Nonlocal and Nonlinear Field Theories

D. I. Blokhintsev

Contents

I. Introduction . . . 137
II. Nonlocal field theory . . . 138
III. Nonlinear field theory . . . 145
IV. Physics of the strong interaction . . . 150

I. Introduction

After the considerable successes of the theory of renormalization, a certain disappointment set in. It has now become clear that this method does not overcome the basic difficulties of modern theory, but rather represents a relatively successful way of circumventing these difficulties, applicable in those cases where especially small scales of space and time do not play an essential role. The most far-sighted physicists never assessed the significance of this method otherwise. Nevertheless, great efforts were usefully expended in order to refine this method to the highest degree.

In the atmosphere of enthusiasm for “renormalization,” a somewhat special place is occupied by the works of those physicists who sought to modify modern theory more profoundly, proceeding from one or another physical idea.

Among attempts of this kind, which already have a considerable history, an especially large place is occupied by nonlocal field theories and nonlinear field theories, which are definite and, in some respects, related attempts to generalize modern quantum field theory.

In the difficult situation in which theory now finds itself, it is timely to examine both these directions critically. In this way, perhaps, the path to truth will become somewhat clearer.

Both in nonlocal theory and in nonlinear theory, a certain elementary length $s_0$ is introduced. Apparently, G. Wataghin[^1] was the first physicist to point out such a possibility for generalizing the theory. However, we shall see from what follows that in reality the root of the difficulties of modern theory goes deeper, and comparatively simple modifications of the theory operating with the concept of an elementary length, in all likelihood, cannot be the basis of a new theory.

In the first chapter of this article we shall consider nonlocal theories, in the second—nonlinear ones. Finally, in the third chapter the limits of applicability of the basic concept of modern theory—the concept of a particle—will be discussed.

II. NONLOCAL FIELD THEORY

The first formulation of a nonlocal field theory was given by M. A. Markov,^2 who proceeded from the physical idea that, at small distances, because of the atomism of charge, the field strength cannot be a measurable quantity.

In accordance with this idea Markov assumed that the potentials of the electromagnetic field \(A_\mu\) do not commute with the coordinates of a test charge \(x_\nu\). If by \(A_\mu(\mathbf{k})\) one denotes the Fourier component of the potential of wave vector \(\mathbf{k}\), then it was assumed that

\[ [x_\nu,A_\mu]= i r_\nu A_\mu , \tag{1} \]

where \(r_\nu\) is a certain four-vector proportional to the elementary length \(s_0\).

It follows from this relation that for \(\Delta x_\nu \simeq r_\nu\), \(\Delta A_\mu \simeq A_\mu\).

This theory leads to the idea of four-dimensionally extended particles, which is expressed formally in the appearance, in the interaction of the electron and the electromagnetic field, of a relativistically invariant cutoff factor.

Later, Yukawa^3 proposed a nonlocal theory, formulating it in such a way that the field potentials \(A\) are nondiagonal matrices in space-time, so that

\[ (x'|A|x'') \ne A(x')\delta(x'-x''). \]

Specifically, it was assumed (for a scalar field \(U\)) that

\[ [x_\mu,[x^\mu,U]]=s_0^2 U . \tag{2} \]

Although an interpretation of this theory was attempted which, as it seems to us, somewhat shifted to the side (through internal degrees of freedom of particles), and although mathematically it was formulated differently from Markov’s theory, nevertheless it rests on the same physical idea and likewise leads to relativistic cutoff form factors.

Another approach to nonlocal field theory was developed in the works of the author,^4 Max Manes,^5 Peierls^6 and others. In this approach the free field is regarded as local, and nonlocality is introduced only in the interaction. The physical idea underlying this direction is the supposition that in small space-time regions other kinds of causal connection are possible than those characteristic of large scales of space and time. In this theory, in its original version, it is assumed that the interaction, for example, of the electron field, described by the current \(J_\mu(x)\), and the electromagnetic field, described by the potential \(A_\mu(x')\), occurs not at one and the same point of space-time, but

“smeared out” by means of a relativistically invariant form factor \(F(x' - x'')\), so that the interaction is described by the function \(W\):

\[ W=\int J_\mu(x)F(x-x')A_\mu(x')\,dx\,dx' \tag{3} \]

instead of the usual form of interaction, where \(F(x-x')=\delta(x-x')\) (“point” interaction). Expression (3) clearly indicates that the interaction can propagate from the point \(P(x)\) to the point \(P(x')\) with any velocity, since the form factor \(F(x-x')\), from considerations of Lorentz invariance, must be a function of the interval \(s^2=(t-t')^2-(x-x')^2\). In order that there be no substantial violation of ordinary causality for large intervals of space-time, it is necessary that the form factor \(F(x-x')\) vanish as the modulus of the interval \(|s^2|\) increases.

For example, one may take \(F\sim e^{-s^2/s_0^2}\), where \(s_0\) is some elementary length. In this case signals propagating with a velocity greater than the velocity of light over large distances will be extremely weak.

Indeed, for the signal to have appreciable strength, it is necessary that \(|s^2|=|t^2-x^2|\leq s_0^2\), but then for macroscopic distances \(|x|\gg s_0\) or times \(|t|\gg s_0\) the signal velocity \(V=\left|\dfrac{x}{t}\right|\) will be close to 1.

This assertion is valid in any coordinate system[^7].

The Markov–Yukawa theory gives nothing new for free fields. The difference from the usual theory is found only in the interaction, which differs from the usual one by the presence of relativistically invariant form factors.

In the theory of nonlocal interaction just described, these form factors are introduced in the simplest and most direct manner. Therefore the Markov–Yukawa theory and the theory of nonlocal interaction, different in their initial physical assumptions, turn out in practice to be equivalent and differ only in the manner of introducing the form factor. In both concepts the form of the form factor remains undetermined. This difficulty, however, would not be an obstacle to the development of the theory, since the form of the form factor could in principle be determined from experiment. Much more substantial difficulties are revealed in the very fundamental scheme of the theory.

M. A. Markov drew attention to the fact that, in the presence of the form factor \(F\), the equations for particles in the many-time formalism of Dirac–Rosen–Podolsky become incompatible.

Indeed, they showed[^8] that Bloch’s condition for the compatibility of these equations,

\[ [H(x_n,t_n),H(x_m,t_m)]=0 \tag{4} \]

for \(|x_n-x_m|^2>|t_n-t_m|^2\), is not satisfied in the case of a nonlocal theory. In this formula \(H(x_n,t_n)\) denotes the Hamiltonian of the \(n\)-th particle, \(x_n,t_n\) are its four-dimensional coordinates, and \(H(x_m,t_m)\) has the same meaning for the \(m\)-th particle. The essence of Bloch’s condition is that measurements performed on two particles \(n\) and \(m\), connected by a spacelike interval, must not influence one another.

In a nonlocal theory, the interaction can also propagate with a velocity greater than the velocity of light in vacuum; therefore it is natural that the Poisson bracket in (4) is in this case nonzero also for space-like intervals. This is a very general situation. The author has noted\(^9\) that a nonlocal theory is in general incompatible with the Hamiltonian method. The Hamiltonian method assumes that each subsequent state is uniquely determined by the preceding state. It is precisely this fact that is expressed in the Schrödinger equation for the wave function of the system \(\Psi\):

\[ i\hbar\,\delta \Psi = H\Psi\,\delta t . \tag{5} \]

Fig. 1. Propagation of signals in local and nonlocal theories: \(\sigma(x)\)—a space-like surface, \(P\)—one of the points on it, \(QPQ'\)—the cone of the past at this point. The double-hatched region is the past. The singly hatched region is the future.

Such an equation, however, cannot exist in a nonlocal theory, where the interaction propagates with an arbitrarily large velocity and therefore the state on some space-like surface \(\sigma(x)\) (Fig. 1) cannot be determined only from the states lying in the past cones (the double-hatched region), but also depends on the future (the hatched region).

W. Pauli showed\(^ {10}\) that, nevertheless, for a nonlocal theory one can find integrals of motion, including an analogue of the Hamiltonian. This also makes it possible to introduce canonical variables. However, one cannot avoid solving equations that contain integrals over time from \(-\infty\) to \(+\infty\), and therefore the theory in Hamiltonian form still turns out to be impossible\(^ {11}\).

One might think, however, that this situation still does not imply a complete incompatibility of a nonlocal theory with quantum mechanics. Indeed, one may assume that the asymptotic quantum theory remains valid. This means that we can assume that, for nonlocal systems, there nevertheless exists at \(t=-\infty\) a wave function \(\Psi(-\infty)\), and at \(t=+\infty\) a wave function \(\Psi(+\infty)\).

At the same time there also exists a scattering matrix \(S(-\infty,+\infty)\), transforming \(\Psi(-\infty)\) into \(\Psi(+\infty)\).

From the time when W. Heisenberg\(^ {12}\) drew attention to the possible significance of this matrix, all variants of nonlocal theory were in one way or another connected with attempts to determine the scattering matrix for a nonlocal theory\(^*)\).

In doing so, one could be guided by the following requirements:

a) the scattering matrix \(S\) must be invariant with respect to Lorentz transformations;

\(^*)\) It should be noted that already in the first paper of M. A. Markov\(^2\), in connection with the proof of the inconsistency of the equations of the many-time formalism, a certain formal scheme of calculation was proposed that was not connected with the Schrödinger equation in its usual understanding.

b) it must satisfy the conditions of causality in macroscopic regions of space-time (i.e., asymptotically, for large intervals of time or space, “acausal” interactions must give a vanishing contribution to transition probabilities);

c) it must be unitary*).

Requirement a) is the requirement, at least, of formal agreement of the nonlocal theory with the theory of relativity; requirement b) means agreement with the ordinary theory, and, finally, requirement c) follows from the assumption of the existence of \(\Psi(-\infty)\) and \(\Psi(+\infty)\) (the total probability of a state must be normalized in the same way both for \(t=-\infty\) and for \(t=+\infty\)).

In constructing a nonlocal theory it was borne in mind that it would be able to eliminate the divergences characteristic of the ordinary theory. However, the development of investigations in the field of nonlocal theory did not lead to successful results. First, it was shown that a nonlocal theory with a two-point form factor \(F(x' - x'')\) is not capable of eliminating the infinities associated with vacuum polarization.

This is evident from the fact that a theory with a form factor of the type \(F(x' - x'')\) is equivalent to replacing the field \(A_\mu(x')\) by the field

\[ B_\mu(x)=\int F(x-x')A_\mu(x')\,dx'. \tag{6} \]

Therefore, for an external field \(A_\mu^0\), the effect of vacuum polarization will be exactly the same as in the ordinary local theory for the field \(B_\mu^0\), i.e., it will be divergent.

In this connection a number of authors\(^{14,15,16,17}\) considered a variant of the nonlocal theory in which the form factor depends not on two, but on three points \(x'\), \(x''\), \(x'''\). Such a variant is possible only in quantum theory and means, for example, that instead of the interaction (3) one assumes an interaction of the form:

\[ W=ec\int \overline{\Psi}(x')\gamma_\mu\Psi(x'')F(x',x'',x''')A_\mu(x''')\,dx'\,dx''\,dx''', \tag{7} \]

i.e., the current \(J_\mu=ec\overline{\Psi}(x')\gamma_\mu\Psi(x')\) is replaced by the quantity \(ec\overline{\Psi}(x')\gamma_\mu\Psi(x'')\). This variant of the theory is not gauge-invariant and for this reason alone should be rejected**).

S. Bloch\(^{17}\) pointed out that it is possible to restore gauge invariance if one introduces into the product \(\overline{\Psi}(x')\Psi(x'')\) a factor of the type \(e^{i\chi(x',x'')}\), where

\[ \chi(x',x'')\sim \int_{x'}^{x''} A_\mu(x)\,dS_\mu. \tag{8} \]

*) The question of the sufficiency of these requirements for determining the scattering matrix in the case of a local theory is discussed in detail in the paper by N. N. Bogoliubov and D. V. Shirkov\(^{13}\).

**) For the electromagnetic field, V. S. Barashenkov showed that a nonlocal theory with such a form factor leads to the occurrence of longitudinal photons\(^{11}\).

One variant of such a theory was investigated by R. Peierls and M. Kretzschmann\(^{18}\). They showed that in this case too the divergences from field theory are not completely eliminated.

Other difficulties were discovered by S. Hayashi\(^{19}\), who drew attention to the fact that, when the Yang—Feldman\(^{20}\) computational scheme is applied, the resulting scattering matrix, beginning with the fourth approximation, becomes nonunitary. In connection with this result, attempts were made to restore this unitarity.

There are two such attempts. The first belongs to S. Hayashi\(^{19}\) and is based on the introduction of additional fields that change the initial field \(\Psi_0\). The meaning of these fields may be seen in the influence of processes in the future on the initial conditions\(^{11}\).

The other was undertaken by B. V. Medvedev, who proceeded from the work of N. N. Bogoliubov and D. V. Shirkov\(^{13}\), in which it was shown that the requirements imposed above on the \(S\)-matrix, together with certain symmetry requirements, still do not fully determine the \(S\)-matrix. This circumstance was used by Medvedev\(^{21}\), who somewhat generalized the causality condition and, using the Bogoliubov—Shirkov method, constructed a unitary scattering matrix for a nonlocal theory in the form

\[ S(g)=T\exp\left\{i\int \Lambda(\xi,g)\,g(\xi)\,d\xi+\int M(\xi,g)\,g(\xi)\,d\xi\right\}, \tag{9} \]

where \(T\) is the generalized \(T\)-product, \(\Lambda\) is the Lagrangian, \(M\) is a certain Hermitian operator ensuring the unitarity of the matrix \(S\), and \(g\) is a quantity indicating the degree of inclusion of the interaction at the four-dimensional point \(\xi\).

As is clear from (9), the operator \(M\) may be regarded as a Hermitian “addition” to the Lagrangian function. The matrix \(S\) in this theory is given, as in all other variants of the theory, in the form of a series in powers of the constant of interaction. The convergence of such series has never been justified.

The difficulties considered above, connected with the absence of gauge invariance, remain in this theory.

The number of works devoted to nonlocal theory is at present very large (see\(^{11}\)). But the line of development of this theory is not ascending. One cannot, of course, deny the possibility of constructing a successful nonlocal theory, but it is difficult to give a favorable prognosis: the development of this theory has not brought encouraging results; on the contrary, many new difficulties have come to light which at first remained unnoticed.

In this state of affairs it is not without interest to consider the question of the possibility of experimentally detecting nonlocality in microphenomena, if it indeed occurs in nature. For this one could use very general dispersion relations. As far as we know, N. N. Bogoliubov was the first to draw attention to this possibility.

Let us consider the essence of the matter in the simplest model. Imagine a one-dimensional particle possessing structure. Let us suppose that it has the form of a dumbbell consisting of two scattering points \(A\) and \(B\) (Fig. 2). Suppose that a signal propagates instantaneously from \(A\) to \(B\), so that if the incident wave

$a_0 e^{-i(\omega_0 t-k_0 x)}$ for $t \gg 0$ set the point $A$ in motion, then a wave scattered by the point $B$ arises instantaneously.

Let $f_A(\omega)$ be the scattering matrix for the element of the particle $A$, and $f_B(\omega)$ the same for the element $B$. Then the wave scattered by the particle $AB$ will be:

\[ \Psi=\int f_A(\omega)a(\omega)e^{-i\omega\left(t-\frac{x}{c}\right)}\,d\omega +\int f_B(\omega)a(\omega)e^{-i\omega\left(t-\frac{x}{c}+\frac{\Delta}{c}\right)}\,d\omega, \tag{10} \]

where

\[ \Delta=AB,\qquad x=AP,\qquad x-\Delta=BP \]

and

\[ a(\omega)=\frac{a_0}{(\omega+i\varepsilon-\omega_0)}. \tag{10′} \]

(This last expression ensures that the incident wave is equal to zero for $t<0$.)

It follows from (10) that the scattering matrix for the particle as a whole will be equal to

\[ f_{AB}(\omega)=f_A(\omega)+f_B(\omega)e^{-i\frac{\omega\Delta}{c}}. \tag{11} \]

For dispersion relations of the type

\[ g(a)=\frac{1}{\pi}\int_{-\infty}^{+\infty}\frac{h(x)\,dx}{x-a} \tag{12} \]

and

\[ h(\omega)=-\frac{1}{\pi}\int_{-\infty}^{+\infty}\frac{g(x)\,dx}{x-a}, \tag{12′} \]

where $f=g+ih$, it is necessary that $f(\omega)$ be an analytic function in the upper half-plane and on the real axis, and vanish on the upper semicircle as $R\to\infty$.

It is not difficult to see that, if $f_A$ and $f_B$ satisfy these requirements, then $f_{AB}$ does not satisfy them because of the factor $e^{-i\frac{\omega\Delta}{c}}$. Moreover, as is not difficult to see from Fig. 2, the sign of the phase of this factor is preserved if the signal from $A$ and $B$ propagates with a velocity greater than the speed of light in vacuum. With a propagation velocity less than $c$, the sign will change and the dispersion relations will be satisfied. From (11) it is seen that if the scattering matrix is multiplied by

\[ e^{i\frac{\omega\Delta}{c}}, \]

then for the quantity

\[ \Phi_{AB}=f_{AB}e^{i\frac{\omega\Delta}{c}} \tag{13} \]

the dispersion relations will hold.

Fig. 2. Propagation of a signal from an extended particle. $Q$ is the source of the wave; $A,B$ are structural elements of the particle $AB$; $P$ is the observation point distant from $AB$.

Thus, we see that in the model under consideration there indeed arises the possibility of detecting nonlocality experimentally and even determining the size of the region of nonlocality. In fact, in the case of a local

in the theory, dispersion relations will be satisfied for the scattering matrix \(f_{AB}(\omega)\), and, in the case of a nonlocal theory, for the quantity \(\Phi_{AB}(\omega)\). Unfortunately, reality turns out to be more complicated, and one can give an example of a nonlocal theory which does not reveal itself in the dispersion relations.

Let us consider, as an illustration, a nonlocal theory of the electromagnetic field with a two-point form factor \(F(P-P')\). The equations of the field and of the particle in this theory have the form\(^4\):

\[ m\frac{dU_\alpha}{d\sigma}+eU_\beta\int dx\,dt\,F(P-P_m)\mathfrak F_{\beta\alpha}(P)=0, \tag{14} \]

\[ \Box^2 A_\alpha+4\pi e\int d\sigma F(P-P_m)U_\alpha=0, \tag{15} \]

where \(m\) is the mass of the particle, \(U_\alpha(\alpha=1,\,2,\,3,\,4)\) are the components of its four-velocity, \(\sigma\) is the proper time, \(\mathfrak F_{\beta\alpha}=\dfrac{\partial A_\alpha}{\partial x_\beta}-\dfrac{\partial A_\beta}{\partial x_\alpha}\) are the components of the electromagnetic field, \(A_\alpha\) are the components of the potential, \(P\) is the point \((x,t)\), \(P_m\) is the point \((x_m,t_m)\) (the position of the particle).

From considerations of Lorentz invariance,

\[ F(P-P_m)=\frac{1}{(2\pi)^4}\int D(\omega^2-k^2)e^{i\omega(t-t_m)-ik(x-x_m)}\,d\omega\,dk. \tag{16} \]

Let us consider the action of an external field \(\mathfrak F^0_{\beta\alpha}\) on the particle. This field is expanded in a Fourier series

\[ \mathfrak F^0_{\beta\alpha}=\int f^0_{\beta\alpha}(\mathbf k)\delta(\omega^2-k^2)e^{i(\omega t-\mathbf kx)}\,d\omega\,d\mathbf k. \tag{17} \]

Substituting (16) and (17) into the integral in (14) and carrying out the integration over \(x\) and \(t\), we find that the force \(K^0_\alpha\) excited by the external field is equal to

\[ K^0_\alpha=eU_\beta\int dx\,dt\,F(P-P_m)\mathfrak F^0_{\beta\alpha} =eD(0)\mathfrak F^0_{\beta\alpha}, \tag{18} \]

i.e., it does not differ from the action of a local field if \(D(0)\) is normalized to 1.

Let us now compute the scattered field. From equation (15) we find

\[ A^s_\alpha=e\int dx'\,dt'\,\mathfrak G(P-P')\,d\sigma F(P'-P_m)U_\alpha, \tag{19} \]

where \(\mathfrak G(P-P')\) is the Green’s function. This Green’s function has the Fourier component

\[ \frac{1}{4\pi}\mathfrak G(\omega^2-k^2)=\frac{1}{k^2-\omega^2}-i\pi\frac{\omega}{|\omega|}\delta(k^2-\omega^2). \tag{20} \]

Substituting into (19) the Fourier expansion for \(\mathfrak G(P-P')\) and for \(F(P-P_m)\), and carrying out the integration over \(dx'\), \(dt'\), we find that this integration reduces to the computation of the integral

\[ \sim\int \mathfrak G(\omega^2-k^2)D(\omega^2-k^2)e^{i\omega(t-t_m)-ik(x-x_m)}\,d\omega\,d\mathbf k. \tag{21} \]

The integral over \(dk\), as is known, has an asymptotic value of order \(|X-X_m|^{-1}\) only on account of the values \(k^2=\omega^2\), i.e., it is proportional to \(D(0)\). Thus, the scattered wave will be the same as in the local theory. It follows from this that the scattering “matrix” will certainly obey the dispersion relations, since it is identical with the scattering “matrix” of the local theory.

At first glance this result may seem paradoxical. However, it should be borne in mind that: a) our proof is valid only without taking into account the reaction of the field, i.e., approximately; b) if one considers the action of a wave limited in time (\(A_a^0=0\) at the point where the particle is located for \(t<0\)), then it is not difficult to see that in the nonlocal theory the particle will begin to move before the wave reaches it; however, this motion will decay for large positive \(t\), i.e., nonlocality manifests itself in the moments of “shaking up” the particle and subsequently disappears. Therefore it does not manifest itself in scattering (for large \(t\) and \(|x|\)).

The example given shows that, despite the absence of causality, the dispersion relations may nevertheless be satisfied. This circumstance complicates the unambiguity of the conclusions that can be drawn from the fact that the dispersion relations are valid. It would undoubtedly be desirable to have a more general analysis of this question.

III. NONLINEAR FIELD THEORY

The first nonlinear theory of the electromagnetic field was proposed by M. Born\(^{22}\).

The general scheme of such a theory may be formulated as follows: the theory is based on the Lagrangian function of the field \(\mathscr{L}\), which depends on the field invariants \(K, J, \ldots\), composed of the components of the field and their derivatives:

\[ \mathscr{L}=\mathscr{L}(K,\ J,\ \ldots). \tag{22} \]

Then from the variational principle

\[ \delta \int \mathscr{L}(K,\ J,\ \ldots)\,dx\,dt \tag{23} \]

there follow field equations which, in the general case, will obviously be nonlinear. From dimensional considerations it is clear that in a nonlinear field theory there will exist an absolute scale of the field \(\varphi_0\). If one now takes into account the existence of the elementary charge \(e\), then one can define an elementary length

\[ s_0=\sqrt{\frac{e}{\varphi_0}}. \]

The existence of this length makes nonlinear theories akin to nonlocal ones.

In the modern canonical field theory, the inclusion of interaction also leads to nonlinear field equations, which, however, are approximate and also contain higher derivatives. In Born’s scheme, nonlinear equations are postulated from the very beginning as the basis of the theory.

So long as such equations are considered within the framework of the classical theory, no fundamental difficulties arise. In particular, one can

choose such variants of the theory as eliminate the infinity of the proper energy of a particle. For example, in one of Born’s variants the Lagrangian is chosen in the form

\[ \mathscr{L}=\left\{1+\frac{\mathscr{E}^{2}-\mathscr{H}^{2}}{\mathscr{E}_{0}^{2}}-\frac{(\mathscr{E}\mathscr{H})^{2}}{\mathscr{E}_{0}^{4}}\right\}^{1/2}, \tag{24} \]

where \(\mathscr{E}\) is the electric field, \(\mathscr{H}\) is the induction of the magnetic field, and \(\mathscr{E}_{0}\) is a certain scale of the field, of order of magnitude equal to \(e/s_{0}^{2}\).* In this case the strength of the electric field of a point charge turns out to be equal to:

\[ \mathscr{E}=\frac{e}{r}\left\{1+\left(\frac{s_{0}}{r}\right)^{4}\right\}^{-1/2}, \tag{25} \]

where \(r\) is the distance from the center of the charge.

The total proper energy of a point charge then turns out to be finite and equal to:

\[ U=1.236\,\frac{e^{2}}{s_{0}}. \tag{26} \]

However, a classical nonlinear theory cannot be the goal of the theoretician, since long before nonlinear deviations begin to play a noticeable role (distances of order \(s_{0}\)), quantum phenomena enter the scene (distances of order \(\hbar/m_{0}c\), \(m_{0}\) being the electron mass). Therefore the nonlinear theory must be quantized. But it is precisely in quantization that fundamental difficulties are revealed. However, before turning to this side of the matter, let us consider the classification of nonlinear equations.

To clarify the essence of the matter we shall restrict ourselves to Lagrangian functions containing derivatives of the field no higher than first order, and, for simplicity, to one dimension and to a scalar field \(\varphi\) [23].

In this case we have two invariants:

\[ K=\frac{1}{2}\left[\left(\frac{\partial\varphi}{\partial t}\right)^{2}-\left(\frac{\partial\varphi}{\partial x}\right)^{2}\right]\quad \text{and}\quad J=\frac{1}{2}\varphi^{2} \tag{27} \]

(the speed of light \(c=1\)). The Lagrange function will be

\[ \mathscr{L}=\mathscr{L}(K,J). \tag{28} \]

From the corresponding variational principle it is not difficult to find the field equation:

\[ A\,\frac{\partial^{2}\varphi}{\partial t^{2}}+2B\,\frac{\partial^{2}\varphi}{\partial t\,\partial x}+C\,\frac{\partial^{2}\varphi}{\partial x^{2}}+D=0, \tag{29} \]

where \(A, B, C, D\) are functions of \(\varphi,\ \dfrac{\partial\varphi}{\partial t},\ \dfrac{\partial\varphi}{\partial x}\).

This equation is formally invariant with respect to Lorentz transformations. However, causality may also be violated, just as it is violated in nonlocal theories. Indeed, let us consider the velocity-

* It was pointed out, as a defect of the theory, that there is arbitrariness in the choice of the Lagrangian; however, in principle it could be determined from experiment.

propagation of interaction in such a nonlinear theory (we shall speak of the “signal velocity”). This velocity is properly understood as the velocity of propagation of weak discontinuities (i.e., such discontinuities where ahead of the signal front \(\varphi=0\), while behind it \(\varphi\ne 0\), but it does not undergo a jump).

It is known that the propagation of such signals takes place along the characteristics of the equation, and the signal propagation velocity is given by the slope of the characteristics,

\[ \xi=\frac{dx}{dt}. \]

The quantity \(\xi\) is determined from the equation

\[ A\xi^2-2B\xi+C=0. \tag{30} \]

Depending on the form of the Lagrange function and on the value of the field and its derivatives, signals may arise that propagate both with a velocity less than the velocity of light and with a velocity greater than the velocity of light (equation (30) may have solutions \(|\xi|>1\)).

For example, for not very large fields we obtain from (30)

\[ \xi=\pm 1 \mp \frac{1}{2}\alpha\left(\frac{\partial\varphi}{\partial t}-\frac{\partial\varphi}{\partial x}\right)^2+\ldots, \tag{31} \]

where

\[ \alpha= \frac{\dfrac{\partial^2\mathcal{L}}{\partial K^2}} {\dfrac{\partial\mathcal{L}}{\partial K}}. \]

Depending on the sign of \(\alpha\) (i.e., depending on the form of the Lagrangian), \(\xi\) will be greater or less than 1. Details are in \(^{23,24}\). It is interesting to note that in such nonlinear theories one cannot exclude the possibility of a situation in which, for certain values of \(\dfrac{\partial\varphi}{\partial t}\), \(\dfrac{\partial\varphi}{\partial x}\), \(\varphi\) (for example, near particles), the characteristics become imaginary, so that the field equations become equations of elliptic type. This would mean that the concept of the causal sequence of events loses its meaning, and we would be dealing with a “lump” of events which mutually condition one another, but do not follow one after another. We are far from asserting that anything of the sort takes place in reality, and consider this case as a purely mathematical possibility*).

For us at present the more important fact is that possible nonlinear theories split into two classes. In the first class (\(\alpha>0\)) the signal propagation velocity is always less than the velocity of light in vacuum, \(|\xi|\le 1\); in the second class (\(\alpha<0\)) \(|\xi|\) may also be greater than 1**).

* More precisely, we think that if anything of the sort is realized, for example inside particles, then it is probably realized in a more subtle way.

** We are now excluding that class of theories in which the characteristics may become imaginary, i.e., we consider the classification only of hyperbolic equations.

Theories of this latter class have much in common with nonlocal theories, and Hamilton’s method is likewise inapplicable to them, as it is to nonlocal theories. They have never been examined in detail by anyone.*)

On the contrary, theories of the first class are not in contradiction with the ordinary understanding of causality; therefore Hamilton’s method is applicable to them, and, consequently, so is the usual scheme of quantization. Among such equations, for example, is the nonlinear equation

\[ \square^{2}\varphi-\chi^{2}(\varphi)\varphi=0, \tag{32} \]

i.e., an equation in which the term determining the mass depends on the field \(\varphi\) itself. For this equation \(|\xi|=1\).

The possibility of a large number of variants of nonlinear theory is often regarded as the principal difficulty of such theories. In reality the difficulties lie in an entirely different point. If it were possible to construct a mathematically consistent nonlinear theory, then one could, by comparison with experiment, seek the uniquely correct variant. Therefore the chief question at the present stage is whether, in principle, such an internally consistent theory can be found. We are now inclined to answer this question in the negative.

Indeed, what would be encouraging in the development of nonlinear equations would be the elimination of divergences in the self-energy of particles. And we have seen that, so long as we remain within the framework of classical theory, such hopes are in fact justified. The situation becomes quite different when we pass to a quantum nonlinear theory. Here we wish in advance to restrict ourselves to theories in which the signal velocity is always less than or equal to the velocity of light. Theories of the second class, or theories with elliptic equations, will contain, besides the difficulties that we intend to consider below, other difficulties that further complicate the matter.

We shall now show that the very simplest divergence, which is easily eliminated in linear theories, acquires an ominous character in nonlinear theory. The issue is the zero-point energy of the field \(E_0\). In a linear theory one may take as the observed energy of the field the quantity

\[ \varepsilon=E-E_0, \tag{33} \]

which, at least for free fields, turns out to be a finite quantity (here \(E_0\) is the infinite zero-point energy of the field, \(E\) is the infinite energy of the excited state of the field, and \(\varepsilon\) is a finite quantity).

In nonlinear theory there is no such additivity. Therefore all field levels prove to be infinite.

Let us examine this aspect of the matter in detail using the example of the simple equation (32). The Hamiltonian function for the field described by equation (32) has the form

\[ H=\frac{1}{2}\int\{\pi^{2}+\nabla\varphi^{2}+F(\varphi)\}\,dx. \tag{34} \]

Put \(F(\varphi)=\chi_{0}^{2}\varphi^{2}+\alpha\varphi^{4}+\ldots\)

*) In one paper by W. Heisenberg\(^{25}\) a similar equation appears.

We shall regard \(\alpha\) as a small parameter. Then the quantity \(\alpha\varphi^{2}\) may be considered as an addition to \(\chi_{0}^{2}\), modulating the mass of the particle:

\[ \chi^{2}(\varphi)=\chi_{0}^{2}+\alpha\varphi^{2}+\cdots . \]

Let us now compute the coefficient \(\chi^{2}(\varphi)\) approximately from the linear theory, replacing its true value by the mean. Thus, one must compute \(\overline{\chi^{2}(\varphi)}\). To this end we represent the field in the form of a Fourier series

\[ \varphi=-\frac{1}{\sqrt{V}}\sum_{k}\sqrt{\frac{\hbar}{2\omega_{k}}}\left(a_{k}e^{ikx}+a_{k}^{*}e^{-ikx}\right), \tag{35} \]

where \(V\) is the smallest volume in which \(e^{ikx}\) is periodic, \(\omega_{k}=\sqrt{k^{2}+\chi_{0}^{2}}\) is the frequency in the zeroth approximation, and \(a_{k}, a_{k}^{*}\) are the creation and annihilation operators of field quanta, obeying the usual quantization rule

\[ [a_{k},a_{k'}^{*}]=\delta_{kk'} . \tag{36} \]

A simple calculation gives

\[ \overline{\varphi^{2}}=\frac{1}{V}\sum_{k}\frac{\hbar}{\omega_{k}}(2N_{k}+1), \tag{37} \]

where \(N\) is the number of quanta of sort \(k\).

Passing from sums to integrals, we have

\[ \overline{\varphi^{2}}=4\pi\int_{0}^{\infty}\frac{\hbar k^{2}\,dk}{\omega_{k}}+ 2\pi\int_{0}^{\infty}\frac{\hbar\rho_{k}k^{2}\,dk}{\omega_{k}}, \tag{38} \]

where \(\rho_{k}\) is the density of quanta \(\left(\rho_{k}=\frac{N_{k}}{V}\right)\). From (38) it is clear that \(\overline{\varphi^{2}}=\infty\). Therefore \(\overline{\chi^{2}(\varphi)}=\infty\).

In other words, in the next approximation, the frequencies \(\omega'_{k}=\sqrt{k^{2}+\chi^{2}}=\infty\) for all \(k\).

Thus, if the eigenvalues of \(H\) are denoted by \(E\), then the difference \(E-E_{0}=\sum N_{k}\hbar\omega'_{k}=\infty\). This is what should have been expected. It is not excluded that the use of special renormalizations might make it possible to get rid of this divergence. In any case, the hope for a nonlinear theory in the sense of an automatic elimination of infinities is completely destroyed*).

*) Schiff\(^{26}\) attempted to quantize the nonlinear theory, defining \(\varphi\) not for a point of space but for a discrete spatial lattice with spacing \(l\). As \(l\to0\), all energy levels, in accordance with our simple calculation, tend to \(\infty\). For \(l\ne0\) they are finite. But it is not difficult to show that in Schiff’s theory there is propagation of a signal with a velocity greater than the speed of light, so that his theory is nonlocal and therefore incompatible with the Hamiltonian method which he uses.

IV. PHYSICS OF STRONG INTERACTION

The preceding analysis of the theory of a nonlocal field and of nonlinear field theory, of course, cannot be regarded as an exhaustive proof of the impossibility of constructing internally consistent theories of the type considered. But this analysis leads to a pessimistic prognosis. The impression is created that both of these directions in the development of field theory miss the mark. They have in common that in both directions the main features of modern quantum theory and relativity theory are preserved; these are merely modifications of modern concepts.

It has now become a generally accepted statement that at small distances modern theory becomes unsuitable. At the same time, the success of renormalization theory shows that, specifically for the electromagnetic field and electrons, this unsuitability appears only in very distant approximations in the constant \(e^2/\hbar c\). On the contrary, for meson theory there is in general no region of applicability, since the constant \(g^2/\hbar c\) is very large. This situation indicates that small distances and small time intervals are not in themselves catastrophic for modern theory; rather, the principal role is played by large interactions.

Let us now consider several simple examples relating to the case of strong interaction.

It is well known that the problem of the motion of an electron in the field of a point charge \(Z > 137\) has no solution. This conclusion, however, is inapplicable to a charge of finite dimensions. In this case, formally, a solution is obtained[^97], leading to a level lying below \(-2m_0c^2\) (\(m_0\) is the electron mass). The difficulty lies in the interpretation of this level, since it mixes with the negative levels of the Dirac sea. The usual interpretation of this level as positronic is apparently untenable. Indeed, from this interpretation it follows that, under an adiabatic bringing together of charges in order to obtain a nucleus with large \(Z\), when the charge passes through some critical value \(Z_0\), an additional charge \(+e\) arises by itself, without the corresponding compensation. Whether another interpretation is possible is an open question, obscured by the circumstance that for \(Z \sim 137\) it is necessary to take into account vacuum polarization. Meanwhile, the calculation of polarization at such values of \(Z\) lies beyond the possibilities of modern theory. Thus, at a binding energy of the order of \(m_0c^2\), the theory encounters fundamental difficulties.

Let us consider another formal example, relating to a scalar field \(\varphi\). Let a scalar particle move in an external meson field \(\varphi_0\). The equation can be written in the form*)

\[ \Box^2 \varphi - \chi^2 \varphi + \alpha \varphi_0^2 \varphi = 0. \tag{39} \]

From this equation it is immediately evident that inside a region where \(\alpha \varphi_0^2 > \chi^2\), it can be rewritten in the form

\[ \Box^2 \varphi + \chi'^2 \varphi = 0, \tag{39'} \]

*) This equation may be regarded as the usual equation with an additional term of the type \(\alpha \varphi^2\).

where

\[ \varkappa'^2=a\varphi_0^2-\varkappa^2. \]

It follows from this that the proper frequencies of such a field, inside the potential well, will be equal to \(\omega=\sqrt{k^2-\varkappa'^2}\). For \(a\varphi_0^2>\varkappa^2\), the group velocity of the particles inside this well turns out to be greater than the velocity of light. This should be regarded as a contradiction, arising once again in the case when the binding energy exceeds the particle’s own energy \(\mu c^2\).

The third example was considered by us earlier\(^{28}\)*), and represents the case of a simple linear interaction of two scalar fields \(\varphi\) and \(\psi\). The Hamiltonian in this case is written in the form

\[ H=\frac{1}{2}\int\{\dot{\varphi}^{2}+\nabla\varphi^{2}+a^{2}\varphi^{2}+\dot{\psi}^{2}+\nabla\psi^{2}+b^{2}\psi^{2}+g\varphi\psi\}\,dx. \tag{40} \]

The quantities \(a^2\) and \(b^2\) determine the masses of the particles belonging to the fields \(\varphi\) and \(\psi\), while \(g\) is the coupling constant of these fields. The eigenvalues of this Hamiltonian are

\[ E=\sum \varepsilon_k N_k+\sum \varepsilon_q M_q+E_0, \tag{41} \]

where

\[ \varepsilon_k=\hbar\sqrt{k^2+A^2}, \]

and

\[ \varepsilon_q=\hbar\sqrt{q^2+B^2}; \]

and \(N_k\) and \(M_q\) are positive integers (or zeros); \(E_0\) is the zero-point energy, equal to \(\frac{1}{2}\sum\varepsilon_k+\frac{1}{2}\sum\varepsilon_q\), \(k,q\) are the momenta of the particles.

\(A\) and \(B\) are quantities determining the masses of the particles, with

\[ A^2=\frac{a^2+b^2}{2}+\sqrt{\frac{(a^2-b^2)^2}{4}+g^2}, \]

\[ B^2=\frac{a^2+b^2}{2}-\sqrt{\frac{(a^2-b^2)^2}{4}+g^2}. \tag{42} \]

From formula (42) it is clear that for \(g^2>a^2b^2\) one of the masses \((B)\) becomes imaginary. This fundamental difficulty, as we see, arises at a large binding energy.**)

We are aware of the extremely schematic character of the examples given. They are artificial in the sense that they concern the motion of a particle in an external field, and not the interaction of particles; nevertheless, they cast serious doubt on the applicability of the concept of a particle in those cases when the interaction energy becomes comparable with the intrinsic energy

*) This example was considered as an illustration of the physical idea that the “nature” of particles may depend on the type of analyzing apparatus. This idea has now found its confirmation in the existence of two kinds of \(\theta\)-particles.

**) The physical case \(g^2>a^2b^2\) means, in the terminology of the theory of oscillations, that the “focus” turns into a “saddle” (loss of stability).

particles. At the same time, the usual scheme of canonical quantization, most deeply connected with the representation in terms of particles, is also called into question.

We do not have the possibility of consistently carrying out calculations for those cases when the particles turn out to be very strongly interacting with one another. However, we can obtain an idea of when this may occur. In doing so, by a strong interaction we shall understand such an interaction of particles in which the energy of their interaction \(W\) exceeds the particles’ own energy. States of such strong interaction may last for a very short time, for example during the time of a collision. But it is clear that only this short time is of importance for the whole process.

Let us first consider the electromagnetic interaction of two electrons. From a comparison of \(W=\dfrac{e^2}{r}\) with the electron energy \(E\) we find that \(W>E>m_0c^2\) when

\[ r < \frac{e^2}{\hbar c}\,\lambda, \tag{43} \]

where \(\lambda \sim \dfrac{\hbar c}{E}\) is the wavelength of the electron.

The size of the localization region of an electron with wavelength \(\lambda\) cannot be smaller than \(\lambda\). Therefore we see that, because of the smallness of the constant \(\dfrac{e^2}{\hbar c}\), the electron is rarely in that region of space where \(W>m_0c^2\). In the case of the interaction of an electron and a charge \(eZ\), we would have \(r<\dfrac{Ze^2}{\hbar c}\lambda\), and for \(Z>137\), \(r\sim\lambda\). Thus a strong interaction of electric charges sets in at \(Z\sim 137\).

In the case of the interaction of nucleons the situation is quite different. For an interaction energy of the form*) \(W\sim g^2\dfrac{e^{-\varkappa r}}{r}\), we obtain the condition \(W>E>Mc^2\) (\(M\) is the mass of the nucleon) in the form

\[ r < \frac{g^2}{\hbar c}\,\lambda . \tag{43'} \]

But in this case \(\dfrac{g^2}{\hbar c}>1\), and the state of strong interaction arises for \(r>\lambda\), i.e., in all cases when \(\lambda<\dfrac{\hbar}{Mc}\).

Therefore it seems very probable that in the process of collision of energetic nucleons, during the time of the collision the nucleons completely lose their individuality as particles, forming states that may be called “compound particles.”

*) If one takes an interaction with a greater degree of singularity,

\[ W=\frac{g^2}{r}\times\left(\frac{a}{r}\right)^n e^{-\varkappa r}, \]

then condition (43') reads:

\[ r<\left(\frac{g^2}{\hbar c}\right)^{\frac{1}{n+1}} \left(\frac{a}{\lambda}\right)^{\frac{n}{n+1}}\lambda \]

and gives the previous result for \(a>\lambda\). Here \(a\) is a certain length.

Nonlocal and Nonlinear Field Theories

A sign of such a state of the “compound particle” is the circumstance that the energy of the particles is concentrated in the interaction energy, and not in the particles’ proper energy.*)

Let us now examine in more detail the conditions for the occurrence of such states. For this purpose consider the energy density \(\varepsilon\) of a system of interacting nucleons and mesons:

\[ \varepsilon=\bar{\psi}D\psi+\frac{1}{2}\left(\dot{\varphi}^{2}+\nabla\varphi^{2}+\omega_{0}^{2}\varphi^{2}\right)+g\bar{\psi}\gamma_{5}\psi\varphi . \tag{44} \]

Here the first term is the energy density of free nucleons, \(D=c\boldsymbol{\alpha}\mathbf{p}+\beta Mc^{2}\) is the Dirac Hamiltonian, \(\bar{\psi},\psi\) is the nucleon field; the second term is the energy density of the meson field \(\varphi\), \(\omega_{0}^{2}=c^{2}\chi_{0}^{2}\); finally, the third term is the interaction energy, which is assumed to be pseudoscalar.

Let, for a given total density \(\varepsilon\), the energy density of the free nucleon field be \(\varepsilon_{1}\), and the energy density of the free meson field \(\varepsilon_{2}\). In the case when the interaction energy is not substantial, \(\varepsilon\simeq\varepsilon_{1}+\varepsilon_{2}\). Let us determine when this case will occur. To do this we shall use dimensional considerations and introduce a certain length scale \(l\), determining the magnitude of the gradients, so that \(\partial/\partial x\sim 1/l\) and \((1/c)\,\partial/\partial t\sim 1/l\). Then we obtain:

\[ \varepsilon_{1}=\bar{\psi}D\psi\sim \bar{\psi}\left(\frac{\hbar c}{l}+Mc^{2}\right)\psi, \quad \text{i.e.}\quad \bar{\psi}\psi\sim \frac{\varepsilon_{1}l}{\hbar c}\left(1+\frac{Mcl}{\hbar}\right)^{-1}, \tag{45} \]

\[ \varepsilon_{2}=\frac{1}{2}\left(\dot{\varphi}^{2}+\nabla\varphi^{2}+\omega_{0}^{2}\varphi^{2}\right)\sim \frac{1}{2}\left(\frac{c^{2}}{l^{2}}+\omega_{0}^{2}\right)\varphi^{2}, \]

i.e.

\[ \varphi\sim \frac{\varepsilon_{2}^{1/2}l}{c}\left(1+\frac{\mu^{2}c^{2}l^{2}}{\hbar^{2}}\right)^{-1/2} \tag{45'} \]

(here \(\mu\) is the mass of the meson; \(M\) is the mass of the nucleon).

*) If, for example, one considers in the second approximation of the Tamm–Dancoff method the interaction of two nucleons, then for their energy we obtain

\[ E=g^{2}\frac{e^{-\chi r_{11}}}{r_{11}}+g^{2}\frac{e^{-\chi r_{22}}}{r_{22}}+2Mc^{2}+\frac{g^{2}e^{-\chi r_{12}}}{r_{12}} . \]

The first two terms are infinite, since \(r_{11}=r_{22}=0\), and represent a contribution to the proper energy of the particles, i.e.

\[ g^{2}\frac{e^{-\chi r_{11}}}{r_{11}}+g^{2}\frac{e^{-\chi r_{22}}}{r_{22}}+2Mc^{2}=2M_{\mathrm{exp}}c^{2}; \]

the last term—the interaction energy \(g e^{-\chi r_{12}}/r_{12}\)—as \(r_{12}\to 0\) has a value no smaller than the first two. Consequently, in those cases when it is substantially large, it must be considered on the same footing as the first two, as a contribution to the proper energy of the two particles.

The interaction will be insignificant if

\[ W = gc \bar{\psi}\gamma_5\psi\varphi \ll \varepsilon_1 \text{ and } \varepsilon_2 . \tag{46} \]

Borrowing from (45) and (45′) \(\bar{\psi}\psi\) and \(\varphi\), we find that (46) will be satisfied if:

\[ \varepsilon_2 \ll \frac{\hbar^2 c^2}{g^2 l^4} \left(1+\frac{Mcl}{\hbar}\right) \left(1+\frac{\mu^2 c^2 l^2}{\hbar^2}\right), \tag{47} \]

\[ \varepsilon_1 \ll \frac{\hbar c}{g l^2}\,\varepsilon_2^{1/2} \left(1+\frac{Mcl}{\hbar}\right) \left(1+\frac{\mu^2 c^2 l^2}{\hbar^2}\right)^{1/2}; \tag{47′} \]

combining both inequalities, we may write:

\[ \varepsilon \ll \varepsilon_{\mathrm{cr}} = \frac{\hbar^2 c^2}{g^2 l^4} \left(1+\frac{Mcl}{\hbar}\right)^2 \left(1+\frac{\mu^2 c^2 l^2}{\hbar^2}\right). \tag{48} \]

Conversely, if \(\varepsilon_1\) and \(\varepsilon_2 \gg \varepsilon_{\mathrm{cr}}\), then the interaction energy will be more substantial than the energy of the free fields. Indeed, by (45) and (45′) \(\varepsilon\) may be represented in the form

\[ \varepsilon \approx \varepsilon_1+\varepsilon_2+ \frac{g l^2}{\hbar c}\,\beta \varepsilon_1 \varepsilon_2^{1/2}, \tag{49} \]

where

\[ \beta = \left(1+\frac{Mcl}{\hbar}\right)^{-1} \left(1+\frac{\mu^2 c^2 l^2}{\hbar^2}\right)^{-1/2}. \]

Let us now require that the sum \(\varepsilon_1+\varepsilon_2\) be much smaller than \(\varepsilon\), while

\[ W=\frac{g l^2}{\hbar c}\,\beta \varepsilon_1 \varepsilon_2^{1/2}\simeq \varepsilon . \]

By direct substitution it is not difficult to verify that this is obtained when \(\varepsilon_1\) and \(\varepsilon_2 \gg \varepsilon_{\mathrm{cr}}\).

In this case the main contribution to the energy of the system is made by the interaction energy, and not by the energy of the free nucleon and meson fields \(\varepsilon_1+\varepsilon_2\). Thus, \(\varepsilon_{\mathrm{cr}}\) (48) indeed separates two physically quite different cases of interaction: for \(\varepsilon \ll \varepsilon_{\mathrm{cr}}\) the interaction is insignificant and the particles retain their meaning; for \(\varepsilon \gg \varepsilon_{\mathrm{cr}}\) the interaction is more substantial than the particles themselves.

Of course, the expression for the critical energy depends on the type of interaction assumed by us. For example, for a pseudovector interaction

\[ W=gca\,\bar{\psi}\gamma_5\gamma_\mu\psi\,\frac{\partial\varphi}{\partial x_\mu} \tag{50} \]

(here \(a\) is a certain length) the value of the critical energy will be

\[ \varepsilon_{\mathrm{cr}} = \frac{\hbar^2 c^2}{g^2 a^2 l^2} \left(1+\frac{Mcl}{\hbar}\right)^2 \left(1+\frac{\mu^2 c^2 l^2}{\hbar^2}\right), \tag{51} \]

i.e., in this case, as \(l\) decreases, the critical energy grows more slowly.

For the electromagnetic interaction

\[ W=ec\bar{\psi}\gamma_\mu\psi A_\mu \tag{52} \]

and in an analogous way we find

\[ \varepsilon_{\mathrm{cr}}=\frac{\hbar^2c^2}{l^2l^4}\left(1+\frac{mcl}{\hbar}\right)^2. \tag{53} \]

Finally, for the beta-decay interaction

\[ W=g^*\bar{\psi}_N\varphi_\nu\varphi_e\psi_N. \tag{54} \]

(here \(g^*\) is the Fermi constant, \(\bar{\psi}_N,\psi_N\) are nucleon functions, \(\varphi_\nu\) is the neutrino function, \(\varphi_e\) is the electron function) the critical energy is equal to

\[ \varepsilon_{\mathrm{cr}}=\frac{\hbar^2c^3}{g^{*2}l^2} \left(1+\frac{Mcl}{\hbar}\right) \left(1+\frac{mcl}{\hbar}\right)^{1/2}. \tag{55} \]

The existence of a critical interaction energy is clear from the fact that the energy of free fields is quadratic with respect to these fields, while the interaction energy is at least cubic. At a high energy density the cubic terms will exceed the quadratic ones. In all cases the critical energy increases as the scale \(l\) decreases, i.e., as the gradients grow. The growth of the gradients means an increase in the relative contribution of free particles, since the energy density of free fields is proportional to \(1/l\) (for fermions) and \(1/l^2\) (for bosons).

Let us now consider when compound particles will actually be formed. We first turn to the case of the pseudoscalar interaction of nucleons and mesons in the nonrelativistic region. Denote the total energy of the colliding particles by \(E=T+2Mc^2\). The scale of the interaction region will be \(l\simeq \dfrac{\hbar}{\mu c}\) (\(\mu\) is the meson mass). The energy density is \(\varepsilon=T/l^3\). Comparing this value of \(\varepsilon\) with \(\varepsilon_{\mathrm{cr}}\) (48), we obtain the condition for the occurrence of compound particles at the moment of collision

\[ T>\frac{\hbar c}{g^2}\,2\mu c^2 \left(1+\frac{M}{\mu}\right)^2, \tag{56} \]

which for \(\dfrac{g^2}{4\pi\hbar c}\sim 15\) gives \(T\sim 0.9\cdot10^8\ \mathrm{eV}\)*).

In the extreme relativistic case it is necessary to take account of the Lorentz contraction, so that the largest gradients will be determined not by the quantity \(l\), but by the quantity

\[ l^*=\frac{\hbar}{\mu c}\sqrt{\frac{2Mc^2}{E}}. \]

The volume of the interaction region will be \(l^2l^*\), and the critical energy is determined from (48) by the same quantity \(l^*\); the energy density \(\varepsilon\) in this case is equal to

\[ \varepsilon=\frac{Mc^2\sqrt{\dfrac{E}{2Mc^2}}}{l^2l^*}. \]

* If the strong-interaction region is assumed to be not \(\dfrac{\hbar}{\mu c}\), but \(\dfrac{\hbar}{Mc}\), then the result will be \(T\sim0.2\cdot10^8\ \mathrm{eV}\).

Comparing this quantity with the critical energy, we find that the state of a compound particle arises for

\[ E<\frac{g^{2}}{\hbar c}\frac{M}{\mu}\,2Mc^{2}, \tag{56'} \]

i.e., for \(E<10^{12}\) eV. The fact that in this case we have obtained not a lower but an upper bound is explained by the fact that, as the energy of the nucleons \(E\) increases, owing to Lorentz contraction the gradients increase greatly, and the relative role of the free fields increases\(^*\). For the pseudovector interaction (50), in the nonrelativistic case, instead of (56) we obtain

\[ T>\frac{\hbar c}{g^{2}}\left(\frac{\hbar}{\mu c a}\right)^{2}2\mu c^{2}\left(1+\frac{M}{\mu}\right)^{2}, \tag{57} \]

i.e., for \(a\sim \dfrac{\hbar}{\mu c}\) the same value of \(T\) as for the pseudoscalar interaction. In the extreme relativistic case, however, instead of (56) we find

\[ \frac{g^{2}}{\hbar c}>\frac{\hbar}{\mu c}\frac{\hbar}{Mc}a^{2}, \tag{57'} \]

i.e., for \(\dfrac{g^{2}}{4\pi\hbar c}\sim 15\) and \(\dfrac{\hbar}{Mc}<a<\dfrac{\hbar}{\mu c}\), there will always be compound states. This is connected with the fact that, in the case of pseudovector interaction, the relative role of the interaction energy increases not only with increasing energy density, but also with increasing gradients. Thus, for the pseudovector interaction, the region of compound particles extends from \(T=10^{8}\) eV without limit toward higher energies\(^ {**}\).

Let us now turn to electromagnetic interactions. Here a certain ambiguity arises in determining the dimensions of the region of interaction. From the expression for the critical energy (53), in the nonrelativistic case we find

\[ T>\frac{\hbar c}{e^{2}}mc^{2}\frac{\hbar}{mcl}\left(1+\frac{mcl}{\hbar}\right)^{2}. \tag{58} \]

Here \(m\) is the electron mass. We see that, for any \(l<\dfrac{\hbar}{mc}\), the quantity \(T\) falls in the region of relativistic energies. In the extreme relativistic case we obtain

\[ E<\frac{e^{2}}{\hbar c}\,2mc^{2}\frac{mcl}{\hbar}, \tag{58'} \]

\(^*\) If one believes in the pseudoscalar interaction, this would mean that, in a hydrodynamic treatment of nucleon collisions, the meson-nucleon fluid\({}^{34}\) could be regarded as a gas.

\(^ {**}\) The existence of compound particles may also be responsible for the formation of above-barrier fragments He, Li, Be, arising in the collision of an energetic nucleon with an atomic nucleus. Such compound states may arise in the nucleus as a result of fluctuations (of course, for a very short time). Then the collision of a nucleon with such a short-lived compound state may lead to the emission of a heavy fragment.

i.e., for \(l<\dfrac{\hbar}{mc}\), the value \(E\) falls in the region of nonrelativistic energies. Therefore, in the case of electromagnetic interactions, because of the smallness of the coupling constant, compound particles do not arise. The same result is obtained also for the decay interaction (54), owing to the smallness of \(g^* = 10^{-48}\ \mathrm{erg}^2\cdot\mathrm{cm}^3\).

From the calculations given we see that the energy \(\varepsilon_{\mathrm{cr}}\) is a physical criterion for deciding the question of the appearance of states of strong interaction, “compound particles.” The concept of a “compound particle” is not altogether new—we give it only a physical definition based on the relative role of the interaction energy and the proper energy of the particles. It has occurred, in one form or another, in various recent works, and in this connection we should like to add a few more remarks.

Suppose that the collision is between two particles \(P\) and \(N\), described in the free state by wave fields \(\Psi_P\) and \(\Psi_N\), and that in this collision a new particle \(\pi\) is produced, described in the free (or nearly free) state by the wave field \(\Phi_\pi\).

By analogy with nuclear reactions one may speak of the entrance channels of the reaction, \(\Psi_P\) and \(\Psi_N\), of its exit channels, \(\Psi'_P\), \(\Psi'_N\), and \(\Phi'_\pi\), and of the region of strong interaction.

Fig. 3. Collision scheme with strong interaction. The dotted line encloses the region of strong interaction. \(\Psi_P\), \(\Psi_N\)—entrance channels; \(\Psi'_P\), \(\Psi'_N\), \(\Phi'_\pi\)—exit channels.

Fig. 3 shows a scheme of such a collision.

The region enclosed by the dotted line is the region of strong interaction. Outside this region, in the entrance channels, we are dealing with almost free fields \(\Psi_P\) and \(\Psi_N\); in the exit channels the fields \(\Psi'_P\), \(\Psi'_N\), and \(\Phi'_\pi\) are also almost free. In the interior region, however, because of the strong interaction, there is neither a field \(\Psi\) nor a field \(\Phi\), but something else, which, when weakened, degenerates into the linear fields \(\Psi\) and \(\Phi\). We know almost nothing at present about this “something else.”

In various modern attempts to consider these states, the most schematic sketches are used. Thus, for example, one may regard the colliding particles as “black,” absolutely absorbing spheres\(^{29,30}\). Such an approach makes it possible to single out diffraction scattering and can give some idea of the probability of formation of a “compound particle” (“sticking probability”). In the case where the lifetime of the “compound state” appreciably exceeds the collision time \(\tau = \dfrac{a}{v}\) (here \(a\) is the radius of the sphere of strong interaction, \(v\) the relative velocity of the particles), one may speak of an “isobaric” state and regard it as a true, new particle\(^{31,32}\). Such representations prove fruitful in many respects, since it is known in advance that especially strong interactions are characterized by quite definite integrals of motion. For example, for the collision of a pair of nucleons, in this respect the state with \(J = 0\) (total angular momentum) and \(T = 1\) (isotopic spin) is essential. For the collision of a \(\pi\)-meson with a nucleon

an important state with \(J=\dfrac{3}{2}\) and \(T=\dfrac{3}{2}\). However, the lifetime of the excited states apparently differs little from the collision time.

Another approach is connected with the application of statistical\(^{33}\) or hydrodynamic methods\(^{34}\). In the statistical method one operates with ratios of the phase volumes “occupied” by each of the possible decay processes of the “compound particle.” It is clear that, by its very nature, this method can describe only the crudest features of this decay. In essence, what is involved is a comparison of the phase volumes of the exit channels; the actual process of particle formation (or, what is the same thing, of the linear fields \(\Psi,\Phi\)) falls outside this consideration.

In the hydrodynamic treatment\(^{34}\) an attempt was made to correct this deficiency of Fermi’s theory in such a way that the “compound state” is regarded as a strongly excited liquid which only in the process of its expansion turns into a gas consisting of particles. The liquid itself is treated as classical, obeying the laws of relativistic hydrodynamics. This is undoubtedly a step forward in comparison with the statistical theory. But this picture too is only a rough scheme. In the first stages of the expansion of this liquid, significant quantum fluctuations\(^{35}\) occur, and therefore the question remains open of describing this essential phase of the expansion.

In considering the “compound state,” we began with an analogy to nuclear reactions. However, this analogy is purely external. A “compound nucleus” is only a special state of the particles participating in a nuclear reaction. In the “compound particle,” the particles from which it was formed, as well as other particles arising within it, interact so strongly that the very notion of the structure of the “compound particle” as consisting of interacting particles loses its meaning. This, however, does not contradict the fact that the “compound particle” can be characterized by integral quantities (spin, charge, mass, etc.) and that its center of gravity will move in accordance with the laws of quantum mechanics. What is at issue is its interior. From the fact that particles, as independent objects, cease to exist because of the strong interaction, it is least of all possible to suppose that such states can be described by any classical methods. Classical theory does not know Planck’s constant, and therefore it is difficult to imagine in what way, from a formation obeying the laws of classical theory, something obeying the laws of quantum theory could arise.

On the contrary, it is more natural to think that “compound particles” are entities governed by laws more general than the laws of quantum theory.

Since the entire apparatus of modern theory is based on the concept of a particle, there is no hope that it will be effective in problems connected with strong interaction. In this, in our view, lies the deepest reason why nonlocal and nonlinear theories, based on the modern method of quantization, which is fundamentally connected with the concept of a particle, do not justify the hopes placed upon them.

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