THE THEORY OF THE MESOTRON AND NUCLEAR FORCES

V. L. Ginzburg

Introduction. § 1. Wave equations for the mesotron. Interaction with the electromagnetic field. § 2. Nuclear forces. § 3. On the difficulties of the theory

INTRODUCTION

By the theory of the mesotron is meant the range of questions connected, on the one hand, with the interpretation of the mesotron observed in cosmic rays and, on the other hand, with the mesotron theory of nuclear forces. Both these divisions of the theory of the mesotron are far from complete and are at a stage of development which, moreover, encounters serious difficulties. It is therefore natural that a finished exposition of the theory of the mesotron cannot be given, and our aim consists only in illuminating the present state of the question*).

At the present time the name mesotron, or meson, is applied not only to the semi-heavy particle observed in cosmic rays, but also to numerous hypothetical particles whose mass is intermediate between the masses of the proton and the electron. We shall use the term “mesotron,” specifying, where necessary, which particle—hypothetical or observed—is meant.

Mesotrons were discovered in cosmic rays in 1937¹; the hard component of cosmic rays at sea level and at small altitudes consists mainly of precisely these particles. At sea level the mesotronic hard component accounts for ~70% of all particles of cosmic radiation. Under laboratory conditions, so far as is known, mesotrons have not yet been obtained. In cosmic rays, however, the study of the properties of mesotrons is impeded by a number of circumstances, first of all by the fact that they contain no large number of slow particles. Therefore, despite intensive experimental work, a whole series of the basic characteristics of the mesotron has not yet been reliably established. Moreover, one cannot even assert that in cosmic rays there is observed only one kind of semi-heavy particles, or even only semi-heavy particles with a single value of the rest mass. The magnitude of the charge, and still more the value of the spin of the me-

*) The article was written in September 1946.

zotron also cannot be considered reliably established experimentally. Nevertheless, setting aside the discussion of the question of the reliability of the available data, we can make the following assertions:

1. There are mesotrons of both signs of charge. The magnitude of the charge is apparently equal to \(\pm e\), where \(e\) is the charge of the electron. In any case the mesotron charge is not equal to \(\pm 2e\), etc.; there are no grounds to suppose that the mesotron charge is close to \(\pm e\) but differs from this value.

2. The mass of the mesotron is approximately \(m = 200\,m_0\), where \(m_0\) is the electron mass. The most frequently occurring values of \(m\) lie between \(150\,m_0\) and \(250\,m_0\). Thus, in any case, the overwhelming majority of the semiheavy particles of cosmic rays at sea level have a mass close to \(200\,m_0\); the assumption that this majority of particles has only one value of the mass does not appear to contradict experiment.

3. The mesotron decays spontaneously, and the lifetime in the coordinate system associated with it is \(\tau_0 \simeq 2 \cdot 10^{-6}\) sec. In the decay of a mesotron an electron (or positron) is emitted. The second emitted particle is most likely a neutrino. However, this has not been proved, and the possibility of mesotron decay into an electron and a photon cannot yet be entirely excluded. If the decay occurs with emission of an electron and a neutrino, then the spin of the mesotron is equal to zero or one, since the spins of the electron and neutrino are one-half, and the total spin must be conserved in the decay. The more probable value of the spin is zero (see § 1). If the decay occurs with emission of an electron and a photon, then the spin of the mesotron is one-half*).

In studying the properties of the mesotron, the chief importance belongs to a quantitative comparison of the processes observed experimentally and caused by mesotrons with theoretical calculations carried out under definite assumptions about the properties of the mesotron. Thus, for example, in order to judge the spin of the mesotron, a comparison is made^2 of the large ionization bursts observed experimentally with calculations performed under the assumption that the mesotron spin is \(0\), \(1/2\), or \(1\).

For the quantitative calculation of various effects due to the interaction of mesotrons with matter, it is necessary to know the initial properties of the mesotron (mass, spin) and the character of its interaction with the electromagnetic field (photons), with light particles (electrons and neutrino), and with heavy nuclear particles (protons and neutrons). With regard to both these questions, theory at the present time cannot make any unambiguous statements. However, if one restricts consideration to particles with definite values

*) The spin is assumed to be expressed in units of \(\hbar\), i.e., if we say that the spin is equal to \(\dfrac{1}{2}\) or \(1\), this means that it is equal to \(\dfrac{1}{2}\hbar\) or \(\hbar\).

spin and rest mass*), the number of equations and expressions for the interaction energy possible from the point of view of the requirements of relativistic invariance turns out to be relatively small. In addition, at least at first, it is natural to restrict oneself to the consideration of particles with spin not exceeding unity. This assumption is not due to the fact that the theory of particles with spin \(>1\) proves to be very complicated, and the significance of spin \(\leq 1\) is clearly distinguished not only by its simplicity but also by certain essential features.^4 Above, speaking of the spin of the mesotron, we have already taken this circumstance into account, assuming that the spin of the neutrino is \(1/2\) and the spin of the mesotron is no greater than unity (if, for example, the spin of the neutrino were \(3/2\), which is possible in principle, then the decay of the mesotron into an electron and a neutrino would be compatible with the assumption that the spin of the mesotron is 2; similarly, the decay of the mesotron into an electron and a photon is compatible with the assumption that the spin of the mesotron is \(3/2\)).

On the basis of what has been said, in the theory of the mesotron and of nuclear forces one considers almost exclusively particles with spin \(0\), \(1/2\), and \(1\).

The simplest is the interaction of the mesotron with the electromagnetic field. This interaction is determined first of all by the presence of an electric charge in the mesotron. The electromagnetic interaction of mesotrons, leading to the formation of \(\delta\)-electrons and to bremsstrahlung, is essential for determining the spin of the mesotron and will be considered in § 1.

More complicated and at the same time important is the question of the interaction of mesotrons with nuclear particles, as well as with electrons and the neutrino. Processes caused by this interaction and essential for cosmic rays are the decay of the mesotron (if it occurs into an electron and a neutrino) and nuclear scattering (for a discussion see in ^5). Further, since mesotrons are unstable, they cannot come from outer space and must be generated chiefly in the upper layers of the atmosphere; the formation of mesotrons by primary cosmic particles, which are most probably protons, apparently is not electromagnetic in character, but is due to nuclear interaction.

The importance of the question of the interaction of the mesotron with nuclear particles is connected, however, not only with processes in cosmic rays, but, to a much greater extent, with the problem of nuclear forces. As is known, after Fermi’s creation of the theory of \(\beta\)-decay, Tamm^5 in 1934 developed a theory of nuclear forces connecting the appearance of these forces with the fact that heavy particles (the proton and neutron) exchange light particles (electrons, positrons, and neutrinos). In this exchange a proton, for example, emits a positron and a neutrino, transforming into a neutron;

*) What has been said means that variants of the theory allowing a change in the spin and mass of particles are not considered (see, for example, ^3). A theory of particles with variable properties is relatively complicated and ambiguous, as a result of which the restriction made is, in any case at first, quite natural.

the neutron, absorbing the same light particles, turns into a proton, etc. As a result of such an exchange of charge, a proton and a neutron located at some distance from one another experience a force interaction.

The situation here is analogous to the interaction, for example, of two moving electrons caused by the exchange of photons. In the electromagnetic case, instead of the concept of the exchange of photons, one may proceed from wave ideas: from this point of view each of the electrons creates around itself a field that acts on the other electron. Similar wave ideas are also applicable to nuclear forces, i.e., one may say that the neutron creates around itself an electron-neutrino field acting on the proton, and so on.

Quantitatively, the theory of electron-neutrino nuclear forces (or the so-called β-forces) proved untenable, since in view of the weakness of the β-interaction the forces turn out to be smaller than necessary by a factor of order \(10^{10}\)—\(10^{12}\) (see\({}^{7}\)).

In order to overcome the difficulties encountered in the theory of β-forces, Yukawa in 1935 proposed the existence of a special field of nuclear forces. Upon quantization this field is associated with certain particles, analogous to photons, which appear upon quantization of the electromagnetic field. Unlike photons, the new particles, which we now call mesotrons, generally speaking may be charged, and in addition their rest mass is not equal to zero. It is easy to show (see § 2) that the mass of the particles \(m\) is directly related to the radius \(r_0\) of action of the forces caused by the exchange of these particles, namely:

\[ r_0 \sim \frac{1}{\varkappa} = \frac{\hbar}{mc}. \tag{1} \]

From experimental data it is known that the radius of action of nuclear forces is of order \(r_0 \sim 2 \cdot 10^{-13}\) cm and, consequently, according to (1) \(m \sim 200\,m_0\). The mass of the new particles thus turns out to be precisely of the same order as the mass of the cosmic mesotron. It is therefore understandable that after the discovery of a semiheavy particle in cosmic rays, the mesotron theory of nuclear forces received a powerful stimulus for further development.

In the mesotron theory of nuclear forces, these forces are due to the fact that a proton and a neutron exchange mesotrons with one another. In this case, if the spin of the mesotron is integral and equal to 0 or 1, the exchange may be effected by a single mesotron, since the spin of a proton (neutron) in transforming into a neutron (proton) may change by 0 or 1. (In the theory of β-forces, however, the exchange occurred by two particles, each of which had spin \(1/2\).) The assumption that the spin of the mesotron is integral is therefore simpler and was adopted by Yukawa. In order to include β-decay in his theory, Yukawa assumed that a mesotron can decay into an electron and a neutrino; furthermore, since there exist both \(\beta^{-}\) (electronic) and \(\beta^{+}\) (positronic)

decays, it was assumed that mesotrons may have both signs of charge. Both of these assumptions (the decay of the mesotron and the existence of mesotrons of both signs of charge) are in agreement with the properties of mesotrons observed in cosmic rays, which still further supports the whole conception of the connection between nuclear forces and the observed mesotrons.

However, the attempt to construct a quantitative theory of nuclear forces that would agree with all experimental data has not yet been crowned with success and encounters serious difficulties. In this connection, no more or less complete theory of nuclear forces exists at the present time, and, strictly speaking, the connection of mesotrons observed in cosmic rays with nuclear forces cannot be considered proven. Nevertheless, the set of qualitative considerations indicated above, as well as the almost certain presence of nuclear interaction of mesotrons in cosmic rays, do not give serious grounds for doubting the common character of the whole circle of questions concerning mesotrons and nuclear forces.

The mesotron theory of nuclear forces is discussed in more detail in § 2.

§ 1. WAVE EQUATIONS FOR THE MESOTRON.

INTERACTION WITH THE ELECTROMAGNETIC FIELD

The form of the equations which the wave function of the mesotron \(\psi\) must satisfy is determined by the assumed value of the spin of this particle. The equations must in this case be relativistically invariant and, consequently, the wave function is a spinor of some rank (or, in the general case, a collection of spinors). The number of independent components of the function \(\psi\) must obviously be connected with the number of possible projections of the spin in some direction—this, indeed, is the idea of describing particles by means of multicomponent functions. What has been said already to a considerable degree determines the character of the wave function and of the corresponding wave equation.

If the spin of the mesotron is zero, the function \(\psi\) has only one component and thus is either a scalar or a pseudoscalar which, as is known, is equivalent to a completely antisymmetric tensor of rank 4, \(\varphi_{iklm}\), having only one independent component, for example the component \(\varphi_{1234}\).*)

*) A pseudotensor is a quantity that behaves in the same way as tensors under all coordinate transformations reducible to rotations. But when the sign of some spatial coordinate changes, the sign of the tensor components and the sign of the pseudotensor components may change differently. For example, a pseudotensor of rank zero, i.e. a pseudoscalar, has only one component, whose sign differs in right-handed and left-handed coordinate systems. A completely antisymmetric tensor of rank 4, \(\varphi_{iklm}\), has the same properties; it has only one independent component,

\[ \varphi_{1234}=\varphi_{2314}=-\varphi_{2341}=-\varphi_{2134}=\cdots \]

The wave equation for a particle with spin zero is as follows:

\[ \left(\Box-\varkappa^{2}\right)\varphi=0, \qquad \Box \equiv \frac{\partial^{2}}{\partial x_{1}^{2}}+ \frac{\partial^{2}}{\partial x_{2}^{2}}+ \frac{\partial^{2}}{\partial x_{3}^{2}}+ \frac{\partial^{2}}{\partial x_{4}^{2}}; \quad x_{4}=ict. \tag{2} \]

In the case where the wave function is pseudoscalar, in equation (2) it is necessary to replace \(\varphi\) by \(\varphi_{iklm}\).

Equation (2), like the other wave equations to be discussed below, represents the equation of a certain field—in the present case, the field of a scalar \(\varphi\). The establishment of the connection between the classical field and the corresponding aggregate of corpuscles is achieved by quantizing this field; under quantization the field \(\Psi\) (in the case of (2), the scalar field \(\varphi\)) is regarded as an operator. We shall not dwell here on the quantum theory of wave fields (see \(^{9,10}\)) and shall confine ourselves to indicating the simplest way in which the mass of a particle can be connected with the quantity \(\varkappa\) appearing in equation (2).

A plane wave that is a solution of equation (2) has the form:

\[ \varphi=Ae^{i(\nu t \pm \mathbf{k}\mathbf{r})}, \qquad \frac{\nu^{2}}{c^{2}}-\mathbf{k}^{2}-\varkappa^{2}=0. \tag{3} \]

At the same time, according to the fundamental proposition of quantum mechanics—the de Broglie relation—the momentum of a particle is \(\mathbf{p}=\hbar\mathbf{k}\), and the square of the energy is

\[ E^{2}=m^{2}c^{4}+\mathbf{p}^{2}=(\hbar\nu)^{2}. \]

Hence, and from (3), it follows that equation (2) describes particles with rest mass \(m\), determined by the equality

\[ \varkappa=\frac{mc}{\hbar}. \tag{4} \]

One of the essential results of quantum field theory is the conclusion that particles with integer spin, described by ordinary tensors, must obey Bose–Einstein statistics. Particles with half-integer spin, described by spinors of odd rank, must obey Fermi–Dirac statistics \(^{11,9}\).

The familiar difference between the cases in which \(\Psi\) is a scalar and a pseudoscalar appears if equation (2) is replaced by a system of first-order equations. For a scalar we have

\[ \frac{\partial\varphi}{\partial x_i}=\chi_i, \qquad \frac{\partial\chi_i}{\partial x_i}=\varkappa^{2}\varphi, \tag{5} \]

where, here and everywhere below, \(i=1,2,3,4\), and summation is performed over indices that occur twice.

V. L. Ginzburg

In the pseudoscalar case

\[ \left. \begin{aligned} \frac{\partial \varphi_{iklm}}{\partial x_l}&=\gamma_{klm},\\ \frac{\partial \chi_{klm}}{\partial x_l} -\frac{\partial \gamma_{lmt}}{\partial x_k} +\frac{\partial \chi_{mik}}{\partial x_l} -\frac{\partial \chi_{ikl}}{\partial x_m} &=\chi^2\varphi_{iklm}. \end{aligned} \right\} \tag{6} \]

Systems (5) and (6) are equivalent to equation (2) for \(\varphi\) or \(\varphi_{iklm}\), as is easily verified by eliminating from (5) or (6), respectively, \(\chi_i\) or \(\gamma_{klm}\). The difference between scalar and pseudoscalar mesotrons, which have the same spin, equal to zero, and the same mass (if the constants \(\chi\) in (5) and (6) are equal to one another), appears only when considering their interaction with particles of half-integer spin (see § 2). With respect to interaction with the electromagnetic field, both schemes (scalar and pseudoscalar) are completely equivalent. Therefore in this paragraph we shall simply speak of a mesotron (particle) with spin zero.

A particle with spin 1 must be described by a wave function with 3 independent components, since the projection of the spin in this case may take the values \(0\) and \(\pm 1\). The simplest, after a scalar, tensor wave function—a 4-dimensional vector—has, however, four components. Nevertheless, a particle with spin 1 is described by a vector wave function \(\varphi_i\) satisfying the equation

\[ (\square-\chi^2)\varphi_i=0. \tag{7} \]

This equation has not three but four solutions, one of which corresponds to a particle with spin zero. In order to exclude this extra solution, the condition

\[ \frac{\partial \varphi_i}{\partial x_i}=0 \tag{8} \]

must also be imposed on \(\varphi_i\).

The system of equations (7)—(8) also describes a particle with spin 1 and a mass determined according to (4).

Instead of equations (7)—(8), in a number of cases it is convenient to use the equivalent system of first-order equations

\[ \left. \begin{aligned} \frac{\partial \varphi_k}{\partial x_l} -\frac{\partial \varphi_l}{\partial x_k} &=g_{ik},\\ \frac{\partial g_{ik}}{\partial x_k} &=-\chi^2\varphi_i. \end{aligned} \right\} \tag{9} \]

A particle with spin 1 may, moreover, be described not by a vector but by a pseudovector wave function or, equivalently, by a wave function \(\varphi_{ikl}\), where \(\varphi_{ikl}=-\varphi_{kil}=-\varphi_{ilk}\). In this case, instead of (9), we have:

\[ \left. \begin{aligned} \frac{\partial \varphi_{ikl}}{\partial x_l}&=g'_{ik},\\ \frac{\partial g_{ik}}{\partial x_i} +\frac{\partial g_{ki}}{\partial x_l} +\frac{\partial g_{il}}{\partial x_k} &=\chi^2\varphi_{ikl}. \end{aligned} \right\} \tag{10} \]

THEORY OF THE MESOTRON AND NUCLEAR FORCES

The distinction between the vector and pseudovector variants of the theory is essential only when considering the interaction with particles of half-integer spin (protons, neutrons, electrons, and neutrinos). Therefore in this section, unless otherwise stated, the wave function of a particle with spin 1 is taken to be a vector.

Particles with spin \(1/2\) obey the well-known Dirac equation

\[ \gamma_k \frac{\partial \Psi}{\partial x_k} + \varkappa \Psi = 0, \tag{11} \]

where \(\gamma_k\) are four-row matrices, \(\Psi\) is a bispinor having four components, and the relation \(\varkappa = \dfrac{mc}{\hbar}\) still holds (for details see, for example, \(^{12}\)).

Equations for particles with spin greater than unity \(^{4}\), and with variable spin \(^{3}\), can also be written down. However, consideration of the interaction of these particles with an external field or with other particles proves to be associated with known difficulties \(^{4,13}\) and has been little studied. Therefore we shall not touch upon this question here.

The introduction of the interaction of particles with spin \(0\), \(1/2\), and \(1\) with the electromagnetic field, described by the vector potential \(A_k\), is achieved by replacing in equations (2), (5), (6), (9), (10), and (11)

\[ \frac{\partial}{\partial x_k} \quad \text{by} \quad \Pi_k = \frac{\partial}{\partial x_k} - \frac{ie}{\hbar c} A_k, \tag{12} \]

where \(e\) is the charge assigned to the particle.

The possibility of the replacement (12) is clear, in particular, from the fact that the variances of \(\dfrac{\partial}{\partial x_k}\) and \(A_k\) are the same; therefore, under such a replacement, the equations remain relativistically invariant. Let us note that in applying (12) to a system of equations, the replacement must be made with a certain caution, namely in such a way as not to make the system contradictory; the latter occurs, for example, if the replacement (12) is made in the system of equations (7)—(8), and not in (9).

The correct introduction of the interaction with the electromagnetic field by means of the replacement (12), and likewise the introduction of interaction with other fields (particles), is automatically achieved when using the variational principle, on which we shall not dwell (see, for example, \(^{9,10}\)).

The transition to the nonrelativistic approximation, or to an equation of the second order, shows that particles with spin \(1/2\) and \(1\), whose interaction with the field is determined only by the charge (replacement (12)), behave as if, in addition, they had a magnetic moment equal to the whole Bohr magneton \(^{10,12}\),

\[ \mu_0 = \frac{e\hbar}{2mc}. \tag{13} \]

Thus, under the indicated conditions, the ratio of the magnetic moment to the spin angular momentum is equal to \(\dfrac{e}{mc}\) for particles with spin \(1/2\), and is equal to \(\dfrac{e}{2mc}\) for particles with spin 1.

However, in addition to the “interaction with the charge,” in the case of spins \(1/2\) and 1 one may also introduce an interaction with the “true” magnetic moment \(\mu_1\). For example, in the case of the Dirac equation, in the presence of such a moment, and also of a charge \(e\), the equation of motion takes the form:

\[ \left(\gamma_k\frac{\partial}{\partial x_k}-\frac{ie}{\hbar c}\gamma_k A_k+ \frac{\mu_1}{\hbar c}\frac{\gamma_k\gamma_l}{2i}F_{kl}+\chi\right)\Psi=0, \tag{14} \]

where

\[ F_{kl}=\frac{\partial A_l}{\partial x_k}-\frac{\partial A_k}{\partial x_l} \]

is the electromagnetic-field strength tensor.

In the nonrelativistic approximation the magnetic moment of the particle described by equation (14) is equal to (see 16):

\[ \left. \begin{aligned} \mu_0+\mu_1&=\gamma\mu_0,\\ \gamma&=\left(1+\frac{\mu_1}{\mu_0}\right). \end{aligned} \right\} \tag{15} \]

The introduction of a term analogous to that present in equation (14), containing \(F_{kl}\), is also possible in the case of equations (9) for spin 1; the total moment in this case can likewise be represented in the form (15). Finally, both in (14) and in (9) one may introduce a term containing \(F_{kl}\) and derivatives of the wave functions; with this, however, there are associated known complicating circumstances.

In the nonrelativistic approximation all the equations presented go over into an equation of the Pauli type:

\[ i\hbar\frac{\partial\Psi}{\partial t} = \left\{ \frac{1}{2m}\left[-i\hbar\nabla-\frac{e}{c}\mathbf{A}\right]^2 +e\varphi-\gamma\mu_0(\mathbf{s}\mathbf{H}) \right\}\Psi, \tag{16} \]

where \(\mathbf{A}\) and \(\varphi\) are the three-dimensional vector and scalar potentials, \(\mathbf{H}\) is the magnetic-field strength, and \(\mathbf{s}\) is the spin operator. For a particle with spin zero, \(\gamma=0\). For the electron, when the constant \(\mu_1\) in (14) is equal to zero, the spin term has the well-known form \(-\mu_0(\boldsymbol{\sigma}\mathbf{H})\), where \(\boldsymbol{\sigma}\) are the two-row Pauli matrices and \(\Psi\) is a two-component wave function (for the Pauli equation for a particle with spin 1, see, for example, in 14). The difference between particles with spin 0, \(1/2\), and 1 appears only in the form of the last term in (16); neglecting this term, we obtain, evidently, the usual Schrödinger equation.

The simplest problem in which the interaction is taken into account is the motion of a particle in a given field; of primary interest is the Coulomb field \(\left(e\varphi=-\dfrac{e^2Z}{r}\right)\). The solution of problems of this type, based on the use of equation (16), constitutes the main content of nonrelativistic quantum mechanics.

The relativistic theory of the hydrogen atom is based on the solution of the problem of the motion of an electron obeying equation (14) with \(\mu_1=0\), \(\mathbf A=0\) and \(e\varphi=-iA_4=\dfrac{e^2Z}{r}\) (see \({}^{12}\)). The agreement of the theory with experiment that obtains in this case is the principal argument in favor of applying the Dirac equation with \(\mu_1=0\) to the electron. The problem of motion in a Coulomb field for a particle with spin zero has also been solved \({}^{12}\). In both these cases the eigenfunctions of the problem form a complete orthonormal system and satisfy the obvious general requirements (they ensure the finiteness of the energy, etc.). In the case of particles with spin 1 and \(\gamma=1\) and \(\gamma\ne1\), as well as particles with spin \(1/2\) and \(\gamma\ne1\) (i.e. \(\mu_1\ne0\)), on the contrary, the problem of motion in a Coulomb field has no solution \({}^{15,16}\), in the sense that the admissible solutions do not form a complete system of functions and there are solutions corresponding to the fall of the particle onto the force center. The reason for the fall consists in the fact that for spin \(1/2\) with \(\gamma\ne1\), and in the case of spin 1 with \(\gamma=1\), the particle has a magnetic moment and in the relativistic approximation* the energy of interaction of this moment with the field of the Coulomb center has the form:

\[ U\sim -\frac{1}{r^3}. \tag{17} \]

In a field of the type (17), both in the classical and in the quantum theory, the motion is limiting, i.e. the particle falls onto the center (for details see § 2). The presence of a moment in the particle leads to difficulties also when considering various radiation processes (light scattering, bremsstrahlung, etc.). The question of the difficulties encountered by the theory will be considered in more detail in § 3.

We shall now dwell on the results of calculating effective cross sections for various electromagnetic processes, carried out for particles with different spins and values of \(\gamma\) (a summary of the results is taken mainly from Pauli’s review \({}^{10}\)**).

All cross sections were calculated in the first nonvanishing approximation of perturbation theory.

Table 1 gives effective cross sections for the scattering of mesotrons by a fixed Coulomb center and Table 2 for scattering by an electron (\(\delta\)-production).

* The Dirac electron, for which \(\mu_1\) in (14) is equal to zero, in the nonrelativistic approximation has a magnetic moment \(\dfrac{e\hbar}{2mc}\); however, in the extreme relativistic approximation the electron behaves as a particle without a magnetic moment \({}^{13,14,15,73}\).

** Some cross sections are also compared in the review by Rossi and Greisen \({}^{27}\).

Spin is everywhere expressed in units of $\hbar$, and the magnetic moment in units of

\[ \mu_0=\frac{e\hbar}{2mc}. \]

Table 1

Scattering of mesotrons by a Coulomb center

$E$ and $m$ are the initial energy and mass of the mesotron; $\theta$ is the scattering angle;

\[ \eta=\frac{E}{mc^2} \]

(the energy $E$ includes the rest energy); $d\Omega$ is the solid angle;

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

	Spin	Magnetic moment (value $\gamma$)	Cross section for scattering	Reference to literature
I	0	0	$\displaystyle \frac{1}{4}r_0^2\frac{\eta^2}{(\eta^2-1)^2}\frac{d\Omega}{\sin^4\frac{\theta}{2}}$	16
II	$1/2$	1	$\displaystyle \frac{1}{4}r_0^2\left[\frac{\eta^2}{(\eta^2-1)^2}-\frac{1}{\eta^2-1}\sin^2\frac{\theta}{2}\right]\frac{d\Omega}{\sin^4\frac{\theta}{2}}$	17
III	$1/2$	$\gamma\ne 1$	$\displaystyle \frac{(\gamma-1)^2}{4}r_0^2\frac{d\Omega}{\sin^2\frac{\theta}{2}};\;(\eta\gg 1)$	16
IV	1	1	$\displaystyle \frac{1}{4}r_0^2\left[\frac{\eta^2}{(\eta^2-1)^2}+\frac{1}{6}\sin^2\theta\right]\frac{d\Omega}{\sin^4\frac{\theta}{2}}$	18, 19
V	1	$\gamma\ne 1$	$\displaystyle \frac{(\gamma-1)^2}{3}r_0^2\eta^2\,d\Omega;\;(\eta\gg 1)$	16

In both tables the cross sections for cases III and IV are higher by one order with respect to the magnitude

\[ \eta=\frac{E}{mc^2}, \]

than for cases I and II. For case V the cross section is higher by still another order. This already reveals the role, noted above, of the magnetic moment, which actively affects the dependence of the cross section on energy. The cross sections given for cases III, IV, and V at high energies prove to be certainly incorrect[^22],[^23]. In the case of Table 1 this is already clear from the fact that the problem of the motion of a mesotron in a Coulomb field (for cases III, IV, and V), when posed rigorously, has no solution; therefore the results obtained by the method of perturbation theory require special investigation in order to determine the domain of their applicability. The cross sections for cases I and II, in any event, are entirely possible, and there is no reason whatever to doubt their validity.

Table 2

Elastic scattering of mesotrons by an electron

\(\epsilon E\) is the energy transferred to the electron. Terms of order \(\dfrac{m}{m_0}\dfrac{mc^2}{\epsilon E}\) and smaller are omitted (\(m_0\) is the electron mass), \(E \gg mc^2\). The remaining notation is as in Table 1.

	Spin	Magnetic moment (value of \(\gamma\))	Cross section per collision (in the coordinate system in which the electron was initially at rest)	Reference to the literature
I	0	0	\(2\pi r_0^2\,\dfrac{m}{m_0}\,\dfrac{mc^2}{E}\,\dfrac{d\epsilon}{\epsilon^2}(1-\epsilon)\)	16
II	\(1/2\)	1	\(2\pi r_0^2\,\dfrac{m}{m_0}\,\dfrac{mc^2}{E}\,\dfrac{d\epsilon}{\epsilon^2}\left(1-\epsilon+\dfrac{\epsilon^2}{2}\right)\)	17, 20
III	\(1/2\)	\(\gamma \ne 1\)	\(\pi(\gamma-1)^2 r_0^2\,\dfrac{d\epsilon}{\epsilon}(1-\epsilon)\)	16
IV	1	1	\(\dfrac{2\pi}{3}r_0^2\,\dfrac{d\epsilon}{\epsilon}\left(1-\epsilon+\dfrac{\epsilon^2}{2}\right)\)	19, 21
V	1	\(\gamma \ne 1\)	\(\dfrac{2\pi}{3}r_0^2\,\dfrac{m}{m_0}\,\dfrac{E}{mc^2}\,d\epsilon(1-\epsilon)\)	16

Table 3 gives the differential and total cross sections for the scattering of light by a mesotron. The values of the initial and final photon energies appearing in the table are connected by the well-known relation

\[ k = k_0 \frac{1}{1+\dfrac{k_0}{mc^2}(1-\cos\theta)} . \]

The effective cross sections for bremsstrahlung and for the production of mesotron pairs by photons are given in Tables 4 and 5. The nucleus is here considered finite and having radius

\[ R=\frac{5}{6}Z^{1/3}\frac{\hbar}{mc}. \]

For case II, the formulas available in \(^{29}\) are given with the corresponding modification required by this assumption.

The cross sections given in Tables 3, 4, and 5 for cases I and II raise no doubts for arbitrarily large energies. On the contrary, for cases III and IV (case V has not been investigated) one obtains cross sections that increase with energy in an impermissible manner and therefore are valid only for not too large energies (see \(^{21,22,23,33}\) and § 3). For example,

in the case of light scattering (the Compton effect), Sections III and IV of Table 3 are valid only if

\[ \left. \begin{aligned} k_0=h\nu &\ll \frac{mc^3}{a}=137\,mc^2,\\ \text{or}\qquad \lambda=\frac{2\pi c}{\nu} &\ll \frac{e^2}{mc^2}. \end{aligned} \right\} \tag{18} \]

We have presented the corresponding cross sections mainly for orientation and in order to illustrate clearly the influence of spin and magnetic moment on various processes.

Table 3

Scattering of light by mesotrons

The scattering mesotron is assumed initially to be at rest. \(k_0\) and \(k\) are the initial and final energy of the photon. The remaining notation—see Table 1.

	Spin	Magnetic moment (value \(\gamma\))	Cross section for scattering through angle \(\theta\). Valid for all energies (except case III)	Total cross section for scattering under the condition \(k_0\gg mc^2\)	Reference to the literature
I	0	0	\(r_0^2\,\dfrac{1}{2}\,\dfrac{k^2}{k_0^2}\cos^2\theta\,d\Omega\)	\(\pi r_0^2\,\dfrac{mc^2}{k_0}\)	24
II	\(1/2\)	1	\(r_0^2\,\dfrac{1}{2}\,\dfrac{k^2}{k_0^2}\left(\dfrac{k_0}{k}+\dfrac{k}{k_0}-\sin^2\theta\right)d\Omega\)	\(\pi r_0^2\,\dfrac{mc^2}{k_0}\left(\dfrac{1}{2}+\ln\dfrac{2k_0}{mc^2}\right)\)	27, 28, 29
III	\(1/2\)	\(\gamma\ne 1\)	\((\gamma-1)^4 r_0^2\,\dfrac{1}{4}\,\dfrac{k}{k_0}\left(\dfrac{k}{mc^2}\right)^2 d\Omega+\cdots\qquad k\gg mc^2\)	\(\dfrac{\pi}{4}(\gamma-1)^4 r_0^2\,\dfrac{k_0}{mc^2}+\cdots\)	33
IV	1	1	\(r_0^2\,\dfrac{1}{2}\,\dfrac{k^2}{k_0^2}\left\{1+\cos^2\theta+\dfrac{1}{48(mc^2)^2}\left[kk_0(28-64\cos\theta+12\cos^2\theta)+(k^2+k_0^2)(29-16\cos\theta+\cos^2\theta)\right]\right\}d\Omega\)	\(\dfrac{5\pi}{36}r_0^2\,\dfrac{k_0}{mc^2}\)	24, 25, 26

An experimental study of the processes produced by the mesotron of cosmic rays may, in principle, make it possible to determine its spin. Up to the present time the only effect which

Table 4

Bremsstrahlung of mesotrons

Initial energy of the mesotron \(E \gg mc^2\); \(\varepsilon E\)—energy of the emitted photon;

\(Z\)—atomic number of the material; \(\displaystyle A=\frac{12(1-\varepsilon)}{5mc^2\varepsilon Z^{1/3}},\quad \alpha=\frac{e^2}{\hbar c}.\)

	Spin	Magnetic moment (value \(\gamma\))	Cross section (in the coordinate system in which the nucleus is at rest)	Reference to literature
I	0	0	\(\displaystyle r_0^2\alpha Z^2\,d\varepsilon\,\frac{16}{3}\left(\frac{1-\varepsilon}{\varepsilon}\right)(\ln A-1/2)\)	31
II	\(1/2\)	1	\(\displaystyle r_0^2\alpha Z^2\,d\varepsilon\,\frac{16}{3}\left(\frac{3\varepsilon}{4}+\frac{1-\varepsilon}{\varepsilon}\right)(\ln A-1/2)\)	29
III	\(1/2\)	\(\gamma\ne1\)	\(\displaystyle r_0^2\alpha Z^2(\gamma-1)^4\,d\varepsilon\left\{\frac{1-\varepsilon}{mc^2Z^{1/3}}E+\frac{\varepsilon}{2}\ln^2 A-\frac{3\varepsilon}{2}\ln A+\ldots\right\}\)	30
IV	1	1	\(\displaystyle r_0^2\alpha Z^2\,d\varepsilon\left\{\frac{E}{mc^2Z^{1/3}}\cdot\frac{\pi}{60}(2-2\varepsilon+7\varepsilon^2)+\frac{\varepsilon}{12}\left(17+\frac{7\varepsilon^2}{2(1-\varepsilon)}\right)\ln^2 A+\left(\frac{16(1-\varepsilon)}{3\varepsilon}+\frac{13\varepsilon}{12}-\frac{5\varepsilon^3}{24(1-\varepsilon)}\right)\ln A+\ldots\right\}\)	31, 32

It has proved possible to use, for this purpose, the formation of large ionization bursts under considerable thicknesses of lead or iron. Assuming that the ionization effect is determined by the electromagnetic bremsstrahlung of mesotrons (the formation of \(\delta\)-electrons proves to be insignificant), one can carry out the corresponding calculations and compare them with experiment \(^{2,34}\). It then turns out that the calculations agree with experiment if it is assumed that the spin of the mesotron is equal to 0 or \(1/2\) (\(\gamma=1\)). It is as yet impossible to distinguish between spins 0 and \(1/2\), since the accuracy of the experiments and of the theoretical calculations is insufficient and does not exceed 100%. However, one also cannot completely exclude the possibility that the spin of the mesotron is equal to 1 (or that the spin is \(1/2\) with \(\gamma\ne1\)). The point is that in the calculations one has to use an effective cross section for bremsstrahlung in the region of high energies, where it is, strictly speaking, not applicable and, furthermore, at a certain energy this cross section actually

Table 5

Production of mesotron pairs by photons

\(E\)—photon energy \((E \gg mc^2)\); \(\varepsilon E\)—energy of the positive mesotron;
\(Z\)—atomic number of the material; \(B=\dfrac{12\varepsilon(1-\varepsilon)}{5mc^2 Z^{1/3}}E\).

	Spin	Magnetic moment (value \(\gamma\))	Cross section (in the coordinate system in which the nucleus is at rest)	Reference to literature
I	0	0	\(r_0^2\alpha Z^2\,d\varepsilon\,\dfrac{16}{3}\varepsilon(1-\varepsilon)\left(\ln B-\dfrac{1}{2}\right)\)	31
II	\(1/2\)	1	\(r_0^2\alpha Z^2\,d\varepsilon\,\dfrac{16}{3}\left[\dfrac{3}{4}-\varepsilon(1-\varepsilon)\right]\left(\ln B-\dfrac{1}{2}\right)\)	29
III	\(1/2\)	\(\gamma\ne 1\)	\(r_0^2\alpha Z^2(\gamma-1)^4\,d\varepsilon\left[\dfrac{\varepsilon(1-\varepsilon)E}{mc^2Z^{1/3}}+\ldots\right]\)	30
IV	1	1	\(r_0^2\alpha Z^2\,d\varepsilon\left[\dfrac{E}{mc^2Z^{1/3}}\dfrac{\pi}{40}(7-2\varepsilon+2\varepsilon^2)+\ldots\right]\)	31, 32

cut off. In such a situation the exclusion of the value of the spin equal to 1 can be convincing only if the effective cross section used is the minimum possible for this spin and, at the same time, leads to the formation of a considerably larger number of showers than is observed experimentally. In the opinion of a number of authors \(^{2,33}\), precisely such a situation obtains. However, in our opinion \(^{35}\), the cross section used \(^{2,33}\) is not minimal, since it is based on applying the formulas for the Compton effect up to energies \(\hbar\nu \sim \dfrac{mc^2}{\alpha}\), which contradicts condition (18). Therefore the indicated comparison of theory with experiment, strictly speaking, shows only that, from the point of view of cosmic-ray experiments, there are no special grounds for the assumption that the mesotron spin is equal to unity. Further, if the spin is nevertheless equal to unity, then calculations based on perturbation theory are inapplicable at energies smaller than is usually assumed \(^{2,34}\). Finally, it may be concluded that bremsstrahlung and other processes caused not by electromagnetic but by nuclear forces do not play a large role, since even the minimum possible electromagnetic bremsstrahlung of a particle with spin zero makes it possible to explain the observed ionization effects.

THEORY OF THE MESOTRON AND NUCLEAR FORCES

§ 2. NUCLEAR FORCES

Between nuclear particles (protons and neutrons) there act special nuclear forces; in the nucleus these forces not only compensate the electrostatic repulsion between protons, but also ensure the stability of the nucleus. The action of nuclear forces also explains the scattering of neutrons by protons and the difference between the observed scattering of protons by protons and that expected in the presence of only a Coulomb interaction. Nuclear forces are short-range forces, and the radius of their action is of the order of \(r_0 \sim 10^{-13}\) cm. At small distances (of the order of \(r_0\)) the interaction energy corresponding to nuclear forces is very large and reaches MeV. Furthermore, nuclear forces depend on the mutual orientation of the spins of the nuclear particles and possess the property of saturation. The latter means that the binding energy of a large number \(A\) of nuclear particles increases proportionally to \(A\), and not proportionally to \(A^2\), as occurs, for example, in the case of the Coulomb interaction of a system of charges. For this reason the volume of the nucleus is approximately proportional to \(A\), in contrast to the atom, whose dimensions depend only weakly on \(Z\).

The task of the theory of nuclear forces is, obviously, to explain the above qualitative properties of these forces and to establish a connection between the various nuclear quantities measured experimentally. For a quantitative test of the theory one may use data relating to the proton, neutron, and deuteron (the calculation of heavier nuclei, in view of its extreme complexity, is at present of no interest from this standpoint). From experiment there are known: the binding energy of the deuteron, equal to \(2.18\) MeV \(^{36}\), the quadrupole moment of the deuteron \(Q = +2.7 \cdot 10^{-27}\) cm\(^2\) (see, for example, \(^{37}\)) and the constants characterizing proton—neutron and proton—proton scattering (for the literature and discussion see \(^{38,39}\)). Understood in a broader sense, the theory of nuclear forces also encompasses questions relating to the individual proton and neutron and to their interaction with other particles. In this area the experimentally known quantities are the magnetic moments of the proton \(^{40}\) and neutron \(^{41}\), which are respectively equal to \(\mu_p = 2.789 \mu_0\) and \(\mu_N = -1.93 \mu_0\), where \(\mu_0 = \dfrac{e\hbar}{2Mc}\) is the nuclear magneton (\(M\) is the mass of the proton)*). Also known are the constants of \(\beta\)-decay of various nuclei, from which, under certain assumptions (see, for example, \(^{7}\)), one can approximately find the lifetime of the free neutron, which ultimately must transform into a proton + electron + neutrino. To this same circle of questions should be assigned the interaction of nuclear particles with mesotrons (scattering, production) and of mesotrons with light particles (mesotron decay).

* The negative sign of the magnetic moment of the neutron means that this magnetic moment is directed opposite to the spin, i.e. to the neutron’s own mechanical moment.

Since nuclear forces also act between uncharged neutrons, it is usually considered obvious that these forces are completely different from electromagnetic ones. Strictly speaking, such a point of view is incorrect, since, for example, it is conceivable to explain nuclear forces by specific features of the motion of particles with spin 1 in an electric field[^42]. However, the existence of non-electromagnetic interactions, evident from the fact of $\beta$-decay, as well as a number of other considerations, make one think that nuclear forces are not reducible to electromagnetic forces and are explained by the mesotron theory, the idea of which was indicated in the introduction.

Especially simple and visual is the classical form of the mesotron theory of nuclear forces, in which the concept of an unquantized mesotron field is used. At the same time, the detailed classical scheme has not only illustrative but also quite real significance, since in the static approximation, when the state of the nuclear particles is considered unchanged, the results of the classical and quantum theories coincide[^37,^43]. The situation here is the same as in electrodynamics, where the Coulomb interaction $-\dfrac{e^2 Z}{r}$ may be taken either from classical theory, as is usually done, or obtained as a result of considering photon exchange[^44]. The use of the static interaction is justified when the nonstatic interaction is neglected, which, generally speaking, is permissible in the theory of the deuteron (since the velocities of the proton and neutron in the deuteron are small in comparison with the speed of light). Of course, for a more complete and rigorous treatment of the problem of nuclear forces it is necessary to use the theory of the quantized mesotron field; the same applies to the calculation of the scattering of mesotrons by nuclear particles, etc.

Our aim in what follows will consist only in explaining the main points of the theory and discussing its results. Therefore we shall dwell in more detail only on the above-mentioned classical theory (for the quantization of the mesotron field as applied to the theory of nuclear forces, see [^9,^45,^46]).

In classical terms, the appearance of nuclear forces is connected with the fact that the proton and neutron are sources of a certain field (the mesotron field), which, acting on other nuclear particles, gives rise to a force interaction. If the field is scalar, then in the absence of sources it obeys equation (2). The presence of sources means that on the right-hand side of the equation there must appear a function playing the role of the charge density or current in electrodynamics. In the latter case, for a point particle the charge density is equal to $e\delta(r-r_0)$, where $\delta$ is the delta function $\left(\int \delta\,dr=1,\ \delta=0\ \text{for}\ r\ne r_0\right)$, and $r_0$ is the position of the charge.

In the static case of interest to us, equation (2) becomes $\Delta\varphi-\varkappa^2\varphi=0$, and the density of the “mesotron charge” is equal to $g\delta(r-r_0)$,

where \(\mathbf r_0\) is the position of the nuclear particle. Therefore the equation for the field takes the form:

\[ \Delta \varphi-\varkappa^{2}\varphi=-4\pi g\delta(\mathbf r-\mathbf r_0). \tag{19} \]

Since the position of the nuclear particle is regarded as fixed, it is clear that it is regarded as sufficiently heavy and is therefore described classically. Let us note that in the quantum theory, for the general case of a nonstatic scalar field, we have

\[ \square \psi-\varkappa^{2}\psi=-4\pi g\beta\delta(\mathbf r-\mathbf r_0), \tag{20} \]

where \(\psi\) must be regarded as an operator and \(\beta\) is a Dirac matrix. The appearance of \(\beta\) is connected with the fact that we consider the nuclear particles to obey the Dirac equation*).

The solution of equation (19) is as follows:

\[ \varphi=g\,\frac{e^{-\varkappa r}}{r}. \tag{21} \]

Using the expressions for the energy of the field, it can be shown\(^9\) that two nuclear particles producing the field \(\varphi\) and located at a distance \(r\) attract one another**), and the energy of their interaction is equal to

\[ U=-g^{2}\,\frac{e^{-\varkappa r}}{r}. \tag{22} \]

The radius of the forces, as is clear from (22), is of the order of \(\dfrac{1}{\varkappa}\). Since in quantum theory \(\varkappa=\dfrac{mc}{\hbar}\) (see § 1), we thus obtain relation (1) between the radius of the forces and the mass of the mesotron. Let us note that above we made no distinction between the proton and the neutron. This can be done only if the field \(\varphi\) is not charged and, consequently, the particles corresponding to it are also not charged (neutral mesotrons or neutrettos). At present this circumstance is not essential for us; it will be discussed further below.

The interaction (22) does not depend on the mutual orientation of the spins of the nuclear particles, which contradicts experiment. In order to explain the question of nuclear forces depending on spin, let us consider the interaction of protons and neutrons with a neutral vector field. The whole theory in this case is extremely closely related to ordinary electrodyna-

*) Let us note that in the right-hand side of equation (20) one more possible term has been omitted, containing derivatives of the \(\delta\)-function and proportional to the constant mentioned above, independent of \(g\).

**) The scalar field \(\varphi\) in the static approximation is analogous to the Newtonian gravitational field, the formal transition to which is achieved by taking \(\varkappa\) equal to zero. Hence it is clear that also in the scalar theory of nuclear forces the particles attract one another (see below the remark that the scalar field is assumed to be uncharged).

... and goes over into this latter one, if one sets \(x=0\). In order that the analogy with electrodynamics appear in a more familiar form, let us rewrite the equations of the vector field (9) in another form, introducing the notation:

\[ \begin{gathered} \varphi_k=A_k,\quad \varphi_4=A_4=i\varphi,\quad (A_1,A_2,A_3)=\mathbf A,\\ \mathcal E_{ik}=F_{ik},\quad (F_{23},F_{31},F_{12})=\mathbf H,\quad (F_{41},F_{42},F_{43})=i\mathbf E. \end{gathered} \tag{23} \]

In the notation (23), equations (9) take the form

\[ \operatorname{rot}\mathbf E=-\frac{1}{c}\frac{\partial \mathbf H}{\partial t},\quad \operatorname{div}\mathbf H=0, \]

\[ \operatorname{rot}\mathbf H+x^2\mathbf A=\frac{1}{c}\frac{\partial \mathbf E}{\partial t},\quad \operatorname{div}\mathbf E+x^2\varphi=0, \tag{24} \]

For \(x=0\), equations (24) go over into Maxwell’s equations for the vacuum. The same applies to equations (7)—(8), which in the new notation have the form

\[ \begin{gathered} \square \mathbf A-x^2\mathbf A=0,\\ \square \varphi-x^2\varphi=0. \end{gathered} \tag{25} \]

Let us now suppose that nuclear particles create a vector field, possessing a “mesotron charge” \(g\) and a mesotron “moment” \(\dfrac{f}{x}\). Then in the general case of quantum theory, instead of (25), one has the equations:

\[ \begin{aligned} \square \mathbf A-x^2\mathbf A &=-4\pi g\,\boldsymbol{\delta}(\mathbf r-\mathbf r_0) +\frac{4\pi f}{x}\left\{ \operatorname{rot}\bigl(\boldsymbol{\beta}\boldsymbol{\sigma}\delta(\mathbf r-\mathbf r_0)\bigr) -\frac{1}{c}\frac{\partial}{\partial t} \bigl(i\boldsymbol{\beta}\boldsymbol{\alpha}\delta(\mathbf r-\mathbf r_0)\bigr) \right\},\\ \square \varphi-x^2\varphi &=-4\pi g\,\delta(\mathbf r-\mathbf r_0) +\frac{4\pi f}{x}\operatorname{div} \bigl(i\boldsymbol{\beta}\boldsymbol{\alpha}\delta(\mathbf r-\mathbf r_0)\bigr), \end{aligned} \tag{26} \]

where \(\boldsymbol{\beta}, \boldsymbol{\alpha}\), and \(\boldsymbol{\sigma}\) are matrices of Dirac’s theory, and \((\varphi,\mathbf A)\) is the quantum field. In the static case of interest to us, \(\varphi\) and \(\mathbf A\) are classical quantities, and moreover:

\[ \begin{gathered} \Delta \mathbf A-x^2\mathbf A=\frac{4\pi f}{x}\operatorname{rot}\bigl(\boldsymbol{\sigma}\delta(\mathbf r-\mathbf r_0)\bigr),\\ \Delta\varphi-x^2\varphi=-4\pi g\delta(\mathbf r-\mathbf r_0). \end{gathered} \tag{27} \]

\[ \text{*} \]

*) The indicated closeness is connected with the fact that electrodynamics is also a theory of a vector field (the 4-dimensional vector is the field potential \(A_k\)).

In (27) both the fields \(\varphi\) and \(\mathbf A\), and the spin vector \(\boldsymbol\sigma\), may be interpreted classically. The solution of the system (27) is

\[ \left. \begin{aligned} \mathbf A&=-\frac{f}{\varkappa}\operatorname{rot}\left(\boldsymbol\sigma\,\frac{e^{-\varkappa r}}{r}\right),\\ \varphi&=\frac{g}{r}e^{-\varkappa r}. \end{aligned} \right\} \tag{28} \]

In electrodynamics the energy of a particle with charge \(e\) and magnetic moment \(\mu\), situated in the field \((\varphi,\mathbf A)\), is equal to \(e\varphi-(\boldsymbol\mu\mathbf H)\). The energy of interaction has the same form in the case of the vector mesotron field, where \(e\) corresponds to \(g\) and \(\mu\) corresponds to \(\dfrac{f}{\varkappa}\boldsymbol\sigma\). Therefore the interaction energy of two identical nuclear particles with spins \(\boldsymbol\sigma_1\) and \(\boldsymbol\sigma_2\), as follows from elementary calculations, proves to be equal to \({}^{37}\):

\[ \left. \begin{aligned} U&=g^2U_1+\frac{2}{3}f^2U_2+f^2U_3,\\ U_1&=\frac{e^{-\varkappa r}}{r},\qquad U_2=(\boldsymbol\sigma_1\boldsymbol\sigma_2)\frac{e^{-\varkappa r}}{r},\\ U_3&=\frac{1}{\varkappa^2}(\boldsymbol\sigma_2\nabla_2)(\boldsymbol\sigma_1\nabla_1)\frac{e^{-\varkappa r}}{r}=\\ &=\frac{1}{\varkappa^2}\left(\frac{1}{r^2}+\frac{\varkappa}{r}+\frac{\varkappa^2}{3}\right) \left(-3\frac{(\boldsymbol\sigma_1\mathbf r)(\boldsymbol\sigma_2\mathbf r)}{r^2}+(\boldsymbol\sigma_1\boldsymbol\sigma_2)\right) \frac{e^{-\varkappa r}}{r}. \end{aligned} \right\} \tag{29} \]

where \(\mathbf r\) is the radius vector of one of the particles relative to the other. The interaction with energy (29) evidently leads to forces depending on the mutual orientation of the spins, and also to noncentral forces depending on the orientation of the spins relative to \(\mathbf r\).

The vectors \(\dfrac{f}{\varkappa}\boldsymbol\sigma_1\) and \(\dfrac{f}{\varkappa}\boldsymbol\sigma_2\) are mesotronic “quasi-magnetic” moments of the nuclear particles; moreover, in the quantum theory the vectors \(\boldsymbol\sigma\) are operators—the well-known Pauli matrices \(\left(\dfrac{\hbar}{2}\boldsymbol\sigma\right.\) is the intrinsic angular momentum of the particle\(\left.\right)\).

Treating the vectors as operators changes nothing in the classical solution (29).

Above we considered the interaction of nuclear particles with scalar and vector fields. Two other cases, where the fields have pseudoscalar and pseudovector character (see § 1), may be considered in an analogous manner and lead to an interaction energy expressed by a linear combination of the terms \(U_1\), \(U_2\), and \(U_3\) (see (29)). Thus the general expression of the mesotron theory for the interaction energy has the form

\[ U=C_1U_1+C_2U_2+C_3U_3, \tag{30} \]

where the constants \(C_1\), \(C_2\), and \(C_3\) are arbitrary.

Until now we have regarded the mesotron field as uncharged; the difference of such a vector field from the electromagnetic field consists only in the fact that the rest mass of the “quantum of the mesotron field”—the mesotron—is equal to \(m=\frac{\hbar\varkappa}{c}\), whereas the rest mass of the photon is zero. We are considering a neutral field not only because of the simplicity of this case, but also for deeper reasons. If the field is charged (in this case, upon quantization, charged mesotrons correspond to it), then for the forces one likewise obtains an expression of type (30), but only in the case of the interaction of a proton with a neutron. In the case of identical nuclear particles (two protons or two neutrons), however, the interaction energy in the approximation considered is equal to zero. This result is quite understandable from the point of view of a quantum scheme operating with the concept of exchange of mesotrons between nuclear particles; since a proton is capable of emitting only a positive mesotron, which can be absorbed by a neutron, but cannot be absorbed by another proton, and so on. Therefore exchange of a single charged mesotron between identical nuclear particles cannot occur, but it can occur between different ones, which explains the indicated character of the interaction energy. Meanwhile, experimental data indicate that the proton—proton and proton—neutron forces are of the same order of magnitude[^38]. Within the framework of the scheme developed here, this fact can be explained only by assuming that there exists a neutral mesotron (neutretto). To avoid the assumption of the existence of the neutretto is in principle possible only in theories operating with exchange of pairs of particles or with excited charge states (see § 3). In general, the arguments in favor of the existence of the neutretto must be regarded as quite weighty. However, in experiment, and first of all in cosmic rays, no definite indications in favor of the presence of the neutretto have yet been obtained. If the neutretto really exists and plays an essential role in nuclear forces, then the mass of this particle must be of the order of the mass of the charged mesotron (this follows from (1)), and this particle must interact relatively strongly with the nucleus. It follows from this that in the earth’s atmosphere neutrettos must be formed in noticeable quantities, just as is the case for charged mesotrons. Further, the inverse process must also be noticeable—the capture of a neutretto by a nucleus, which will lead to nuclear disintegration. What has been said compels one to suppose that the nuclear disintegrations (“stars”) observed in cosmic rays may, to a considerable extent, be caused precisely by neutrettos. The existing experimental data do not contradict this supposition[^48].

Clarification of the question of the existence of the neutretto is extremely urgent; from this point of view, apparently, investigation of “stars” in cosmic rays is of primary interest[^48].

The explanation of nuclear forces by exchange of neutral mesotrons alone (the “neutral” theory) appears unsatisfactory,

because in doing so the connection is lost between nuclear forces and the behavior of charged mesotrons in cosmic rays, as well as the connection with β-decay*). Meanwhile, precisely this connection is one of the most attractive features of the mesotron theory of nuclear forces. Therefore, various variants of a mixed theory are usually considered, in which both charged and neutral mesotrons figure. A particularly popular theory of the mixed type is the so-called “symmetric” theory ⁴⁹, ⁹, ³⁷, in which the nuclear forces proton—proton and proton—neutron are exactly equal (in a state symmetric with respect to the “charge coordinate” ³⁹).

In the general theory, taking into account both charged and neutral mesotrons, the static interaction energy has the form (30), while the constants \(C_1, C_2, C_3\) also depend on the “charge state” of the nuclear particles, i.e. on whether they are in the “state” of a proton or in the “state” of a neutron.

The “exchange” character of the nuclear forces caused by charged mesotrons, which is connected with the continuous exchange of charge between nuclear particles (whence the term “exchange” force), also ensures saturation of the nuclear forces (see above and in more detail in ⁷).

The solution of problems of nuclear physics in the nonrelativistic approximation is reduced to integrating the Schrödinger equation for protons and neutrons with potential energy (30). The principal problem here, of course, is that of the deuteron and the consideration of proton—proton and proton—neutron scattering. However, the investigation of these questions encounters an essential difficulty at the very first steps. The point is that the nuclear energy has the form (17), i.e. is proportional to \(-\dfrac{1}{r^3}\), and in this case the Schrödinger equation has inadmissible solutions corresponding to the falling of particles onto one another; in other words, if the potential has a pole of order higher than \(\dfrac{1}{r^2}\), the problem of finding the complete system of stationary states has no solution. This result has, to a certain extent, a classical character, since in classical mechanics the potentials \(-\dfrac{1}{r^{2+\varepsilon}}\) \((\varepsilon \ge 0)\) also lead to the fall of the particle to the center (see, for example, ⁵⁰). The same conclusion is easily reached quantum-mechanically. A particle can fall to the center if its mean kinetic energy, as it approaches the center, grows faster than the mean potential energy decreases. Further, the mean kinetic energy of a particle located in a region of order \(r\) from the center is equal to \(T=\dfrac{p^2}{2m}\sim\dfrac{\mathrm{const}}{r^2}\), since,

* The interaction of nuclear particles with charged mesotrons also makes it possible, in principle, to indicate a path for explaining the anomalous magnetic moments of the proton and neutron (as we have seen above, these moments are not equal to the nuclear magneton for the proton and to zero for the neutron, as follows from Dirac’s theory) ⁴⁶.

by virtue of the uncertainty relation, \(p^2 \gtrsim \dfrac{\hbar^2}{r^2}\). Hence it is clear that if the mean potential energy as \(r \to 0\) decreases more slowly than \(-\dfrac{1}{r^2}\), then falling-in is impossible; if, however, \(U \sim -\dfrac{1}{r^{2+\varepsilon}}\) \((\varepsilon \geq 0)\), then the lower level will not exist, since upon diminution of the region in which the particle is located its energy tends to \(-\infty\). The same, of course, applies to the two-body problem, which, as is known, in relative coordinates reduces to the problem of the motion of a single particle in the field of a force center.

Thus, if in (30) \(C_3 \ne 0\), then the deuteron problem has no solution. Nor may one simply put \(C_3=0\) without more ado, since in all variants of the theory with one kind of meson the constant \(C_3\) is proportional to \(C_2\) \({}^{45}\). Therefore, setting \(C_3=0\), we leave in (30) only the term \(C_1 U_1\), which gives no spin dependence of the forces, in contradiction with experiment. It is possible to put \(C_3=0\), while at the same time preserving \(C_2 \ne 0\), only by assuming that there are at least two kinds of mesons. It is precisely this variant of the theory, in which vector and pseudoscalar mesons are introduced, that has acquired a certain currency \({}^{43,51}\). In this case a “symmetric” theory is used, and as a result mesons of four kinds altogether are introduced: neutral (vector and pseudoscalar) and charged (vector and pseudoscalar). The masses of the vector and pseudoscalar particles may be different \({}^{51}\). Quite apart from the fact that the introduction of mesons of several kinds gives rise to a feeling of dissatisfaction, the theory leads to difficulties that make its success (the exclusion of the term with \(U \sim -\dfrac{1}{r^3}\)) entirely illusory. First, a term of the type \(\dfrac{1}{r^3}\) is excluded only in the static approximation and appears, with corresponding complications, in a nonstatic treatment \({}^{52}\). Secondly, the theory leads to one result that directly contradicts experiment; namely, it follows from the theory \({}^{53,54}\) that the scattering of neutrons by protons should be stronger, for example, at an angle \(\pi/2\) than at an angle close to zero (in the coordinate system in which the proton was initially at rest). In experiments with neutrons of energy \(>10\) MeV, however, when the asymmetry effect becomes appreciable, the opposite dependence is observed \({}^{55}\).

Finally, thirdly, even if the indicated way of excluding the term with \(1/r^3\) were to lead to the goal in the theory of nuclear forces, it would not make it possible to remove another, no less important difficulty connected with the first. The point is that consideration of the scattering of mesotrons by a proton—neutron leads to the conclusion that if the heavy particle has a “quasi-magnetic” moment \(f/\varkappa\) (see above), then the effective cross section for scattering increases without bound with the energy \({}^{46,35}\), which is inadmissible*).

*) More precisely, an unbounded increase of the cross section with energy contradicts the general principles of the theory only under certain additional conditions \({}^{22}\), which, however, are fulfilled in the cases of interest to us.

This very substantial difficulty, which we shall discuss further in § 3, is not removed by introducing two kinds of mesotrons, since mesotrons of each kind can scatter independently of one another and, owing to the fact that \(f \ne 0\) (although \(C_3=0\)), this scattering will increase without bound with energy. Thus, the “mixed, symmetric” theory of Møller–Rosenfeld\(^{47}\), Schwinger\(^{51}\), and others is unsatisfactory for a whole series of reasons.

Another group of variants of the theory of nuclear forces is based on a “cutoff” of the inadmissible potential of the type \(1/r^3\). This means that the expression for the potential \(U_3 \sim -1/r^3\) is regarded as valid only up to some distance \(r_0\). For \(r<r_0\) this potential is “cut off,” i.e. is replaced by some other potential not containing the inadmissible singularity, for example by the potential \(U=\mathrm{const.}\) (for \(r<r_0\)). The operation of “cutting off” is of a formal character, is nonrelativistic, and can be justified only by the hope that a more complete and exact theory will automatically lead to some change of the potential (or even to a deeper change of the entire usual scheme for introducing nuclear forces) which is equivalent to a certain “cutoff” (see \(^{37}\) and § 3). The “cutoff” involves the introduction of a new constant \(r_0\), and more precisely even of a new function \(U(r)\) for \(r<r_0\). At first glance it may seem that, with an arbitrary choice of \(U(r)\), one can obtain any results. This, however, is not true, since the quantity \(r_0\) must not exceed the radius of the nuclear forces \(\hbar/mc\) (otherwise the use of the expressions of mesotron theory becomes meaningless, and also for other reasons\(^{17}\)); furthermore, under reasonable assumptions, the form of the function \(U(r)\) does not strongly affect the results\(^{37}\). Therefore, by introducing a “cutoff” and comparing the calculations with the experimental data on the deuteron, one can exclude certain theoretical possibilities. Thus, the “symmetric” theory with vector—charged and neutral—mesotrons\(^{37}\) proves unsatisfactory, since in order to obtain correctly the deuteron level and the cross section for neutron—proton scattering, one has to assume that \(r_0 \gtrsim \hbar/mc\), and, most importantly, the sign of the quadrupole moment of the deuteron proves to be wrong\(*\), while its magnitude is greater than that observed by about a factor of 10. On the contrary, in the case of the “neutral” vector theory the deuteron data can be made to agree with it\(^{37}\). However, as has already been pointed out, the use of neutral mesotrons alone appears unsatisfactory. It is, however, evidently quite possible to introduce, in this scheme, an additional relatively weak interaction of the proton—neutron with a charged mesotron. A similar variant of the “asymmetric” theory (vector neutretto + charged mesotrons), na—

* The quadrupole moment of the deuteron has a positive sign\(^{40}\), which corresponds to an elongated, cigar-like shape of the deuteron.

as far as we know, has not been calculated. An analogous, although in some respects simpler and more attractive, variant of the “asymmetric” theory has recently been considered by Holten[^54], but only very incompletely. In this scheme the neutral mesotron is a scalar, while the relatively weakly interacting charged mesotron is chosen to be pseudoscalar. The term of type \(l/r^3\) is present for the pseudoscalar mesotron, and thus “cutting off” is necessary.

Whereas in the “symmetric” theory the neutron—proton and proton—proton forces in the \({}^1S\) state are strictly equal, in the “asymmetric” theory this equality has only an approximate character, which does not contradict experiment (see[^54],[^38],[^39]). In addition, in the “symmetric” theory with charged mesotrons of one type, difficulties are encountered in comparing data on \(\beta\)-decay in the nucleus and on mesotron decay in cosmic rays[^51],[^53]. In the “asymmetric” theory this difficulty is alleviated[^54].

Further, the conclusion indicated above—that neutron scattering by protons must be weaker in the forward direction than at large angles—is very general and, apparently, inherent in any theory in which the main part of the nuclear forces has an “exchange” character, i.e. is due to the “exchange” of charged mesotrons[^53]. The point is that, in an exchange interaction, the proton and neutron in the act of scattering change places. More precisely, because of charge exchange, the particle that was formerly a proton becomes a neutron, and conversely. In a collision, it is in general most probable that the particles undergo only a small deflection and, thus, scattering through small angles predominates; the scattering particle, in the case of a fast incident particle, flies off mainly at an angle \(\pi/2\) to the latter. However, in exchange scattering the scattered and scattering particles are interchanged in the indicated sense, which explains the predominance of neutron scattering through \(\pi/2\): in this case, in essence, one observes the proton that was initially at rest, which has transferred its charge to the incident neutron. This general argument, as well as calculations[^53], shows that if the scattering experiments[^55] are correct, then the main nuclear interaction is not of an “exchange” character. The simplest theory of non-exchange nuclear forces is based on the introduction of the neutretto, which already in itself is a certain argument in favor of this very introduction and of the consideration of the “asymmetric” theory. One of the main tasks facing the “neutral” as well as the “asymmetric” theories consists in explaining the saturation of nuclear forces. To explain saturation in these cases is very difficult[^37], and in most cases, in particular in Holten’s[^54], saturation has no place.1 However, at the present time, even independently

apart from the question of saturation, it still cannot be said whether the “unsymmetrical” theory with a “cutoff” is capable of explaining all the available data. As we have seen, despite the introduction of a “cutoff,” it is by no means easy to satisfy all these data, which will lend attempts of this kind a certain interest.

At the same time, it should be borne in mind that in theories with a “cutoff” the difficulty connected with the unlimited growth of the cross sections for mesotron scattering is fully preserved, which would seem already by itself to make these theories unsatisfactory. However, in this question one may reason in the same way as in the case of the cutoff of the potential \(1/r^3\), namely, one may assume that a more complete theory will lead to a “cutoff” of the cross sections. Such a point of view may be regarded as admissible if the “cutoff” of the cross sections is to be made for wavelengths smaller than the cutoff radius for the potential \(r_0 \sim \hbar/mc\), i.e. for mesotron energies \(E=\hbar \nu > mc^2\).

In the case of the “charged” and “symmetrical” theories this question does not arise, and the cross section turns out to be larger than that observed already at \(E \sim mc^2\) ^{35, 58}. In the “unsymmetrical” theory of Holthen, in view of the relative weakness of the interaction with charged particles, the indicated difficulty apparently disappears (\(v^{54} f^2/\hbar c \sim 0.01\), whereas, for example, in the “symmetrical” theory \(f^2/\hbar c \sim 0.1\)). At the same time, of course, even a mutually consistent cutoff of the expressions for the potential and for scattering is only a minimal success and, in the main, merely shifts the center of gravity of the question into the region of justifying the “cutoff” operations. Within the framework of the general scheme of the theory of nuclear forces discussed above, there is also another tempting possibility ^{59}, based on consideration of non-static forces, i.e. on taking relativistic effects into account. The theory is “unsymmetrical,” and, just as in ^{53}, the neutral mesotron is regarded as scalar and the charged one as pseudoscalar. The essential difference, however, is that the interaction of the pseudoscalar mesotron with the proton–neutron is chosen so that in the nonrelativistic approximation it is absent, i.e. \(C_3\) in (30) is equal to zero, and the “difficulty \(1/r^3\)” disappears.

In the relativistic approximation, however, the charged mesotron gives rise to an interaction which proves to be very substantial. In qualitative respect this theory of Tamm ^{59} is in agreement with the principal experimental data and, at the same time, is a unified scheme of the type considered here, devoid of deep internal difficulties (“cutoff” of the potential and of the cross sections). It should not be forgotten, however, that the quantity and accuracy of the currently available data on the system of two nuclear particles are such that any theory of nuclear forces now faces a serious quantitative test. Therefore, until quantitative calculations have been carried out, which has not yet been done, a more detailed discussion of Tamm’s theory is premature.

In addition to the theories considered above, based on the idea of exchange by a single mesotron with integer spin, attempts were made to construct “pair” theories. In the latter the proton and neutron exchange at once a pair of particles of different sign, with spin \(1/2\) and mass of order \(200m_0\) \(^{60,61}\). Such theories, which also lead to difficulties, would acquire, in our opinion, interest only in the event that the spin of the mesotron in cosmic rays were equal to \(1/2\). Meanwhile, at present it is more probable that the spin of the mesotron is integral and that in its decay an electron and a neutrino are emitted. A final experimental clarification of this question appears to be very important.

There also exist “pair” theories operating with the exchange of a pair of particles with integral spin (see, for example, \(^{62}\)). No interesting results have been obtained along this path.

Theories of nuclear forces connected with the introduction of excited spin and charge states of nuclear particles will be discussed in § 3.

§ 3. ON THE DIFFICULTIES OF THE THEORY

As we have seen in §§ 1 and 2, the theory of the mesotron and of nuclear forces encounters substantial difficulties, manifested in the appearance of a potential of the type \(1/r^3\) and in the unbounded growth of cross sections for the scattering of light by the mesotron and for the scattering of mesotrons by a proton—neutron. We shall conventionally call all these difficulties “difficulties of mesotron theory,” or “difficulties of the second kind.” In the Dirac theory of the electron, or in the theory of a particle with spin zero and a scalar wave function, analogous difficulties do not arise. At the same time, as is well known, the relativistic quantum theory of the electron and of all other particles also encounters fundamental difficulties, which we shall call “difficulties of the first kind,” and which are connected with the infinity of the self-energy of elementary particles in the existing quantum theory of any field. The “difficulties of the first kind,” or, in other words, the absence of a theory of elementary particles, lead to the fact that, in a strict formulation, the description of the motion of the electron and of other particles is at present impossible. There is no space here to dwell in detail on the discussion of the “difficulties of the first kind,” and we shall confine ourselves to referring to the literature in which this discussion is given \(^{9,10,29,63,64,65}\). It is extremely important, however, to emphasize the well-known fact that the presence of “difficulties of the first kind” does not yet make a theory sterile. Indeed, the problem of the motion of the Dirac electron in a Coulomb field has a solution which, moreover, agrees with experiment; in exactly the same way, the calculation of effective cross sections for various radiative processes involving the electron, carried out by the method of perturbation theory, leads to results that are not only internally consistent but also agree with experiment.

In the presence of “difficulties of the second kind,” on the contrary (and it is precisely in this that the difficulties consist), already the first non-vanishing approximation of the theory

of perturbations leads to incorrect results (an unbounded growth of the cross sections) and, moreover, either even the problem of the motion of a particle in a Coulomb field has no solution (see § 1), or else a “forbidden” potential of the type \(1/r^3\) appears.

Analysis shows that the appearance of “difficulties of the second kind” is connected either with the presence in the particles of a magnetic (“quasimagnetic”) moment, or with the fact that the scattered particles are charged\(^{66}\). In § 1 we saw that the cross section for the scattering of light by a particle with spin \(1/2\) increases without bound if this particle has a “true” magnetic moment, i.e. \(\mu_1 \ne 0\) (see (14)). The growth of the cross section for the scattering of light by a particle with spin one is likewise connected with the presence in it of a magnetic moment in the relativistic approximation\(^{14,73}\). Further, a growth of the cross section for the scattering of mesotrons by a proton—neutron takes place if the heavy particle possesses a “quasimagnetic” moment described by a term quite analogous to the term with \(\mu_1\) in (14). The falling of particles with spin 1 and with spin \(1/2\) and with \(\gamma \ne 1\) onto a Coulomb center is also caused by the presence in these particles of a “true” magnetic moment*), as a result of which the effective potential proves to have the form \(-1/r^3\). Finally, the appearance of the potential \(-1/r^3\) in the theory of nuclear forces is connected with the “quasimagnetic” moment; this is clear already from the fact that the energy of interaction of two magnetic moments \(\mu_1\) and \(\mu_2\) in magnetostatics is equal to

\[ U=-\frac{3(\mu_1 r)(\mu_2 r)}{r^5}+\frac{(\mu_1\mu_2)}{r^3}, \tag{31} \]

where \(r\) is the radius vector of one of the particles relative to the other. The “forbidden” potential \(U_3\) in (29) passes into (31) if one sets \(\chi=0\), i.e. \(m=0\), which precisely corresponds to the transition to electrodynamics.

As has already been mentioned in passing, “difficulties of the second kind” also appear in the scattering of charged mesotrons, for example vector ones, not by a moment but by the “quasielectric” charge of the heavy particle. In this case the unbounded growth of the cross sections is caused by a decrease in the number of intermediate states in the scattering. The latter is connected with the fact that the proton can emit only positive mesotrons, and the neutron only negative mesotrons\(^{68,69,9}\).

It is easy to see that, at least in their basis, the “difficulties of the second kind” connected with the presence of a magnetic (or “quasimagnetic”) moment have a classical nature\(^{66,14,67,3}\). We shall dwell on this question in somewhat more detail, beginning with the case of the scattering of light by a magnetic moment.

*) We speak of a “true” moment as distinct from the magnetic moment of the Dirac electron, which in the extreme relativistic approximation does not manifest itself\(^{13,14,15,73}\).

The classical nonrelativistic equation of motion for the moment has, as is known, the form:

\[ \frac{d\mathbf S}{dt}=\dot{\mathbf S}=[\boldsymbol{\mu}\mathbf H]=\delta[\mathbf S\mathbf H], \tag{32} \]

where \(\mathbf S\) is the angular momentum of the particle and \(\boldsymbol{\mu}\) is its magnetic moment, which, as usual, is taken equal to \(\delta \mathbf S\), where \(\delta\) is a constant.

Considering the scattering of light and putting the magnetic field \(\mathbf H\) equal to \(\mathbf H_0 e^{i\nu t}\), we readily find\(^3\) that the effective cross section for this process is equal to

\[ \sigma=\frac{8\pi\delta^4}{3c^4}\frac{([\mathbf S\mathbf H_0])^2}{H_0^2}\,\nu^2=\mathrm{const.}\,\nu^2, \tag{33} \]

i.e. it increases without bound with the frequency as \(\nu^2\), exactly as in the quantum calculation carried out by the method of perturbation theory. The reason for this situation is quite clear. The classical calculation indicated above is carried out by taking the field \(\mathbf H\) in (32) to be equal to the external field of the incident wave, which fully corresponds to the quantum-mechanical calculation in the first nonvanishing approximation of perturbation theory. Meanwhile, in the sense of (32), by the field \(\mathbf H\) one must understand the total field, equal to the sum of the external field and the proper field of the magnetic moment. Taking the proper field into account shows\(^ {23}\) that if in (32) \(\mathbf H\) is understood as the external field \(\mathbf H_{\mathrm{ext}}\), then this very equation must be written in the form:

\[ \dot{\mathbf S}=\delta[\mathbf S\mathbf H_{\mathrm{ext}}]-\frac{\delta^2}{c^2 r_0}[\mathbf S\ddot{\mathbf S}]+\frac{2\delta^2}{3c^3}[\mathbf S\dddot{\mathbf S}], \tag{34} \]

where \(r_0\) is the effective radius of the particle possessing the moment \(\delta\mathbf S\) (in the classical electron theory, as is known, a point particle cannot be considered, since for a point particle the second term on the right-hand side of formula (34) becomes infinite, similarly to what occurs in connection with the electromagnetic mass of a point charge).

The last term in (34), which represents the analogue of the well-known radiation-friction force, violates the conservative character of the equation, and we shall not consider it. Equation (34), even without the last term, leads to a cross section which at low frequencies has the form (33), while at high frequencies is constant:

\[ \sigma=\frac{8\pi}{3}r_0^2\frac{([\mathbf S\mathbf H_0])^2}{S^2H_0^2}=\mathrm{const.} \tag{35} \]

Moreover, if, in the spirit of the correspondence principle, one puts \(S\sim \hbar\), \(\delta\sim \frac{e}{mc}\), and \(r_0\sim \frac{e^2}{mc^2}\), then the condition of smallness of the frequency means that

\[ \hbar\nu\ll mc^2\quad\left(\lambda=\frac{2\pi c}{\nu}\gg \frac{\hbar}{mc}\right). \tag{36} \]

The frequency may be regarded as large when the opposite inequality is satisfied \((\hbar \nu \gg mc^2)\). Thus, taking into account the self-field of the magnetic moment leads, in the classical theory, to the removal of the “difficulty of the second kind” associated with the scattering of light by this moment.

The interaction energy of two magnetic moments has the form (31), i.e., is of the type \(1/r^3\). From what was said in § 2 it is clear that in the classical theory as well the motion of a pair of magnetic moments will be limitational, i.e., they will fall onto one another, unless some energy is taken into account in addition to the potential energy and the kinetic energy of the orbital motion. If the action of the self-field is neglected, there is in fact no other energy depending on \(r\). However, taking account of the self-field by using equation (34), without the last term, leads to the fact that, as the moments approach one another and their precession becomes ever faster, the energy associated with this precession increases and has the form2:

\[ T_{\mathrm{pr}}=\frac{c^2 r_0}{2\delta^2 S^2}\,\mathbf K^2, \qquad \mathbf K=\mathbf S+\frac{\delta^2}{c^2 r_0}\,[\mathbf S\dot{\mathbf S}]. \tag{37} \]

It is not difficult to verify that, as \(r\to 0\), \(T_{\mathrm{pr}}\) grows as \(1/r^6\), and thus the general argument of § 2, which indicates the inevitability of falling for \(U\sim -1/r^3\), no longer applies.2

If one sets \(r_0\sim e^2/mc^2\), \(S\sim \hbar\), and \(\delta\sim e/mc\), then \(|U|\sim T_{\mathrm{pr}}\) and

\[ \frac{\partial(U+T_{\mathrm{pr}})}{\partial r}=0 \]

for

\[ r\sim \sqrt[3]{\frac{e^2}{\hbar c}}\,\frac{\hbar}{mc}. \tag{38} \]

From what has been said it is clear that, in the case of a magnetic (and “quasimagnetic”) moment, the “difficulties of the second kind” have a classical nature and are connected with neglect of the self-field.

There arises, however, the question why neglect of the self-field in the case of a particle with charge (but without moment) does not lead to difficulties of the second kind. In order to answer this question, it is enough to recall the well-known situation with the inclusion of the self-field in classical electron theory. In the equation of motion \(m\ddot{\mathbf r}=e\mathbf E\), the field \(\mathbf E\) must be understood as the sum of the external field \(\mathbf E_{\mathrm{ext}}\) and the self-field

particles; moreover, exclusion of the self-field leads to the equation

\[ m\ddot{\mathbf r}=e\mathbf E_{\mathrm{ext}}-m_{\mathrm{el}}\ddot{\mathbf r}+\frac{2e^2}{3c^3}\dddot{\mathbf r}, \tag{39} \]

where the electromagnetic mass \(m_{\mathrm{el}}\sim \dfrac{e^2}{r_0c^2}\) (\(r_0\) is the radius of the particle). Taking the self-field into account, in addition to the dissipative term \(\dfrac{2e^2}{3c^3}\dddot{\mathbf r}\), leads to the appearance of the electromagnetic mass or, in other words, of a term \(-m_{\mathrm{el}}\ddot{\mathbf r}\) of the same form as the inertial term \(m\ddot{\mathbf r}\) assumed in advance. Moreover, if \(m\) in (39) is taken to be the total mass of the particle measured experimentally, then taking the electromagnetic mass into account is not only unnecessary but simply inadmissible. The difference in the case of the magnetic moment consists in the fact that here the conservative term, allowing for the influence of the self-field and proportional to \([\mathbf S\ddot{\mathbf S}]\), has a form quite different from the inertial term \(\dot{\mathbf S}\) introduced at the outset (for more detail see \(^{3}\)).

An analogous situation occurs not only in nonrelativistic but also in relativistic quantum theory. For example, in the Dirac equation the term with the mass appears from the very beginning (the term \(x\psi\) in (14)). Therefore the use of perturbation theory, equivalent to neglecting the self-field, leads to correct results. On the contrary, perturbation theory leads to the appearance of “difficulties of the second kind” in those cases in which, in the initial equations, i.e. so to speak already in the zeroth approximation, the conservative part of the reaction of the self-field has not been taken into account.

In the classical theory, if we do not wish to take into account the self-field of the magnetic moment, then instead of the equation

\[ \dot{\mathbf S}=\delta[\mathbf S\mathbf H_{\mathrm{ext}}] \tag{40} \]

one must use the equation

\[ \dot{\mathbf S}=\delta[\mathbf S\mathbf H_{\mathrm{ext}}]-\frac{\delta^2}{c^2r_0}[\mathbf S\ddot{\mathbf S}]. \tag{41} \]

In nonrelativistic quantum theory one may proceed in exactly the same way, treating the vector \(\mathbf S\) as an operator. From equation (40), both in the quantum and in the classical cases, it follows that the particle’s intrinsic angular momentum (i.e. its spin) \(\mathbf S\), both in a field and in the absence of a field, remains unchanged in magnitude, i.e. \(S^2=\mathrm{const}\).

Thus, in the case of equation (40) and the corresponding Pauli equation (16), the spin cannot change, and therefore it is permissible to consider a particle with one definite value of the spin. On the contrary, in the case of equation (41) the angular momentum of the particle

is equal to \(\mathbf K\) (see (37)), and which is the integral of motion of the energy, has the form:

\[ E=\frac{c^2 r_0^2}{2\delta^2 S^2}\mathbf K^2-\delta(\mathbf{SH}). \tag{42} \]

In this case, in the field the intrinsic moment (spin) is not conserved. This means that in the quantum theory one cannot restrict oneself to considering a single value of the spin, for example, equal to \(\hbar/2\). Instead it is necessary to allow that the particle may be in states with other values of the spin, equal to \({}^{3}/_{2}\hbar\), \({}^{5}/_{2}\hbar\), \({}^{7}/_{2}\hbar\), etc. In the case of integral spin it must be assumed that states with spin \(0, 1, 2\), etc. are possible. To higher values of the spin there corresponds a larger intrinsic energy. Thus, taking the intrinsic field into account, we arrive at the idea of the existence of excited spin states of elementary particles.

The calculation of light scattering with allowance for excited states leads \(^{69, 67}\) to the cessation of the growth of the cross section for scattering. The question of the “\(1/r^3\) difficulty” in the quantum theory with excited states has not been considered correctly. Considerations based on correspondence with the classical theory (see above) compel one to think that the introduction of excited spin states will remove this difficulty as well.

As it turns out \(^{69, 68}\), the elimination of the unbounded growth of the cross section for scattering of vector charged mesotrons by the “quasi-electric” charge can be achieved by introducing excited charge states of the proton and neutron, i.e. by the assumption that the charge of these particles may also be equal to \(+2e, +3e\ldots\) and \(-e, -2e\ldots\), etc. The intrinsic energy of the proton—neutron in states with charge not equal to \(+e\) or \(0\) is greater than in these normal states, which also explains the insignificance of the new states under ordinary conditions.

The introduction of excited charge states may be regarded as the result of taking into account the reaction of the intrinsic charged field of a heavy particle on its motion. Further, the introduction of excited states may be justified not only in the manner set forth above \(^{3}\), but also as the result of a detailed quantum consideration of the intrinsic field of particles. This consideration is possible, however, only under a certain extreme assumption about the energy of interaction of the particle with the field. Namely, this energy must be large in a certain sense, so that, for example, in the case of “quasi-charge” interaction the inequality \(g^2/\hbar c \gg 1\) would be satisfied, where \(g\) is the “quasi-electric” charge of the particle (if \(g=e\), where \(e\) is the elementary electric charge, then \(e^2/\hbar c = 1/137\), and thus in electrodynamics the indicated inequality is, of course, not satisfied). Such a theory, the so-called theory of “strong coupling,” has recently been subjected to detailed development \(^{69-72,\ 79-88}\).

The existence of excited charge states to a certain extent erases the distinction between the proton and the neutron, since, for example, both these particles can emit a positive mesotron, while the neutron will thereby pass into a state with charge \(-e\), which is excited. As a result, taking account of excited states, it is at least possible in principle to explain the closeness of the proton—proton and proton—neutron forces \(^{85}\), as well as the preferential scattering of neutrons through small angles \(^{80}\) without introducing neutral mesotrons. At the same time, as has already been said, the difficulties associated with scattering disappear. The question of the “\(1/r^{3}\) difficulty” remains unclear, but it seems to us that there is serious hope for its elimination.

The real touchstone for theories introducing the idea of excited proton—neutron states will, of course, be an experimental clarification of the question of the very existence of these states. The corresponding excitation energy should be of the order of 10–30 MeV and thus lies in a region already accessible at the present time. However, no special experiments in this direction have been performed, and experimentally the question remains completely open (for possible experiments see \(^{87,88}\)).

At the same time one must not forget that the path indicated above for eliminating “difficulties of the second kind,” by introducing excited states, as well as the theory of “strong coupling” that leads to the same result, have a nonrelativistic character (with respect to the heavy particles). This is connected with the fact that the particle is regarded as extended, having a radius \(r_{0}\) (see, for example, (41)). For \(r_{0} \to 0\) divergent expressions are obtained, which in this case is a manifestation of the fundamental “difficulties of the first kind.” A nonrelativistic theory, however, in its very foundation can have only a limited, chiefly heuristic, value, approximately the same as that of a “successful” theory of nuclear forces with a “cutoff” (see § 2). A genuine theory of nuclear forces must undoubtedly be relativistic or, more precisely, must admit a relativistic formulation (with a subsequent transition to the nonrelativistic approximation for solving nonrelativistic problems). Only in this case can one remove the almost complete indefiniteness and arbitrariness in the choice of expressions for the energy of interaction of a heavy particle with a field and obtain some guarantee of the reliability of all constructions.

The existence of “difficulties of the first kind” does not at present permit a relativistic treatment of the proper field of elementary particles. Therefore the possibilities of the theory are now limited to the consideration of radiation processes by the methods of perturbation theory and to the solution of mechanical problems without taking radiation into account; the “internal” properties of a particle, such as, for example, its mass, are taken into account in the equations of motion by introducing constants that are arbitrary from the theoretical point of view. In view of what has been said, until the fundamental problems of the theory of elementary particles are solved, the only conceivable path for a relativistic consideration of excited states consists in constructing a theory,

not trying to take detailed account of the particles’ own field, but introducing new degrees of freedom and new constants^3,74. In the case, for example, of a particle with an “intrinsic” moment, these degrees of freedom and constants correspond to coordinates determining the position of the moment, and to moments of inertia^74. The construction of such a relativistic scheme encounters a number of difficulties and unexplored questions and has not yet been sufficiently advanced^3,74. Further work in this direction seems to us especially urgent.

Summarizing everything set forth in this review, we see that the theory of the mesotron and nuclear forces has not yet solved the basic problems before it and is in a stage of energetic search for ways of eliminating the various difficulties it encounters. The attempts made recently (September 1946) to overcome these difficulties may be grouped, with some reservation, around three directions.

1. There is still hope^59 for the success of an “asymmetric,” relativistic theory not connected with the necessity of a “cutoff,” which was discussed in § 2. The question of the possibility of such a path will be resolved quickly as a result of comparing quantitative calculations with experimental data.

2. Consideration of various theories with a “cutoff” continues. Of special interest here would be attempts somehow to justify this “cutoff” of cross sections and of the potential and to introduce it into a consistent relativistic framework; the latter is also connected with efforts aimed at at least partially eliminating the “difficulties of the first kind”^64,78. The calculations of radiation processes with damping taken into account^75,77 that belong here do not seem to us consistent^3 and, in any case, are not convincing (see also^89). Their substantial defect is also the actual absence of a connection with the theory of nuclear forces and with the elimination of its difficulties.

3. The considerations set forth above make, it seems to us, the notions of excited spin and charge states very attractive. The development of the corresponding nonrelativistic theory, both in the usual form^3,67,68 and in the approximation of “strong coupling”^69–72,79–88, must necessarily be based on a relativistic consideration of the question (see above and^3,74). At the same time there is no guarantee that the solution of the problem will be found in at least one of the directions listed. Moreover, the opinion is very widespread that genuine success of the theory of the mesotron and nuclear forces will be achieved only as a result of a fundamental revision and development of the existing quantum theory. Which point of view is correct here will, of course, become clear only in the course of further work.

In conclusion it should be emphasized that, for the development of any variants of the theory, it is extremely important and essential to supplement and refine the experimental data, and first of all to determine definitively the spin of the mesotron in cosmic rays and to clarify the question of the existence of a neutral mesotron and of excited spin and charge states.

References

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4. M. Fierz and Hely, Phys. Acta 12, 3 (1939).
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1. I owe the indication of this circumstance to Prof. I. E. Tamm, whom I take this opportunity to thank for reading the manuscript and for his comments. ↩

2. If the magnetic moments are parallel to one another and to the line joining them, then \(U\) is still \(\sim -1/r^3\), while precession of the moments is absent and falling must take place. However, the existence of such an exceptional position is not of special significance, since in classical theory, for example, even in a central Coulomb field, the charge falls to the center if the orbital angular momentum of this charge is zero. ↩↩