INTRODUCTION TO THE THEORY OF ELEMENTARY PARTICLES

D. Ivanenko

Submitted 1947 | SovietRxiv: ru-194701.81086 | Translated from Russian

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INTRODUCTION TO THE THEORY OF ELEMENTARY PARTICLES

D. Ivanenko

§ 1. Historical introduction. Table of elementary particles . . . . . 149
§ 2. Electron . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 151
§ 3. Positron . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 158
§ 4. Electromagnetic field and photons . . . . . . . . . . . . . . . . . . . 160
§ 5. Neutrino. Fermi’s theory of beta decay . . . . . . . . . . . . . . . 162
§ 6. Proton . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 167
§ 7. Neutron . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 169
§ 8. Model of the atomic nucleus . . . . . . . . . . . . . . . . . . . . . . . 170
§ 9. Mesotron . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 175
§ 10. Gravitational field . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 179
§ 11. Cosmic rays. § 12. Abundance of particles and elements.
§ 13. General questions of relativistic quantum mechanics. § 14. Difficulties of the theory.

§ 1. HISTORICAL INTRODUCTION. TABLE OF ELEMENTARY PARTICLES

In 1895 the first elementary particle—the electron—was discovered. The discovery of the atomic nucleus and the creation of the planetary model of the atom (1911) led to the discovery of a second particle—the proton. The discovery of a third particle—the neutron—in 1932 and the determination of the composition of nuclei showed that all matter consists essentially of electrons, protons, and neutrons, to which must be added the electromagnetic field, or the particles corresponding to it—photons—and the gravitational field. Later, in cosmic rays, three new particles were discovered—the positron (1932) and the positive and negative mesotrons (1937). The existence of the neutrino particle is also very probable, and the presence of neutral mesotrons (neutretto) is plausible.

Here “elementary” denotes the simplest particles and fields, not composed of other known particles and fields. Thus elementary particles now play the role of Democritus’ indivisible atoms. It is important to note that the study of matter over the last half-century has so far in no case led to any further fragmentation into, say, “subparticles,” but “to a considerable extent” has been connected with the discovery of new elementary particles and of ever newer properties of them.

TABLE OF ELEMENTARY PARTICLES

Along with the mass, the atomic weight on the scale $ {}_{8}\mathrm{O}^{16}=16.000 $ is indicated; the value of the spin $s$ is given in fractions of $h/2\pi$;

\[ e_{0}=4.80\cdot 10^{-10}\ \text{esu};\quad \mu_{B}=\frac{eh}{4\pi mc}\ \text{(Bohr magneton)};\quad \mu_{0}=\frac{eh}{4\pi Mc}\ \text{(nuclear magneton)},\ \mu\text{-mass of the mesotron}; \]

\[ f\sim g'\chi_{0};\quad f'\sim g'\chi_{0};\quad \chi_{0}=\frac{2\pi\mu c}{h};\quad g\sim 5e;\quad g'\sim 4\cdot 10^{-17}\sim 10^{-8}e \]

Particle	Mass	Type of wave function, spin, equation of motion, statistics	Electric charge, magnetic moment	Mesotron charge and moment	Abundance
1. Graviton (quantum of the weak gravitational field)	$0$	Symmetric tensor of rank 2; Einstein equation; $s=2$; Bose statistics	$0$ $0$	$0$ $0$	The gravitational field is a curvature of space-time caused by all kinds of matter
2. Photon $(\gamma)$ (quantum of the electromagnetic field)	$0$	Vector potential; Maxwell equations $s=1$ Bose statistics	$0$ $0$	$0$ $0$	Emitted and absorbed in various processes. The density of radiant energy in the known part of the universe is $\rho \simeq 10$
3. Neutrino? $(\nu)$	$m_{\nu}<0.2\,m;$ $m_{\nu}=0?$	Bispinor; $s=1/2$; Dirac equation; Fermi statistics	$0$ $\mu_{\nu}<\dfrac{1}{7000}\,\mu_{B}$	$g'$ $f'$	Produced in $\beta$-decay, $K$-capture, and mesotron decay; owing to their penetrating ability they are practically not absorbed by matter. They have not been discovered with final certainty.
4. Electron $(e_{-})$	$m=9\cdot 10^{-28}$ $Ae=5.48\cdot 10^{-4}$	Bispinor; $s=1/2$. Dirac equation; Fermi statistics	$-e_{0}$ $-\mu_{B}$	$g'$ $f'$	Enters into the composition of all atoms. In the part of the universe known to us there are approximately $N_{e}\sim 10^{75}$ electrons. They are absorbed and produced in various processes of high energy in atomic nuclei, cos...

No., particle	Mass	Spin, equation, statistics	Electric charge / magnetic moment	Interaction	Remarks
					mic rays and in various interactions of elementary particles. ($\beta$-decay, $K$-capture, decay of mesotrons; creation and annihilation of pairs)
5. Positron ($e_{+}$)	$m$	Bispinor; $s=1/2$; Dirac equation; Fermi statistics.	$e_{0}$; $\mu_{B}$	$s'$; $f'$	Produced in various high-energy processes, in cosmic rays and nuclear processes. Owing to the excess of electrons, with which $e_{+}$ can annihilate, the concentration of positrons is negligible.
6. Neutral mesotron? ($\mu_{0}$)	$\mu \sim 20\,m?$	Scalar, pseudoscalar, vector? Accordingly spin $s=0$ or $1$? Accordingly the de Broglie equation, pseudoscalar equation, or Proca equation? Bose statistics.	$0$	–	Not yet reliably discovered. Apparently may be produced in collisions of high-energy nucleons.
7. Negative mesotron ($\mu_{-}$)	$\sim 200\,m$	Scalar, pseudoscalar, vector? Accordingly spin $s=0$ or $1$? Accordingly the de Broglie equation, pseudoscalar equation, or Proca equation? Bose statistics.	$-e_{0}$	–	Charged mesotrons constitute the hard component of cosmic rays. Produced in cosmic rays. Radioactive. Lifetime $\tau_{0}=1.5\cdot 10^{-6}$ sec.; $\mu$ are absorbed by nuclei. The concentration of mesotrons is negligible.
8. Positive mesotron ($\mu_{+}$)	$\sim 200\,m$	Scalar, pseudoscalar, vector? Accordingly spin $s=0$ or $1$? Accordingly the de Broglie equation, pseudoscalar equation, or Proca equation? Bose statistics.	$+e_{0}$	–	Charged mesotrons constitute the hard component of cosmic rays. Produced in cosmic rays. Radioactive. Lifetime $\tau_{0}=1.5\cdot 10^{-6}$ sec.; $\mu$ are absorbed by nuclei. The concentration of mesotrons is negligible.
9. Proton ($p$)	$M=1836.5\,m$; $A_{p}=1.00758$	Bispinor; $s=1/2$; Dirac equation; Fermi statistics.	$\mu_{p}=+2.7896\,\mu_{0}$	$g$; $f$	Protons and neutrons constitute all atomic nuclei. The number of nucleons in the known part of the universe is approximately $N\sim 10^{75}$. Apparently, primary cosmic rays consist mainly of protons. In various reactions protons and neutrons transform into one another. Free neutrons are radioactive. Lifetime $\tau \sim 10^{4}$ sec.
10. Neutron ($n$)	$M_{n}=1839\,m$; $A_{n}=1.008941$	Bispinor; $s=1/2$; Dirac equation; Fermi statistics.	$\mu_{n}=-1.985\,\mu_{0}$	$g$; $f$	Protons and neutrons constitute all atomic nuclei. The number of nucleons in the known part of the universe is approximately $N\sim 10^{75}$. Apparently, primary cosmic rays consist mainly of protons. In various reactions protons and neutrons transform into one another. Free neutrons are radioactive. Lifetime $\tau \sim 10^{4}$ sec.

Thus, according to modern physical views, all known matter on the Earth, the Sun, the stars, and other astronomical objects, in interstellar space and in the flux of cosmic rays—in short, in the entire known part of the universe—consists of elementary particles and fields of the following few types:

a) Fields in the narrow sense, or particles devoid of rest mass: 1) the electromagnetic field, or photons; 2) the weak gravitational field, or gravitons.

b) Light particles: 3) electrons, 4) positrons, 5) neutrino.

c) Intermediate particles: 6) positive mesotrons, 7) negative mesotrons, 8) neutral mesotron (neutretto).

d) Heavy particles, or nucleons: 9) protons, 10) neutrons.

At the same time, despite the fact that a number of properties of particles have not yet been studied, and the discovery of new particles is by no means excluded, while theory is in many respects far from completion, nevertheless it is beyond doubt that, in the main, physical phenomena reduce to processes involving the ten types of matter just enumerated. All known processes consist in the motion of elementary particles, their interactions and the formation of systems of particles (nuclei, atoms, etc.), and in the mutual creation and annihilation of particles.

The carriers of the basic properties of matter—mass, charge, magnetic moment, etc.—turned out to be elementary particles moving in space, which itself is subject to the action of matter (in what follows we shall often write “particle” without further qualification, i.e. also meaning a field in the narrow sense).

Although in many respects we are still far from a natural system of elementary particles analogous, in its regularity, to Mendeleev’s “natural periodic system of chemical elements,” and are evidently living through a period roughly analogous to the epoch of Döbereiner’s triads, nevertheless now, thanks to the establishment of a number of profound correlations, there is of course no longer any question of some disorderly accumulation of particles.

The number of all possible reactions and types of motion connected with elementary particles is truly enormous. Investigations of recent decades have shown that all known effects are explained by the presence in particles, besides mass and electric charge, of a number of new properties: magnetic moment, mesotron charge, angular momentum (spin), and statistical properties. Of very great significance for all particles has proved to be the quantum character of their motion, described by wave functions. Since elementary particles very often move with high velocities—of the order of the velocity of light—their kinematics must also be relativistic. Relativistic quantum mechanics has achieved many remarkable results, but in a number of points it remains an unfinished theory.

In what follows it will be explained that the type of wave functions, or spin, and the equations of motion associated with this determine a number of fundamental

properties of particles: statistics, kinematic magnetic moment, character of interactions and lifetime. True relativistic quantum equations of motion describe the form of existence of elementary particles, we may say, paraphrasing Engels’ well-known thesis.

The properties of elementary particles can be divided into three main classes: a) proper mass (determining, so to speak, the basic individuality of the particle), b) the type of wave functions (or spin) and the equation of motion, c) charges (constants determining the intensity of the coupling of particles with one another).

If we wish to give a description of matter in a certain part of the universe as a whole, then it is necessary to consider the abundance of the chemical elements, cosmic rays, and also to establish the connection with cosmological problems. It is advisable from the very beginning to present the basic information about particles in the form of a table, since we shall often have to consider the entire set of particles simultaneously. Explanations will be given in the corresponding paragraphs.

§ 2. ELECTRON

The existence of an elementary portion of electric charge was predicted by Stoney and Helmholtz in 1874–1885. The carrier of negative electricity—the electron—was discovered by J. J. Thomson in 1895, who established that cathode rays in a discharge tube are nothing other than a stream of negatively charged particles—electrons.

Repeated attempts to find a particle of smaller but non-vanishing mass, or of smaller charge, led to no results; in particular, the experiments of Ehrenhaft, who announced the (supposed) discovery of a “subelectron,” proved to be erroneous. Recognition of the elementary nature of the electron quickly entered science.

In 1925–1926 a number of new properties of the electron were discovered. Ehrenfest’s pupils—Goudsmit and Uhlenbeck, on the basis of spectral data, proved the presence in the electron of an intrinsic mechanical angular momentum (rotational moment), or, as it came to be called at Dirac’s suggestion, “spin,” the component of which in any direction can take only two values: $S_z=\pm \dfrac{1}{2}\dfrac{h}{2\pi}$. As soon became clear, there can be no question here of a real “rotation” of the electron. The electron spin should be regarded as the manifestation of a new internal degree of freedom, and the electron should be characterized by four degrees of freedom: three ordinary, external ones, connected with motion in space, and a fourth, spin one. Therefore the state of the electron is specified by four quantum numbers.

At the same time it became clear that the electron possesses a magnetic moment, closely connected with the spin and equal in absolute magnitude to the Bohr magneton.

Next, in 1925 Pauli established the fundamental principle according to which, in each energy state inside the atom, characterized by four quantum numbers, there can be no more than one electron.

In the language of modern wave mechanics, the validity of the Pauli principle means the necessity of describing particles by functions that are antisymmetric with respect to the interchange of the coordinates and spins of any pair of particles. Let us emphasize that obedience, or non-obedience, to the Pauli principle at first erroneously seemed to be a special property of electrons, protons, or other particles, which had to be explicitly postulated. Fermi and Dirac, applying the Pauli principle to an electron gas, constructed a new quantum statistics, different from the classical one.

On the other hand, a gas of photons, helium atoms, or argon atoms, etc., obeys Bose–Einstein statistics, which permits an unlimited number of particles in any quantum state. Bose particles are described by symmetric wave functions.

Finally, in those same years the most important new property of the electron, as well as of the proton and of all other elementary particles, nuclei, and atoms, was discovered; figuratively speaking, it consists in the presence of special wave properties. The wave properties of particles were predicted in 1924–1925 by de Broglie. The de Broglie wavelength $\lambda$ for a body moving with momentum $p$ is equal to $\lambda=\frac{h}{p}$, or, at low velocities,

\[ \lambda=\frac{h}{mv}, \]

where $m$ is the mass of the body at rest. In other words, the motion of an electron, as of other particles, is described by waves (usually denoted by the letter $\psi$), which propagate in space according to the laws of a special quantum or wave mechanics. It turned out that de Broglie waves have a probabilistic character; namely, the quantity $\rho\,d\tau=\psi^{*}\psi\,d\tau$ gives the probability of finding the particle in the volume element $d\tau$.

Quantum mechanics explained in a remarkable way the phenomena of electron diffraction, the homeopolar chemical bond, ferromagnetism, radioactive $\alpha$-decay, and many other phenomena involving free and bound electrons and other particles, which were entirely inaccessible to Bohr’s approximate quantum theory; nevertheless, in 1913–1923 the latter had succeeded, in the main, in correctly describing the motion of intra-atomic electrons and in explaining the periodic system of the elements. There is not the slightest doubt that the nonrelativistic quantum mechanics of de Broglie–Schrödinger–Heisenberg–Born–Jordan gives an exact description of motions at not too high velocities.

Quantum-mechanical, or wave, properties of motion manifest themselves especially strongly in those cases when $\lambda$ is of the order of the dimensions of the region in which the processes take place (for example, for electrons in an atom, $\lambda \simeq 10^{-8}\ \text{cm}$). In other words, quantum mechanics is essential for all processes in which the magnitude of the “action” (energy $\times$ time) is of the order of the quantum of action $h=6.6\cdot10^{-27}\ \text{erg}\cdot\text{sec}$, which is precisely

has a place for elementary particles and the simplest systems: nuclei, atoms, etc., possessing small mass.

Classical Newtonian and nonquantum relativistic mechanics are characterized by their universalism, i.e., the description of the motion of any objects by the same equations. In exactly the same way, nonrelativistic quantum mechanics describes the motion of the waves of any particles and systems by the same Schrödinger equation.

The questions of kinematics should be singled out in particular, since the transition to a synthesis of relativity and quantum mechanics—these two great theories of the first half of the twentieth century—is extremely characteristic of the contemporary understanding of elementary particles. One may say that the theory of elementary particles is, to a considerable degree, relativistic quantum mechanics.

The construction of quantum kinematics satisfying the requirements of the theory of relativity led to stunning successes, but at the same time almost every step along this path of synthesis is achieved with difficulty, recalling the stages in the construction of nonrelativistic quantum theory or the relativistic generalization of the electrodynamics of moving bodies.

A very substantial step in the theory of elementary particles was made in 1928 by Dirac, who established the relativistic quantum equation of motion of the electron. In doing so, in order to describe spin, it was necessary to introduce a wave function with four components, possessing special transformation properties—the so-called spinor (more precisely, bispinor); the four components of the spinor $\psi$-function obey a system of four differential equations, which can be written in the form of a single matrix equation (see § 13.5).

All the relative complexity of Dirac’s theory is repaid by its successes. First, the Dirac equation automatically leads to the value of the spin $1/2$, or, better said, it is precisely the Dirac equation that corresponds to spin $1/2$. Second, as Pauli showed, the fulfillment of his principle and obedience to Fermi statistics follow from the half-integral value of the spin. Thus it became clear for the first time that in relativistic quantum mechanics there is not, and cannot be, a universal equation of motion suitable for all particles; rather, each particle obeys its own individual equation, determined by its spin and by the character of its wave functions.

Third, according to the theory, the electron acquires a magnetic moment of the required magnitude without any innate magnetism having to be assigned to it in advance. Figuratively speaking, one may say that the motion of a particle described by spinor functions is so complicated that it as if “rotates” and, owing to the presence of charge, acquires an effective magnetic moment, which should be called “kinematic.” Let us note that relativistic quantum mechanics of charged particles with spin different from zero always leads to the appearance of a kinematic magnetic moment equal to the Bohr magneton taken with the corresponding mass. Thus,

spin, magnetism, and the statistics of electrons proved to be explained by relativistic quantum mechanics.

The Dirac equation, in excellent agreement with experiment, was able to describe completely the interaction of electrons with the electromagnetic field and with other charged particles, and to explain the motion of electrons in various concrete cases, clarifying all possible details of the structure of spectral lines, to give a complete explanation of the Compton effect, the photoelectric effect, the bremsstrahlung emission of photons by electrons in their collisions with nuclei, etc., etc.

Further, analyzing his equation, Dirac discovered that, along with the electron, it can describe “anti-electrons,” particles of the same mass and spin, but of positive charge, never previously observed.

Dirac predicted that the new particles could be created and annihilated simultaneously with electrons. Referring for details to § 4, we shall emphasize here that, indeed, in accordance with Dirac’s hypothesis, a new property was discovered for the electron, and then for other particles as well: the possibility of being created and annihilated, like photons.

As regards the connection of the electron, as of other elementary particles, with the gravitational field, one should point out the insignificance for them of all effects of gravitation, in view of the small mass of the particles. The Newtonian energy of interaction of the electron with the proton will be approximately $10^{40}$ times smaller than their Coulomb binding energy

\[ \frac{\varkappa m M}{r} : \frac{e^{2}}{r} \sim 10^{-40}. \]

Therefore, gravitational effects for individual elementary particles may be neglected (see § 10).

Finally, in recent years, new specific properties have been discovered in electrons, as in other particles, due to their connection with a special—mesotron—field, in other words, with mesotrons, new particles with mass most often $\mu = 200\,m$, discovered in the free state in cosmic rays. In far-reaching analogy with the charge—electric—and dipole—magnetic—properties of the electron, a mesotron quasi-electric charge $g'$ has been found, considerably smaller in magnitude than $e$, (and in nature, of course, entirely different from electric charge). Apparently, the electron also possesses a quasi-magnetic moment $f' \sim \dfrac{g'}{\varkappa_{0}}$, approximately $10^{10}$ times smaller than the Bohr magneton.

Owing to its connection with the mesotron field, the electron can be produced (together with a neutrino) in the decay of a negative mesotron $\mu_- \to e_- + \nu$, and can also be produced in the $\beta$-decay of radioactive nuclei (see § 5). The reverse process of absorption of an electron by a nucleus or by a nuclear proton, called $K$-capture (since most often electrons from the $K$-shell of atoms are absorbed), has also been observed. It should be foreseen that the specific mesotron properties of electrons will lead

to the emergence of attractive forces between them and neutrons. The presence of such forces should help elucidate the nature of the so-called isotopic shift of spectral lines in medium elements (Ivanenko). At the same time, the theory predicts that nuclear effects should somewhat diminish the Coulomb electric attraction between the proton and the electron and cause a shift of the fine-structure levels in the same direction as that observed experimentally (Heitler–Frohlich–Kahn). The magnitude of the coupling with the electromagnetic and mesotronic field can be characterized not by charges, but by the corresponding “fine-structure” constants:

\[ \alpha=\frac{2\pi e^{2}}{hc}=\frac{1}{137.02} \]

(Sommerfeld’s constant),

\[ \beta'=\frac{2\pi g'}{hc}\sim 10^{-18} \]

(the nuclear fine-structure constant). The smallness of $\alpha$ and $\beta'$ plays a very significant role, making it possible to regard the action of the electromagnetic field or the mesotronic field, generally speaking, as a weak perturbation, and to apply the theory of successive approximations. The same circumstance makes it possible to draw a comparatively sharp boundary between charged particles and the electromagnetic field. If $\alpha$ were comparable with 1, the coupling with the field would be significant, and the transformation of electrons and positrons into photons would be a very frequent phenomenon. In that case the distinction between the electromagnetic field and the particles generating this field would, to a considerable extent, be erased.

On the other hand, it is essential that in the domain of relativistic quantum effects the problem of a single isolated body, i.e. a single elementary particle, strictly speaking loses its meaning, since particles transform into one another, are created and annihilated. Thus the theory of elementary particles suggests the desirability of considering the entire aggregate of particles simultaneously.

In summary, one may conclude that the principal quality of the electron is its electric charge, since its magnetic and mesotronic properties are expressed much less vividly.

According to present-day data, the irreducible properties of the electron are: mass, electric and mesotronic charges, and also spin, or the character of the wave functions, which at the same time determines the equation of motion. The theory of the electron belongs among the most successfully developed chapters of elementary-particle physics. However, the relativistic quantum mechanics of the electron, as of all other particles, is far from free of defects of various kinds; we shall list only the principal ones, partly interconnected: first, difficulties connected with the infinite energy of the electromagnetic field produced by the electron; second, difficulties due to the new treatment of the vacuum and to the nonlinear generalization of electrodynamics in connection with the theory of the positron and of pairs; and third, difficulties with the infinities that arise in higher approximations of perturbation theory.

As for all other particles, the values of the electron’s mass and charges are taken from experiment and as yet have no explanation. Thus

Thus, the task of a future, more complete theory will be, above all, the elimination of all the difficulties enumerated and the derivation of the constants of mass and of the electric and mesotronic charges.

§ 3. POSITRON

Within experimental accuracy, all the constants of the positron coincide in absolute value with the electronic ones. The theories of the positron and of the electron are exactly the same and, to a considerable extent, form parts of a single whole. Let us recall the circumstances of the discovery of the positron, which was the first elementary particle predicted theoretically. In constructing the equation for the electron, Dirac noted in it a difficulty connected with the presence, alongside positive values, of negative values of the energy. Dirac at first left open the question of eliminating this “plus-minus” difficulty. Later it became clear that it is impossible simply to discard the negative values of the energy. Moreover, it turned out that any relativistic quantum kinematics leads to the same “plus-minus” difficulty, rooted already in the classical relativistic relation between energy and momentum:

\[ E^2 = c^2p^2 + m^2c^4 \]

(which is a generalization of the more familiar formula $E = mc^2 + p^2/2m$, or $E_{\mathrm{kin}} = \dfrac{p^2}{2m}$, suitable only at small velocities). In a nonquantum theory, particles that at the beginning of their motion possessed positive energies cannot, by continuously changing their energy, pass into a state of negative energy. Likewise, particles possessing negative energy remain unobservable for us. Therefore states with negative energy can simply be discarded.

A different state of affairs obtains in quantum mechanics, which admits jumps from one state to another. In order to forbid transitions into states with negative energy, Dirac made the assumption that all these states are already occupied by electrons, one on each level with a given momentum and value of spin. In this case an infinite number of electrons on the levels of negative energy is postulated to be unobservable; however, under the influence of certain external actions, electrons may pass into states of positive energy. Then, simultaneously with the birth of an ordinary electron, there arises an unoccupied state, or “hole,” in the distribution of levels of negative energy, which will behave as an antielectron with the opposite sign of charge. Conversely, the fall of an electron into a “hole” will signify its annihilation together with a positron. In this process two photons will be emitted.

At first Dirac supposed that the antielectron was a proton, but it was soon shown that the mass of the antielectron is equal to the electronic one. From the point of view of the modern method of second quantization this circumst—

... is obvious (see § 13.6). The initial interest in Dirac’s hypothesis, owing to the absence of positive electrons, died away rather quickly, and the presence of solutions with negative energy in Dirac’s equation continued to be regarded by many authors, headed by Bohr and Pauli, as a certain difficulty, allegedly testifying even to a “crisis” in quantum mechanics and requiring removal by no means with the aid of Dirac’s hypothesis. Without yielding to the general doubts and to attempts to modify the equation—for example, to exclude from it, in accordance with Schrödinger’s proposal, the “dangerous” terms leading to new effects—Dirac calculated a number of phenomena involving the new particle and pointed out, in particular, the possibility of the simultaneous creation and annihilation of a pair: an electron and an antielectron.

Quite independently of Dirac’s prediction, the positron was discovered in 1932–1933 in cosmic rays by Anderson and Blackett. Blackett observed the tracks of pairs $e_-, e_+$ and even showers of these particles. Soon the creation and annihilation of pairs $e_-, e_+$ was detected under laboratory conditions, in exact agreement with Dirac’s prediction. The most effective for the formation of pairs proved to be collisions of photons of high energy $h\nu \gg mc^2$ with nuclei of heavy elements [the interaction of the electromagnetic field, with which grows proportionally to the square of the nuclear charge $(Ze)^2$]. The energy of the photon is completely converted into the rest and kinetic energy of the particles of the pair, while the atomic nucleus, serving as a sort of “catalyst,” takes up the excess momentum. A single photon by itself, of course, cannot transform into a pair. Alikhanov succeeded in observing the creation of pairs from $\gamma$-rays in the field of the nuclei producing them. Pairs are also observed in collisions of electrons with nuclei and, in principle, may arise in the most diverse interactions of photons and charged or even merely “magnetized” particles with one another in the region of high energies.

Of great theoretical interest is the creation of pairs in the collision of two photons, which has an extremely small probability and has not been observed experimentally up to the present time*).

The simultaneously discovered reverse process of destruction or annihilation of an electron and a positron proceeds with the emission of two photons:

\[ e_- + e_+ \to \gamma + \gamma'; \]

\[ 2mc^2 + E_{\rm kin}^{e_-} + E_{\rm kin}^{e_+} \to h\nu + h\nu'. \]

In addition to the creation of positrons as components of a pair in various processes connected with the electromagnetic field, they may arise, like electrons, in effects of nuclear character, owing to the presence in positrons of a mesotronic quasicharge $g'$ (and, probably, quasi-

*) Let us note here, incidentally, the curious possibility of the formation of a stable system: a “positronium” (Ruark) from an electron and a positron, rotating about a common center of gravity for a certain time equal in order of magnitude to $\tau \sim \alpha^{-5} \sim 10^{-10}$ sec (Ivanenko and Sokolov).

magnetic moment $f'$). These phenomena include the emission of positrons in the $\beta$-decay of artificially radioactive nuclei $\left({}_{7}\mathrm{N}^{13},\ {}_{8}\mathrm{O}^{15}\ \text{and others}\right)$, as well as in the decay of positive mesotrons: $\mu_{+}\to e_{+}+\nu$. The prediction and discovery of positrons and of the creation and annihilation of pairs was, undoubtedly, one of the greatest triumphs of all modern science and may boldly be placed in the same rank as the discovery of the planet Neptune or of new chemical elements in the Mendeleev system. Later it became clear that all elementary particles, in accordance with relativistic quantum mechanics, are capable of being created and annihilated, passing into other particles, under strict observance of the laws of conservation of energy, momentum, angular momentum (spin), and electric charge. Therefore the separation of particles from one another is to some extent conventional; this applies especially to pairs of particles and antiparticles with opposite signs of charge.

From the theoretical point of view, the positron, possessing spin $1/2$, is described by spinor $\psi$-functions of the electronic type, obeys the Dirac equation, and, in accordance with this, satisfies Fermi statistics. Moreover, the complete solution of the Dirac equation always takes into account the possibility of positrons participating in processes connected with electrons, and conversely the participation of electrons in positron effects:

\[ \psi=\psi_{\mathrm{el}}+\psi_{\mathrm{pos}}. \]

Only in the region of small energies can $e_{-}$ be separated from $e_{+}$. The interaction of positrons with all fields and particles is in every respect completely analogous to the interaction of electrons. Despite its enormous successes, the theory of the electron and positron still cannot be regarded as complete, owing to the presence of a number of difficulties inherent in relativistic quantum mechanics as a whole (see § 14).

§ 4. ELECTROMAGNETIC FIELD AND PHOTONS

The theory of the electromagnetic field, along with the theory of electrons, belongs to the most highly developed branches of the physics of elementary particles. To an elementary wave or “portion of the field” there corresponds a particle: a light quantum or photon. In view of the absence of rest mass in the photon, the corpuscular properties in this case are expressed much less sharply than in particles possessing rest mass. It is not surprising, therefore, that historically the wave theory of the electromagnetic field was developed first (Huygens–Fresnel–Maxwell).

On the other hand, the peculiar course of the development of physics led to the quantum properties of motion being discovered for the first time not in electrons, but in the phenomena of radiation, which at that time had been studied more deeply. In 1900, in considering thermal radiation, Planck discovered the quantum of action $h$ (dimension $\mathrm{erg}\cdot\mathrm{sec}$). In 1905, in explaining the photoelectric effect, Einstein arrived at the concept of photons. Decisive

a stage in the recognition of the corpuscular nature of light was Compton’s discovery in 1923 of the special scattering of X-rays, which was interpreted as the “collision” of a photon with an electron.

The dual character of electromagnetic phenomena then served de Broglie as the basis for predicting the wave properties of electrons. The further deepening of the analogy between light and particles of finite rest mass proved very fruitful. Electrons and other particles turned out to be capable of being annihilated and created, like photons; the theory of nuclear forces, carried by mesotrons possessing finite mass, is likewise constructed by analogy with the theory of electromagnetic forces carried by photons.

One must distinguish, as in the case of other fields, first, the propagation of a free electromagnetic field; second, the emission or creation of a field by charges; third, the action of the field on charges, its absorption by charges, and the creation of charged particles by the field.

The free electromagnetic field and the creation of the electromagnetic field by charges and magnets are described by the Maxwell–Lorentz equations. The wave function of the electromagnetic field consists of 4 components of the potential (the scalar potential $A_0$ and the vector potential $\mathbf{A}$) and forms a four-dimensional vector. The electromagnetic field itself (the electric field $\mathbf{E}$ and the magnetic field $\mathbf{H}$) represents certain combinations of derivatives of the potentials. The presence of several components of the potentials or of the field strengths of the electromagnetic field makes it possible to describe the polarization of waves, or the spin of the photon (see §§ 13.3 and 13.4). The value of the photon spin is equal to $1^*)$.

According to Pauli’s theorem (see § 13.6), photons, as particles possessing integral spin, obey Bose statistics, which leads to Planck’s formula for the distribution of energy in the spectrum of equilibrium (“black-body”) radiation at different temperatures.

In Maxwell’s theory the following circumstances should be noted: 1) the absence of magnetic poles; 2) the real character of all quantities, which corresponds to the neutrality of the field; 3) the pronounced, in a certain sense, character of Maxwell’s equations, corresponding to the vanishing of the photon rest mass. A generalization of Maxwell’s equations for the case of finite mass is given by the Proca equations (see § 9 and § 13.5).

The action of the electromagnetic field on charged particles is taken into account by additional terms in the equations of relativistic quantum mechanics, corresponding to the energy of interaction of the particles with the field (see § 13.6).

It remains, finally, to take into account the connection of the electromagnetic field with the gravitational field. Ehrenfest and Tolman analyzed the gravitational field,

$^*)$ From the pictorial point of view this is quite clear on the basis of the selection rule according to which, for example, a change of the quantum number $l$, characterizing the angular momentum in the transition of an electron in an atom, is equal to $\Delta l=\pm 1$. Obviously, this change in the angular momentum of the atom is transferred to the emitted photon or is acquired by the atom at the expense of the absorbed photon.

created by a narrow beam of light. In this way it was made very clear that the electromagnetic field produces gravitational effects, like any other kind of matter; the gravitational effects of photons, like those of all elementary particles, are extremely weak.

Let us also note an attempt, not without interest from the point of view of the reduction of particles to one another, by de Broglie to construct a neutrino theory of light, also developed by Jordan, Kronig, and Sokolov. In the simplest version of this hypothesis the photon was represented as composed of two neutrinos. The development of this idea did not lead to any definitive results.

Maxwell’s equations—together with the equations of quantum mechanics for charged and “magnetic” particles—fully explain the most varied phenomena of emission, absorption, and scattering of light by free electrons and by electrons bound in atoms, molecules, etc.; the emission and absorption of $\gamma$-rays by protons and nuclei; the bremsstrahlung emission of photons by electrons, protons, mesotrons, etc.; all spectral effects, and the fine and hyperfine structure of the spectral lines of atoms.

Among electromagnetic effects, a special place is occupied by the emission of photons in the annihilation of $e_{-}, e_{+}$, as well as the inverse phenomenon of the creation of an $e_{-}, e_{+}$ pair by photons. As has already been indicated, the possibility of such effects means the absence of a sharp boundary between the electromagnetic field and particles. Analogous effects may occur with pairs of oppositely charged mesotrons. Moreover, the connection with pairs of particles leads to the necessity of generalizing ordinary electrodynamics in a nonlinear manner (see § 14.2).

One of the central points of the theory of the electromagnetic field is the correct correspondence of corpuscles (photons) to waves. This was done by Dirac, who in 1927 gave the so-called method of second quantization (see § 13.7).

To summarize, it may be said that we have a deeply developed quantum theory of the electromagnetic field, which answers all the fundamental questions concerning the spin and statistics of photons, the correspondence of photons to waves and the general structure of the equations of the electromagnetic field, and which, moreover, in remarkable agreement with experiments, describes the most varied phenomena connected with radiation. However, quantum electrodynamics, just like the relativistic quantum mechanics of the electron and of other particles, cannot be regarded as complete, in view of a number of difficulties, the methods for eliminating which are still far from obvious (see § 14).

§ 5. NEUTRINO. THEORY OF BETA DECAY

The neutrino hypothesis arose from attempts to explain $\beta$-decay. It is known that, in contrast to $\alpha$-particles emitted by the nuclei of certain elements (U, Ra, Th, etc.) with certain definite energies, electrons and positrons are emitted by all naturally and artificially radioactive elements with all possible energies from zero

up to a certain maximum, characteristic for each emitter, energy, for example: in ${}_{83}\mathrm{Ra}^{210}$, $E_{\max}=1.17\mathrm{MeV}$; in ${}_{7}\mathrm{N}^{13}$, $E_{\max}=0.92\mathrm{MeV}$. Directly, it is as though a violation of the conservation of energy takes place. The state of affairs is illustrated by the so-called curves of $\beta$-spectra, which give the number of $e_-$ (or $e_+$) decays as a function of energy. At some energy, approximately $0.3E_{\max}$, the number of emitted electrons reaches its greatest value. The number of particles with energy close to the maximum is, however, very small.

Two points of view were expressed on the question of $\beta$-decay. Bohr assumed that the apparent nonconservation of energy is real and that energy is in fact not conserved in $\beta$-decay. The few attempts to develop this idea (Landau, Beck) yielded no results.

On the other hand, Pauli proposed explaining the diffuseness of the $\beta$-spectrum by the emission, simultaneously with the electron (or $e_+$), of a new hypothetical particle—the neutrino. The electron and the neutrino together always carry away from $\beta$-radioactive nuclei of a given type an energy equal to the maximum, but upon emission from each individual nucleus one or the other particle, according to the law of chance, receives different portions of energy. At the same time neutrinos, possessing enormous penetrating power, are not registered in any way by modern instruments and therefore, directly, have not up to now been observed.

An excellent proof of the necessity of admitting the neutrino in order also to satisfy the law of conservation of momentum was recently provided by Allen, who observed recoil nuclei in the $K$-capture reaction:

\[ {}_{4}\mathrm{Be}^{7}+e_k\longrightarrow {}_{3}\mathrm{Li}^{7}+\nu . \]

The momentum of the nuclei, within the limits of accuracy, corresponds to the emission of a neutrino with rest mass close to zero.

The further development of the theory of the neutrino reduces to a more exact study of $\beta$-decay, and also of $K$-capture and the decay of mesotrons. Under the influence of the successes of the theory of $\beta$-decay, Bohr abandoned the idea of nonconservation of energy, and there is now no doubt either of the universality of the laws of conservation of energy, momentum, and angular momentum, or of the general fruitfulness and correctness of the neutrino hypothesis.

Independently of this or that refinement of the theory, the neutrino is characterized by the following properties:

1) the neutrino must possess spin ${}^{1}/_{2}$ and therefore be described by the Dirac equation, just as the electron or positron. Indeed, a nucleus of integer or half-integer spin preserves this feature in $\beta$-decay or $K$-capture, which may be interpreted as the transition of a neutron into a proton or of a proton into a neutron in the corresponding nuclei. For example:

\[ {}_{7}\mathrm{N}^{13}\longrightarrow e_{+}+\nu+{}_{6}\mathrm{C}^{13}; \qquad (7_p+6_n\longrightarrow e_{+}+\nu+6_p+7_n), \]

i.e.

\[ p\longrightarrow n+e_{+}+\nu \]

\[ {}_{19}\mathrm{K}^{40}\longrightarrow e_{-}+\nu+{}_{20}\mathrm{Ca}^{40}; \qquad (19_p+21_n\longrightarrow e_{-}+\nu+20_p+20_n); \]

i.e.

\[ n\longrightarrow p+e_{-}+\nu. \]

Since both nucleons $p, n$ and the emitted $e_{-}, e_{+}$ have half-integer spin, in order to conserve angular momentum it is also necessary to emit a neutrino with half-integer spin. There are no grounds whatever in favor of the integer spin $3/2$.

Neutrinos, as particles possessing half-integer spin, must obey Fermi statistics.

2) With regard to the rest mass of the neutrino, no final conclusion can yet be drawn. The mass of the neutrino can be determined by calculating the difference of the energies of $\beta$-emitters, for example
${}_{1}\mathrm{H}^{3}-{}_{2}\mathrm{He}^{3}=e_{-}+\nu+E_{\max}$,
or by taking into account the conditions of stability of isobaric nuclei adjacent in the periodic system (according to Mattauch and Sizo), and also by analyzing the form of the curves of $\beta$-spectra. In any case the mass of the neutrino is much smaller than the mass of the electron and is possibly equal to zero: $m_{\nu}<0.2m$. However, in its properties as a spinorial Dirac particle of spin $1/2$, the neutrino is much closer to electrons and positrons than to photons. This example forces one to treat the distribution of particles by masses with caution and speaks rather in favor of classification by spin values*).

3) The neutrino in all probability does not possess a magnetic moment. In any case the penetrating power of the neutrino indicates that its possible magnetic moment cannot exceed $1/7000$ of the magnetic moment of the electron $\left(\mu_{\nu}<\frac{\mu_{\mathrm{B}}}{7000}\right)$.

4) The neutrino, together with the electron and the positron, possesses an insignificant specific mesotronic charge $g'$, allowing it to interact with the mesotronic or with the gravitational field, similarly to $e_{-}$ and $e_{+}$. The decay of mesotrons is due to the presence in light particles of the charge $g'$.

We shall now briefly set forth the foundations of the theory of $\beta$-decay, which rests on two propositions: 1) in view of the fact that, as we have indicated, $e_{-}, e_{+}, \nu$ cannot exist in nuclei as such (ready-made) in a stable state, $\beta$-decay is nothing other than the creation of the indicated particles; here there is an exact analogy with the emission of photons. 2) According to Pauli’s hypothesis, simultaneously with the electron (or $e_{+}$) there is emitted a neutrino, not directly observed and characterized by the features indicated above. On the basis of these two propositions, Fermi in 1934 constructed a theory of $\beta$-decay which explains well—

* It is not excluded, however, that $e_{-}, e_{+}, \nu$ form a “triad” of light particles, characterized by different values of an additional internal coordinate of the type of isotopic spin, analogously to the proton, neutron, and hypothetical antiproton. Then the difference in proper masses will not be so substantial within the triad.

...embracing both the basic facts and a number of details and predicting new phenomena discovered later.

The very fact of the emission of two particles qualitatively at once explains, according to F. Perrin, the form of the $\beta$-spectra.

The general form of the spectral curve and the dependence of the lifetime on the energy are confirmed by experiment. True, the experimental spectra give the greatest number of decay electrons at an energy equal not to one half of the maximum, as in Perrin’s simple calculation, but to approximately one third of the limiting energy. A complete explanation of this fact presents great difficulties for the theory. The best agreement with experiment is obtained if the neutrino mass is taken to be zero.

In order to proceed further, it is necessary to establish the energy of interaction of nucleons with the field of pairs $(e_-, \nu)$ or $(e_+, \nu)$ emitted in $\beta$-decay. In analogy with the binding energy of a charge at rest with the electromagnetic field $U=eA_0$, Fermi took as a basis the following expression for the interaction of nucleons with the pairs $(e_-, \nu)$ and $(e_+, \nu)$

\[ U_F=g_F(\psi^{*}\psi+\psi\psi^{*}) \]

($\psi$ contains the wave functions of electrons and positrons). Here the constant $g_F$ is determined by comparison with experiment:

\[ g_F=4\cdot10^{-50}\ \text{erg}\cdot\text{cm}^{3}. \]

Subsequently it became clear that this simple expression should be generalized by introducing terms depending on the spins of the light particles, which does not change the general structure of the theory. The most general expression for the energy of interaction of nucleons with the field of pairs of light particles is determined by invariance requirements (see § 13.6). Comparison with experiment makes it possible to choose from the general expression a number of terms best corresponding to the description of $\beta$-decay. If the interaction energy is specified, quantum mechanics unambiguously derives the expression for the probability of $\beta$-decay and of the mean lifetime of the $\beta$-emitter. In a more exact calculation one should introduce an additional factor taking into account the interaction of $e_-$, $e_+$ with the electric field of the nucleus. Indeed, $e_-$ produced by the nucleus are attracted to it, while $e_+$ are repelled.

One of the greatest successes of the theory is the explanation of the groups of $\beta$-emitters discovered by Sargent in a purely empirical way. It turns out that, at one and the same energy $E_{\max}$, the probability of decay may take different values for different $\beta$-emitting nuclei, sharply differing from one another (by approximately a factor of 100). The theory of $\beta$-decay naturally leads to such a division into groups, in view of the fact that the emission of $(e,\nu)$ may be associated either with “allowed” (${}_{7}\mathrm{N}^{13}$, ${}_{29}\mathrm{Cu}^{64}$, ${}_{6}\mathrm{C}^{11}$, etc.) or with “forbidden” to various degrees (${}_{15}\mathrm{P}^{32}$, ${}_{11}\mathrm{Na}^{24}$, ${}_{83}\mathrm{RaE}^{210}$, etc.) transitions of nucleons into one another. The probability of forbidden transitions of different order will be smaller than for allowed transitions.

In addition to nuclear reactions, neutrinos should arise in the spontaneous decay of free mesotrons, observed in cosmic rays. Proceeding from the weighty assumption of the mesotron’s integral spin, we obtain for the neutrino again a half-integral spin from the decay reaction (see § 9):

\[ \mu_{\pm}\to e^{\mp}+\nu . \]

On the other hand, the same decay process may occur on mesotrons that transmit the interaction in atomic nuclei between nucleons. According to Yukawa’s hypothesis, this is the explanation of β-decay as a secondary process. Thus the phenomenological theory of β-decay receives a model foundation. What had appeared to be the direct emission of light particles by nucleons turns out to be due, first, to the virtual emission of mesotrons by nucleons (owing to the quasicharge $g$) and, second, to the decay of mesotrons into $(e_{\pm}, \nu)$ (owing to the quasicharge $g'$).

Neutrinos, having no electric charge and no appreciable magnetic moment, practically devoid of rest mass, not interacting electromagnetically with other charges and fields, and connected only with the mesotron field by an insignificant quasicharge, possess enormous penetrating power. A neutrino has a chance of splitting a nucleus only in passing through a layer of matter with a thickness of approximately the terrestrial globe. Thus, the modern theory satisfactorily explains the failure of attempts at direct observation of the neutrino.

Summarizing the theory of the neutrino, one may say that relativistic quantum mechanics has been able to give a satisfactory explanation of the basic properties of the hypothetical particle, while the successes of the theory of β-decay and $K$-capture, which are among the splendid achievements of modern nuclear physics, serve as the best confirmation of the neutrino hypothesis. However, the state of affairs is, of course, far from satisfactory, in view of the absence of direct experimental confirmation of the existence of the neutrino. In the end, the possibility of various unexpected outcomes on this point is not excluded—for example, that the neutrino will in some essential way differ from other particles, just as, say, the weak gravitational field differs from other types of matter.

Furthermore, the theory of β-decay contains a number of difficulties; in particular, the question of the decay of free mesotrons is not fully consistent with the decay of mesotrons transmitting nuclear forces, i.e. with β-decay. The lifetime of free mesotrons, according to observations in cosmic rays, is $\tau_0 = 2\cdot 10^{-6}$ sec, whereas the theory of β-decay, according to Yukawa, gives for the lifetime $\tau \sim 10^{-8}$ sec. One of the possible ways out of this difficulty is the repeatedly discussed hypothesis according to which nuclear forces are transmitted chiefly by mesotrons of one definite (for example, vector) type, whereas in cosmic rays there are observed pre-

predominantly mesotrons of another, for example pseudoscalar, type (see § 9).

Finally, let us note that attention has recently been drawn to the possible role of neutrinos in stellar and cosmic phenomena. In the formation of $\alpha$-particles in stars from hydrogen, neutrinos carry away up to 6% of the energy. On the other hand, Batagin has noted that under conditions of ultrahigh temperatures $T \sim 10^{12}$ degrees (which may possibly occur, for example, in a “special” pre-stellar state; see § 12), the processes of production of mesotrons, and consequently also of decay of mesotrons with emission of a neutrino and an electron, begin to play an essential role. Since neutrinos, possessing great penetrating power, leave the system practically without interacting with other particles, the inverse processes of neutrino absorption may be disregarded. Thus neutrinos do not take part in establishing and maintaining statistical equilibrium. This may indicate that the very concept of temperature, characteristic of thermodynamic equilibrium, loses its meaning at such high temperatures.

§ 6. PROTON

After the consideration of light particles, it is more convenient, while preserving a certain historical sequence, to pass to heavy particles. The proton, as the nucleus of the hydrogen atom, was discovered when the Rutherford planetary model of the atom was established in 1911. The basic properties of the proton are well known. The mass of the proton is 1836.5 times greater than the mass of the electron.

The spin of the proton is equal to $s=\dfrac{1}{2}\dfrac{2\pi}{h}$; protons obey Fermi statistics.

The value of the proton spin $1/2$ uniquely determines its relativistic quantum kinematics. Namely: the motion of protons, as of electrons, positrons, and neutrinos, must be described by spinor wave functions obeying the Dirac equation. Here, however, a number of additions must be made. First, the Dirac equation automatically leads to the existence, for any charged particle, of a kinematic magnetic moment equal in absolute value to the Bohr magneton taken with the corresponding proper mass. Consequently, for the proton we would obtain the so-called nuclear magneton $\mu_0=\dfrac{eh}{4\pi Mc}$. Experiment, however, gives a magnetic moment equal to $\mu_p=+2.7896\mu_0$, which evidently is not completely reducible to the kinematic one, but contains a certain fraction of an intrinsic magnetic moment. Second, the Dirac equation, like any relativistic quantum kinematics, leads to the prediction of the existence of antiparticles of opposite charge sign, in the present case negative antiprotons $(p_-)$. Despite

despite intensified searches, antiprotons have not been discovered up to now. It should be noted that the systematic study of heavy particles in cosmic rays was begun only in the very last years, and so far it has been possible to detect with certainty only secondary protons and neutrons knocked out of nuclei, whereas the presence of primary protons falling upon the earth’s atmosphere has been proved only indirectly. Thus one cannot say with complete certainty that antiprotons are absent from the flux of cosmic rays. As for obtaining antiprotons under terrestrial conditions, there are as yet no technical means—for example, accelerators giving energies of the order of two billion electron-volts, required for the production of a pair $p, p_{-}$.

Thus, although there are no special grounds for doubting the possibility of the existence of antiprotons, one must nevertheless allow for the possibility of a rather radical change of kinematics in the event that proof is obtained of the absence of antiprotons. This caution is dictated by a third, most important circumstance, which must be borne in mind when applying the Dirac equation to protons.

Atomic nuclei are composed of protons and neutrons; moreover, in $\beta$-decay and other processes these two elementary particles transform into one another (see §§ 5 and 8). In addition, the proton and the neutron have equal spins, obey the same statistics, possess nearly equal masses, and have close or equal values of mesotronic charges (see § 8). In short, the two nucleons are very similar to each other. According to Heisenberg’s idea, the proton and the neutron should be regarded as two states of one and the same particle, the “nucleon,” the neutron being the excited, higher state. From the mathematical point of view one may assign to the nucleon a new, fifth, internal coordinate, characterized by the so-called isotopic spin $\tau$, whose value, for example, $\tau_z=+1$, indicates that we are dealing with a proton, while the value $\tau_z=-1$ indicates that a neutron state is in question.

It is not difficult to combine the Dirac equations for the proton and the neutron (each for the four components $\psi$) into one equation for a wave function with 8 components. However, this will be only a superficial unification, in no way taking account of the deep kinship of the proton and neutron and, at the same time, not making it possible to describe the change of mass in the transition of one particle into the other.

The absence of any indications of such a deep connection between the two nucleons is, in our opinion, the weakest point of the generally accepted hypothesis of the possibility of describing nucleons by the Dirac equation (without any more substantial modifications of it).

The nuclear properties of protons and neutrons must evidently be characterized by comparatively large values of certain specific nuclear quasi-charges and, possibly, dipole moments, since in atomic nuclei protons and neutrons are very strongly bound to one another. According to modern conceptions, protons and neutrons should be consid—

assign to the mesotron a quasielectric charge $g$, in absolute value approximately five times greater than the electric charge, if they are measured by quantities of the same dimension, and, in addition, an intrinsic mesotron quasimagnetic moment of magnitude $f \sim \dfrac{g}{x_0}$, where

\[ x_0=\frac{2\pi\mu c}{h} \]

($\mu$ is the mass of the mesotron)*). In absolute value $f$ exceeds the nuclear magneton by several tens of times. Owing to the presence of mesotron charges and moments, the proton (and neutron) can interact with the mesotron field, in other words, absorb and emit mesotrons, just as, owing to the presence of electric charge and magnetic moment, protons are capable of emitting and absorbing photons.

At the same time, in contrast to other particles: $e_{-}, e_{+}, \nu, \mu_{-}, \mu_{+}, \gamma$, which can be created and annihilated, protons and neutrons in all known processes are either conserved or transform into one another, so that their total number does not change. However, the theory points to the possibility of the creation and annihilation of $p$ and $n$ in a variety of processes with the participation of the electromagnetic field or of mesotrons. If antiprotons $p_{-}$ exist, then possible, for example, are processes of conversion of the pair $(p,p_{-})$ into two photons or of creation of the pair $(p,p_{-})$ in the collision of two photons.

Analogous processes could also take place with the emission of mesotrons

\[ p+p_{-}\rightleftarrows \mu_{+}+\mu_{-}. \]

§ 7. NEUTRON

Searches for a neutral particle of the proton type were carried out in Rutherford’s laboratory as early as the 1920s, but without success. Analysis of the behavior of electrons under the conditions of the atomic nucleus led to the idea that electrons cannot exist in nuclei. According to the original views, electrons were supposed to be very tightly bound to protons. However, the neutron, independently of any assumptions, was discovered by Chadwick at the beginning of 1932, who boldly admitted the emission of new neutral particles possessing a rest mass, in order to explain the contradictory properties of the radiation discovered as early as 1928 by Bothe in the bombardment of beryllium with $\alpha$-particles

\[ {}_{4}\mathrm{Be}^{9}+{}_{2}\mathrm{He}^{4}\to{}_{0}n^{1}+{}_{6}\mathrm{C}^{12}. \]

The mass of the neutron was determined from analysis of the splitting of various nuclei. After it had become clear that neutrons are knocked out from

*) The corresponding fine-structure constant

\[ \beta=\frac{2\pi g^2}{hc}\simeq 0.2. \]

of the most diverse nuclei, there remained no doubt that neutrons are present in all nuclei. However, only after the establishment of the nucleon model of the nucleus did it become clear that it was necessary to recognize the neutron as an elementary particle possessing spin $1/2$ and, in accordance with this, obeying Fermi statistics.

The neutron possesses a magnetic moment, the value of which was determined with certainty by Bloch: $\mu_n=-1.985\,\mu_0$. It is unnecessary to emphasize the non-kinematic, intrinsic character of the neutron’s magnetic moment.

The intrinsic part of the magnetic moment of the neutron, as of the proton, according to modern views, is due to the mesotron field (see § 14.1).

In contrast to the proton, the free neutron is not stable, but is radioactive and must spontaneously decay into a proton, an electron, and a neutrino: $n \to p + e_- + \nu$. This is connected with the fact that the rest mass of the neutron is greater than the sum of the masses of the proton and the electron. A similar decay of nuclear neutrons appears as ordinary $\beta$-decay.

Experimental determination of the lifetime of the free neutron and observation of its decay have so far not been successful, since the neutron has a very high probability of being absorbed by one of the surrounding atomic nuclei before it has time to decay.

Radioactive decay excludes neutrons as possible primary cosmic particles, since they would undoubtedly have had time to decay before reaching the earth’s atmosphere.

The kinematics of the neutron and its coupling to the fields of other particles are entirely analogous to those of the proton. The neutron, as a particle possessing spin $1/2$, is described by spinor wave functions and obeys the Dirac equation.

In view of the absence of electric charge, the hypothetical antineutrons predicted by theory must differ from neutrons only by the sign of the magnetic moment.

The nuclear properties of neutrons are likewise in all respects similar to the properties of protons. It is clear that the neutron, to an even greater degree than the proton, is predominantly a nuclear particle. As is known, neutrons are very effective agents for the splitting of all atomic nuclei, and thanks to reactions with neutrons nuclear processes for the first time acquired technical significance.

§ 8. MODEL OF THE ATOMIC NUCLEUS

A. Composition of the nucleus

Since the construction of the model of the nucleus has not yet been completed, it is useful to compare it with the model of the atom. After the discovery of electrons as constituent parts of all atoms, the nature of intra-atomic forces was clarified by Rutherford and Bohr in 1911–1913. It turned out that $e_-$ are attracted to the nucleus by electric forces according to the law

Coulomb’s law, known from macroscopic physics, while the charge of the nucleus and the number of $e_-$ are determined by the number of the element in the periodic system. On the other hand, the construction of quantum mechanics even in the nonrelativistic approximation presented great difficulties and was completed only in 1927–1928.

The model of the nucleus developed in a different sequence. The most difficult part of atomic theory, namely quantum kinematics, is taken in ready-made form for nuclear particles. In this connection, for nucleons the nonrelativistic approximation is apparently, in the main, sufficient (although a further analysis of relativistic corrections is highly desirable). The second part of the model—the composition of nuclei—was finally clarified in 1931–1933.

Let us briefly recall the justification for the structure of the nucleus from nucleons alone. Formerly nuclei were considered to consist of both kinds of elementary particles then known: $p$ and $e_-$. However, such a conception could not explain a number of properties of nuclei. The contradictions appeared especially clearly in the analysis of spin, statistics, and magnetic moment. For example, the magnetic moments of all nuclei are hundreds of times smaller than electronic ones. It was completely incomprehensible how the magnetic moments could have turned out to be completely compensated. The discovery of the neutron hastened the resolution of all difficulties and fairly soon—although, contrary to not infrequent assertions, by no means immediately—led to the final formulation.

It should be emphasized that the detection of $n$ in various, or all, nuclei by itself did not yet settle the question. The simplest supposition was that nuclei contain all three types of elementary particles: electrons, protons, neutrons. Such a model of the nucleus was indeed proposed by F. Perrin and Auger soon after Chadwick’s discovery. Obviously, such a phenomenological model could in no way remove the deep difficulties listed above.

Obviously, the question had to be approached from a principled point of view, by considering all particles and finally determining which of them are capable of entering into the composition of the nucleus.

After a more thorough analysis of the very possibility of the existence of $e_-$ inside the nucleus, undertaken by us jointly with Ambartsumian, it became clear that $e_-$ apparently lose here their individuality. The decisive argument is the following: in order to have any possibility at all of being in a system of such small dimensions as the nucleus, a particle must be subject to the action of sufficiently large attractive forces.

However, if the interaction energy exceeds the proper energy of, for example, an electron, $E = mc^2$, then the latter obviously cannot exist in the nucleus while preserving its individuality. The same argument applies to all light and intermediate particles: $e_-$, $e_+$, $\gamma$, $\mu$, $\mu_+$, $\mu_0$. In fact, whereas the binding energy of an electron in an atom is of the order of $10$—$100$ electron-volts, which is much less than the proper energy $E = mc^2 = 0.5 \cdot 10^6\ \mathrm{eV}$, the mass defect (energy

...of binding) in nuclei, calculated per particle, reaches values of several million electron-volts. On the other hand, in a nucleus composed of nucleons, per one $p$ or $n$ there falls a binding energy of $7$–$8$ MeV (except for the very light nuclei, where the mass defect is smaller), which is much less than the proper energy of the nucleons $E = 1837mc^2 \sim 900$ MeV. Therefore nucleons can exist in nuclei while retaining their individuality.

Thus, there are no electrons, positrons, or mesotrons in nuclei. The number of $p$ is equal to the nuclear charge $Z$, and the number of $n$ is equal to the difference between atomic weight and charge, $A - Z$.

From empirical considerations it is clear that the neutron has spin $1/2$ and obeys Fermi statistics (see § 7). Of course, $n$ must be recognized as an elementary particle, by no means consisting of either $p + e_- + \nu$ or $p + \mu_-$, etc. Such notions of a composite $n$ would leave all the earlier difficulties unresolved, merely transferring them from the whole nucleus to the neutron.

The emission of $e_-$ and $e_+$ by nuclei in $\beta$-decay and the emission of mesotrons in collisions of nucleons with one another must then be treated in precise analogy with the emission of light by an atom or nucleus, which is undoubtedly one of the most important and nontrivial consequences of the new theory of the nucleus. Although it would occur to no one to speak of the real existence of a photon in an atom or nucleus before its emission, nevertheless it proved less easy to become accustomed to the idea of the production of electrons or positrons in $\beta$-decay, or annihilation upon entry into the nucleus ($K$-capture). It should be recalled, moreover, that the nuclear model was proposed before the discovery of the positron and the observation of annihilation and pair production.

The removal of the contradictions of the old model, the discovery of empirical regularities to which the new conception led, and, finally, the discovery of the production of $e_+$ and $e_-$—all this rather quickly made the new model universally accepted. No subsequent attempts to return to a composite neutron or a composite proton were successful.

B. The Problem of Nuclear Forces

The nucleon model of the nucleus leads to the question of the forces acting between the proton and the neutron. This third, dynamical, part of the nuclear model proved to be the most difficult and to this day remains not completely solved.

First of all it is easy to be convinced that no ordinary forces known up to now are suitable for explaining nuclear interactions. Indeed, gravitational forces are ruled out for atomic and nuclear particles because of their extreme smallness. Electrical forces between nucleons are ruled out because of the neutrality of $n$. The idea arises of using magnetic forces. However, these forces too are insufficient.

All interactions in atoms, molecules, solid and liquid bodies, and gases ultimately reduce to electric and, in part, magnetic forces. Let us recall that electric Coulomb forces, owing to quantum kinematics and the complexity of systems of particles which, in the case of electrons, obey Fermi statistics, lead to the appearance of effective exchange, dispersion van der Waals, and chemical forces. For large accumulations of atoms, gravitation begins to play a role. Thus, we are faced with the fundamental task of understanding the nature of, or constructing a model for, nuclear forces conditioned by some field, or by particles or combinations of particles of a new type. In the eighteenth century such a problem would probably not have caused particular embarrassment, and purely phenomenologically new “nuclear” forces (or a corresponding nuclear “fluid”) would have been introduced into science alongside the electric, magnetic, thermal, and light (ether) fluids and phlogiston. The problem would have consisted in choosing a successful mathematical expression for the new forces (like Newton’s law of gravitation). Without rejecting the selection of successful expressions for forces, at present, however, we must connect the nuclear field with some new combination of known particles, or admit the existence of new particles matched to the nuclear field. Let us begin with the model of electric forces. Two charges $e$ and $e'$ are connected with one another by virtue of the fact that one of them generates an electric field around itself, while the other absorbs this field and, conversely, the first charge absorbs the electric field generated by the second charge.

Quantum theory formulates the same state of affairs somewhat differently, speaking of the virtual emission of photons, i.e. quanta of the electromagnetic field, by one particle and their absorption by another particle. Although the quantum derivation in this case gives nothing new, it possesses enormous heuristic power. Indeed, one may in general draw the conclusion, as was first pointed out by the author of this article and by Ambartsumian, that all possible interactions may also be carried by different particles of finite mass, and not only by quanta of a field with zero rest mass, such as photons. This idea lies at the foundation of the whole modern theory of interactions, in particular the theory of nuclear forces.

Let us formulate the properties of the new forces:

1) Nuclear forces are specific forces, irreducible to other previously known forces.

2) Nuclear forces, in accordance with data on the dimensions of nuclei, etc., must be short-range forces, with a range of action of order $\sim 10^{-13}$—$10^{-12}$ cm; at greater distances they rapidly decrease and become negligible in comparison with electric forces.

3) Nuclear forces, on the basis of data on mass defects, etc., correspond to binding energies of nucleons of the order of several million electron-volts per particle.

4) Nuclear forces depend essentially on the spin of the particle.

5) Nuclear forces depend essentially on the orientation of the spins relative to the mutual distance. In other words, nuclear forces have a noncentral character, similarly, for example, to magnetic forces. This is manifested especially convincingly in the presence, in the deuteron (the nucleus of heavy hydrogen), of an electric quadrupole moment in the ground state (with effective area $Q=+2.7\cdot10^{-27}\ \text{cm}^2$).

6) Nuclear forces must lead to a certain saturation, expressed by the fact that the binding energy, calculated per nucleon, remains approximately constant ($7$—$8$ MeV) over an enormous range of isotopes, from the $\alpha$-particle to the end of the system of elements. This property of nuclear forces brings them close to chemical forces, which are due to the quantum exchange effect. In this connection we shall point out that nuclear matter, or the nucleon “liquid,” is incompressible to a good approximation.

7) An important property of nuclear forces consists in the equality of the forces: proton—proton and proton—neutron, and also, apparently, neutron—neutron.

8) The theory of nuclear forces must explain the basic regularities of $\beta$-decay.

9) Since, as has become clear at the present time, the nuclear field turns out to be connected with mesotrons, the theory must clarify the questions of the interaction of mesotrons with nucleons and light particles.

Since all possibilities for describing nuclear forces by known fields proved unsuccessful, we first proposed explaining the interaction between nucleons by the exchange of pairs of particles, an electron and a neutrino: $(e_{-}, \nu)$ and $(e_{+}, \nu)$, which may not only be virtually emitted and absorbed by nucleons, but are actually emitted in $\beta$-decay (Tamm, Ivanenko, 1934). Since the field of pairs of Fermi particles has no classical analogue, the theory of pair forces can only be quantum. Taking as a basis the Fermi binding energy of nucleons with the pair field (see § 5), we obtain in the second approximation the interaction energy between nucleons

\[ V=a\,\frac{g_F^2}{hcr^5}\,P . \]

(where $a$ is a numerical coefficient $\sim 1$, depending on various refinements of the theory; the operator $P$ indicates the transition of $p,n$ into each other in the interaction, i.e. exchange of charge or the equivalent exchange of coordinates and spins). At large distances $V$ rapidly decreases, especially in view of the additional factor $e^{-\chi r}$; $\chi=\dfrac{2\pi mc}{h}$. In its construction, the theory of pair nuclear forces, in which the possibility of the transfer of interaction by particles of finite mass was first proved, undoubtedly constituted a considerable success. Exchange forces of short range were obtained; subsequently spin and noncentral forces also proved naturally inclu-

valued in the theory. However, the magnitude of the interaction of nucleons due to the exchange of pairs of light particles proved to be negligible, and the pair $\beta$-forces are only a “copy” of the principal nuclear forces. The original sin of the theory of pair forces evidently consisted in its connection with $\beta$-decay, a phenomenon, so to speak, “geologically” rare in the life of nuclei, whose probability is determined by the small constant $g_F$. Attempts to explain nuclear forces by fields corresponding to other combinations of known particles, for example $e_-$, $e_+$, likewise had no success.

§ 9. MESOTRON

A. Prediction of the Mesotron and Mesotron Nuclear Forces

The second stage of the theory of nuclear forces is connected with the work of Yukawa, who, while preserving the basic idea of the transfer of forces by particles of finite mass, proposed explaining nuclear interactions by a new field, hypothetical at that time, or particles: mesotrons. Despite the unfinished character of the theory of nuclear forces, there is now no doubt that the interaction between nucleons is indeed due to mesotrons, either charged or neutral. A proton, transforming into $n$, emits $\mu_+$, absorbed by a neutron, and conversely, $n$ emits $\mu_-$, absorbed by $p$:

\[ \begin{aligned} &1)\quad p \to \mu_+ + n,\qquad n + \mu_+ \to p,\\ &2)\quad n \to \mu_- + p;\qquad p + \mu_- \to n. \end{aligned} \]

Assuming that $\mu_{\pm}$ have integral spin and, consequently, can carry the interaction “singly,” we obtain, in the simplest case of scalar neutral mesotrons of spin 0, described by the de Broglie equation, for the interaction energy of two nucleons:

\[ V=-\frac{g^2}{r}e^{-\chi_0 r}. \]

At small distances we again obtain an interaction of the Coulomb type:

\[ V \simeq -g^2/r. \]

On the basis of empirical information on the magnitude of the nuclear forces we determine $g$:

\[ g \simeq 5e. \]

At large distances $r \gg \frac{1}{\chi_0}\left(\chi_0=\frac{2\pi \mu c}{h}\right)$, $V$ rapidly decreases; hence we determine the mass of the new particle, i.e. of the mesotron, taking $\frac{1}{\chi_0}$ equal to the radius of action of the nuclear forces

\[ \left(\frac{1}{\chi_0} \simeq 10^{-13}\ \text{cm} \simeq \frac{e^2}{mc^2}\right); \]

\[ \mu \sim 137\,m. \]

In addition, Yukawa assumed that $\mu_-$, $\mu_+$ may be connected with light particles in the same way as with heavy ones, i.e., that $e_-$, $e_+$, $\nu$ possess quasicharges $g'$ ensuring their coupling with the mesotron field (see § 5). This makes it possible for mesotrons to decay into pairs of particles according to the scheme:

\[ \mu_- \longrightarrow e_- + \nu;\qquad \mu_+ \longrightarrow e_+ + \nu . \]

The decay of charged mesotrons, which carry the nuclear interaction, will be perceived as $\beta$-decay. In the theory of $\beta$-decay itself nothing is changed by such a model, but in a remarkable way it predicts the possibility of the decay of a free mesotron according to the same scheme (let us note that the hypothetical neutralettoes probably decay according to the scheme $\mu_0 \longrightarrow e_- + e_+$, which should lead to the possibility of emission of pairs $e_-$, $e_+$ by nuclei).

Thus, proceeding from the theory of nuclear forces, there is predicted a new particle of intermediate mass, of integer spin and, consequently, of Bose type, produced by nucleons as well as by light particles and acting upon them, and at the same time radioactive.

Hardly anyone, apart from the very narrow circle of theorists, paid attention to this rather fantastic prediction, in its bold refinement of details, which, however, in all its main points was confirmed after the discovery of the mesotron in cosmic rays in 1937 by Anderson and Neddermeyer.

Careful observations of the hard component showed that the penetrating particles do indeed possess a mass $\mu$ much greater than that of $e_-$, $e_+$, and consequently do not lose as much energy in the bremsstrahlung emission of photons as light particles.

Let us list the principal properties of mesotrons:

Charge. The $\mu_-$, $\mu_+$ of both signs of charge have been discovered, with $\mu_-$ occurring more often in cosmic rays. Repeated attempts to detect $\mu_0$ have not yet led to definitive success. Grechinger, Lloyd-Smith and Kruger announced a year ago the discovery of neutralettoes under laboratory conditions, as a result of bombardment of various nuclei by deuterons; this, however, requires additional confirmation.
Mass. The majority of mesotrons have mass $\mu \sim 200\,m$. Recently, lighter and heavier mesotrons with masses in the range $100$—$700$, even $990\,m$, have been reliably detected (Nishina, Lukirsky, Yuz, Alikhanian, Leprince-Ringuet). The question of the mass spectrum of $\mu$ is still far from clear.
The spin of mesotrons is apparently integer and equal to 0 or 1, as may be concluded from the satisfactory explanation of $\beta$-decay and of the decay of cosmic $\mu_-$, $\mu_+$, and also on the basis of the theory of nuclear forces, which suggests the greater naturalness of integer spin. Thus the statistics of $\mu^{\pm}$ must be Bose. As for the kinematics, in view of the absence of more precise information on the character of the $\psi$-functions of mesotrons, one has to test all variants of description suitable-

... for integer spin: 1) scalar $\psi$ (spin 0), the de Broglie equation; 2) pseudoscalar $\psi$ (spin 0), a pseudoscalar equation; 3) vector $\psi$ (spin 1), the Proca equation; or, finally, 4) pseudovector $\psi$, obeying the corresponding equation (spin 1). There are no grounds for ascribing to mesotrons a spin greater than 1.

The radioactive decay of the mesotron has been determined in cosmic rays by many methods. For a mesotron at rest the lifetime is

\[ \tau_0 = 2 \cdot 10^{-6}\ \text{sec}, \]

while for fast $\mu_{\pm}$, in accordance with the theory of relativity, the lifetime increases, depending on the velocity,

\[ \tau = \frac{\tau_0}{\sqrt{1 - v^2/c^2}}. \]

Rasetti was able to determine $\tau_0$ in studying the decay of $\mu_{-}, \mu_{+}$ stopped in a block of lead under laboratory conditions.

In a Wilson chamber the tracks of the products of mesotron decay have not been detected, apparently owing to the diffusion of slow $\mu_{\pm}$ to the place where they decay, escaping observation; it is possible that in this process a peculiar “mesotronium” atom is formed owing to recombination of $\mu_{+}$ and $e_{-}$ (Pomeranchuk and Migdal, Ivanenko). The discovery of mesotrons and the confirmation of Yukawa’s hypothesis have appeared as a new triumph of elementary-particle physics and of the theory of nuclear forces.

The action of the electromagnetic field on the mesotron is described in the usual way by adding the corresponding components of the potentials to the momentum and energy,

\[ \mathbf{P} \longrightarrow \mathbf{p} \mp \frac{e}{c}\mathbf{A}; \qquad E \longrightarrow E \mp e\varphi . \]

On the other hand, Maxwell’s equations are supplemented on the right-hand sides by terms describing the distribution of charges and currents of mesotrons which generate the electromagnetic field. Experimentally nothing is known about the magnetic moment of mesotrons. Gravitational interactions for mesotrons are, obviously, negligible.

B. Modern theory of nuclear forces

The description of nuclear forces by means of scalar mesotrons, charged or neutral, proved to be too simplified. In order to obtain spin and noncentral nuclear forces, it is necessary to couple the nucleons with a mesotron field possessing dipole spin properties and described by several components. The best method proved to be the description of $\mu$ by means of Proca’s equations (spin 1) or a pseudoscalar equation (spin 0). In this way one succeeds in arriving at spin and noncentral forces, in particular in calculating the quadrupole moment of the deuteron. The equality of the forces $p—p$, $n—n$, $p—n$ insistently requires the introduction, along with the charged mesotrons, also of neutral mesotrons as particles carrying the interaction. Therefore the theory is usually developed on the basis of using a mixture of $\mu_{-}, \mu_{+}, \mu_{0}$, most often in equal proportion (Kemmer). Moreover,

it is natural to suppose that nuclear forces are carried by mesotrons of different masses. Finally, possible mixtures have repeatedly been discussed, for example, of vector and pseudoscalar mesotrons, which simultaneously realize, according to the hypothesis of Møller and Rosenfeld, the nuclear field. Other authors used only charged $\mu_-, \mu_+$ (Yukawa, Heitler) or only neutral $\mu_0$ (Bete, Ivanenko and Sokolov), or took $\mu_0$ with a small admixture of $\mu_-, \mu_+$ (Hulthén, Tamm). However, it has not been possible to satisfy all the requirements of the theory of nuclear forces. In the theory of nuclear forces there are two fundamental difficulties:

1) The divergence of higher approximations, which makes it necessary, in particular, to admit hypothetical $\mu_0$ in order to ensure interaction between identical nucleons $(p—p;\ n—n)$.

2) The effective dipole character of the mesotron field (vector and pseudoscalar) leads to the appearance, in the interaction energy, calculated classically in the case of neutral $\mu_0$ and quantum-mechanically in the case of mesotrons of any sign of charge, of terms having a quasi-magnetic form

\[ V \simeq -\frac{f^2}{r^3} e^{-\chi_0 r} \simeq \begin{cases} -\left(\dfrac{g}{\chi_0}\right)^2 \dfrac{1}{r^3} & \text{for small } r,\\[6pt] 0 & \text{for large } r. \end{cases} \]

With such a rapid increase of the interaction energy at small distances, nucleons would fall onto one another and stable orbits would be impossible. The dipole character of mesotrons is also manifested in various effects of scattering of $\mu$ by nucleons and of various particles and photons by mesotrons. In this case the effective cross sections increase without bound with increasing energy, which, of course, is absurd. In the vector case such difficulties arise both for electromagnetic and for specifically nuclear interactions, whereas in the pseudoscalar case only nuclear effects lead to difficulties.

Thus, despite the general success in the understanding of the mesotron, connected with the prediction of the particle itself, its role in the nucleus and spontaneous decay, and also with the successful description of certain effects, the general state of the theory of nuclear forces and of mesotron theory is far from satisfactory. In view of this situation, various nuclear effects have to be calculated with artificially selected interaction energies that reproduce the basic features of nuclear forces, in particular their short-range character. Fortunately, many phenomena do not depend on a more exact form of the interaction; these include, for example, $\alpha$-decay and the fission of uranium.

The problem of constructing nuclear forces is, obviously, one of the most important in all of contemporary physics. Knowledge of the exact law of interaction between nucleons, analogous to the Newtonian or Coulomb law, would make it possible to calculate any nuclear effects with

any isotope. It is still unclear whether the difficulties of the theory of nuclear forces are rooted chiefly in insufficient knowledge of the properties of the mesotron, or whether they are connected with difficulties inherent in all relativistic quantum mechanics (see § 14).

§ 10. THE GRAVITATIONAL FIELD

The gravitational field occupies a special place in the theory of elementary particles. At first sight it seems that gravitation has no place here at all, since the gravitational actions of particles upon one another are negligibly small.

However, we must give, as far as possible, the fullest description of the structure of matter; and since the gravitational field is a certain, albeit special, kind of matter, it is necessary to decide the question to which elementary particles gravitation can be reduced, and whether this is possible at all.

Further, in a rather unexpected way, Pauli and his collaborator Fierz have found that the field equations of particles of spin 2, deprived of rest mass, coincide exactly with Einstein’s equations for a weak gravitational field.

In understanding gravitation one should distinguish three stages. The first of them is connected with Newton’s classical theory, the second with Einstein’s general theory of relativity; with the third stage we associate all the problems of the quantum general-relativistic theory.

Newton himself, as is well known, emphasized the absence of any model of gravitational forces and put forward his law of interaction as a phenomenological formula (see § 13.6). Repeated attempts by Lesage, Lomonosov, and many other scientists of the 18th and 19th centuries to explain gravitation by impacts of some particles and by other means proved fruitless and had no influence in science.

One of the essential features of classical physics was the circumstance that gravitation remained apart from the unification of the various branches of science in the 19th century (light—electricity—magnetism, etc.) and continued to be regarded as a property not connected with other manifestations of matter and, alongside all other properties, in no way connected with space and time.

Let us pass to the second stage in the understanding of gravitation. According to the general theory of relativity, established by our great contemporary in 1916, in order to describe the gravitational field completely one must generalize Newtonian theory in three directions. First, gravitation, like all other phenomena, must be described in the four-dimensional world, i.e. in space—time.

Second, Newton’s theory of gravitation is suitable only as an approximation for a weak field. To describe the gravitational field it proved insufficient to have a single component of the potential; it was necessary to introduce 10 components of the potential. In other words, the wave function

of the gravitational field has 10 components. The Newtonian potential itself $\varphi$ turns out to be only an addition to one of the 10 components of the gravitational potential, though the most essential addition:

\[ g_{44}=1+\frac{2\varphi}{c^2}. \]

The totality of the 10 components of the gravitational potential, which transform in a definite way upon transition to various coordinate systems, constitutes the so-called symmetric tensor of rank 2:

\[ g_{\mu\nu}=g_{\nu\mu}\quad(\nu,\mu=1,2,3,4). \]

This point is considerably more characteristic of gravitation, since other fields are described by spinors, scalars, vectors, etc., but still represent nothing that goes beyond the framework of a wave-relativistic generalization of the theory of different fields.

The third point, however, signifies a radical breaking of all known notions of space, time, and gravitation; in it lies the main essence of Einstein’s gigantic achievement. Namely, it turns out that the components of the gravitational potential coincide with the components of the so-called metric tensor $g_{\mu\nu}$, which characterizes the geometry of the 4-dimensional world, i.e. of space-time*). Deviations of the metric quantities from their constant Galilean or pseudo-Euclidean values, i.e. from

\[ g^0_{44}=+1,\qquad g^0_{rs}=-\delta_{rs} \]

($\delta_{rs}=0$ for $r\ne s$ and $\delta_{rs}=1$ for $r=s;\ r,s=1,2,3$), on the one hand, characterize the curvature of space-time, which in this case is described by Riemannian geometry, and, on the other hand, correspond to the appearance of gravitation. Thus the presence of gravitation is completely reduced to the curvature of four-dimensional space-time. Since, according to Einstein, the curvature of geometry is caused by any substance, space-time thereby for the first time proves to be connected with other kinds of matter and subject to their action.

Along with the equations describing the generation of the gravitational field by matter or, in the particular case, the free field, it is necessary to have a law of action of the gravitational field on the motion of matter, i.e. a generalization of Newtonian force. In view of the above-mentioned coincidence of the components of the gravitational potential with the components of the metric, the solution of this part of the problem is ultimately relatively simple: it is necessary only to rewrite all equations in the so-called generally covariant tensor form, replacing, in particular, ordinary derivatives by covariant ones, which take into account the curvature of space.

Additional investigation was required by the Dirac equation for particles of spin $1/2$, since the covariant derivative of a spinor (or semi-vector) had previously not been known. As was shown by us (Fock and Ivanenko, 1929), the concept of covariant differentiation can also be generalized to spinors, more precisely—to bi-spinors. Thus it was possible to write the Dirac equation in arbitrary curvilinear—

*) Namely, the “interval” between two infinitely close events

\[ ds^2=g_{\mu\nu}\,dx_\mu\,dx_\nu \]

(summation over the indices $\mu,\nu$ from 1 to 4 is understood).

... coordinates and at the same time generalize it to the case in which gravitation is present. It is interesting that in this process the electromagnetic field is included without any additional distortion of the Riemannian metric. This circumstance seems very significant from the standpoint of criticizing overly simplified attempts to construct a “unified” field theory, which, after gravitation, sought also to reduce the electromagnetic (and perhaps even the mesotron) field to certain geometric properties of space. Since the transition from pseudo-Euclidean flat geometry to a Riemannian curved metric explained the forces of gravitation, at the moment of such a triumph of the general theory of relativity it seemed almost obvious that electromagnetism too was subject to geometrization, which would require a further generalization or, if one likes, distortion of the Riemannian geometry itself—for example, taking account of the torsion of space along with curvature, and so on.

First Weyl (in 1918), then Eddington, Einstein, Schouten, and many others proposed an enormous number of possible generalized geometries, attempting to construct a unified or even triple (taking the mesotron field into account) (Schrödinger, 1944) field theory. All these attempts, without exception, despite their often mathematical elegance and seductive generality, proved physically completely fruitless. After 1927–1928, when the interests of our science focused on quantum mechanics and the nucleus, cosmic rays, and elementary particles, the appearance of new variants of the “unified” field theory almost entirely ceased. It is now clear that all these attempts were quite premature. Moreover, it became clear that any possible future unification could occur only after the gravitational field itself had been properly quantized.

Be that as it may, our result of 1929, for its part, indicated the possibility of incorporating electromagnetic forces geometrically without any distortion of Riemannian geometry.

The complicated nonlinear equations of Einstein’s gravitational field have been solved only in a few cases.

Of greatest importance is Schwarzschild’s centrally symmetric solution, which found, from Einstein’s equations, the curvature of the metric under the influence of a body of mass $M$, for example the Sun (see § 13.4 f). This makes it possible to refine the motion of the planets in comparison with Newtonian theory and to predict two other effects: the deflection of light and the red shift of spectral lines in a gravitational field*).

*) The corrections, generally speaking, lie beyond the limits of observational accuracy, with the exception of Mercury, for which Einstein’s theory explains the small displacement of the perihelion of the orbit by $42''.9$ per century. The deflection of a light ray passing at the edge of the Sun by the Sun’s gravitational field is $1''.75$; in the order of magnitude it agrees with subsequent observations; the red shift of spectral lines in a gravitational field has been confirmed especially convincingly by observations of superdense stars (for example, of the companion of Sirius).

It cannot be left unnoticed that the three predictions of the general theory of relativity concern very small effects. Moreover, the deflection of light is also obtained from Newtonian theory, but there it is half as large, which is plainly refuted by experiment. In essence, only the red shift is a completely new effect. One is again compelled to marvel at the exceptional accuracy of Newtonian theory. Much, very much, has changed in physics over the 250 years since the appearance of the Principia; the understanding of gravitation has been profoundly modified, but the concrete, observable consequences of the new—Einsteinian—conceptions have turned out to be extremely insignificant and few in number.

The confirmations of the three insignificant Einsteinian effects proved sufficient to become convinced of the agreement with experiment of the grand picture of the general-relativistic theory, without which we cannot imagine the modern physical picture of the world, almost as we cannot without quantum theory, despite the incomparably greater, tangible, everyday significance and applicability of the latter. Here, of course, one must also bear in mind that the general theory of relativity is, in essence, the only theory capable of advancing us also in the solution of cosmological problems (see § 12).

We are now interested in finding the gravitational field produced by elementary particles. It is clear that, for the latter, quantum effects will play an essential role; therefore the solutions obtained with the aid of Einstein’s equations for elementary particles can have only the most approximate significance, though they probably convey certain aspects of reality. For a point particle one again obtains the Schwarzschild solution.

Almost always gravitational fields turn out to be weak, corresponding to an insignificant distortion of the pseudo-Euclidean metric. It can be shown that, according to the rules of second quantization, waves of a weak gravitational field correspond to particles—gravitons—just as photons correspond to the electromagnetic field (see § 13.5). Like the photon, gravitons have no rest mass. The spin of the graviton is equal to 2. Gravitons, evidently, must obey Bose statistics.

With the aid of gravitons one can first of all solve the problem of the emission of gravitational energy by moving matter*).

In view of the smallness of the gravitational constant $ \varkappa $ and of the masses of elementary particles, the radiation of gravitational energy is negligible even at high frequencies of oscillation. On the other hand, for astronomical

*) It is interesting that, owing to the value 2 of the graviton spin, the radiation of gravitational waves is not dipole, as for the photon, or a vector meson, but quadrupole.

The magnitude of the radiated energy is therefore determined not by the second but by the third derivative with respect to time. Per second there is emitted gravitation-

of objects, despite the large magnitude of their masses, because of the smallness of the frequencies of oscillation, the emission of gravitational energy is likewise negligible. For binary stars, in the course of a year there is emitted approximately $10^{-12}$ of their total energy, i.e. the loss of energy by gravitational radiation becomes noticeable only over cosmological intervals of time of the order of $10^9$—$10^{10}$ years.

The action of an individual graviton is practically impossible to detect, in contrast to the photon, since the number of high-frequency gravitons is negligibly small.

With the aid of gravitons one can also solve the problem of the interaction of particles. The quantum calculation of the interaction energy of two particles is entirely analogous to the calculation of the interaction of two charges through photons. Let particle $A$ emit gravitons and particle $B$ absorb them, and conversely; as a result of such a two-act process of exchange of gravitons, an interaction arises between $A$ and $B$, of precisely Newtonian form.

The value of the results obtained so far still lies exclusively in the domain of principle; real gravitational effects manifest themselves only for bodies of large mass, for which a quantum treatment is unnecessary.

In this connection one cannot fail to point out that the inverse effect of absorption of the energy of the gravitational field, as well as the creation of particles at the expense of the energy of the gravitational field and the annihilation of particles with the emission of gravitons, have hitherto not been considered at all. Despite the very small probability of such processes, they are undoubtedly of fundamental interest. The consideration of such effects may still further erase the difference between the gravitational field, understood, according to Einstein, as a property of space and time, and other types of elementary particles and fields.

The history of physics teaches us the fruitfulness of the rapprochement and unification of ideas about different types of matter, toward which one should strive in the present case as well; at the same time it is only necessary to take into account all the unsuccessful experience of the “unified” field theory, on the one hand, and the successes of relativistic quantum mechanics, which unifies all kinds of particles and fields, on the other.

It should be noted that if for electrons and the electromagnetic field, and also for nucleons and the mesotron field, in the region of high energies the distinction between field and particles is to a certain extent lost,

energy:

\[ \frac{dW}{dt}=\frac{\varkappa^2}{c^5}\left|\dddot{J}_{\mu\nu}\right|^2, \]

where $J_{\mu\nu}$ is the tensor of the quadrupole moment.

Gravitational radiation friction, or damping (expressed by a term containing the 5th time derivative of the quadrupole moment), is entirely negligible.

in view of the relatively large value of the fine-structure constants $\alpha, \beta$ and the transformations of particles into one another, then in the case of gravitation the particles and the gravitational fields produced by them differ extremely sharply. In view of the smallness of the gravitational fine-structure constant $\gamma=\dfrac{2\pi x m^{2}}{hc}$ for all masses $m$ of elementary particles, particles and systems of particles—nuclei, atoms, molecules—may be regarded as “immersed” in space-time and as distorting the latter only very slightly. It should not, however, be forgotten that this insignificant distortion, for macroscopic and astronomical objects, leads to effective forces exceeding all the others.

Since the gravitational field, in the general case, cannot differ fundamentally from a weak field, it too should be regarded as a special kind of matter, although no particles can be directly associated with a nonlinear field.

Let us note that the repeatedly expressed idea of a kinship between neutrinos and gravitons cannot be directly correct, in view of the difference in spins. In the spirit of de Broglie’s neutrino theory of light, which, as was indicated, has not led to completed results, one would have to construct the graviton, at least, from four neutrinos or from quadruplets of neutrinos.

Thus, in summarizing the question of gravitation, one may say that the treatment of the weak gravitational field is naturally included in the theory of elementary particles. At the same time, gravitons belong to a special kind of matter, since the gravitational field is a curvature of space-time. The problems of the quantum geometry of elementary particles, on the one hand, and the quantum treatment of the general theory of relativity, i.e. of a strong gravitational field, probably partly connected with cosmological questions, are still at the stage of the very earliest development.

(To be concluded in the next issue.)

Submission history

[v1] 1947

Particle	Mass	Type of wave function, spin, equation of motion, statistics	Electric charge, magnetic moment	Mesotron charge and moment	Abundance
1. Graviton (quantum of the weak gravitational field)	\(0\)	Symmetric tensor of rank 2; Einstein equation; \(s=2\); Bose statistics	\(0\) \(0\)	\(0\) \(0\)	The gravitational field is a curvature of space-time caused by all kinds of matter
2. Photon \((\gamma)\) (quantum of the electromagnetic field)	\(0\)	Vector potential; Maxwell equations \(s=1\) Bose statistics	\(0\) \(0\)	\(0\) \(0\)	Emitted and absorbed in various processes. The density of radiant energy in the known part of the universe is \(\rho \simeq 10\)
3. Neutrino? \((\nu)\)	\(m_{\nu}<0.2\,m;\) \(m_{\nu}=0?\)	Bispinor; \(s=1/2\); Dirac equation; Fermi statistics	\(0\) \(\mu_{\nu}<\dfrac{1}{7000}\,\mu_{B}\)	\(g'\) \(f'\)	Produced in \(\beta\)-decay, \(K\)-capture, and mesotron decay; owing to their penetrating ability they are practically not absorbed by matter. They have not been discovered with final certainty.
4. Electron \((e_{-})\)	\(m=9\cdot 10^{-28}\) \(Ae=5.48\cdot 10^{-4}\)	Bispinor; \(s=1/2\). Dirac equation; Fermi statistics	\(-e_{0}\) \(-\mu_{B}\)	\(g'\) \(f'\)	Enters into the composition of all atoms. In the part of the universe known to us there are approximately \(N_{e}\sim 10^{75}\) electrons. They are absorbed and produced in various processes of high energy in atomic nuclei, cos...

Full Text

INTRODUCTION TO THE THEORY OF ELEMENTARY PARTICLES

CONTENTS

§ 1. HISTORICAL INTRODUCTION. TABLE OF ELEMENTARY PARTICLES

TABLE OF ELEMENTARY PARTICLES

§ 2. ELECTRON

§ 3. POSITRON

§ 4. ELECTROMAGNETIC FIELD AND PHOTONS

§ 5. NEUTRINO. THEORY OF BETA DECAY

§ 6. PROTON

§ 7. NEUTRON

§ 8. MODEL OF THE ATOMIC NUCLEUS

A. Composition of the nucleus

B. The Problem of Nuclear Forces

§ 9. MESOTRON

A. Prediction of the Mesotron and Mesotron Nuclear Forces

B. Modern theory of nuclear forces

§ 10. THE GRAVITATIONAL FIELD

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INTRODUCTION TO THE THEORY OF ELEMENTARY PARTICLES