THE MYSTERIOUS NUMBER 137*

Max Born

1. Introduction

Two fundamental theories dominate modern physics: Einstein’s theory of relativity and Planck’s quantum theory. Both of these theories represent a generalization of the classical laws of mechanics and electrodynamics. The theory of relativity deals with the influence of relative motion on physical phenomena; the classical laws are valid in the limiting case of small velocities (small in comparison with the speed of light \(c = 3 \cdot 10^{10}\ \mathrm{cm/sec}\)), while new phenomena, for example the variation of mass with velocity, are measurable only in those cases when the velocity of particles approaches \(c\). On the other hand, quantum theory deals with the finest structure of light and matter; the classical laws prove to be valid in the limiting case of macroscopic bodies. There exists a definite quantity having the dimension of energy multiplied by time (the dimension of “action”), and equal to \(6.5 \cdot 10^{-27}\ \mathrm{erg \cdot sec}\). This quantity (Planck’s constant \(h\)) is characteristic of quantum phenomena. We encounter quantum phenomena whenever the “action” has a magnitude of the order of \(h\), or even smaller. Both theories—quantum theory and the theory of relativity—have been developed to a high degree of perfection. But they have not merged into one. Quantum theory is a remarkable mathematical construction; with astonishing success it predicts natural phenomena in all cases when the velocity of the particles under consideration is small in comparison with the speed of light, i.e. when relativistic effects may be neglected. Fortunately, this limitation is indeed fulfilled in most problems with which atomic physics is concerned. The velocity of the electrons of which the outer parts of atoms consist is for the most part small in comparison with \(c\). Even in nuclear physics the motion of protons and neutrons, owing to their large mass (1840 times greater than the mass of the electron), is sufficiently slow that relativistic corrections may be neglected. This circumstance has been used in the most recent attempts to construct a theoretical model of the nucleus based on quantum mechanics.

* Lecture delivered at the South Indian Science Association (Bangalore, India) on November 9, 1935; Proc. Indian Acad. Sci. (A), 2, 533, 1935. Translated from English by A. Sveshnikov, edited by M. Bronstein.

However, there are phenomena in which fast particles are involved, and in which, consequently, the nonrelativistic quantum theory ceases to be applicable. The first phenomenon of this kind—the fine structure of the spectral lines of hydrogen—was investigated by Sommerfeld in 1916. He discovered that in the formula which describes the fine structure of the lines there appears the number

\[ \frac{2\pi e^2}{hc} \]

—a dimensionless combination of the fundamental physical constants: the velocity of light \(c\), Planck’s quantum constant \(h\), and the charge of the electron \(e\). The dimensionless quantity \(\frac{2\pi e^2}{hc}\) (the “fine-structure constant”) is precisely that mysterious number mentioned in the title. The fine-structure constant plays an important role in a very large range of phenomena, far beyond the bounds of that special question in the study of which this constant was first discovered. The present lecture has as its aim to clarify the role of the fine-structure constant in the laws of atomic physics and to set forth certain considerations concerning the deeper physical meaning of this dimensionless quantity.

2. Bohr Orbits

We shall begin with a brief review of the old quantum theory of Niels Bohr, since Sommerfeld’s work was based on this theory.

Consider a system consisting of a nucleus with positive charge \(Ze\), and an electron with negative charge \(-e\), where \(e\) is the elementary unit of electricity \((e = 4.77 \cdot 10^{-10}\) electrostatic units). These particles move about their common center of gravity in Keplerian ellipses, since, according to Coulomb’s law, the force acting between them is equal to \(-\frac{e^2 Z}{r^2}\), where \(r\) is their mutual distance. The energy \(E\) of this motion does not depend on the eccentricity of the ellipse, but is determined only by its major semiaxis \(a\). If the energy of a stationary electron, infinitely removed from the nucleus, is taken as the zero level of energy, then the energy will have the negative value:

\[ E=-\frac{Ze^2}{2a}. \]

Kepler’s third law establishes a relation between the major semiaxis \(a\) and the period of revolution \(T\) (or, what is the same thing, between \(a\) and the quantity \(\nu=\frac{1}{T}\), i.e., the number of revolutions per unit time):

\[ \frac{a^3}{T^2}=a^3\nu^2=\frac{Ze^2}{4\pi^2 m}, \]

where \(m\) may be equated to the true mass of the electron, since the heavy nucleus may be regarded as practically immobile. (The influence

of the motion of the nucleus, which, generally speaking, is small, we shall not consider here.)

According to classical laws, the energy \(E\) can take a continuous series of values. Applying Planck’s quantum theory to the system under consideration, Bohr assumed that in reality only certain values of \(E\), forming a discrete series, are admissible. The simplest way of calculating these values consists in applying the quantum principle, according to which the angular momentum of a rotating system is always an integral multiple of the quantity \(\dfrac{h}{2\pi}=\hbar\) (the symbol \(\hbar\), introduced by Dirac, will henceforth be used throughout). In the case of a circular orbit of radius \(a\), the circumference is \(2\pi a\), the velocity is \(\dfrac{2\pi a}{T}=2\pi a\nu\), the momentum is \(2\pi a\nu m\), and the angular momentum is \(2\pi a\nu m\cdot a=2\pi a^2\nu m\). Therefore the quantum condition has the form:

\[ 2\pi \nu m a^2=\hbar n,\quad n=1,\ 2,\ 3,\ldots \]

From the last three equations we find:

\[ a=\frac{\hbar^2}{m e^2}\frac{n^2}{Z},\qquad E=-\frac{m e^4}{2\hbar^2}\frac{Z^2}{n^2}. \]

These expressions also remain valid in the case of elliptic orbits with major semiaxis \(a\).

We have derived these well-known formulas here because the combinations of constants occurring in them play a very important role later on. Let us introduce the following notation:

\[ a_1=\frac{\hbar^2}{m e^2}=0.532\cdot 10^{-8}\ \text{cm}. \]

\[ E_1=\frac{m e^4}{2\hbar^2}=2.15\cdot 10^{-11}\ \text{erg}. \]

The stationary states of the hydrogen atom and of similar one-electron systems are characterized by the following formulas:

\[ a_n=a_1\frac{n^2}{Z},\qquad E_n=-E_1\frac{Z^2}{n^2}, \]

where \(a_1\) and \(-E_1\) are the radius and energy of the first orbit of the hydrogen atom \((Z=1)\).

\(E_n\) is called the “Balmer term”; applying Bohr’s frequency condition

\[ h\nu=E-E' \]

For light emitted or absorbed by the hydrogen atom, we find:

\[ h\nu = E_1\left(\frac{1}{n'^2} - \frac{1}{n^2}\right). \]

For \(n' = 2,\ n = 3, 4, 5, 6,\ldots\) this formula gives the well-known hydrogen Balmer series. Taking other values of \(n'\), we obtain all the other known series of this atom. For \(Z = 2, 3,\ldots\) the corresponding series of one-electron ions \(He^+\), \(Li^{++},\ldots\) are obtained.

A detailed investigation of these lines shows that they are not simple, but complex (the fine structure of spectral lines); the investigation of fine structure—this was the problem which Sommerfeld began to solve. However, before recounting this further development of the theory, we shall consider the physical meaning of the two constants \(a_1\) and \(E_1\) introduced by Bohr.

3. Atomic units

The theory of Bohr orbits, based on a combination of classical mechanics with quantum conditions, is logically unsatisfactory. Therefore Bohr’s orbits were replaced by more abstract theories—various formulations of quantum mechanics. However, the fundamental constants \(a_1\) and \(E_1\), introduced by Bohr, have been retained in the new theory as well. In all cases where the velocities of the electrons are small in comparison with \(c\), these constants are natural units of length and energy for all atoms. As Hartree showed, the properties of all atoms, expressed in these “atomic units” \(a_1\) and \(E_1\), turn out to be abstract mathematical numbers or functions. The Hamiltonian function of an atom, i.e. its energy considered as a function of the electronic momenta \(p_1, p_2,\ldots\) and coordinates \(x_1, y_1, z_1, x_2, y_2, z_2,\ldots\), is equal to

\[ H = \frac{1}{2m}\sum_k p_k^2 - \sum_k \frac{Ze^2}{r_k} + \sum_{kl}' \frac{e^2}{r_{kl}}, \]

where \(r_k\) is the distance of the \(k\)-th electron from the nucleus, and \(r_{kl}\) is the mutual distance of the \(k\)-th and \(l\)-th electrons. The atom is described by the Schrödinger wave equation

\[ (H - E)\psi = 0, \]

where \(H\) is the differential operator obtained by replacing the components of the vector \(p_k\) by

\[ \frac{\hbar}{i}\frac{\partial}{\partial x_k}, \qquad \frac{\hbar}{i}\frac{\partial}{\partial y_k}, \qquad \frac{\hbar}{i}\frac{\partial}{\partial z_k}. \]

It is easy to show that, after introducing \(a_1\) and \(E_1\) as units, the wave equation takes the following form:

\[ \left\{ -\frac{1}{2}\sum_k\left(\frac{\partial^2}{\partial x_k^2}+\frac{\partial^2}{\partial y_k^2}+\frac{\partial^2}{\partial z_k^2}\right) -\sum_k \frac{Z}{r_k} +\sum_{kl}' \frac{1}{r_{kl}} -\frac{1}{2}E \right\}\psi=0. \]

This equation contains only numerical constants. If the number of electrons is equal to 1, then we obtain the wave equation of the hydrogen-like atom; the energies of the stationary states turn out to be exactly equal to the Balmer terms, which in atomic units have the following form:

\[ E_n=-\frac{Z^2}{n^2}. \]

In the same way, the energies and geometrical dimensions of all atoms, expressed in atomic units, also turn out to be dimensionless mathematical constants.

The same is true with respect to molecules, if the motion of the nuclei is neglected, which is entirely permissible, since the masses of the nuclei are very large in comparison with the mass of the electrons. Having arbitrarily chosen the arrangement of the nuclei, we can determine the motion of the electrons from Schrödinger’s equation, which in atomic units contains no other constants (apart from abstract mathematical numbers), except the coordinates of the nuclei. Then the energy of the electrons will be a function of these nuclear coordinates \(E(X_1,Y_1,Z_1,X_2,Y_2,Z_2,\ldots)\), and the minima of this function determine the possible arrangements of the nuclei. If \(X_1,Y_1,Z_1,X_2,\ldots\) are expressed in atomic units, then it is clear that the relative positions of the nuclei in the molecule are determined by purely mathematical numbers. It follows from this that all molecules must have dimensions of the order of \(a_1\) and energies of the order of \(E_1\). Crystals are nothing other than very large molecules. If the number of atoms is given, then the energy and dimensions of a crystal at low temperatures must be determined by quantities of the order of \(a_1\) and \(E_1\).

There are many cases of atomic collisions that also belong to the class of phenomena governed by the constants \(a_1\) and \(E_1\). These include all phenomena in the consideration of which the motion of the nuclei may be neglected, provided only that the velocity of the electrons is small in comparison with \(c\); for example, the phenomena of ionization and excitation of atoms by slow electrons. The effective cross section of such a process and the angular distribution can be calculated with the help of the same wave equations as the stationary states; therefore here, too, everything will be expressed in purely mathematical terms, if \(a_1\) and \(E_1\) are chosen as units.

We see that a large domain of physics is governed by the atomic units \(a_1, E_1\). However, there are many cases in which these units are insufficient.

The first, to some extent trivial, exception concerns the motion of nuclei. Whenever this motion plays any role (for example, in the thermal motion of gases or solids, in optical phenomena, etc.), the masses of the nuclei become the determining factor.

Empirically, the masses of nuclei are determined by means of “atomic weights.” However, we know from Aston’s experiments with the mass spectrograph that chemically determined atomic weights refer not to pure substances, but to mixtures of isotopes, i.e. nuclei with the same charge \(Ze\), but with different masses. The masses of pure isotopes turn out, almost exactly, to be whole multiples of a certain minimal mass \(M\)—the mass of the hydrogen nucleus (the proton). The small deviations encountered from these whole numbers are easily explained by the assumption that nuclei are not a simple sum of protons, but mechanical systems with a certain binding energy. Since, according to Einstein’s law, energy and mass are equivalent to one another, the release of energy in the formation of nuclei from protons leads to a decrease of mass. Therefore, when atomic nuclei must be included in the discussion, one should introduce yet another new physical constant—the proton mass \(M\), approximately 1840 times greater than the electron mass. The number 1840—the ratio of masses \(\frac{M}{m}\)—is the second mysterious number of physics. It determines the order of magnitude of all nuclear motions, for example the velocity of gas molecules, the rotation and vibration of molecules, the vibrations of crystal lattices, and all properties of matter depending on these motions: specific heat, thermal conductivity, diffusion, absorption of infrared rays, the Raman effect, rates of chemical reactions, etc.

However, I think that the problem of the number 1840 is closely connected with the problem of the number 137, which is the subject of this lecture. We shall return to this question later.

4. Sommerfeld’s fine-structure formula

Let us now turn to the second condition limiting the use of the atomic units \(a_1\) and \(E_1\), namely: let us consider the assumption that the velocities of the electrons are small in comparison with \(c\). Whenever this assumption ceases to be true, we must take the theory of relativity into account.

This was first done by Sommerfeld in order to explain the fine structure of the hydrogen lines. As was mentioned, he derived his formula starting from Bohr’s model. The method he used clearly shows in precisely which cases relativistic corrections must be taken into account. Therefore we shall begin with a brief consideration of Sommerfeld’s arguments. It was indicated above that all elliptical orbits with the same major semiaxis have the same energy, independently of their eccentricities. However, in the case of very elongated orbits the perihelion

very close to the nucleus, and therefore the velocity of the electron near perihelion becomes very large. In these cases the laws of classical mechanics cease to hold and must be replaced by the laws of relativistic mechanics. This means that the mass no longer remains constant, but increases with increasing velocity. If this change in mass is taken into account, the motion will no longer proceed along an ellipse with fixed axes, but along a curve having the form of a rosette. Such motion may be represented as the result of a superposition of the classical motion along an ellipse and the precession of the perihelion. The period of this precession depends on the degree of elongation of the ellipse and is quantized by means of a new quantum number \(k\), which may take the values \(0, 1, 2\ldots\) The energy becomes a function of the number \(k\), which, in a first approximation, is determined by the formula:

\[ E=-E_1\frac{1}{n^2}\left\{1+\left(\frac{e^2}{\hbar c}\right)^2 Z^2\left(\frac{n}{k}-\frac{3}{4}\right)\right\}. \]

This formula leads to a much larger number of stationary states, since to each value of \(n\) there correspond several states with different \(k\). As a result, each spectral line is split into a number of separate components.

The experimental confirmation of Sommerfeld’s formula for the fine structure may be regarded as brilliant proof both of Einstein’s principle of relativity and of Planck’s quantum theory.

Sommerfeld’s formula was the first attempt to unite these two great theories. As was said above, each of these theories is characterized by its own universal constant: the theory of relativity by the constant \(c\), and quantum theory by the constant \(h\). Therefore it is natural that in Sommerfeld’s formula the correction term depends on both of these constants, having as coefficient the square of the quantity

\[ \alpha=\frac{e^2}{\hbar c}, \]

which Sommerfeld called the fine-structure constant. This constant is equal to unity divided by the mysterious number 137, which appears in the title of my lecture. What, then, is mysterious about this number? And why is it worth speaking about this number at all?

The most remarkable thing about the quantity \(\alpha=\frac{e^2}{\hbar c}\) is that this combination of three universal constants turns out to be a dimensionless number. This is easy to show. The electric force acting between two particles with the same charge \(e\), located at a distance \(r\) from one another, according to Coulomb’s law is equal to \(\frac{e^2}{r^2}\); the energy necessary to separate these particles is equal to \(\frac{e^2}{r}\). Conseque-

Thus, \(e^2\) can be measured in units of \(\text{erg}\cdot\text{cm}\). On the other hand, \(\hbar\) has the dimension \(\text{erg}\cdot\text{sec}\), and the dimension of \(c\) is \(\text{cm}/\text{sec}\); consequently, \(\hbar c\) is measured in \(\text{erg}\cdot\text{cm}\), i.e. has the same dimension as \(e^2\). Substituting the known values of \(e\), \(\hbar\), \(c\), we find:

\[ \alpha = 0.00734 \cong \frac{1}{137}. \]

The fact that this number is many times smaller than unity not only gives us the right to regard relativistic effects as small (and, consequently, to consider their influence as “fine structure”), but also leads to very important consequences in the question of the structure of matter in general.

The mysterious character of the number mentioned in the title will become especially clear if we compare the two numerical coefficients in the formula for the fine structure, namely: \(\left(\frac{1}{137}\right)^2\) and \(\frac{3}{4}\). The latter coefficient is an arithmetical quantity derived from mathematical reasoning. The first coefficient, however, has a different nature: it is derived as a combination of dimensional physical quantities measured in arbitrary units, and therefore our knowledge of this coefficient depends on the accuracy of measurements.

Is such a state of affairs satisfactory? I think, in no case. We must require that numerical coefficients in physical laws always be mathematical numbers, like \(\frac{3}{4}\) or \(\pi\), and so on. If in the present case the matter seems to be otherwise, then this, one must suppose, has occurred because of the imperfection of the theory. A more perfect theory ought to derive the number \(\alpha\) with the help of purely mathematical reasoning, without referring to the results of measurements. But in that case we arrive at a remarkable conclusion: if \(\alpha\) is a mathematical number, then the constants \(e\), \(\hbar\), \(c\) cannot be regarded as independent! In other words, if two of them are measured, the third can be derived by calculation. If, for example, \(\hbar\) and \(c\) are taken as the fundamental quantities, then an ideal theory should give us the possibility of calculating the charge of the electron! But it is perhaps too risky to reason about a theory which does not yet exist, and therefore, before we pass on to a further analysis of these questions, we shall first consider the role that the number 137 plays in the existing theory.

5. Electronic Units

We see that in the nonrelativistic approximation the properties of atoms and molecules can be expressed by abstract mathematical numbers if length and energy are expressed in Hartree atomic units \(a_1\) and \(E_1\). Sommerfeld’s results show that these units are insufficient for describing all properties of atoms, since relativistic effects depend on the constant \(\alpha\). But \(a_1\) and \(E_1\) are expressed through the quantities \(e\) and \(m\), characterizing the electron, and also through the quantum constant \(\hbar\). The latter quantity also enters into \(\alpha\),

and since \(\alpha\) is an abstract number, \(\hbar\) can be eliminated from \(a_1\) and \(E_1\):

\[ a_1=\frac{\hbar^2}{me^2}=\frac{e^2}{mc^2}\left(\frac{\hbar c}{e^2}\right)^2 =\frac{e^2}{mc^2}\frac{1}{\alpha^2}=(137)^2\frac{e^2}{mc^2}. \]

\[ E_1=\frac{me^4}{2\hbar^2}=\frac{1}{2}mc^2\left(\frac{e^2}{\hbar c}\right)^2 =\frac{1}{2}mc^2\alpha^2=\frac{1}{2}\frac{1}{(137)^2}mc^2. \]

Now \(a_1\) and \(E_1\) are expressed only in terms of quantities characterizing the electron: through the length \(a_0=\dfrac{e^2}{mc^2}\) and the energy \(\varepsilon=mc^2\). What, then, is the physical meaning of these quantities?

With respect to the energy \(\varepsilon\), the answer is given by Einstein’s law, which follows from the theory of relativity. This law states that mass and energy are physically identical and that the ratio of energy to mass is equal to \(c^2\). Consequently, the quantity

\[ \varepsilon=mc^2 \]

is the energy of the electron.

In an analogous way, the length \(a_0=\dfrac{e^2}{mc^2}\) may be regarded as the radius of the electron. In this connection it is necessary to make several remarks.

At the present time physicists for the most part regard the electron as a point charge, and reject any attempt to determine the dimensions and form of the electron as fundamentally inadmissible and meaningless. But this was not always so. A quarter of a century ago it was considered good form to regard the electron as a body of finite dimensions, for example as a sphere of radius \(a_0\), possessing a charge distributed over its surface or throughout its entire volume. Starting from these ideas, the electromagnetic energy of the electron and its electromagnetic momentum \(p\) can be expressed, by means of Maxwell’s equations, in terms of the charge \(e\) and the radius \(a_0\). In the case of a spherical form of the electron and for small velocities the result has the following form:

\[ \varepsilon=A\frac{e^2}{a_0}, \qquad p=B\frac{e^2}{a_0}\frac{v}{c^2}, \]

where \(A\) and \(B\) are numerical coefficients depending on the distribution of the charge.

Both these expressions tend to infinity when \(a_0=0\). Therefore the hypothesis that the electron is a point charge proves inconceivable, if one proceeds solely from the assumption of the strict correctness of Maxwell’s equations.

The second formula, having the form \(p=mv\), suggests that the coefficient \(m=B\dfrac{e^2}{a_0c^2}\) may be interpreted as the electromagnetic mass of the electron.

According to Einstein’s law (\(mc^2=\varepsilon\)) one should expect that \(mc^2=B\dfrac{e^2}{a_0}\) will be equal to \(\varepsilon=A\dfrac{e^2}{a_0}\), i.e., that \(A=B\). However, calculations by means of Maxwell’s equations show that these coefficients are not equal to one another. Thus, for example, in the case of a surface charge \(A=\dfrac{2}{3}B\).

In order to explain this contradiction, physicists assumed that the charge is held within the volume of the electron by some very large cohesive forces, the nature of which is unknown to us. To these forces there corresponds an additional energy, which leads to the appearance of the missing quantity \(\dfrac{1}{3}B\).

However, this explanation looks somewhat artificial and not entirely satisfactory. The form of the electron and the distribution of charge in it have, within the framework of such a theory, just as arbitrary a character as these cohesive forces. Therefore physicists in general began to refrain from any hypotheses concerning the internal structure of the electron. On the other hand, however, the relation between the mass or energy of the electron and its radius appeared so attractive that physicists did not decide to abandon it. Therefore the equation

\[ \frac{e^2}{a_0}=mc^2=\varepsilon=8.1\cdot 10^{-7}\ \text{erg}, \]

where the coefficient \(A=1\) has been chosen arbitrarily, began to be regarded as a conventional definition of the electron radius:

\[ a_0=\frac{e^2}{mc^2}=2.82\cdot 10^{-13}\ \text{cm}. \]

The radius and energy of the first orbit of the hydrogen atom are expressed through \(a_0\) and \(\varepsilon\):

\[ E_1=\frac{\alpha^2}{2}\,\varepsilon =\frac{\varepsilon}{2(137)^2} =\frac{\varepsilon}{2\cdot 18770}, \]

\[ a_1=\frac{1}{\alpha^2}\,a_0=(137)^2a_0=18770a_0. \]

Now the important significance of the number 137 becomes obvious: its square determines the ratio of the energy and radius of the hydrogen atom (\(E_1\) and \(a_1\)) to the energy and radius of the electron (\(\varepsilon\) and \(a_0\)).

Let us call \(\varepsilon\) and \(a_0\) the electron units of energy and length. Since \(e^2=a_0\varepsilon\), the value of \(e\) in these units is equal to 1. The circumstance that

\[ \frac{1}{\alpha^2}=(137)^2=18770\gg 1, \]

confirms our right to treat electrons in atoms and molecules as point charges with constant masses. If one increas-

...scale the electron up to the size of a pea (diameter \(0.5\ \mathrm{cm}\)), then the hydrogen atom will turn into a sphere with a diameter of \(100\ \mathrm{m}\). The binding energy of the atom is smaller than the electron’s own energy by the same factor.

We shall now show that all the laws of atomic physics, expressed in electronic units, turn into dimensionless equations containing only numerical coefficients. These coefficients, however, are not purely mathematical numbers, since among them occurs the constant \(\alpha\). The large numerical value

\[ \frac{1}{\alpha}=137 \]

determines the order of magnitude of all physical phenomena expressed in electronic units.

6. Relativistic wave equations

The explanation of the fine structure given by Sommerfeld was an important first attempt which, however, had the same shortcomings as Bohr’s theory as a whole. It therefore became necessary to replace Sommerfeld’s theory by another theory, based on the modern methods of quantum mechanics.

We shall therefore turn to the consideration of attempts to construct a wave equation for the electron satisfying the principle of relativity.

The first relativistic wave equation for one electron in a given electromagnetic field was proposed by Gordon and Klein; it has the following form:

\[ \left\{ -\left(\frac{W}{c}+\frac{eA_0}{c}\right)^2 + \left(\mathbf{p}+\frac{e\mathbf{A}}{c}\right)^2 + m^2c^2 \right\}\psi=0, \]

where the operator \(W\) denotes

\[ -\frac{\hbar}{i}\frac{\partial}{\partial t} \]

(by analogy with the relations

\[ p_x=\frac{\hbar}{i}\frac{\partial}{\partial x},\ldots); \]

here \(A_0\) is the scalar potential, and \(\mathbf{A}\) is the vector potential of the external field. Since this field is caused by electrons or nuclei, we may factor out from the potentials the multiplier

\[ \frac{e}{a_0}, \]

equal to the potential of a point charge \(e\) at the distance \(a_0\). We obtain

\[ A_0=\frac{e}{a_0}\varphi_0,\qquad \mathbf{A}=\frac{e}{a_0}\bar{\varphi}, \]

where \(\varphi_0\) and \(\bar{\varphi}\) are dimensionless functions of space and time.

If \(a_0\) is chosen as the unit of length, and

\[ \frac{a_0}{c} \]

as the unit of time, then the wave equation takes the form:

\[ \left\{ -\left(\frac{1}{i}\frac{\partial}{\partial t}-\alpha\varphi_0\right)^2 + \left(\frac{1}{i}\nabla+\alpha\bar{\varphi}\right)^2 + \alpha^2 \right\}\psi=0. \]

It contains only a dimensionless constant \(\alpha\). However, this equation does not express certain important properties of the electron, namely: the electron spin, i.e. the fact that the electron possesses an angular momentum equal to \(\frac{1}{2}\hbar\), i.e. half the angular momentum corresponding to the first Bohr orbit, and at the same time possesses a magnetic moment \(\frac{e\hbar}{2mc}\), equal to the magnetic moment of this orbit. Moreover, theoretical difficulties are associated with the interpretation of this equation, since it contains the second derivative of \(\psi\) with respect to time. The statistical interpretation of quantum mechanics assumes that \(|\psi|^2 dx\,dy\,dz\) is the probability of finding the electron in a definite volume element \(dx\,dy\,dz\). In this case, knowledge of the distribution of the function \(\psi\) at the instant \(t=0\) should be sufficient in order to be able to calculate this distribution at any subsequent instant of time; and hence the differential equation must be of first order with respect to \(\frac{\partial}{\partial t}\). But since the equation given above turns out to be not of first but of second order, the specification of \(\frac{\partial \psi}{\partial t}\) at the instant \(t=0\) must enter among the initial conditions, which is difficult to interpret physically.

On the basis of these considerations Dirac found his famous wave equation of first order with respect to \(x, y, z, t\):

\[ \left\{\left(\frac{W}{c}+\frac{e}{c}A_0\right)+\bar{\gamma}\left(\mathbf{p}+\frac{e}{c}\mathbf{A}\right)+\gamma_0 mc\right\}\psi=0, \]

which contains as coefficients four numerical (dimensionless) operators, namely: the matrices \(\gamma_0,\ \bar{\gamma}=(\gamma_1,\gamma_2,\gamma_3)\).

The most important property of Dirac’s equation is that it leads to an explanation of the electron spin. This property of the equation becomes clear if one tries, as a first approximation, to derive from it the Gordon–Klein equation: in this derivation an additional term appears, which has the form

\[ \frac{\hbar e}{2mc}\,\bar{\sigma}\mathbf{H}, \]

where \(\mathbf{H}\) is the magnetic-field strength, and \(\bar{\sigma}\) is a vector operator composed of the matrices \(\gamma\). In addition to this term, an analogous electric term appears (purely imaginary and having no immediate physical meaning). The magnetic term may be interpreted as the interaction energy of the field \(\mathbf{H}\) with the magnetic moment of the electron, which is equal to \(\frac{\hbar e}{2mc}\) (the so-called magneton).

If electronic units are introduced, Dirac’s equation assumes the form:

$$ \left\{\left(-\frac{1}{i}\frac{\partial}{\partial t}+\alpha \varphi_0\right)+\bar{\gamma}\left(\frac{1}{i}\nabla+\alpha\bar{\varphi}\right)+\gamma_0\right\}\psi=0. $$

It contains only the constant \(\alpha=\dfrac{1}{137}\).

The expression for the magneton can be transformed as follows:

$$ \frac{\hbar e}{2mc}=\frac{1}{2}\,e\,\frac{e^2}{mc^2}\,\frac{\hbar c}{e^2}=\frac{ea_0}{2\alpha}, $$

where \(ea_0\) is the dipole moment of two elementary charges separated from one another by a distance \(a_0\), which in electronic units is equal to 1. In these units the magneton is equal to \(\dfrac{1}{2\alpha}=68.5\).

If the potentials remain constant in time, then stationary states exist, which can be determined by taking the function \(\psi\) proportional to \(e^{-\frac{i}{\hbar}Et}\), or, in electronic units, \(e^{-i\alpha Et}\). After reduction by \(\alpha\), the wave equation takes the form:

$$ \left\{E+\varphi_0+\bar{\gamma}\left(\frac{1}{i\alpha}\nabla+\bar{\varphi}\right)+\gamma_0\right\}\psi=0. $$

In the case of a hydrogen-like atom or ion, the field of the nucleus is described by the scalar potential \(A_0=\dfrac{Ze}{r}\), whereas \(\mathbf{A}=0\), i.e., in electronic units \(\varphi_0=\dfrac{Z}{r}\), \(\bar{\varphi}=0\). The equation in this case can be solved exactly: the stationary states are determined by the formula

$$ E=\frac{1}{\sqrt{1+\dfrac{\alpha^2 Z^2}{\left\{\,n-k+\sqrt{k^2-\alpha^2 Z^2}\,\right\}^2}}}, $$

where \(n\) and \(k\) are the same quantum numbers as in Sommerfeld’s theory. The very same formula was also derived by Sommerfeld, starting from Bohr’s atomic model. It is easy to show that the approximate formula which was given above can be obtained from the exact formula by expanding in a series in powers of \(\alpha^2\). The zeroth-order term is the rest energy of the electron (i.e. 1 in electronic units); it is not written in Sommerfeld’s formula. The first-order term with respect to \(\alpha^2\) corresponds to the Balmer term, and the next term gives the Sommerfeld correction.

The complete agreement of the exact formula with the structure of the X-ray terms found experimentally in all elements (up to the very

...of a very high atomic number) shows that the stationary states of the electron in the field of the nucleus obey, with great accuracy, a law connected with the number 137.

7. Interaction of Electrons with Radiation

The most important of all those cases in which the constant \(\alpha\) enters into the formulation of the laws of physics is the interaction of electrons with electromagnetic radiation.

Above we dealt with the stationary states of atoms and molecules, with the sizes of atoms and their energies in these states. In doing so we did not take into account that excited states are unstable, since the surrounding ether can take energy from the atom and carry it away in the form of radiation (light). This process depends on the constant \(\alpha\), as can easily be shown by means of semiclassical arguments.

An electron oscillating with frequency \(\nu\) radiates energy in the form of electromagnetic waves according to the equation

\[ -\frac{dE}{dt}=\frac{1}{\tau}E, \]

where the constant \(\tau\), having the dimension of time, is defined as follows:

\[ \tau=\frac{3}{8\pi^{2}}\frac{mc^{3}}{e^{2}\nu^{2}}. \]

This classical law can be carried over into quantum theory. In that case we have not a continuous loss of energy, but an instantaneous transition from excited states to states of lower energy. Each excited state has a certain lifetime; it is determined by the matrix element of the dipole moment of the system, which can be calculated with the aid of Schrödinger’s wave equation.

The theory leads to the result that all lifetimes have the form \(f\tau\), where \(\tau\) is the quantity indicated above, and \(f\) is a mathematical number characterizing the state under consideration.

Let us now calculate the lifetime of the hydrogen atom in the first excited state and compare it with the period of oscillation corresponding to the transition to the normal state. In this particular case the number \(f\) turns out to be approximately equal to 1. Consequently, the lifetime is equal to \(\tau\), and since the period \(T=\frac{1}{\nu}\), we obtain:

\[ \frac{\tau}{T}=\nu\tau=\frac{3}{8\pi^{2}}\frac{mc^{3}}{e^{2}\nu}. \]

Substituting here the value \(\nu\) from Balmer’s formula,

\[ \nu=\frac{1}{h}E_1\left(\frac{1}{1^2}-\frac{1}{2^2}\right)=\frac{3}{4h}E_1=\frac{3}{8h}mc^2\alpha^2, \]

we find:

\[ \frac{\tau}{T}=\frac{2}{\pi}\frac{1}{\alpha^3}=\frac{2}{\pi}(137)^3=1.64\cdot 10^6. \]

The quantity inverse to this number is the relative width of the spectral line:

\[ \frac{\Delta\nu}{\nu}=6.1\cdot 10^{-7}. \]

This fraction may also be regarded as the ratio of the energy of interaction of the electronic orbit with the radiation field \((h\Delta\nu)\) to the emitted energy \((h\nu)\).

The enormous value of the lifetime, measured in periods of oscillation of the emitted light (or, what is the same thing, the extreme smallness of the energy of interaction between the atom and the ether), plays a very great role in the world of phenomena around us and determines our method of describing this world. It gives us the possibility of considering atoms separately from the surrounding field and of ascribing stationary states to them. If \(\alpha\) were larger than it actually is, we could not draw a boundary between matter and ether, and the problem of finding the laws of nature would be hopelessly difficult for us. But the circumstance that \(\alpha\) has the value \(\frac{1}{137}\), and not some other value, is of course not a matter of chance, but a law of nature. It is clear that the explanation of the number \(\alpha\) is one of the central problems of natural science. However, before discussing this question, it is necessary to mention certain other phenomena that are determined by the magnitude of \(\alpha\).

One of the simplest phenomena in the fundamental respect (even simpler than the radiation of a bound electron just considered) is the interaction of free electrons with the radiation field. Introducing into the Dirac equation the variable field of a light wave, one can consider this phenomenon in an exhaustive way. In this problem the potential \(\varphi_0\) is a periodic function of \(\nu t-\mathbf{k}\mathbf{r}\), forming a plane wave with frequency \(\nu\) and with wave vector \(\mathbf{k}\) (the wave vector is the vector directed perpendicular to the wave front and having length \(\frac{1}{\lambda}\), where \(\lambda\) is the wavelength).

The action of a light wave on free electrons consists in the scattering of light. If the frequency is large, as for example in the case of X-rays, then the scattered wave has a lower frequency than the primary wave. This is the Compton effect. It is known,

that this phenomenon can be explained by applying the laws of conservation of energy and momentum, if one assigns to the light quantum the energy \(h\nu\) and the momentum \(\dfrac{h\nu}{c}\). By a simple calculation one obtains the result:

\[ \Delta \lambda = 2\lambda_0 \sin^2 \frac{\varphi}{2}, \]

where \(\varphi\) is the scattering angle of the light quantum, and

\[ \lambda_0=\frac{h}{mc}=2.42\cdot 10^{-10}\ \text{cm} \]

is the so-called Compton wavelength. This length is characteristic of purely quantum effects, just as the electron radius

\[ a_0=\frac{e^2}{mc^2}=2.82\cdot 10^{-13}\ \text{cm} \]

is characteristic of purely electronic effects. Their ratio, i.e.

\[ \frac{a_0}{\lambda}=\frac{e^2}{hc}=\frac{1}{2\pi}\frac{e^2}{\hbar c}=\frac{\alpha}{2\pi}, \]

turns out to be equal to the fine-structure constant divided by \(2\pi\).

Thus the existence of the mysterious number is reduced to the fact that in the laws of nature there exist two different “natural” units of length—the larger unit \(\lambda_0\), arising from quantum theory, and the smaller unit \(a_0\), connected with the electron.

The statistical laws of light scattering, namely the law of angular distribution and the total probability of scattering, can be expressed with the aid of the effective cross section \(q\). For long waves the effective cross section can be derived with the aid of Maxwell’s equations and classical mechanics; the result of this derivation is the well-known formula of J. J. Thomson:

\[ q=\frac{8\pi}{3}a_0^2. \]

For shorter waves the Dirac equation was used. The calculation, first carried out by Klein and Nishina, gives for the cross section an expression that differs from Thomson’s by a factor depending on \(\eta=\dfrac{\lambda_0}{\lambda}\), where \(\lambda\) is the wavelength of the incident light, and \(\lambda_0=\dfrac{h}{mc}\) is the Compton wavelength:

\[ q=\frac{8\pi}{3}a_0^2 f(\eta), \]

where

\[ f(\eta)=\frac{3}{4}\left\{\frac{1+\eta}{\eta^{2}}\left[\frac{2(1+\eta)}{1+2\eta}-\frac{1}{\eta}\lg(1+2\eta)\right]\right\}+ \]

\[ +\frac{1}{2\eta}\lg(1+2\eta)-\frac{1+3\eta}{(1+2\eta)^2}. \]

Since

\[ \lambda_0=\frac{2\pi a_0}{\alpha}=2\pi\cdot137a_0, \]

then

\[ \eta=\frac{2\pi}{\alpha\lambda}, \]

if \(\lambda\) is measured in electronic units. In the same units the cross section

\[ q=\frac{8\pi}{3}f\left(\frac{2\pi}{\alpha\lambda}\right) =\frac{8\pi}{3}f\left(\frac{2\pi\cdot137}{\lambda}\right) \]

turns out to be a numerical function of \(\lambda\), containing the constant

\[ \alpha=\frac{1}{137}. \]

There exist many other possible types of interaction of light quanta with charged particles. An important case of such an interaction is the scattering of a beam of electrons passing by nuclei and emitting light in the process. It turns out that this process is the most effective cause of absorption of an electron beam, more effective than, for example, excitation or ionization, provided only that the energy is appreciably greater than \(\varepsilon=mc^2\) (i.e., greater than 1 in electronic units); this process determines the absorption of cosmic rays in passing through matter. As an example let us give the formula of Bethe and Heitler, which determines (in the case of electrons with energy greater than \(\varepsilon\), but less than \(\frac{\varepsilon}{\alpha}=137\varepsilon\)) the loss of energy per unit length in a medium containing \(N\) atoms with atomic number \(Z\) per unit volume:

\[ -\frac{dE}{dx}=NqE, \]

where the effective cross section \(q\) in electronic units is equal to

\[ q=Z^{2}\alpha\left(4\lg 2E-\frac{4}{3}\right), \]

i.e., it is equal to a certain numerical function depending on \(\alpha=\frac{1}{137}\). These examples, the number of which could easily be increased, show that the constant 137 plays an exceptionally important role in all phenomena of nature.

8. Unitary theory of the electromagnetic field

Nevertheless, this number 137 itself is not explained by existing theories. Such a state of affairs is entirely unsatisfactory. It is necessary to look for ways of correcting it. For this purpose let us try to discover the weak points of the theory—contradictions between it and experiment, as well as its internal logical contradictions.

As regards discrepancies between theory and experiment, the general opinion of physicists is that such discrepancies do indeed exist. Very fast particles are absorbed many times more weakly than the theory predicts. This is proved by the enormous penetrating power of cosmic rays. The braking of electrons by nuclei, accompanied by the emission of light (this was mentioned in the preceding paragraph as the most effective cause of absorption), would force cosmic radiation to be absorbed within a length of 1 m of water. In reality, however, this radiation is detected in the water of deep lakes almost at a depth of 500 m. In several cases the braking of individual electrons in passing through a lead plate has been directly measured. In a Wilson chamber placed in a strong magnetic field, curved paths are obtained, and the braking can be calculated from the increase in the curvature of the path after passage through a lead plate. For particles possessing an energy greater than $100 e$, the decrease in velocity measured in this way proved to be distinctly less significant than the theory predicted.

Passing to internal logical contradictions, let us recall the prejudiced way in which the concept of the “radius of the electron” was introduced into the theory. Being unable to reconcile the assumption of finite dimensions of the electron with Maxwell’s equations, physicists tacitly agreed to use the concept of the “electron radius,” putting it equal to

\[ a_0=\frac{e^2}{mc^2} \]

and at the same time to regard the electron as a point charge!

Obviously, the field equations must be altered in such a way that the electron has a finite radius and that the “cohesive forces” necessary to keep all the elements of the electronic charge in equilibrium satisfy the principle of relativity.

Since the constant $\hbar$ does not enter into $a_0$, it seems very unlikely that this problem has any relation to quantum theory. Therefore attempts have repeatedly been made to solve it within the framework of the classical theory. It is necessary to point out two guiding ideas on which such attempts were based. One of these ideas, belonging to Einstein, is that gravitation is regarded as the source of the internal cohesive forces of the electron; attempts to solve the problem of the electron are based on uniting the gravitational field with the electromagnetic field into a certain single scheme. Many “unified field theories” were published (by Einstein, Weyl, Eddington, Veblen, and others), but without success. This seems to me quite understandable. Gravitation is a very small force, experi-

THE MYSTERIOUS NUMBER 137

experimentally studied only in the case of macroscopic, electrically neutral bodies. It is permissible to doubt whether it makes any sense at all to speak of gravitational forces acting between two electrons. The gravitational attraction between two electrons at a distance \(r\) is equal to

\[ \chi \frac{m^2}{r^2}, \quad \text{where } \chi = 6.664 \cdot 10^{-8}\ \mathrm{cm}\cdot \mathrm{g}^{-1}\cdot \mathrm{sec}^{-2} \]

(the gravitational constant), whereas the electrical repulsion is equal to \(\dfrac{e^2}{r^2}\). The ratio of the gravitational force to the electrical force is

\[ \chi \frac{m^2}{e^2} = 2.4 \cdot 10^{-43}. \]

The extremely small magnitude of this ratio convincingly shows that attempts to explain the existence of elementary particles by gravitation, just like all attempts to construct a unified field theory, are absolutely erroneous.

Another path—the modification of Maxwell’s equations—was first tried by Gustav Mie in 1912. Mie showed that Maxwell’s equations can be subjected to a substantial generalization without thereby losing either their outward form or their invariance properties. He created a remarkable construction—a nonlinear field theory—but at the same time introduced a number of unnecessary assumptions, which led him into such great difficulties that this path was soon abandoned, and only recently have I revived this theory again.

The direct introduction of a length (the radius of the electron) into any field equations appears impossible if charges are regarded as pointlike. For these equations, according to Mie, have, generally speaking, the following form:

\[ \dot{\mathbf{D}} = \operatorname{rot}\mathbf{H}, \qquad \operatorname{div}\mathbf{D} = 0, \]

\[ \dot{\mathbf{B}} = -\operatorname{rot}\mathbf{E}, \qquad \operatorname{div}\mathbf{B} = 0, \]

where the components of the vectors \(\mathbf{D}, \mathbf{H}\) are functions of the components \(\mathbf{B}, \mathbf{E}\); since the equations are homogeneous with respect to \(x, y, z, ct\), they cannot in any way be simplified by introducing some “natural” unit of length. But since the relation between \(\mathbf{D}, \mathbf{H}\) and \(\mathbf{B}, \mathbf{E}\) is not linear in character, there exists the possibility of introducing a natural unit of field. It turned out that the following assumption is successful:

\[ D_x = -\frac{\partial L}{\partial E_x}, \ldots, \qquad H_x = \frac{\partial L}{\partial B_x}, \ldots, \]

where

\[ L = b^2\left\{\sqrt{1+\frac{1}{b^2}(\mathbf{B}^2-\mathbf{E}^2)-\frac{1}{b^4}(\mathbf{B}\cdot \mathbf{E})^2}-1\right\}. \]

The function \(L\) is called the Lagrangian function. In the case of the ordinary Maxwell equations for empty space the Lagrangian function is equal to \(L=\frac{1}{2}(B^2-E^2)\), whence it follows that \(\mathbf D=\mathbf E,\ \mathbf H=\mathbf B\). The new function \(L\) is a generalization of this Lagrangian function. It contains a constant \(b\), which is called the “absolute field”; it is assumed that \(b\) is very large. If \(\mathbf B\) and \(\mathbf E\) are small in comparison with \(b\), then \(L\), in the first approximation, turns into the Maxwellian expression \(L=\frac{1}{2}(B^2-E^2)\). Together with Infeld I have shown that \(L\) is the simplest function satisfying the requirements of the principle of relativity.

As Mie had already shown, the conservation laws of electromagnetic energy and momentum can be derived from the field equations and written in the form of equations of the type “the four-dimensional divergence is equal to zero.” For example, the law of conservation of energy has the form

\[ \frac{\partial U}{\partial t}+\operatorname{div}\mathbf S=0, \]

where \(U=L+\mathbf E\mathbf D\) (energy density) and \(\mathbf S=\mathbf E\times\mathbf H\) (momentum density). There exists a static solution (\(\mathbf H=0,\ \mathbf B=0\)) corresponding to a point charge. The equation \(\operatorname{div}\mathbf D=0\) has the solution with center of symmetry \(D_r=\frac{e}{r^2}\), and since

\[ D_r=-\frac{\partial L}{\partial E_r} = \frac{E_r}{\sqrt{1-\frac{1}{b^2}E_r^2}}, \]

then

\[ E_r= \frac{D_r}{\sqrt{1+\frac{1}{b^2}D_r^2}} = \frac{e}{\sqrt{r_0^4+r^4}}, \]

where \(r_0\) is determined by the equality

\[ b=\frac{e}{r_0^2}. \]

\(E_r\) remains finite everywhere; at distances \(r\) large in comparison with \(r_0\), we have \(E_r\to\frac{e}{r^2}\), i.e. the field \(E_r\) obeys Coulomb’s law; in the case of small \(r\), however, the field \(E_r\) does not tend to infinity, but takes the value \(\frac{e}{r^2}=b\) at \(r=0\). Thus, the constant \(b\) is equal to the field at the center of the electron. Owing to the fact that this theory is relativistically invariant, Einstein’s law \(E=mc^2\) is fulfilled exactly. A simple calculation gives for the energy

of rest, the following expression:

\[ E = mc^2 = \int U\,dx\,dy\,dz = 1.236\,\frac{e^2}{r_0}. \]

This formula may be regarded as an exact definition of the “electron radius” \(r_0\), which differs from the “conventional” radius \(a_0\) by a numerical factor:

\[ r_0 = 1.236\,\frac{e^2}{mc^2} = 1.236\,a_0. \]

Let us note the fundamental features of the new theory: it regards matter and field not as two distinct realities, but as one and the same reality. A particle is only a special point of the field, and its mass is the energy of the field associated with the special point. Such a theory is a unitary field theory, in contrast to the accepted dualistic theories, in which masses are introduced separately for each type of particle.

We cannot enter into a discussion of the other consequences of this theory, and shall confine ourselves only to considering the question of the mysterious number 137. It cannot be hoped that this new theory of the electromagnetic field will be able to help us directly explain this number; for the constant \(\hbar\) enters into the number 137, and therefore it must be connected with quantum ideas and can never be derived from a purely classical theory. The new field theory removes the difficulties connected with the “electron radius” (or, what is the same thing, with the infinite self-energy of the electron considered as a point charge), but, of course, this theory is only a limiting case of quantum electrodynamics, just as classical mechanics is a limiting case of quantum mechanics (Bohr’s correspondence principle).

9. Quantum Electrodynamics

The idea of light quanta (photons), in the form in which it was proposed by Einstein for explaining the photoelectric effect, was the first indication that quantum principles should be applied to the electromagnetic field in empty space.

For this purpose, by means of Fourier analysis, the field is decomposed into a set of harmonic waves; each coefficient of the Fourier expansion, representing the amplitude of a monochromatic wave, may be regarded as a mechanical coordinate to which quantum conditions should be applied. In this way it proved possible to give an exact meaning to the concept of the “photon” and to derive the laws of photon statistics, i.e., to obtain Planck’s formula and to calculate radiation fluctuations. Dirac succeeded in constructing a consecu-

tive theory of emission, absorption, and scattering of light, in which the atom and the field surrounding it are considered as a quantized mechanical system.

Subsequently Pauli and Heisenberg gave the theory a form in which the components of the field vectors are no longer expanded in a Fourier series, but are at once treated as quantum variables. The Pauli–Heisenberg method can also be applied to the unitary theory of the field. This is done in the following way: the components of the vectors D and B are chosen as the basic variables; just as in quantum mechanics the coordinate \(q\) and the momentum

\[ p=\frac{\hbar}{i}\frac{\partial}{\partial q} \]

do not commute with one another, so here it is assumed that certain components of the vectors D and B are noncommutative. D and B are functions of the spatial vector \(\mathbf r(x,y,z)\) and of the time \(t\); the variables \(x,y,z,t\) are regarded as commuting. It is assumed that any components of D and B* commute if they refer to two points at a finite distance from one another, but that this ceases to hold when the two points approach one another. The following permutation relations are introduced:

\[ D_y(\mathbf r_1,t)B_z(\mathbf r_2,t)-B_z(\mathbf r_2,t)D_y(\mathbf r_1,t)= \]

\[ =\hbar c\,\frac{\partial}{\partial x}\,\delta(\mathbf r_1-\mathbf r_2), \]

where

\[ \delta(\mathbf r_1-\mathbf r_2)=\delta(x_1-x_2)\delta(y_1-y_2)\delta(z_1-z_2) \]

(the product of three Dirac “improper” functions**).

If D and B are expressed in field units equal to the constant \(b\), then one obtains

\[ D_y^{(1)}B_z^{(2)}-B_z^{(2)}D_y^{(1)} =\frac{\hbar c}{b^2}\,\frac{\partial}{\partial x}\,\delta(\mathbf r_1-\mathbf r_2). \]

\[ *\qquad (pq-qp)f(q)=\frac{\hbar}{i}\left[\frac{\partial}{\partial q}(qf)-q\frac{\partial f}{\partial q}\right]=\frac{\hbar}{i}f, \]

whence

\[ pq-qp=\frac{\hbar}{i}. \]

\[ **\ \delta(x)=0\ \text{for }x\ne 0,\ \text{while for }x=0\ \delta(x)\ \text{is made infinite in such a} \]

way that

\[ \int_{-\infty}^{\infty}\delta(x)\,dx=1. \]

Since \(b=-\dfrac{e}{r_0^2}\), the coefficient is equal to

\[ \frac{\hbar c}{b^2}=\frac{\hbar c}{e^2}r_0^4=\frac{r_0^4}{a}. \]

The factor \(r_0^4\) before the \(\delta\)-function appeared because the \(\delta\)-function is not a dimensionless quantity, but has the dimension \(\dfrac{1}{r^3}\); indeed, its integral over all space is equal to unity; consequently the expression \(r_0^4\dfrac{\partial}{\partial x}\delta\) has no dimension.

These commutation relations must be supplemented by differential equations determining the change of any operator \(F\) in space and in time:

\[ \frac{\hbar}{i}\frac{\partial F}{\partial t}=WF-FW, \]

\[ \frac{\hbar}{i}\frac{\partial F}{\partial x}=-(p_xF-Fp_x), \]

where

\[ W=\int U\,dv,\qquad \mathbf{p}=\int \mathbf{S}\,dv \]

(the total electromagnetic energy and the total momentum). The energy density \(U\) may be any function of the vectors \(\mathbf{D}\) and \(\mathbf{B}\). In the special case of the Lagrangian function \(L\) introduced in the preceding paragraph, we obtain (if \(b=1\) is put)

\[ U=\sqrt{1+\mathbf{D}^2+\mathbf{B}^2+\mathbf{S}^2}-1,\qquad \mathbf{S}=\mathbf{D}\times\mathbf{B}. \]

Conversely, these expressions for \(U\) and \(\mathbf{S}\) may be taken as the basic assumptions and the generalized field equations derived from them.

It can also be shown that an isolated system as a whole satisfies the ordinary laws of quantum mechanics. In particular, the total angular momentum of such a system

\[ \mathbf{M}=\int(\mathbf{r}\times\mathbf{S})\,dv \]

is quantized and always turns out to be equal to an integral multiple of the constant \(\hbar\). This result shows that in this form the theory is incomplete, since the spin phenomena do not follow from it: for we know that electrons (and protons) have an angular momentum equal to \(\dfrac{1}{2}\hbar\). Therefore my collaborator Pryce proposed

such a variant of the theory in which the coordinate \(q\) and the momentum \(\boldsymbol{\xi}\) of the point charge are adopted as the new quantum variables. These variables are connected with the field by the quantum relations:

\[ [D_x,\ \xi_x]=e\delta(\mathbf r-\mathbf q)\ldots \]

The expressions for the energy and momentum are modified in such a way that the phenomena of spin follow from them; in the case of one special point the energy and momentum have the following form:

\[ W=\int U\,dv+\mathbf a\boldsymbol{\xi} \]

\[ \mathbf p=\int \mathbf S\,dv+\boldsymbol{\xi}. \]

Here \(\mathbf a\) is a vector whose components are equal to the Dirac matrices \(\alpha_1,\alpha_2,\alpha_3\). It can be shown that the field equations in this case turn out to be as follows:

\[ \dot{\mathbf D}-\operatorname{rot}\mathbf H =4\pi e\mathbf a\delta(\mathbf r-\mathbf q), \]

\[ \operatorname{div}\mathbf D =4\pi e\delta(\mathbf r-\mathbf q), \]

\[ \dot{\mathbf B}+\operatorname{rot}\mathbf E=0,\qquad \operatorname{div}\mathbf B=0. \]

These equations now contain on the right-hand side \(\delta\)-functions corresponding to a point charge located at the point \(\mathbf q\) and moving with velocity \(\mathbf a\).

The total momentum of the closed system now turns out to be equal to the vector sum of the electromagnetic momentum

\[ \int(\mathbf r\times\mathbf S)\,dv \]

and the spin momentum \(\frac12\hbar\). Price’s method apparently makes it possible to connect Dirac’s theory of spin with the general ideas of the nonlinear field theory, in which particles are regarded as singular points.

However, on the other hand, Price’s equations are not fully suitable for our problem of explaining the number 137. Since the commutation relations for the field components already contain the constant \(\hbar\), the ideal theory should not have introduced \(e\) as a new independent constant; but this is precisely what is done in Price’s commutation relations

\[ [D_x,\ \xi_x]=e\delta(\mathbf r-\mathbf q),\ldots \]

Nevertheless, Price’s theory, as we shall see below, is after all a certain step in the right direction.

10. The Mass of the Proton

As we have seen, the total angular momentum of a closed system is equal to the vector sum of the real electromagnetic angular momentum \(l\hbar\), with \(l=0,1,2,\ldots\), and the spin angular momentum \(\frac{1}{2}\hbar\), belonging to the singular point. Therefore the quantum number \(j\), by which the resultant angular momentum is determined, takes the values \(\frac{1}{2}, \frac{3}{2}, \frac{5}{2},\ldots\), and to each value of \(j\) there correspond two states: \(l+\frac{1}{2}\) and \((l+1)-\frac{1}{2}\). All this is well known to us from the theory of spectra, where the azimuthal quantum number of the orbital angular momentum \(l\) is combined with spin \(\frac{1}{2}\), thus forming the so-called inner quantum number \(j\). But in the theory of spectra two states with the same \(j\) have approximately the same energy, and therefore the presence of the spin expression leads only to a fine structure of the lines, whereas here the application of the very same representation to a point charge, which is regarded as a singular point of the nonlinear field equations, must lead not to a “fine” but to a “coarse” structure, i.e. it must explain the enormous difference between the internal rest energies of the proton and the electron.

This idea was put forward by Pryce. He proceeded from the following experimental fact: both kinds of particles, protons and electrons, have the same angular momentum. Since the presence of spin is not connected with the accumulation of energy, the state \(0+\frac{1}{2}\) without electromagnetic angular momentum (\(l=0\)) must have only electrostatic energy, whereas the state \(1-\frac{1}{2}\), having a finite electromagnetic angular momentum (\(l=1\)), must also have additional electromagnetic energy, by which one could explain the large difference in mass.

We shall show that an approximate calculation of the energy confirms this hypothesis. A rigorous calculation, of course, presupposes the existence of an exact solution of our field equations, and moreover not only within the framework of the classical theory, but also with allowance for quantum relations. For the present we are still very far from such an exact solution. But this is not required. For the required estimate it is sufficient to compare with one another certain well-established facts.

We know from Stern’s experiments the magnetic moment of the proton. The correct order of magnitude will be obtained if, in the expression for the Bohr magneton \(\mu=\frac{\hbar e}{2mc}\), the electron mass is replaced by the proton mass \(M\). However, the experimentally measured value of the magnetic moment of the proton is not exactly equal to the expected magnitude (the “nuclear magneton” \(\frac{m}{M}\mu=\frac{1}{1840}\mu\)), but proves to be approximately \(2^{1/2}\) times larger,

This apparently means that the magnetic moment of the proton is not a primary phenomenon, as is the magnetic moment of the electron, but is obtained as the result of the superposition of two approximately compensating phenomena. This agrees well with Proca’s ideas, according to which the proton should be regarded as a state of an elementary charge \(j = 1 - \dfrac{1}{2}\). We know that in the atom the magnetic moments of the spin and of the orbital motion of the electron around the nucleus are both equal to the Bohr magneton, despite the difference in the corresponding angular momenta. It is natural to suppose that the very same thing also holds for the magnetic moment of the electromagnetic field of the elementary charge. The magnetic moments corresponding to the quantum numbers \(l = 1\) and \(s = -\dfrac{1}{2}\) would compensate each other if there were no interaction between them. Such an interaction is precisely the possible cause of the appearance of a small difference, amounting to about \(2 \tfrac{1}{2}\) nuclear magnetons; however, this question is not of interest to us at present.

Let us now try to estimate the energy of a rotating electromagnetic field with magnetic moment \(\mu\). Since an exact calculation cannot be made, we shall use an analogous electrostatic problem. We already know that in the unitary theory one obtains for the energy of a point charge an expression \(A \dfrac{e^2}{a_0}\)—the same as in the classical theory for the energy of a charged sphere, but only with a somewhat different numerical coefficient \(A\). One may boldly assume that the same is true in the magnetic case. The magnetic energy of a magnetized sphere of radius \(a_0\), having moment \(\mu\), is equal to \(\dfrac{1}{2}\dfrac{\mu^2}{a_0^3}\). One may be certain, at least as regards the order of magnitude, that this relation is preserved also in the unitary theory. Let us substitute into it the quantity \(\mu = \dfrac{e a_0}{2\alpha}\). As we shall presently see, the value of the magnetic energy following from this is large in comparison with the electrostatic energy, and therefore it may be identified with the total energy of the rotating charge \(Mc^2\). Thus:

\[ Mc^2 = \frac{1}{2}\frac{e^2 a_0^2}{(2\alpha)^2}\frac{1}{a_0^3} = \frac{1}{8\alpha^2}\frac{e^2}{a_0}. \]

On the other hand, the electrostatic energy of a point charge is

\[ mc^2 = \frac{e^2}{a_0}. \]

It follows from this that

\[ \frac{M}{m} = \frac{1}{8\alpha^2} = \frac{(137)^2}{8} = \frac{18\,770}{8} = 2340. \]

This is in fair agreement with the actual mass ratio of the proton and the electron, which is equal to 1840: after all, the numerical coefficients used by us were to some extent arbitrary. Only the factor \(\frac{1}{\alpha^2}\) appears in a completely unambiguous way, and it is this that determines the order of magnitude.

If the views set forth are correct, then the positron, the proton, the electron, and the (hypothetical) negative proton are states of one and the same elementary charge; hence one should expect that transitions are possible, for example, of a proton into a positron with the emission of light quanta. The energy of these quanta must be of the order \(Mc^2\), i.e. \(1840\,mc^2\), and since \(mc^2\) is approximately equal to \(1/2\) million electron-volts, the emitted radiant energy will be of the order of a billion volts.

The following objection is possible: why, in that case, does the proton exist at all? If the proton is formed from the positron by absorbing energy, why does it not spontaneously pass back into its initial state?

This can be answered if one takes into account that a rotating symmetric distribution of elements of charge of one and the same sign has no dipole moment; as a consequence, the proton can emit not dipole, but only quadrupole radiation (of magnetic type). But it is known that in this case the rotational quantum number changes not by 1, but by 2. Therefore the state \(l = 1\) turns out to be metastable and cannot pass into the state \(l = 0\) by emitting radiant energy.

However, such transitions can be induced by collision with fast electrons. Thus one may expect that transitions of protons into positrons will occur in matter bombarded by cosmic rays. Such a transition should be accompanied by a violent emission of light quanta from one point of space—something like a powerful explosion, which can tear electrons away from surrounding atoms or produce many electron pairs (a phenomenon of which we shall speak in the next paragraph).

Explosions of this kind were observed by Hoffmann as “jolts” (“Stösse”) in an ionization chamber; it is possible that the “showers” of particles (electrons and positrons) discovered by Blackett and Occhialini are also nothing other than small “ionization jolts,” although this, however, has not yet been finally established. The generally accepted theories do not give a satisfactory explanation of the explosions (“ionization jolts” and showers). I am inclined to think that in these explosions we are dealing with the phenomenon of the transformation of protons into positrons.

The hypothesis of the identical nature of light and heavy particles, which are different states of rotation of one and the same special point of the electromagnetic field, reduces the mysterious number 1840 to another mysterious number—137. But the problem of the number 137 has not only not been solved, but has not even approached

to its resolution. It is extraordinarily difficult to understand the large numerical coefficient in the formula for the magneton: \(\mu = \dfrac{e a_0}{2a} = 68.5 e a_0\). It is very unlikely that classical considerations could be of any use here. The general formula for the magnetic moment of rotating charges is:

\[ \mu = \frac{1}{2c}\sum e(\mathbf r \times \mathbf v). \]

If for \(\mathbf v\) we take the maximum value \(c\) and distribute the whole charge at one and the same distance \(R\), then we obtain \(\mu = \dfrac{1}{2} eR\). Consequently, the effective radius must be equal to \(\dfrac{a_0}{\alpha} = 137a_0 = \dfrac{\lambda_0}{2\pi}\), where \(\lambda_0\) is the Compton wavelength. This argument reveals to us the source of the difficulty—the existence of two natural units of length: the electrostatic unit \(a_0\) and the quantum unit \(\dfrac{\lambda_0}{2\pi}\), which are in the ratio \(\alpha\) to one another.

11. Attempts to Explain the Mysterious Number

We assigned to the quantity \(\dfrac{1}{\alpha}\) the integer value 137. We did this for the simple reason that the accuracy of the measurement of \(\alpha\) is not sufficient to determine the next decimal place with certainty. There are two methods for determining \(\alpha\): first, measurement of the quantities \(e\), \(\hbar\), and \(c\), from which one can then calculate \(\alpha = \dfrac{e^2}{\hbar c}\), and, second, direct measurement of the fine structure of hydrogen-like atoms. The latter method has recently been improved through the use of heavy hydrogen (deuterium). By comparing the fine structure of H and D one can reduce the determination of \(\alpha\) to a relative measurement. The results obtained undoubtedly indicate that \(\dfrac{1}{\alpha}\) is greater than 137 (approximately \(\dfrac{1}{\alpha} = 137.2\)). I think that we have no grounds for considering \(\dfrac{1}{\alpha}\) to be an integer.

Several years ago, when the numerical value of \(\alpha\) was known still less accurately than now, Eddington proposed a theory according to which \(\alpha\) should be exactly equal to the integer 136. This number was obtained (for \(n = 4\)) from the expression \(\dfrac{1}{2}n^2(n^2 + 1)\), representing something like the “number of degrees of freedom” of the Dirac electron (which is described by means of matrices with four

lines and columns). Later, when it became clear that \(\frac{1}{\alpha}\) is closer to 137 than to 136, Eddington altered his theory in such a way that one was added. He also derived a formula for the ratio of the masses of the proton and the electron: this ratio, according to Eddington, is the root of a certain quadratic equation in which, alongside the number 136, the number 10 plays a role \(\left[\text{the value } \frac{1}{2}n^2(n^2+1) \text{ for } n=2\right]\).

I personally have never been able to understand this theory. It seems mystical to me.

We must now return to the question of how the number 137 can be explained from the point of view of the theories presented here.

Since the number \(\alpha=\frac{e^2}{\hbar c}\) relates \(e\) and \(\hbar\) to one another, one may either assume the existence of a smallest electric charge \(e\) and try to derive from it the quantum constant \(\hbar\), or conversely.

We now know that charges can be created and can disappear. This fact, first predicted by Dirac’s theory of the positron and obvious from the standpoint of the unitary field theory, has been established experimentally with complete certainty. A light quantum, passing through the field of a nucleus, can form an electron pair consisting of a positive and a negative electron; such a pair can disappear and be neutralized, emitting light quanta.

As we shall now see, the laws of these phenomena can be derived from Dirac’s quantum-mechanical theory. It therefore seems quite natural to regard \(\hbar\) as the fundamental quantity and to try to explain the origin of elementary electric charges. Let us first consider this process from the point of view of the unquantized unitary field theory. In this theory the electron and positron are represented by two identical point charges of opposite sign, surrounded by a static field. If the distance between the singular points is large compared with \(a_0\), then the field of each charge may be considered separately. But when the electrons approach one another, their fields merge, since the equations are nonlinear. In all details this process has been traced in a certain special case, namely in the case of two dimensions and infinitely slow motions. My collaborator Pryce derived an exact solution for this case. His formulas describe the gradual concentration and disappearance of the field when the charges approach one another up to complete coincidence. However, in order to carry out this quasistatic process, forces would be required to hold both charges in their instantaneous positions. If the charges are made free, they will accelerate toward one another, creating a magnetic field. Since this field has a tendency to persist (electromagnetic inertia), the charges will rush through the position of neutralization and will appear with opposite signs. A kind of oscillator or dipole is formed, the force of which

connection of which is determined by the pole itself. Since energy will be radiated in the form of spherical waves, the oscillations must die out, and the amplitudes must decrease until the neutralization becomes complete and all the energy is radiated.

Consequently, the annihilation of the pair is connected with oscillations “around nothing.” The frequency of these oscillations must be determined by the constants of field theory \(c, b, e\). But \(c\sqrt{\dfrac{b}{e}}=\dfrac{c}{r_0}\) is the only quantity having the dimension of reciprocal time that can be constructed from these constants. Consequently, the frequency of neutralization must have the form \(\nu=\gamma \dfrac{c}{r_0}\), where \(\gamma\) is a numerical constant.

If there exists any correspondence between these classical considerations and the actual quantum process of neutralization, then the frequency \(\nu\) must be the Compton frequency

\[ \nu_0=\frac{c}{\lambda_0}=\frac{mc^2}{h}. \]

For, owing to the conservation of momentum, the emitted light must consist, at least, of two photons; the energy \(2mc^2\) disappears, hence \(mc^2=h\nu_0\). Substituting here \(mc^2=1.236\,\dfrac{e^2}{r_0}\) and \(\nu_0=\gamma \dfrac{c}{r_0}\), we find:

\[ \alpha=\frac{e^2}{\hbar c}=\frac{2\pi\gamma}{1.236}. \]

Thus we obtain a new interpretation of the constant \(\alpha\): it is determined by the frequency of neutralization of an electron pair.

The only real attempt made so far to calculate \(\alpha\) is closely connected with these considerations.

As was said above, Dirac explained the existence of positive electrons and the neutralization of pairs from the point of view of his wave equation. This equation has one property which was formerly regarded as a great defect: the Dirac equation is satisfied by states with negative energy. The energy of a free electron is determined by the relativistic formula \(E=c\sqrt{m^2c^2+p^2}\), where \(p\) is the momentum; here the square root may have either sign. Consequently, the possible values are positive numbers \(>mc^2\) and negative numbers \(<-mc^2\). To overcome this difficulty, Dirac assumed that all states with negative energy are ordinarily filled with (negative) electrons, and, according to the Pauli principle, no state can contain more than one electron (this principle has been verified from the laws of atomic spectra). But from time to time, for example under the action of a photon, an electron may be ejected from a state with negative energy. Then it appears as an ordinary electron, but at the same time in the reservoir of electrons with negative energy a “hole” is formed, which behaves exactly as a positive electron would behave.

an electron with positive energy. If the electron falls back into the hole, emitting photons, then the pair disappears again.

This strange idea proved to be very fruitful, since it made it possible to calculate the probabilities of the formation and disappearance of pairs, and the results of these calculations agree excellently with experiment. The phenomena of the formation and disappearance of pairs may be treated as ordinary acts of emission or absorption, except that one of the states under consideration has negative energy. The theory makes it possible to calculate the effective cross section for the formation of pairs by a photon \(h\nu\) in the field of a nucleus with charge \(Ze\):

\[ q=\frac{1}{2\pi} r_0^2 Z^2 \alpha^2 \frac{28}{9}\left(\lg \frac{2h\nu}{mc^2}-\frac{218}{27}\right). \]

This formula describes the observed phenomena with astonishing accuracy.

However, on the other hand, Dirac’s theory leads to serious internal contradictions. It is forced to proceed from the assumption that an infinite number of electrons possessing negative energy produces no field whatever, although they themselves are subject to the action of a field (otherwise light, after all, could not create pairs). A sharp boundary between positive and negative values of energy exists only in the case of free electrons; in the presence of a field this boundary ceases to be clear. Finally, the difficulty of the “infinite self-energy” of the electron, inherent in any theory based on Maxwell’s equations, is complicated here by the presence of still other infinities, arising as a consequence of the infinite number of electrons in states with negative energy.

Dirac and Heisenberg attempted to overcome these difficulties by subtracting the charge density of the electrons in negative states from the total density. However, since both these densities are infinitely large, their difference cannot be defined unambiguously, and therefore Dirac and Heisenberg were forced to introduce a whole series of arbitrary assumptions.

In one way or another, there exist certain equations, derived by these authors and claiming to describe the motion of positrons and electrons. These equations have one property in common with unitary field theory: they are nonlinear. The nonlinear terms in these equations lead to one phenomenon which can also be derived from unitary field theory (but not from Maxwell’s equations)—to the scattering of light by light. The scattering of light by light may be interpreted from the point of view of Dirac’s theory in the following way. Two photons can be temporarily absorbed, forming an electron pair, which immediately thereafter disappears again, emitting two other photons. Something like a collision of two photons occurs. But the mechanism of this collision is closely connected with the creation and annihilation of pairs. Thus Dirac’s theory turns out to be connected with the unquantized unitary field theory. Although from the point of view of such a unitary theory

and it is impossible to compute the creation of pairs, but the scattering of light by light can be computed, and comparison of the results obtained with the aid of both theories makes it possible to find the desired relation between \(e\) and \(\hbar\).

This was in fact done by two of Heisenberg’s students—Euler and Kockel. They calculated the scattering of light by light according to Dirac–Heisenberg’s theory of “holes” and showed that this scattering can be expressed as a consequence of a nonlinear correction term to Maxwell’s equations, corresponding to the Lagrangian function

\[ L=\frac{1}{2}(\mathbf{B}^{2}-\mathbf{E}^{2}) -\frac{1}{90\pi}\left(\frac{e^{3}}{m^{2}c^{4}}\right)^{2}\frac{1}{\alpha} \{(\mathbf{B}^{2}-\mathbf{E}^{2})^{2}+7(\mathbf{B}\cdot\mathbf{E})^{2}\}. \]

This Lagrangian function may be compared with the Lagrangian function of the unitary field theory, expanded in a power series:

\[ L=b^{2}\left\{\sqrt{1+\frac{1}{b^{2}}(\mathbf{B}^{2}-\mathbf{E}^{2})-\frac{1}{b^{4}}(\mathbf{B}\cdot\mathbf{E})^{2}}-1\right\}= \]

\[ =\frac{1}{2}(\mathbf{B}^{2}-\mathbf{E}^{2}) -\frac{1}{8b^{2}}\{(\mathbf{B}^{2}-\mathbf{E}^{2})^{2}+4(\mathbf{B}\cdot\mathbf{E})^{2}\}+\ldots \]

There is no complete agreement between the two correction terms, since the coefficients of \((\mathbf{B}\cdot\mathbf{E})^{2}\) are different.

Disregarding this circumstance, let us equate the coefficients:

\[ \frac{1}{90\pi}\left(\frac{e^{3}}{m^{2}c^{4}}\right)^{2}\frac{1}{\alpha} =\frac{1}{8b^{2}}. \]

Above we had:

\[ b=\frac{e}{r_{0}^{2}}, \qquad mc^{2}=1{,}236\,\frac{e^{2}}{r_{0}}, \]

and therefore

\[ \frac{1}{\alpha}=\frac{45\pi(1{,}236)^{4}}{4}=82{,}4. \]

The number obtained is \(1.66\) times smaller than the experimental value \(137\), but nevertheless it has the correct order of magnitude. If one recalls the arbitrary assumptions made in the Dirac–Heisenberg theory, the discrepancy should not appear discouraging. In any case, the Dirac–Heisenberg theory gives correction terms which, in the first approximation, agree with those given by the unitary field theory. Let us also note that the numerical factor \(1.236\) in the relation between \(mc^{2}\) and \(\dfrac{e^{2}}{r_{0}}\) is based on a classical calculation, which in reality is an unjustified extrapolation of the classical theory to the internal structure of the electron.

To observe the scattering of light by light is practically impossible because of the smallness of the effect. The cross section of this phenomenon has the order of magnitude

\[ q = r_0^2 \frac{1}{\alpha^4}\left(\frac{r_0}{\lambda}\right)^6, \]

i.e., approximately \(10^{-30}\ \mathrm{cm}^2\) for \(\gamma\)-rays and \(10^{-70}\ \mathrm{cm}^2\) for visible light.

12. Conclusion

In proposing a unitary field theory, I know that I am going against the views of the most outstanding physicists—Bohr, Pauli, Heisenberg. They apparently believe that the final theory must express the laws of motion of elementary particles by means of wave equations in which a definite charge and mass are assigned to each particle. The electromagnetic field, on the other hand, is to play only the role of a fiction introduced in order conveniently to describe the finite speed of propagation of the interaction of particles. Against this program I put forward the following objection: I cannot understand how it is possible in this way to explain the existence of heavy and light particles with their mass ratio equal to the number 1840, and also the connection between charge and the quantum constant, characterized by the number 137.

Nevertheless, it may turn out that both paths will lead to the same result, since in the preceding paragraph a close connection between them was shown. Just as quantum mechanics may be regarded as a synthesis of the idea of particles and the idea of waves, achieved through a critical analysis of the basic concepts of space-time and energy-momentum, so too the physics of the future will perhaps prove to be a synthesis of the idea of the quantum and the idea of elementary charge, to be achieved through a critical study of the concepts used in the description of the field and of its singular points.