} \]

Thus, the wave function representing a solution of this equation has the form

\[ \psi = \exp\left[\frac{2\pi i}{h}(p_x x + p_y y + p_z z)\right]\cdot \exp\left(\frac{i\pi p^2}{hm}t\right). \]

The form of the wave functions corresponding to motions in electromagnetic fields is not obtained directly from experiment. Therefore the wave equation that determines the behavior of these functions,

must be generalized in the corresponding way, which can be done in full agreement with the influence of fields on a charged particle known from electrodynamics. In order to generalize the equation to motion in force fields, one makes use of the fact that the form of the wave equation is closely connected with the relation (1) between energy and momentum. Namely, if the quantities \(E\) and \(p\) are regarded as “operators”:

\[ E \to -\frac{h}{2\pi i}\frac{\partial}{\partial t},\quad p_x \to \frac{h}{2\pi i}\frac{\partial}{\partial x},\quad p_y \to \frac{h}{2\pi i}\frac{\partial}{\partial y},\quad p_z \to \frac{h}{2\pi i}\frac{\partial}{\partial z}, \tag{4} \]

acting on the wave function, then relation (1) directly becomes the wave equation. But the relation between energy and momentum in the presence of electromagnetic fields is known from the classical theory of electrons, so that the wave equation can be constructed accordingly. In an electrostatic field determined by the potential \(V\), the energy of the electron is given by the expression:

\[ E=\frac{p^2}{2m}+eV, \tag{5} \]

where \(e\) denotes the charge of the electron.

If the electric field \(\mathbf{E}\) is not static and if magnetic fields \(\mathbf{H}\) are also present, then to describe them it is necessary to add the vector potential \(\mathbf{A}\) with three components \(A_x, A_y, A_z\), whence we obtain the field strengths:

\[ \mathbf{E}=-\operatorname{grad} V-\frac{1}{c}\frac{\partial \mathbf{A}}{\partial t},\quad \mathbf{H}=\operatorname{rot}\mathbf{A} \]

The relation between energy and momentum in the nonrelativistic mechanics of the electron has, for this case, the form:

\[ E=\frac{1}{2m}\left(p-\frac{e}{c}\mathbf{A}\right)^2+eV. \tag{6} \]

If we again take as the basis the above-mentioned principle for constructing the wave equation, then from (5) we obtain the following relation for static fields:

\[ -\frac{h^2}{8\pi^2 m}\Delta\psi+eV\psi=\frac{h}{2\pi i}\frac{\partial\psi}{\partial t}. \tag{7} \]

This is nothing other than the well-known Schrödinger wave equation. The wave functions obtained from it must represent the various states of motion that the electron can assume in the force field \(V\). Each of these states may be represented as the superposition of a series of so-called stationary states, just as in the case of membrane vibrations they can always be represented in the form of a superposition of normal vibrations. Stationary states are characterized by the fact that their wave function executes a monochromatic oscillation in time,

On the basis of the relation between energy and frequency, one can ascribe to them a definite energy, which is obtained by multiplying the frequency by \(h\), whereas those states which represent a superposition of stationary states of different frequency cannot possess a definite energy.

If we write the wave function of a stationary state in the form:

\[ \psi=\varphi\cdot \exp 2\pi i\nu t=\varphi\cdot \exp \frac{2\pi i}{h}Et, \]

where the monochromatic character of the dependence on time is clearly visible, then, substituting this function into (7), we obtain for \(\varphi\) the following equation:

\[ \Delta\varphi+\frac{8\pi^2 m}{h^2}(E-eV)\varphi=0. \]

This is the well-known Schrödinger equation, independent of time. It determines the energy of stationary states and the form of the corresponding wave functions.

Proceeding from the relation \(E=h\nu\), we would have to suppose that the energy of stationary states must always be positive, if by the frequency \(\nu\) one understands, as usual, the number of oscillations per second, which, of course, cannot be negative. However, with a suitable normalization of the potential energy, states are possible whose total energy is negative; thus, for example, the energy of the bound states of the hydrogen atom is for the most part regarded as negative. In reality, however, the frequency of the matter wave has rather an abstract meaning: the oscillations of this wave are never observed, and only the difference of the frequencies of two stationary states manifests itself in the form of the frequency of a spectral line. The meaning of “negative frequency” consists in the following: the oscillation of the proper function is represented with the aid of the factor \(\exp 2\pi i\nu t\); in this, as usual in the theory of oscillations, one writes \(e^{2\pi i\nu t}\) instead of \(\cos 2\pi\nu t\) only for the sake of simplifying calculations, taking into account only the real part of the written expression.

In wave mechanics, however, the factor \(\exp 2\pi i\nu t\) has an independent significance. A single real oscillation \(\cos 2\pi\nu t\) or \(\sin 2\pi\nu t\) would not satisfy the wave equation. The wave function must be complex, which becomes understandable only if it is regarded merely as an auxiliary quantity for calculations, having no direct physical significance. In doing so, one must strictly distinguish the cases when the dependence on time is given by the factor \(e^{+2\pi i\nu t}\) and \(e^{-2\pi i\nu t}\), whereas for an ordinary oscillation in both cases one would obtain essentially one and the same thing. In this sense we shall also, for the case \(e^{-2\pi i\nu t}\), speak of a “negative frequency” and thereby define such a dependence on time which in the given case physically has the meaning of a negative ...

energy. If one assumes that the potential energy is equal to zero, i.e., considers a freely moving particle, then one should expect the purely kinetic energy always to be positive. And indeed, for this case Schrödinger’s wave equation also admits wave functions only with positive frequency, since in the present case

\[ \nu=\frac{p^{2}}{2mh}. \]

The immediate physical result that can be obtained from the wave function is the probability of finding the electron at a definite point in space. This probability is determined directly by the intensity of the wave at the given point; by intensity, just as in the theory of mechanical vibrations, one should understand the square of the modulus of the amplitude of the wave. The charge of the electron is often imagined as distributed over the whole region of the wave function (“smeared out”) in the sense that the magnitude of the charge at any point of the region must correspond to the probability of finding the electron at that point. Such a “charge cloud” in many respects (though not in all) conveys the physical behavior of the electron state under consideration. The charge density \(\rho\) of the cloud at a definite point is equal to the electron charge \(e\) multiplied by the probability of finding the electron at the given point: \(\rho=e\psi^2\). The charge density must satisfy the condition that the sum of the charges present in space always remain constant. Since \(\psi\) in the present case is proportional to the intensity of the wave function, this requirement means that the total intensity of the wave function (the integral of the intensity, extended over all space) does not depend on time. This requirement is in fact satisfied for an arbitrary wave function by virtue of the wave equation.

Along with the charge density \(\rho\), the concept of current density \(\mathbf{i}\) is also introduced; it is determined by the probability of finding the electron moving in a definite direction.* The expression for the current density is determined by the requirement that, when the charge density increases at one point at the expense of its decrease at another point, there should result a definite current density corresponding to the transfer of charge (this relation is quantitatively established by the so-called continuity equation for charge and current). This reasoning shows that the current density \(\mathbf{i}\) is obtained from the wave function \(\psi\) by means of the relation

\[ \mathbf{i}=\frac{e}{m}\frac{h}{4\pi i}\left(\psi\,\operatorname{grad}\psi^{*}-\psi^{*}\operatorname{grad}\psi\right), \tag{8} \]

* More precisely: \(\frac{|\mathbf{i}|}{e}\) represents the probability that the electron, in the course of 1 sec, passes through a unit area in the direction perpendicular to \(\mathbf{i}\).

where by \(\psi^*\) is denoted the quantity complex-conjugate to \(\psi\). As is easy to see, the current density depends on the gradient of the wave function. When the value of \(\psi\) changes rapidly in space (small wavelength), the velocity of the electron is very large. The expressions for the charge density and the current density are not only closely connected with one another, but also follow directly from the wave equation itself. No relation of another kind could satisfy, on the one hand, the constancy of the total charge and, on the other hand, the indicated connection between the two expressions on the basis of the wave equation.

Only the action of a static electric field enters into Schrödinger’s wave equation (7). In order to generalize this equation also to motions taking place under the influence of time-variable electromagnetic perturbations, in constructing the wave equation one should start from the relation (6) between energy and momentum, which also contains the vector potentials. In this way we arrive at the generalized wave equation:

\[ -\frac{h^2}{8\pi^2 m}\Delta\psi -\frac{h}{2\pi i}\frac{e}{mc}\,\mathbf A\cdot \operatorname{grad}\psi +\frac{e^2}{2mc^2}\mathbf A^2\psi +eV\psi =-\frac{h}{2\pi i}\frac{\partial\psi}{\partial t}. \]

With the aid of this equation one can also consider the influence of light on the electron.

This influence is for the most part expressed by calculating the probability per unit time with which an electron, under the action of a light wave, passes from one stationary state \(k\) into another state \(l\). This transition probability is determined by the expression

\[ W=\frac{8\pi^3 e^2}{3h^2}\,s(\nu)\,A_{kl}^{\,2}, \tag{9} \]

where \(s(\nu)\) denotes the density of radiation of that frequency which corresponds to the difference of the energies of the two stationary states, and \(A_{kl}\) is the so-called transition matrix element, which can be calculated from the wave functions \(\varphi_k\) and \(\varphi_l\) of both states.

As is clear from these considerations, the electron can change its state in the atom only when electromagnetic fields act on it. Thus we obtain only absorption and emission of light under the influence of incident radiation. To explain the spontaneous emission of a light quantum it is necessary to generalize the theory further, and in such a way as to apply quantum theory not simply to the electron, but also to the electromagnetic field. The simplest consequence obtained from this is the existence of light quanta: the energy of an electromagnetic oscillation of frequency \(\nu\) can be only an integral multiple of the quantum \(h\nu\). Further, quantum theory requires the presence of so-called “zero-point oscillations”; thus every oscillator in the state of least

energy is not at absolute rest, but oscillates about the position of equilibrium. Therefore the amplitudes of the electromagnetic oscillations likewise cannot be completely annihilated; at the very least they must undergo zero-point oscillations. The quantum nature of the electromagnetic field has as its consequence zero-point oscillations of the field strength in space free of light quanta, representing the state with the least energy.

Zero-point oscillations act on an electron in the same way as ordinary electric oscillations, and can change its state, but only in the direction of its transition to a state with a lower energy level, since empty space can only absorb, but not give up, energy. Thus spontaneous emission of light is obtained as a direct consequence of the existence of these peculiar zero-point fields. Spontaneous emission of light is the emission of a light quantum caused by the zero-point oscillations of empty space. The amplitudes of the electromagnetic zero-point oscillations must correspond to the radiation density

\[ s_0(\nu)=\frac{8\pi h\nu^3}{c^3}, \]

so that, substituting this density into formula (9), we obtain the probability of spontaneous emission.

Schrödinger’s wave mechanics can be applied without particular difficulty to systems consisting of several electrons. The wave functions describing the state of motion of such systems depend on the position coordinates of all the \(n\) electrons making up the system, and therefore they are represented very non-visually, since they pertain to a space of \(3n\) dimensions. Here and in what follows we shall not dwell on the special methods for solving many-body problems in quantum mechanics, since they are not necessary for a fundamental understanding of quantum mechanics.

One of the essential results of the quantum mechanics of many bodies consists in a change in classical statistics. The complete indistinguishability of electrons from one another leads to the fact that all those states which differ from one another only by a permutation of the individual electrons are represented by exactly the same wave function, so that they should be regarded as one single state. However, wave mechanics in principle permits two possibilities, and only experiment can decide in favor of one of them: namely, wave functions may be expressed symmetrically or antisymmetrically in the coordinates of the particle. The former means that an arbitrary number of particles may be in one and the same state of motion, which leads to Bose statistics; the latter has the consequence that all electrons must have different states of motion, which leads to Fermi statistics. Experiment has shown that electrons satisfy Fermi statistics; they obey the so-called Pauli exclusion principle: in one and the same quantum state one cannot find more than one electron. If one takes into account,

considered in § 4, the electron’s intrinsic rotation, which can occur in two mutually opposite directions; it then follows that in one and the same state of motion there can simultaneously be at most two electrons with oppositely directed spins.

One of the very substantial shortcomings of quantum mechanics is that the theory itself cannot determine whether the electron obeys the Pauli exclusion principle or is subject to Bose statistics. Only experiment can decide the matter in favor of one of the possibilities. Nevertheless, the brilliant experimental confirmation of all those phenomena that follow from the Pauli principle—the construction of the periodic system of the elements, the theory of metallic conduction, etc.—must be regarded as a great achievement of wave mechanics.

3. The Relativistic Schrödinger Equation

The Schrödinger wave equation does not satisfy the requirements of the theory of relativity, since it proceeds from the classical mechanics of the electron, which, as is well known, is applicable only at small velocities. Thus, the relation between kinetic energy and momentum for rectilinear motion of an electron free from the action of forces, at high velocities, is expressed no longer in the form (1), but in the form

\[ E^2 = c^2 p^2 + m^2 c^4 . \tag{10} \]

The expression for the energy also includes the energy \(mc^2\) of the electron’s mass. And indeed, for momenta \(p\) small in comparison with \(mc\), this expression passes over into \(E = mc^2 + \frac{p^2}{2m}\), i.e. into the sum of the mass energy and the nonrelativistic kinetic energy. The frequencies of the free waves of electrons must therefore be related, in the relativistic wave equation, to the wavelength \(\lambda\) by the relation

\[ \nu^2 = \frac{c^2}{\lambda^2} + \frac{m^2 c^4}{h^2}, \tag{11} \]

if one starts from the fundamental relations \(E = h\nu\) and \(p = \frac{h}{\lambda}\). Hence, again, the following should necessarily be taken as the relativistic wave equation for an electron moving in space free from the action of forces \(^{1,2}\)

\[ \Delta \psi - \frac{4\pi^2 m^2 c^2}{h^2}\,\psi = \frac{1}{c^2}\,\frac{\partial^2 \psi}{\partial t^2}. \tag{12} \]

This equation is called the “scalar relativistic wave equation,” since the wave functions \(\psi\) here represent scalar functions of space, in contrast to Dirac’s wave equation, whose functions are quantities possessing, in a certain sense, a direction (spin).

The generalization of the equation to the case of motion in electromagnetic force fields is made on the basis of the generalized relativistic relation between energy and momentum, known from the theory of electrons; namely, instead of (6), at high velocities one has the relation:

\[ (E-eV)^2=c^2\left(p-\frac{e}{c}A\right)^2+m^2c^4. \]

Applying again the operators (4), we obtain from this the complete scalar wave equation. However, in what follows we shall not need this special form of the equation.

The wave functions obtained are very similar to those which correspond to Schrödinger’s equation. However, an essential difference consists in the fact that in the present case the number of states of motion is twice as large. Thus, for the case of motion without the action of forces, according to the nonrelativistic formula (2), to each wavelength there corresponds only one positive value \(\nu\), whereas the relativistic expression (11) admits one positive and one negative frequency (here we attach to the sign of the frequency the formal meaning indicated in the preceding section). It would be natural to ascribe negative energies to states with negative frequency. However, a more detailed investigation shows that the energy in the relativistic Schrödinger equation depends only on the absolute magnitude, and not on the sign, of the frequency * . Thus, in contrast to nonrelativistic wave mechanics (and also to the relativistic theory of the rotating electron of Dirac), here we have \(E=h\nu\). The negative sign of the frequency here signifies only that the corresponding states are represented by electrons with negative charge \(-e\) ** . This assertion can be proved by considering the corresponding behavior of these wave functions in electric fields; one may also apply the expression for the charge density and verify whether it gives, for wave functions with negative frequency, a charge of the opposite sign. Indeed, the scalar relativistic wave equation leads to a new expression for the charge density, and here we no longer have proportionality between the charge density and the intensity of the wave function, namely:

\[ \rho=\frac{e}{mc^2}\frac{h}{4\pi i}\left(\psi^*\frac{\partial\psi}{\partial t}-\psi\frac{\partial\psi^*}{\partial t}\right). \]

It is easy to see from this that the charge density depends on the sign of the frequency in the sense that states with negative frequency pro-

* This result can be obtained by quantizing the wave equation (12). Moreover, a similar relation also holds for light waves for which there exists an analogous wave equation.

** By the letter \(e\) we everywhere denote the actual negative charge of the electron, so that the opposite charge, \(-e\), is positive.

are in the form of motions of the electron with a charge of the opposite sign. From the relativistic wave equation it would therefore directly follow that there exists an electron with positive and negative charges.

Here the charge density can no longer be connected with the intensity of the wave function; indeed, when, for example, a positively charged wave is superposed on a negatively charged one, a large intensity may result without an increase in the charge density. The wave equation, however, requires only the conservation of the total charge, so that the total intensity of the wave functions may change under certain perturbations. Since the total intensity of matter waves may be regarded as an expression of the entire amount of matter present, it follows from this that the total amount of matter need not remain constant.* It turns out that these changes can in essence occur only under the influence of light of high frequency, and that equal amounts of positive and negative matter must simultaneously disappear or arise. This phenomenon proves to be identical with the creation and annihilation of electron pairs in the absorption or emission of \(\gamma\)-rays\(^3\). More detailed calculations also make it possible to obtain quantitative results, which agree completely with those results that we can at present obtain experimentally.

The scalar relativistic wave equation plainly does not give a correct representation of the electron, since it does not contain the spin of the electron. It gives incorrect values for the terms of the fine structure of the hydrogen atom and an incorrect formula for the scattering of \(\gamma\)-rays by electrons. Moreover, it turns out that it is applicable only to such particles as obey Bose statistics. At the same time, however, it shows that the simplest relativistic generalization of Schrödinger’s equation leads inevitably to the existence of oppositely charged electrons and to their creation and annihilation under the action of light; thus these processes are contained more deeply in the formalism of wave mechanics than might have seemed on the basis of the theory of holes, which is associated with so many difficulties.

4. The Spin of the Electron

Having considered the physical content of the simple relativistic generalization of Schrödinger’s wave equation, we shall now turn to attempts to construct this wave equation with spin taken into account. It turned out that one cannot consider the influence of spin on the mechanics of the electron independently of the relativistic generalization. Already in the spectral terms it is evident that the perturbations caused by spin (multiplet splitting) have the same order of magnitude as the shifts—

* This self-evident relation is also obtained by quantizing wave equation (12)\(^3\).

...of terms required by the relativistic correction. In heavy elements, for which the high velocities of the electrons already cause strong deviations from nonrelativistic kinematics, splittings of the fine structure are also observed, having the order of magnitude of the gross structure. In Schrödinger wave mechanics, up to now we have had wave functions of a scalar nature, as, for example, sound waves in gases, which describe the propagation of a scalar quantity (the change in air pressure). The presence of the electron’s intrinsic rotation leads to the necessity of representing wave functions as quantities with many components, as, for example, electromagnetic waves, which represent the propagation of both fields with six components.

In order to understand how the spin of the electron is described in quantum mechanics, it is necessary to consider the properties of the angular momentum of the particles of an atom. The magnitude of the angular momentum can take only the values \(\sqrt{l(l+1)}\,\frac{h}{2\pi}\), where \(l\) is an integer or half-integer. If an arbitrary direction in space is taken as the \(Z\)-axis, then only those states of rotation are possible for which the \(Z\)-component of the angular momentum assumes one of \(2l+1\) values:

\[ l\,\frac{h}{2\pi},\quad (l-1)\,\frac{h}{2\pi},\ldots,-l\,\frac{h}{2\pi} \]

or combinational states obtained by superposing these states. Every state of rotation of a particle with angular momentum \(\sqrt{l(l+1)}\,\frac{h}{2\pi}\) can be represented as a superposition of the indicated \(2l+1\) “elementary states of rotation.” For the angular momentum of the electron’s intrinsic rotation, as follows from many experimental data, one must always put \(l=\frac{1}{2}\). Hence we obtain for it the magnitude \(\sqrt{\frac{3}{4}}\,\frac{h}{2\pi}\). Thus every spin state can be obtained by superposing \(2l+1=2\) elementary states, and they should be chosen so that, in a definite direction, one component of the spin is equal to \(+\frac{1}{2}\,\frac{h}{2\pi}\), and the other to \(-\frac{1}{2}\,\frac{h}{2\pi}\). One also says that one state has parallel spin and the other antiparallel spin in the given direction.

The composition of an arbitrary spin from two elementary states is formally analogous to obtaining an arbitrary state of polarization of a light wave by superposing two mutually perpendicular polarized light waves. The analogy consists in the fact that in both cases the two elementary oscillations are not fixed in advance; in the case of a light wave one may orient in space the two mutually perpendicular directions of polarization arbitrarily with respect to the direction...

direction of the wave; by rotating both of these directions, one can obtain different decompositions of one and the same oscillatory state. In exactly the same way, in decomposing a spin state one may arbitrarily choose the indicated directions and decompose the general spin state into these elementary components, which in the chosen direction have the components \(+\dfrac{h}{4\pi}\) and \(-\dfrac{h}{4\pi}\).

However, this analogy must not be carried too far. The decomposition of light into two mutually perpendicular oscillations is a simple decomposition of a vector into two components. The decomposition of the electron’s intrinsic rotation into two components is already of an entirely different nature, because here the components must be oppositely directed. In order to express this state of affairs clearly, a special quantity is introduced, corresponding to the electron’s intrinsic angular momentum and called a spinor, and the two components are called the components of the spinor. Thus, even formally, the very important rules for transforming the components of a spinor will be different from the rules for transforming the components of a vector. Let us give one example: when the coordinate system is rotated about the \(X\)-axis through an angle \(\alpha\), the \(Z\)-component of a vector initially parallel to the \(Z\)-axis will be diminished in the ratio \(\cos \alpha\). The component of a spinor initially situated parallel to the \(Z\)-axis, the spin of which is parallel to the \(Z\)-axis, will decrease in the ratio \(\cos \dfrac{\alpha}{2}\).

Owing to the presence of spin it is no longer possible to describe the state of the electron by a simple scalar wave \(\psi\). If spin represented a rotation admitting a classical description, i.e. if its angular momentum were an ordinary vector with three assignable components, then it would be natural to represent the wave associated with the electron as a vector wave, the vector \(\psi\), with three components \(\psi_x\), \(\psi_y\), \(\psi_z\), having to lie in the direction of the angular momentum. But a rotation with angular momentum

\[ \sqrt{\frac{3}{4}}\,\frac{h}{2\pi} \]

can be described only by a spinor with two spinor components; hence the appropriate representation for the electron wave will be a “spinor wave” \(\psi\), where the “spinor \(\psi\)” consists of both spinor components \(\psi_\alpha\), \(\psi_\beta\). The value of the components for a definite spinor wave depends, of course, on the chosen coordinate system, i.e. on the direction (the \(Z\)-axis) in which the elementary states are taken. Thus, for example, for an electron wave whose spin has a \(Z\)-component exactly equal to \(+\dfrac{h}{4\pi}\), the spinor component \(\psi_\beta\), corresponding to the opposite spin, vanishes. In a coordinate system with another direction of the \(Z\)-axis, both components will be different from zero.

The wave equation for a wave function with many components consists of several equations for each component, which may be coupled with one another in such a way that the behavior of any one component will not be independent of the remain-

ones. Despite the fact that the spin properties of electrons cannot be considered without taking relativistic mechanics into account, it is possible to generalize the nonrelativistic Schrödinger equation in such a way that it will, at least approximately, describe the behavior of two-component spinor wave functions. We shall briefly consider such an extension here.

The wave equation for motion in the absence of forces must be identical for both components with the former Schrödinger equation, since in a space free from the action of forces the spin cannot influence the motion of the electron. We shall express this by the symbolic wave equation

\[ \Delta \psi = -\,\frac{4\pi i m}{h}\,\frac{d\psi}{dt}, \]

where in place of \(\psi\) one must substitute both \(\psi_a\) and \(\psi_b\). In the presence of electric or magnetic fields, the motion of the electron is affected by the magnetic moment associated with its intrinsic rotation. Any rotation of electric charges, directed in an arbitrary way, gives rise to a magnetic moment proportional to the rotational impulse (mechanical moment); the coefficient of proportionality depends entirely on the distribution of the rotating charge. If a point charge revolves about some center, then the magnetic moment \(\mathbf{M}\) is equal to the product of the rotational impulse by

\[ \frac{e}{2mc}. \]

This relation is satisfied, for example, for the motion of an electron in an orbit around an atomic nucleus. The corresponding factor for the intrinsic rotation of the electron, according to classical ideas, would have to depend on the internal structure of the electron. We borrow it from experiment, which has shown that here this factor is twice as large as the coefficient of proportionality for orbital motion. Thus the magnetic moment of the electron is equal to

\[ \frac{e}{mc}\,\mathbf{s}, \]

where \(\mathbf{s}\) is the rotational impulse of the spin. The magnitude of the magnetic moment is therefore equal to

\[ \sqrt{\frac{3}{4}}\;\frac{e}{mc}\,\frac{h}{2\pi}, \]

and the \(Z\)-component in some direction taken as the \(Z\)-axis is always

\[ +\frac{e}{mc}\,\frac{h}{4\pi} \quad \text{or} \quad -\frac{e}{mc}\,\frac{h}{4\pi}. \]

Only the Dirac wave equation leads to these values, without appealing to any experimental data.

The magnetic moment of the electron becomes noticeable when it moves in a magnetic field \(\mathbf{H}\), since it then gives rise to an additional potential energy

\[ V^* = \mathbf{M}\cdot\mathbf{H}, \]

which is determined by the scalar product of \(\mathbf{H}\) and the magnetic moment of the spin. But an electric field also acts on the spin, which is quite understandable, since every electric field \(\mathbf{E}\) has a magnetic component for a moving electron (this also makes clear the splitting of terms due to spin in atoms where there are no magnetic forces). Both actions must be correspond-

...are taken into account in the corresponding way in the wave equations of both components \(\psi_\alpha\) and \(\psi_\beta\).

The influence of an electric field on the spin cannot be consistently obtained in the nonrelativistic theory, since the magnetic action of the field on a moving electron is in itself a relativistic effect. However, this action can be described approximately by adding to the potential energy the additional term
\[ V_E=-\frac{2}{2mc}\,\mathbf{p}\times\mathbf{E}\cdot\mathbf{M}, \]
which represents the influence of the field \(\mathbf{E}\) on the spin of an electron that is moving and possesses momentum \(\mathbf{p}\). Then the influence of the spin can be taken into account in the Schrödinger wave equation (7), if the usual potential energy \(V\) is increased by adding the influence of the magnetic field \(V^*\) and of the electric field \(V_E\). Finally, if the magnetic moment \(\mathbf{M}\) is expressed through \(\mathbf{s}\)—the angular momentum of the spin: \(\mathbf{M}=\frac{e}{mc}\mathbf{s}\), then we obtain the so-called Pauli wave equation for both components of the spinor \(\psi_\alpha\) and \(\psi_\beta\).*
\[ -\frac{h^2}{8\pi^2 m}\Delta\Psi +e\left\{V+\frac{e}{mc}\mathbf{H}\cdot\mathbf{s} +\frac{e}{2m^2c^2}\mathbf{p}\times\mathbf{E}\cdot\mathbf{s}\right\}\Psi =\frac{h}{2\pi i}\frac{\partial\Psi}{\partial t}. \tag{13} \]

The meaning of \(p\), entering into \(V_E\), is well known. The components of this vector \(p_x,p_y,p_z\) should be replaced in the wave equation by the operators (4). It remains now to interpret the corresponding meaning of the spin vector \(\mathbf{s}\) in the wave equation.

Just as, instead of the product \(p_x\psi\), a new function is introduced, namely
\[ \frac{h}{2\pi i}\frac{\partial\psi}{\partial x}, \]
so also the product \(s_x\Psi\) gives rise to a new spinor function which, however, is not obtained from \(\psi\) by means of a mathematical operation, but is obtained by permuting the components of the spinor. Thus, we have:
\[ s_x\psi_\alpha=\frac{h}{4\pi}\psi_\beta,\qquad s_y\psi_\alpha=-i\,\frac{h}{4\pi}\psi_\beta,\qquad s_z\psi_\alpha=\frac{h}{4\pi}\psi_\alpha, \]
\[ s_x\psi_\beta=\frac{h}{4\pi}\psi_\alpha,\qquad s_y\psi_\beta=+i\,\frac{h}{4\pi}\psi_\alpha,\qquad s_z\psi_\beta=-\frac{h}{4\pi}\psi_\beta. \]

In this way a connection is obtained between the two wave equations for \(\psi_\alpha\) and \(\psi_\beta\), with \(\psi_\beta\) entering into the first equation and \(\psi_\alpha\) into the second. The wave functions obtained from the Pauli wave equation are very similar to the functions of the Schrödinger equation, since the additional term arising as a result of spin is in general very small. But here every stationary state is doubled owing to the two possible orientations of the spin. The wave functions of the two states of such a doublet differ essentially only in the different ratio of the amplitudes between the components \(\psi_\alpha\) and \(\psi_\beta\).

* The magnetic terms of the Schrödinger equation have been omitted here for simplicity.

One should not expect that the Pauli wave equation will quantitatively correctly reflect all spin effects, since relativistic perturbations always have the same order of magnitude. However, this is approximately taken into account by the fact that the influence of the electric field on the spin is reduced by half (the Thomas factor). The progress made by this equation in comparison with the Schrödinger equation consists in a qualitative explanation of the multiplet structure and, above all, in a complete accounting of the anomalous Zeeman effect in weak magnetic fields and of the Paschen–Back effect in strong fields. Indeed, the Pauli wave equations explain, from the point of view of wave mechanics, the known addition of the angular-momentum vectors from the motion in the orbit and from spin into one total angular momentum and lead, moreover, to the Landé factor \(g\), as the coefficient of proportionality between the mechanical and magnetic moments of the state of an atom whose angular momentum is composed partly of orbital moments and partly of the spin moment.

It should be pointed out, however, that the magnetic spin effect is not obtained directly from the wave equation itself, as is the case for the magnetic moment obtained as a result of circulation in the orbit around the nucleus; this latter moment follows necessarily from the behavior of the current density of the corresponding wave function. The magnetic spin moment is taken from experiment and is added as an additional term to \(v\) in the wave equation, whereas in an exhaustive theory it should follow directly from the spinorial properties of the electron wave. Only Dirac’s relativistic theory of the electron spin satisfies this requirement.

5. Dirac’s theory of the electron

A consistent relativistic wave equation for the wave functions of a spinor has such a strange and unusual character that its understanding is perhaps possible only with the help of an analogy, which is easiest to give by using the structure of Maxwell’s equations.

As is known, the vectors of the electric and magnetic fields are kinematically closely related to one another. A purely electric field, considered by a moving observer, ceases to be such and also contains magnetic components. Therefore the two fields are considered not as fields characterized by two different vectors, but as parts of one and the same quantity, the so-called six-component field tensor, characterized by the vector components of the electric and magnetic field vectors. In passing to a moving system the electric-field vector partly also gives a component of the magnetic field, just as a field vector parallel to the \(X\) axis, upon rotation about the \(Z\) axis, also gives a component along the \(Y\) axis.

The wave equation for the propagation of the electric and magnetic fields in a vacuum, as is known, has the form

\[ \Delta F=\frac{1}{c^2}\frac{\partial^2 F}{\partial t^2}, \]

where, in place of \(F\), one may substitute each of the components of the field vector separately. It follows from this that each component propagates in the form of a wave with velocity \(c\); however, from this one cannot derive the mutual connection between the fields. For this, Maxwell’s equations for the vacuum are necessary:

\[ \frac{1}{c}\frac{\partial \mathbf E}{\partial t}=\operatorname{rot}\mathbf H, \]

\[ \frac{1}{c}\frac{\partial \mathbf H}{\partial t}=-\operatorname{rot}\mathbf E. \]

These equations establish a connection between the two fields and, in particular, indicate that every electric wave must be accompanied by a magnetic one. Maxwell’s equations contain only first derivatives of the field vectors; these equations are broader than the wave equation; the latter can be derived from them.

Let us now return to the two-component spinor wave functions of the electron. It turned out that a complete description of the electron wave by two components of a spinor is impossible. Alongside the wave \(\psi\), there must also be another spinor wave \(\varphi\), with components \(\varphi_\alpha\) and \(\varphi_\beta\), which stand in the same relation to the \(\psi\)-wave as the magnetic-field vector does to the electric-field vector. In passing to a frame of reference possessing another motion, part of the \(\psi\)-wave passes into the \(\varphi\)-wave. Therefore both pairs of components \(\psi\) and \(\varphi\) are united into one “relativistic spinor” with four components \(\psi_\alpha\), \(\psi_\beta\), \(\varphi_\alpha\), \(\varphi_\beta\), which constitute the analogue of the six-component field tensor. However, for electron velocities small in comparison with the speed of light \(c\), the accompanying spinor wave \(\varphi\) is very weak in comparison with \(\psi\), which, perhaps, may be considered analogous to the fact that for slowly moving charges the magnetic-field vector is likewise always small in comparison with the electric one. Therefore, in the nonrelativistic wave equation of the spinning electron, it was possible to neglect the second spinor wave entirely. But for fast particles, to which nonrelativistic mechanics is no longer applicable, the order of magnitude of the \(\psi\)-wave and the \(\varphi\)-wave is the same. We shall consider the physical significance of the \(\varphi\)-waves below.

The wave equation for an electron moving in space free from the action of forces is determined, as usual, by the fundamental relations between the quantities characterizing the particle \((E,p)\) and the wave parameters \((\nu,\lambda)\):

\[ h\nu=E,\qquad \frac{h}{\lambda}=p. \]

The relativistic relation between momentum and energy \(E^2=c^2p^2+m^2c^4\) leads directly to the relation

\[ \nu^2=\frac{c^2}{\lambda^2}+\frac{m^2c^4}{h^2} \tag{11} \]

between the frequency and the wavelength of electron waves. The only wave equation leading to such waves is the equation already indicated in § 3:

\[ \Delta \Phi - 4\pi^2\frac{m^2c^2}{h^2}\Phi = \frac{1}{c^2}\frac{\partial^2\Phi}{\partial t^2}, \tag{14} \]

which must be satisfied if, in place of \(\Phi\), one substitutes any of the four components \(\psi_\alpha,\psi_\beta,\varphi_\alpha,\varphi_\beta\). This wave equation is similar to the wave equation of the electromagnetic field; it differs by the term \(4\pi^2 \dfrac{m^2c^2}{h^2}\Phi\), owing to which relation (11) is satisfied, whereas for light waves the relation \(\nu=\dfrac{c}{\lambda}\) always holds. If \(\Phi\) were a scalar, then this second-order wave equation would completely determine the behavior of the wave function. In fact, from equation (14) one can obtain only the behavior of each component by itself, but not the relations between them. These relations must be established only by Dirac’s special equations:

\[ \frac{1}{c}\frac{\partial\psi_\alpha}{\partial t} = -\frac{\partial\varphi_\beta}{\partial x} -i\frac{\partial\varphi_\beta}{\partial y} +\frac{\partial\varphi_\alpha}{\partial z} +\frac{2\pi imc}{h}\psi_\alpha, \]

\[ \frac{1}{c}\frac{\partial\psi_\beta}{\partial t} = -\frac{\partial\varphi_\alpha}{\partial x} +i\frac{\partial\varphi_\alpha}{\partial y} -\frac{\partial\varphi_\beta}{\partial z} +\frac{2\pi imc}{h}\psi_\beta, \]

\[ \frac{1}{c}\frac{\partial\varphi_\alpha}{\partial t} = -\frac{\partial\psi_\beta}{\partial x} -i\frac{\partial\psi_\beta}{\partial y} +\frac{\partial\psi_\alpha}{\partial z} -\frac{2\pi imc}{h}\varphi_\alpha, \]

\[ \frac{1}{c}\frac{\partial\varphi_\beta}{\partial t} = -\frac{\partial\psi_\alpha}{\partial x} +i\frac{\partial\psi_\alpha}{\partial y} -\frac{\partial\psi_\beta}{\partial z} -\frac{2\pi imc}{h}\varphi_\beta, \]

however, the form of these equations, as well as the existence of \(\varphi\)-waves, necessarily follows from the requirement that the electron wave represent a four-component relativistic spinor wave and that the wave equation (14) be satisfied for the components. The construction of these equations is similar to the construction of Maxwell’s equations. The time derivative of one wave \(\psi\) is determined by a certain combination of the spatial derivatives of the second wave \(\varphi\). Dirac’s equations are more general than the wave equation (14), which can be derived from Dirac’s equation, which, besides the wave equation, also contains relations between the components, for example the requirement that every \(\psi\)-wave be accompanied—

was a certain $\varphi$-wave. However, the analogy indicated here has its limits: Maxwell’s equations establish relations between two three-component vector quantities $\mathbf E$ and $\mathbf H$, which can be represented in the form of one six-component field tensor; Dirac’s equations give relations between two two-component spinor quantities $\psi$ and $\varphi$, which combine together into one four-component relativistic spinor.

Dirac’s equations can be combined if one makes use of symbolic vectors and quantities which transform the functions standing next to them into others (similarly to the vector $\mathbf s$ in the preceding section). Then all four equations can be written in the form:

\[ \frac{1}{c}\frac{\partial \Phi}{\partial t} = \alpha_x \frac{\partial \Phi}{\partial x} + \alpha_y \frac{\partial \Phi}{\partial y} + \alpha_z \frac{\partial \Phi}{\partial z} + \frac{2\pi i m c}{h}\,\beta \Phi . \tag{15} \]

Instead of $\Phi$ one should substitute each of the components $\psi_\alpha$, $\psi_\beta$, $\varphi_\alpha$, $\varphi_\beta$, while the symbolic components $\alpha_x$, $\alpha_y$, $\alpha_z$ of the vector $\boldsymbol{\alpha}$ and the symbolic quantity $\beta$ have the following meaning: multiplication by $\alpha_x$ leads to the transformation:

\[ \left( \begin{array}{c} \psi_\alpha \to \varphi_\beta \\ \psi_\beta \to \varphi_\alpha \\ \varphi_\alpha \to \psi_\beta \\ \varphi_\beta \to \psi_\alpha \end{array} \right) \]

In exactly the same way

\[ \begin{array}{ccc} \text{for } \alpha_y: & \text{for } \alpha_z: & \text{for } \beta: \\[6pt] \left( \begin{array}{c} \psi_\alpha \to -\,i\varphi_\beta \\ \psi_\beta \to i\varphi_\alpha \\ \varphi_\alpha \to -\,i\psi_\beta \\ \varphi_\beta \to i\psi_\alpha \end{array} \right) & \left( \begin{array}{c} \psi_\alpha \to \varphi_\alpha \\ \psi_\beta \to -\,\varphi_\beta \\ \varphi_\alpha \to \psi_\alpha \\ \varphi_\beta \to -\,\psi_\beta \end{array} \right) & \left( \begin{array}{c} \psi_\alpha \to \psi_\alpha \\ \psi_\beta \to \psi_\beta \\ \varphi_\alpha \to -\,\varphi_\alpha \\ \varphi_\beta \to -\,\varphi_\beta \end{array} \right). \end{array} \]

The symbolic equation (15) is the Dirac wave equation for an electron in a space free from the action of forces.

The generalization of this equation to motions in force fields is again carried out with the aid of the relation between the energy and momentum of the electron, known from the relativistic dynamics of the electron:

\[ (E-eV)^2 = c^2\left(\mathbf p-\frac{e}{c}\mathbf A\right)^2 + m^2c^4 . \tag{16} \]

The influence of the electromagnetic field is taken into account here by adding to the energy the term $-eV$, and to the momentum the term $-\dfrac{e}{c}\mathbf A$. In order that—

to obtain the corresponding change of the wave equation, one must add the potentials \(eV\) and \(-\dfrac{e}{c}\mathbf A\) to those expressions which correspond to the energy and momentum by virtue of relation (4). Thus the following substitutions must be made:

\[ \frac{h}{2\pi i}\frac{\partial \Phi}{\partial t} \;\longrightarrow\; \frac{h}{2\pi i}\frac{\partial \Phi}{\partial t}-eV\Phi, \]

\[ \frac{h}{2\pi i}\frac{\partial \Phi}{\partial x} \;\longrightarrow\; \frac{h}{2\pi i}\frac{\partial \Phi}{\partial x} -\frac{e}{c}A_x\Phi \]

and so on.
After these substitutions we obtain the complete Dirac wave equation:

\[ \frac{1}{c}\frac{\partial \Phi}{\partial t} -\frac{2\pi i}{hc}eV\Phi = \alpha_x\left( \frac{\partial}{\partial x} -\frac{2\pi i e}{hc}A_x \right)\Phi \]

\[ +\alpha_y\left( \frac{\partial}{\partial y} -\frac{2\pi i e}{hc}A_y \right)\Phi + \]

\[ +\alpha_z\left( \frac{\partial}{\partial z} -\frac{2\pi i e}{hc}A_z \right)\Phi + \]

\[ +\beta\frac{2\pi i mc}{h}\Phi . \]

However different the outward form of the Dirac and Schrödinger equations may be, there is nevertheless an internal kinship between them. Indeed, if the velocity of the particles is very small, then, owing to the smallness of the functions \(\varphi_\alpha\) and \(\varphi_\beta\), these two functions can easily be eliminated; as a result, for \(\psi_\alpha\) and \(\psi_\beta\) one obtains precisely the Schrödinger equation with small additional terms which take into account the magnetic action of spin; in essence, we then have the equation in the form of the Pauli equations (13), described in the preceding section. This shows that the Dirac equations really are a generalization of the Schrödinger wave equation. In the limiting case of small velocities, the small functions \(\varphi_\alpha\) and \(\varphi_\beta\) can be calculated directly from \(\psi_\alpha\) and \(\psi_\beta\). In this case, approximately, we shall have:

\[ \varphi_\alpha = -\frac{h}{4\pi imc} \left( \frac{\partial \psi_\beta}{\partial x} -i\frac{\partial \psi_\beta}{\partial y} + \frac{\partial \psi_\alpha}{\partial z} \right), \]

\[ \varphi_\beta = \frac{h}{4\pi imc} \left( \frac{\partial \psi_\alpha}{\partial x} +i\frac{\partial \psi_\alpha}{\partial y} - \frac{\partial \psi_\beta}{\partial z} \right). \tag{17} \]

From this it is clear that \(\varphi_\alpha\) and \(\varphi_\beta\) are proportional to the derivatives of the wave functions \(\psi\), which in turn are proportional to the velocities of the particles.

How, then, is the charge density and the current density calculated from the four-component wave functions? As we showed in the raz-

PROBLEMS OF THE NEW QUANTUM THEORY OF THE ELECTRON

case 2, with the aid of Schrödinger’s equation these expressions are necessarily obtained from the wave equation, if one takes into account all those properties of these quantities which follow from the conservation of the quantity of electricity. The charge density again proves to be proportional to the intensity of the waves, namely to the sum of the intensities of the four components:

\[ \rho=e\left(|\psi_{\alpha}|^{2}+|\psi_{\beta}|^{2}+|\varphi_{\alpha}|^{2}+|\varphi_{\beta}|^{2}\right). \tag{18} \]

The total charge (the integral of the charge density \(\rho\), extended over all space) here is also proportional to the total intensity of the wave function. Therefore here, from the constancy of the total charge, there also follows the constancy of the quantity of matter, in contrast to what was the case for the scalar relativistic equation. Thus the Dirac wave equation could not take account of the occurrence of pairs of electrons through absorption of light. From the charge density \(\rho\) one can find an expression for the current density if one takes into consideration that every increment of charge at one point at the expense of some other point causes a current between the two points. The three components of the current density have the form:

\[ \begin{aligned} i_x&=ec\left(\psi_{\alpha}^{*}\varphi_{\beta}+\psi_{\beta}^{*}\varphi_{\alpha} +\varphi_{\alpha}^{*}\psi_{\beta}+\varphi_{\beta}^{*}\psi_{\alpha}\right),\\ i_y&=ec\left(i\psi_{\alpha}^{*}\varphi_{\beta}-i\psi_{\beta}^{*}\varphi_{\alpha} -i\varphi_{\alpha}^{*}\psi_{\beta}-i\varphi_{\beta}^{*}\psi_{\alpha}\right),\\ i_z&=ec\left(\psi_{\alpha}^{*}\varphi_{\alpha}-\psi_{\beta}^{*}\varphi_{\beta} +\varphi_{\alpha}^{*}\psi_{\alpha}-\varphi_{\beta}^{*}\psi_{\beta}\right). \end{aligned} \tag{19} \]

In these expressions the component \(\psi\) is everywhere multiplied by a component \(\varphi\). Thus we see that the remarkable \(\varphi\)-components of the wave function, introduced only on the basis of the requirements of the theory of relativity, also appear in physically measurable quantities. In virtue of (18) they contribute their part to the charge density and even determine the components of the current in a very essential way.

The expressions for the current density differ greatly from Schrödinger’s expressions:

\[ i_x=\frac{he}{4\pi mi}\left(\psi\,\frac{\partial \psi^{*}}{\partial x} -\psi^{*}\frac{\partial \psi}{\partial x}\right), \]

given in Section 2 (formula 8). First of all, it seems, there are no derivatives of the wave function, which had the chief significance for the velocity of the particle. In actual fact, however, these derivatives are contained in a known way in the functions \(\varphi\), which, by virtue of (17), are approximately obtained from the spatial derivatives of the functions \(\psi\).

In order to be convinced of the difference between the new current densities and Schrödinger’s, it will be very instructive to decompose the newly obtained expressions into two parts\({}^{6}\), in such a way that one part \(i'\) is identical with the current density of Schrödinger’s equation.

of the Schrödinger theory, and would contain Schrödinger currents for all four components, while the other, \(f''\), would correspond to a current of rest. Whereas the first part in all respects corresponds to the actual mechanical displacement of the electron, the second part leads to closed circular currents accompanying every motion of the electron. These circular currents produce a magnetic moment whose magnitude is exactly equal to the actual moment of the electron spin. Thus the magnetic moment of the electron spin follows directly from the structure of the Dirac equation and from the expressions, corresponding to this equation, for the current density. Consequently, the requirement that the magnetic properties of spin follow directly from the spinorial nature of the wave functions is here satisfied.

The solution of the Dirac equation for the case of the motion of an electron in a force field is a more difficult mathematical problem than the solution of the Schrödinger equation, owing to the presence here of four components. Here, too, however, all states of motion can be obtained by superposing stationary states, characterized by the fact that their wave functions, i.e. all four components, perform monochromatic oscillations of one and the same frequency. The frequencies \(\nu\) that form the stationary states give the allowed values of the energy \(h\nu\) of the electron in the given force field. In this way, in calculating the hydrogen atom, one can obtain the spectral terms and their fine structure in exact quantitative agreement with experiment.

A quantitative verification of Dirac’s wave equation was also carried out by means of the scattering of light by free electrons (the Compton effect); namely, the calculation gives for the intensity of the scattered light the so-called Klein–Nishina formula, which was exactly confirmed when tested also for light of very short wavelength\({}^{7}\). A test was also possible by means of experiments with “polarized” electron beams (by “polarized” beams one understands beams of electrons with identical spins). Namely, by scattering in force fields an ordinary electron beam, in which the two possible spin orientations occur equally often, one can achieve the attenuation of one component, as a result of which a “partially polarized” beam is obtained. However, the quantitative determination of the regularities obtained here has not yet been sufficiently verified theoretically and experimentally.

If one compares the state of an electron corresponding to the Schrödinger and Dirac wave equations, one can see that to each state possible for the Schrödinger equation there correspond in the Dirac equation two states with oppositely directed spins. But besides this doubling of states, which is in complete agreement with experiment (the doublet structure of the terms of atoms with one optical electron), the Dirac wave equations correspond to still other superfluous states, which lead to great difficulties. These are such states in which the motion

PROBLEMS OF THE NEW QUANTUM THEORY OF THE ELECTRON

of the electron in electric fields takes place as if the charge of the electron were not \(e\), but \(-e\). They are quite analogous to those which occurred in the scalar relativistic wave equation (§ 3) and which there one could indeed attribute to electrons with the opposite charge \(-e\). However, such a way out of the situation is impossible for the Dirac wave equation, since the charge density according to formula (18) has, for all states, one and the same sign. Indeed, it is proportional to the intensity of the waves, and the intensity, as the square of the amplitude, is always positive. Unfortunately, the behavior of the electron opposite to the usual one is explained here in a very unsatisfactory way, namely by the fact that such states must be assigned a negative kinetic energy, or, what is the same thing, a negative mass.

The wave functions of these states with negative mass differ from the ordinary states above all in the form of their dependence on time, namely their oscillations are determined by the quantity \(e^{-2\pi i\nu t}\). The appearance of a “negative frequency” is connected with negative kinetic energy. It should only be noted that “negative frequency” by itself does not yet give the right to draw the conclusion of negative energy—in the scalar relativistic wave equation it implies only the presence of the opposite charge; that here the energy is in such a direct relation to the frequency of the wave function can be shown only by a more detailed investigation of the wave equation. The wave functions of states with negative mass are characterized, in addition, by the fact that in them the components \(\psi\) and \(\varphi\) exchange roles. Thus, in the case of slow motions the \(\psi\)-waves are very weak, while the \(\varphi\)-waves satisfy an approximate Schrödinger equation with negative electron mass.

Let us now consider, as an explanatory example, the energy spectrum of a free electron obtained from the Dirac wave equation (Fig. 1). The lowest energy level for ordinary states with positive mass is the state of rest; in this case the energy is determined exclusively by the rest energy \(mc^2\). Adjoining this level in the upward direction is a continuous spectrum containing the energy values of rectilinear motions in all directions and with all possible velocities. Each of these states must be counted twice in accordance with the two possible orientations of the spin. But, in addition, there are also states with negative mass, which represent an exact mirror reflection of the states with positive mass and which, beginning with \(mc^2\), the rest energy of an electron with negative mass, extend to \(-\infty\). The analogous energy spectrum of an electron in a Coulomb field (the hydrogen atom) (Fig. 2) differs from the spectrum of a free electron by the appearance of discrete states having energies less than the rest energy \(mc^2\), since bound states have smaller energies than the rest energy of a free electron. The very lowest

of these states only a very small amount—approximately by \(\dfrac{1}{(137)^2}mc^2\)—differs from the rest energy. In Fig. 2 this distance is taken large. The continuous states above \(mc^2\) correspond to an ionized atom. The bound states form narrow doublets, since the two spin orientations, owing to the presence of the Coulomb field, do not have exactly the same energy. The spectrum

Fig. 1. Energy levels of a free electron according to Dirac’s wave equation.

Fig. 2. Energy levels of an electron bound to a nucleus according to Dirac’s wave equation (schematic representation). (The distance between the discrete spectrum and the boundary of the continuous spectrum \(+mc^2\) is greatly enlarged; the doublet nature of the lines is not shown.)

of negative energies has no discrete states, since an electron with negative mass (just like a positively charged electron) is repelled by the atomic nucleus, and therefore cannot find itself in a bound state. The possibility of states of the electron with negative mass is, of course, in contradiction with experience. Since to these states there corresponds an energy lying deeper than the energy corresponding to the normal states, spontaneous transitions to states with negative energy should have been observed, with simultaneous emission of radiation. The calculation gives a considerable transition probability for such a process; consequently, an electron with positive mass ought, emitting light, to turn into an electron with negative mass.

Many attempts have been made to exclude states with negative mass from the wave equation. However, this has so far not been possible without substantially affecting the remaining results. The appearance of critical states is most closely connected with Dirac’s representation of the electron spin; nevertheless, in view of the exceptionally good agreement with experiment of many of the conclusions, one must admit that this theory coincides to a significant degree with reality.

6. The Theory of the Positron

Both for Schrödinger’s wave equation and for Dirac’s equation, it is characteristic that the charge density is proportional to the intensity of the wave function. In this case the theorem on the conservation of the total charge must also have as its consequence the conservation of the total intensity. Physically this means that the quantity of matter, under any external influences, remains unchanged, i.e. that the number of elementary particles is constant. This circumstance, which before the discovery of the positron was considered a necessary premise, must at present be regarded as a serious deficiency, since, in consequence of the occurrence in nature of the formation of electron pairs, only such a wave equation as is connected with a variable number of particles can correctly reflect reality. As an example of such an equation we cited in Section 3 the scalar relativistic wave equation, for which the total intensity of the wave functions is not necessarily conserved as a constant. However, a scalar wave function in no way corresponds to reality; the existence of spin compelled one to consider wave functions as having several components, as a result of which we necessarily arrived at Dirac’s wave equations for four components, so that the constancy of the number of particles and the existence of states with negative mass appear inseparably linked with the wave-mechanical representation of the electron spin.

Dirac wished to eliminate simultaneously both contradictions with experiment by his bold theory of positrons[^9]. He assumed that all states with negative mass are occupied, each by one electron. According to the Pauli principle, a transition from a state with positive mass to an occupied state with negative mass is then impossible, since in each such state there can be only a single electron. But electrons with negative mass must be physically unobservable, and the electric field produced by them must in some way be compensated. Conversely, when some electron passes from the region of negative energies into the region of positive energies, the “hole” thereby produced in the set of electrons with negative mass must appear in the form of a missing negative charge, i.e. in the form of a positive charge. This hole obeys the laws of motion of a positive charge, since

a multitude of electrons “surrounding” the hole, owing to the negativity of their mass, should behave in force fields in the opposite way from the usual one, i.e., should exhibit the properties of a positive charge.

To illustrate this we may make use of the diagrams of terms shown in Figs. 1 and 2, where all negative energy levels must be occupied. Then it is easy to see that a quantum of light with energy greater than \(2mc^2\) would be able to jump across the free space between the upper and lower terms and raise one of the electrons from a negative energy level to a positive one, thereby forming one hole and one ordinary electron with positive energy. In the first section it was shown that, owing to the conservation of total momentum in space free from the action of forces, this is possible only under the action of two light quanta, the sum of whose energies is at least equal to \(2mc^2\). In the vicinity of strongly charged atomic nuclei the creation of an electron pair may also occur under the action of only one light quantum possessing sufficient energy.

On the basis of such a representation it is possible to calculate, using Dirac’s wave equation, the probability of the production of one electron and one hole under the action of a \(\gamma\)-ray. This probability is identical with the probability of transition from some level with negative mass to a level with positive mass, which can be calculated according to firmly established rules, proceeding from the wave functions corresponding to both levels. Such a calculation was carried out for the formation of an electron pair near heavy nuclei and proved to be in strikingly exact agreement with the experimental results obtained thus far.\(^{10}\) Thus, if a light quantum of frequency \(\nu > \dfrac{2mc^2}{h}\) penetrates into the Coulomb field of an atomic nucleus with charge \(Ze\), then for the process of producing an electron pair with absorption of the quantum we obtain approximately:

\[ q=\left(\frac{e^2}{mc^2}\right)\frac{Z^2e^2}{hc}\left(\frac{28}{9}\ln\frac{2h\nu}{mc^2}-\frac{218}{27}\right). \]

In an analogous way one can convince oneself that the theory of holes gives the annihilation of one electron and one positron with the emission of light. In fact, an electron with positive mass* can always, emitting light, pass into an unoccupied state with negative mass—“occupy a hole”; in doing so the hole is filled, and at the same time both the electron and the hole disappear. Here too the theorem of momenta for space free from the action of forces requires the emission of two light quanta,

* Not to be confused with a positive electron, which in the theory of holes represents a missing negative electron with negative mass.

The problematic aspect of the hole theory runs up against the notion that all states with negative mass, of which there are infinitely many, must be occupied by electrons. This infinitely large number of electrons must, of course, create at every point of space an infinite charge density and a field of infinitely great intensity. The requirement of the hole theory that these charges not be observable cannot be reconciled with the consistent application of Dirac’s wave equation, according to which a definite charge must also be assigned to states with negative mass. Still more problematic, however, is the possibility of taking this into account quantitatively. If by this one understands that electrons in negative states produce no electromagnetic action at all, then they should also fail to react to a light wave, so that a transition to positive energy states with absorption of light would be impossible. Another explanation consists in leaving the electrons their electromagnetic properties, while eliminating the infinite charge density by simply always subtracting this density from every result of the theory. Such attempts, however, have not led to any unambiguous result, since the subtraction of infinite quantities one from another cannot be formulated mathematically in a unique way.

Every external field—for example, the electric field of an atomic nucleus—also exerts a certain action on electrons with negative mass and, consequently, changes the distribution of their charges in space. The question arises whether this change in the distribution of charges can be observed through the fact that it produces, near the nucleus itself, an additional electric field that alters the behavior of the nuclear field. The hole theory is not in a position to answer this question unambiguously, since it cannot be decided whether the total distribution of the charges of electrons with negative mass remains unobserved, or whether at least the changes caused by external fields will be observable. The latter conclusion is more probable because holes in the charge distribution should be regarded as observable in the form of positive particles; consequently, small changes caused by electric fields must obviously also be observable.

This problem leads, moreover, to one further substantial complication, namely: calculation shows that all changes in the charge distribution in states with negative energy, which are caused by small external fields, are also infinitely large[^11]. Of course, the change in each individual state is very small, but the sum of such changes, taken over an infinitely large number of electrons, is infinitely large. This difficulty, perhaps, is a difficulty of a different order, since we still know very little about the interactions between electric fields and particles with very high negative or positive energy (this will be discussed in the next section). If our assumptions concerning this interac-

actions prove unsuitable, then all the same one may think that these changes are finite, though perhaps not large.

Thus one must acknowledge that the hole theory has by no means yet overcome those difficulties which are connected with the fundamental premise of this theory—the existence of an infinitely large number of states with negative energy. Nevertheless, the indisputable quantitative agreement between the theoretical calculations of the transformation of an electron with negative mass into an electron with positive mass and the actual occurrence in nature of the creation of pairs of electrons shows that the hole theory must in some way be connected with reality.

As has already been indicated, in considering an infinite density of charge, the hole theory takes into account, although not unambiguously, a definite influence of electric fields on “empty space.” According to the conception of the hole theory, “empty space” is filled with those electrons which occupy states with negative mass. External electromagnetic fields change the distribution of the charges of the electrons, as a result of which phenomena may arise that are called the dielectric polarization of the vacuum. Dielectric phenomena in ordinary matter are obtained in a similar way when charge is displaced within matter, caused by external fields acting on this matter.

It should be noted that the dielectric constant of the vacuum, the same for all points, could not be detected experimentally, since it would simply reduce all charges in one and the same ratio and, consequently, there would be no possibility of determining the true charges. Therefore only a dielectric constant depending on the field strength or on the frequency of the variation of the field in time can be of physical interest. The first would lead to changes in the spatial structure of force fields, while the second would manifest itself in radiation, in the form of a dependence of the effective charges of electrons on the frequency of light.

The possibility of such an influence of the vacuum is not connected exclusively with the perhaps somewhat doubtful conception of the hole theory, but may be justified from a more general point of view. The creation of pairs of electrons from the vacuum under the action of light waves of high frequency is an experimentally established interaction between the electromagnetic field and the “vacuum”; hence one may freely suppose that such an interaction is not limited only to light waves of high frequency. If a beam of light of high frequency is absorbed in a strong electric field, producing pairs of electrons, then it is quite probable that strong fields must also act on light of lower frequency, at least by deflecting light rays, which should lead to the scattering of light in electric fields. With this process there would also be associated the scattering of one light ray by another, so that it would no longer be possible to consider that two light waves can pass, one through the other, without any changes.

The calculation of such processes on the basis of present-day ideas about the creation of electron pairs, connected with the theory of holes, presents difficulties, since this theory cannot be formulated without contradictions. Recently Dirac and Heisenberg have tried to find an expression of the most general form for the new properties of the vacuum, so that it should not be in contradiction with the experimental data now known.[^12] If one requires that the creation of electron pairs be determined by the same formulae as in the theory of holes—since these formulae are well confirmed by experiment—then one can arrive formally at results which, perhaps, in the most general form will be correct, although such a method is ambiguous and by no means unconditionally obligatory. By such methods, for example, the effective cross-section for the scattering of a light quantum by another quantum[^13] was determined, and the value obtained was

\[ q \sim \left( \frac{e^2}{mc^2} \right)^2 \cdot \left( \frac{e^2}{hc} \right)^2 \cdot \frac{\lambda_0^{\,6}}{\lambda^4}, \]

where \(\dfrac{\lambda_0}{\lambda}\) is the ratio of the Compton wavelength* to the mean wavelength of the scattering and scattered light quanta. Thus the numerical value of this area is approximately \(10^{-50}\,\mathrm{cm}^2\) for \(\gamma\)-rays and \(10^{-70}\,\mathrm{cm}^2\) for rays of visible light.

7. Interaction between matter and the electromagnetic field

The interaction between the electromagnetic field and the electron is described in quantum mechanics by the “correspondence method.” This means that in quantum mechanics one borrows the relation between the field strength and the charge obtained from Maxwell’s equations and from the classical theory of electrons, and then modifies this relation to the extent required by the wave-mechanical nature of charges and the quantum nature of light. The action of the field on charges and the creation of the field under the action of charges are the same, as is known from the old theory of electrons; only the state of motion obtained as a result differs from the corresponding states of motion of an ordinary particle as a consequence of the wave nature of electrons.

On this basis it proved possible in principle to construct quantum electrodynamics, which would contain not only the relation between the electromagnetic field and a single electron, but would also take into account the interaction of many electrons with one another. In fact, the latter can always be decomposed into the creation of a field under the action of electrons and the action of the created field on electrons. This program, first carried out by Heisen-

berg and Pauli, encounters profound internal difficulties, which will be considered below.

Even before the appearance of quantum theory, the electrodynamics of the electron contained a number of unresolved problems, connected above all with the question of the spatial dimensions of the electron. The electric field of the electron possesses a certain amount of energy, which is always associated with the electron. This energy is called the electron’s proper electrical energy. The field strength at a distance \(r\) from a resting electron, as is known, is equal to \(\frac{e}{r^2}\), and, consequently, it is the greater the closer one approaches the electron. If the electron is regarded as a point charge, then infinitely large fields arise, which lead to an infinitely large proper energy. But according to the theory of relativity every energy \(E\) must possess a mass \(\frac{E}{c^2}\), so that it is necessary to ascribe to the electron a finite radius if only in order that it should possess a finite mass. Since the field energy \(E_{\text{field}}\) of an electron at rest with radius \(r_0\) is equal to \(\frac{e^2}{2r_0}\), the radius of the electron is usually taken so large that the mass obtained from its field energy, \(m \sim \frac{e^2}{r_0 c^2}\), would be exactly equal to the observed mass of the electron \(m = 9 \cdot 10^{-28}\). It follows from this that the radius of the electron must have a value of the order \(r_0 \sim \frac{e^2}{mc^2}\).

However, the introduction of a finite extension of the electron leads in wave mechanics to new difficulties. It would seem that the notion of an “electron cloud”—a charge spread out over the whole region of the wave function—solves this problem, since in wave mechanics the charge apparently always has a finite extension and therefore does not lead to any excessively large fields. However, it is incorrect to identify directly the wave function, spread over a finite region, with the charge. In fact, this wave function represents only the probability of finding the electron as a whole at some point of the given region.

If, however, one ascribes to the electron a finite extension, one may arrive at a contradiction with the simplest laws of superposition of electron waves. Namely, if there are waves with sufficiently small wavelength, then one can always, by superposition, construct a wave packet whose intensity is concentrated in an arbitrarily small region. And since the wavelengths of electrons are inversely proportional to the electron momentum, one can find waves of arbitrarily small wavelength for electrons with sufficiently large momenta. Therefore a finite radius of the electron fundamentally contradicts quantum mechanics, so that all attempts to introduce it without obstruction into the theory have so far remained unsuccessful.

It is now clear that the calculation of the proper energy of the electron leads to infinitely large values also in quantum

mechanics. This energy must be added to the energy of the electron’s state. Since the self-energy in different states of motion of the electron must be very different, the introduction of the self-energy completely changes the energy differences between the various stationary states of the atom, and we ought to have an infinite displacement of the spectral lines. In reality, the displacement of spectral lines caused by the additional self-energy must be very small, since it is not detected experimentally. Thus the theory seems to suffer complete collapse when one attempts to take into account the field created by the electron. On the contrary, the results of quantum mechanics in calculating the states of motion of electrons in atoms show that the action of external fields on the electron can be correctly taken into account. Apparently, the failure of the theory should be attributed mainly to the interaction between the radiation field and wave functions with a very short wavelength, which apparently cannot be represented by the ordinary “correspondence method.” Here one is dealing with waves shorter than the classical value of the electron radius \(\frac{e^2}{mc^2}\), possessing energies exceeding \(7\cdot 10^7\ \mathrm{V}\).

It is interesting to point out here that, in general, only because of the weak coupling between charge and field was it possible to construct the quantum mechanics of matter without a clear understanding of this coupling. Observation of stationary energy states in the atom is possible only because the time spent in one state, owing to the weak coupling with the radiation field, is very large in comparison with the periods of the other oscillatory processes in the atom. As a measure of the coupling between the electron and the field one may use the well-known Sommerfeld fine-structure constant

\[ \alpha = 2\pi \frac{e^2}{hc} = \frac{1}{137}. \]

Here \(\alpha\) is a dimensionless quantity, representing the ratio of the purely electrical quantity \(e^2\) to the product \(hc\), which has the same dimension, of the fundamental constants of quantum mechanics and relativistic mechanics. Thus, for example, the lifetime of the first excited state of the hydrogen atom is approximately equal to \(\left(\frac{1}{\alpha}\right)^3\) periods of the light emitted in the transition to the unexcited state. Thus, the smaller \(\alpha\) is (i.e. the coupling between the electron and the radiation field), the longer the electron, unperturbed by the radiation field, remains in a stationary state. A further example of precisely the smallness of the ratio \(\frac{2\pi e^2}{hc}\) facilitating the possibility of broad applications of quantum mechanics without solving the problem of self-energy is as follows. As already mentioned, the difficulties connected with the appearance of an excessively large self-energy are caused precisely by those electron waves whose wavelength is smaller than the classical value of the electron radius \(\frac{e^2}{mc^2}\). But those encountered in the motions of the electron in the atom

wavelengths usually have a magnitude at least

\[ \frac{h}{2\pi mc}, \]

i.e., at least \(\frac{1}{\alpha}\) times greater than the critical wavelength. But if the ratio \(\alpha\) were not so small, then quantum mechanics would already be inapplicable to the inner electrons of heavy atoms.

A consistent transfer of classical electrodynamics into quantum mechanics has proved impossible, since it encounters internal contradictions in connection with the infinitely large field energies that result. The new discoveries of the creation of matter under the action of light apparently indicate that classical electrodynamics, in its very foundations, is only an approximation. Indeed, Maxwell’s equations require the existence of electromagnetic waves in empty space, which can be superposed and propagated quite independently of one another. In the preceding section we have already pointed out that the possibility of the creation, under the action of light, of an electron pair limits the freedom of passage of light rays through space. The scattering of light in strong electric fields or by means of other light rays means that Maxwell’s equations for the vacuum can be correct only approximately. And indeed, the additional terms to Maxwell’s equations represent certain deviations in the behavior of fields well only insofar as the frequencies involved are so small that there can be no question of the creation of electron pairs.^13

It is interesting to note that, with such a representation, we arrive at a closer connection between the field and matter. Whereas one cannot conceive of matter without an electromagnetic field, because it consists of charged particles, until now it has been assumed that fields can exist in empty space independently of matter. However, the situation resulting from the discovery of the positron shows that such independence also cannot be considered valid, since the fields themselves either directly create matter or else produce states of polarization similar to matter.

Many experiments have been performed in order to establish a closer connection between the field of matter waves and the electromagnetic field. The essential difference between the two kinds of waves is that electron waves are four-component spinors, whereas light waves are obtained from two field vectors. De Broglie attempted to establish a formal connection, based on the fact that quantities possessing the properties of vectors^15 can easily be composed from products of two spinors. One example of such quantities is the current density defined by formulas (19). In a similar way one can construct quantities possessing properties analogous to those of the six components of the electromagnetic field. The circumstance that, formally, a vector can be constructed from two spinors, but cannot

to represent a spinor by means of vectors led de Broglie to assert that the electromagnetic field of the vectors E and H can be obtained from spinor waves; in doing so, however, de Broglie wanted to make use of the wave function of the so-called “neutrino.” These are uncharged particles, of smaller mass than electrons, which play a certain role in nuclear physics. According to this conception, one light quantum should be obtained from two neutrinos; but it could perhaps be obtained in exactly the same way from the wave functions of one positive and one negative electron, which, under this coupling, would give an uncharged photon. However, one of the essential difficulties in such a construction of the field from the wave functions of matter is that spinor particles—electrons, neutrinos—obey Fermi statistics, so that in the same state there can be no more than two such particles with oppositely directed spins. Their combination—the photon—must, however, satisfy Bose statistics. But, as is known, a system of two bound particles which themselves obey Fermi statistics will obey Bose statistics only if these particles are bound by a very large force. We have no point of departure for assumptions about the existence, still less about the behavior, of such forces, and therefore it has so far not been possible fruitfully to continue the attempt conceived by de Broglie; thus this idea apparently has at present only formal, and not physical, significance.

An attempt was made to proceed also in the reverse direction—to construct matter out of the electromagnetic field^16. Even before the appearance of wave mechanics, Mie expressed the idea of changing the fundamental equations of electrodynamics in order to resolve the question of the structure and proper energy of the electron. The attempts of Mie, which in recent times have been successfully continued by Born and Infeld, proceed from the fact that the material electron must not be set in opposition to the electromagnetic field as something essentially different, and that the laws of the field must be changed in such a way that the field would possess certain features, nodes, whose motion and action on the field would in essence coincide with the corresponding actions of the electron, but in such a way that the difficulties indicated above would be eliminated. In the electrodynamics of Born and Infeld this problem is solved in its main outlines, and it should be noted that the changes made to Maxwell’s equations lead to the scattering of light by electromagnetic fields in empty space, just as was obtained from the theory of holes. However, the application of wave mechanics to such electrons and the explanation of all phenomena associated with them (for example, spin and the occurrence of electron pairs) apparently encounters insurmountable difficulties.

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1. By the Compton wavelength is meant the wavelength entering into the Compton effect, \(\lambda_0 = \dfrac{h}{mc}\). ↩