ON NONLOCAL FIELDS AND THE COMPLEX NATURE OF “ELEMENTARY” PARTICLES

(Dynamically Deformable Form Factor)

M. A. Markov

At the present time there are two sharply different directions in field theory. One direction is connected with various kinds of “subtraction operations”*), with various methods of regularizing the known divergent expressions.

The other direction is connected with the ideas of a nonlocalized field, with various attempts to introduce into consideration the extension of elementary particles.

In contrast to the second conception, the circle of ideas of the first direction is not connected with consideration of the extension of elementary particles. At the basis of all the subtraction prescriptions known so far, of methods of regularization and of the renormalizations of world constants associated with it, there lies a purely physical hypothesis of fundamental significance. We have in mind the assumption of the pointlike nature of elementary particles. In other words, in these theories the notion of the pointlike nature of the interaction of fields, so characteristic of all modern theory, is preserved.

Therefore all the subtraction operations known so far, all formal methods of regularization, leave unchanged all finite results of ordinary perturbation theory. For example, the expression for the cross section of the Compton effect in its first approximation remains unchanged.

Conversely, all theories connected with notions of extended elementary particles inevitably change the expressions of all finite effects for small wavelengths and small relative distances between particles. In the latter case, for example, the cross section for the Compton effect will contain, as a factor, an expression that rapidly

*) We have in mind the Heitler integral equation, the Dirac λ-process, in particular the known methods of regularization, etc.

decreasing with the frequency of the $\gamma$-quantum. This factor reflects the dimensions of the particle in momentum space.

The “elementary lengths” (i.e., the “dimensions” of elementary particles) that have ever been discussed in such theories are lengths connected with known combinations of universal constants:

\[ \frac{\hbar}{\mu c};\qquad \frac{e^2}{mc^2};\qquad \frac{\hbar}{Mc};\qquad \frac{g^2}{\mu c^2};\qquad \frac{g^2}{Mc^2}, \tag{1} \]

where $\mu$ is the meson mass, $e$ the electric charge, $g$ the nuclear charge, $m$ the electron mass, and $M$ the nucleon mass. All these lengths are of the order $10^{-13}$—$10^{-15}\ \text{cm}$.

The next conceivable “boundary of elementary lengths” is many orders of magnitude removed from this one*).

It is essential to note that at the present time experiment is passing precisely through this first boundary (1) of possible “elementary lengths.” In experiments on the production of mesons by $\gamma$-quanta the photon wavelength is $\lambda < 10^{-13}\ \text{cm}$; in experiments on the collision of fast nucleons the “impact parameters” are $< 10^{-13}\ \text{cm}$, and so on.

Thus, the choice between two classes of possible theories (i.e., theories that treat elementary particles as pointlike or as extended) is at present becoming also an experimental question.

The latter circumstance substantially changes the entire situation in the theory of elementary particles and, at the same time, casts the whole problem of the search for a new theory in a somewhat different light.

Indeed, until very recently, the essentially only criteria for evaluating the theories under consideration were their internal consistency and logical completeness while satisfying the general requirements of relativistic covariance.

With the availability of the corresponding experimental possibilities, an entirely different situation is now being created, one that it is expedient to use rationally.

In § 3 we shall develop and make more concrete these last considerations. As for the internal logical completeness of the two competing directions in contemporary theory that we are considering, with respect to the first of them we shall confine ourselves to brief remarks, while we shall examine in greater detail the present state and possibilities of the second direction.

The formal prescription techniques of regularization have, admittedly, attained a high degree of perfection, but they continue to raise a number

\[ \text{*) We have in mind the gravitational radius of an elementary particle,} \]
or a possible electromagnetic radius of the electron when vacuum polarization is taken into account, $r_0 \sim 10^{-58}\ \text{cm}$.

doubts and a feeling of dissatisfaction even in a purely internal logical aspect.

Indeed, one takes an equation which in this theory is itself considered meaningless,*)

\[ \frac{\partial \varphi}{\partial t}=\infty, \tag{2} \]

then formal solutions of this equation are written down. Naturally, these formal solutions are also devoid of meaning: they diverge. Next, by formal devices, this “solution” is given the meaning of a convergent expression. The original “solution” changes so sharply that it no longer satisfies the original equation.

Here there is at least a logical leap which requires comprehension.

It may very well be that it would have been more consistent to abandon the original equations and learn to compose the necessary “solutions” without them, precisely by a purely recipe-like route. But, unfortunately, in considering a number of questions we are forced to return again and again to the original differential equations.

In other words, it is not possible to break away completely from the original differential equations. In any case such a program has not been carried out.

Perhaps, on the contrary, one should seek such equations whose solutions would be precisely those expressions which are obtained, with the aid of regularization methods, from the divergent solutions of existing equations.

But so far such equations have not been found. Perhaps in our assessment of the formal recipes of regularization a certain conservatism of thought plays some role, and this whole system of recipe-like recommendations should be viewed from some other angle and seen in it a closed system of new propositions; but as yet attempts of this kind, i.e. attempts to comprehend the whole situation from this point of view, are also absent.

The real successes of formal regularization methods are not so considerable as to ignore all these circumstances that do not satisfy us. The point is that the recipes of regularization are still organically adapted only to perturbation theory.

As is known, there is a suspicion that the corresponding series after regularization and renormalizations diverge even in electrodynamics. But, most importantly, the direction under discussion has so far not produced practically important results in the field of meson effects.

*) Unfortunately, up to now the existence of divergences in the strict sense has not been proved, since we are unable to solve the strictly basic equations of the field, and here surprises are possible.

M. A. MARKOV

§ 1. EXISTING ATTEMPTS TO REGARD ELEMENTARY PARTICLES AS EXTENDED

(a dynamically nondeformable form factor)

All known attempts to construct a field theory free of the difficulties associated with divergences, by regarding elementary particles as extended, lead ultimately to the use of a certain form factor characterizing the extension of the elementary particle[^1]. Attempts in this direction are characterized by the following feature: all the form factors proposed in them are given by functions whose form does not change under the influence of the acting forces*). If one may put it so, the extended particle in these theories possesses an absolutely rigid structure.

It is natural that a signal in such a medium (inside the particle) propagates, in contradiction with the theory of relativity, with infinite velocity, and it is precisely here that the source of the failures of the various attempts in this direction is to be found.

There are various attempts at a mathematical formulation of the same physical idea of the extension of elementary particles. Let us consider some of them.

a) Extension of particles within the framework of Hamilton’s method

The infinite-time formulation of Hamilton’s method completes the process of improving the successive relativistic notation of equations. The well-known Tomonaga–Schwinger equation:

\[ i\hbar c\,\frac{\delta \psi(\sigma)}{\delta \sigma(x)}=H(x)\psi(\sigma) \tag{3} \]

is a concise notation for the infinite system of Tomonaga equations:

\[ i\hbar c\,\frac{d\psi}{dt_{xyz}}=H(xyzt)\psi, \tag{4} \]

where \(H\) is the interaction energy density. The infinite system of equations (4) is compatible if, as is known, the integrability conditions are satisfied**):

\[ H(x)H(x')-H(x')H(x)=0. \tag{5} \]

Here \(x\) and \(x'\) are space-time points. In the case, for exam-

*) Of course, kinematically, in accordance with Lorentz transformations, the form factor is deformed.

**) For simplicity of discussion we assume that the function \(H(x)\) contains no derivatives of the field.

... for example, a scalar field \(U\), interacting with the spinor field \(\psi_1\), \(H(x)\) has the form

\[ H=g\psi^{+}(x)U(x)\psi(x). \tag{6} \]

The integrability condition (5) is always reduced to conditions imposed on Poisson brackets of the form \([\psi^{+}(x)\psi(x')]\) and \([U(x)U(x')]\). These Poisson brackets are expressed in terms of known \(\Delta\)-functions, which indeed vanish on the spacelike surface \(\sigma\), as a consequence of which, instead of the infinite system of equations (4), the possibility arises of a compact notation (3).

The consecutively relativistic notation of the equations in the form (3) is incompatible with any known attempt to introduce into the theory some relativistically invariant form factor. In these cases the integrability conditions of equations (4) turn out not to be satisfied. Indeed, the interaction of the scalar field \(U\) with an extended source in the form*)

\[ H(x)=g\int \psi^{+}(x)U(x'')\psi(x)F(xx'')\,dx'', \tag{7} \]

where \(F(xx'')\) is a function invariant with respect to the full Lorentz transformation group.

In the case (7) the brackets \([U(x)U(x')]=\Delta(x-x')\) are replaced by brackets of the quantities

\[ \left[\int U(x'')F(xx'')\,dx'';\ \int U(x''')F(x'x''')\,dx'''\right]\ne \Delta(x-x'). \tag{8} \]

The purpose of introducing the form factor \(F(x'x''')\) is to make the known divergent expressions finite. The appearance of divergences is connected with the singularities of \(\Delta(x-x')\) on the light cone.

If the form factor \(F(x'x''')\) eliminates the divergences of known integrals, then this same form factor inevitably eliminates the pole of the \(\Delta(x-x')\)-function. “Smearing out” a point particle, the form factor also smears out the \(\Delta(x-x')\)-function**). In other words, on the right-hand side there now stands an expression that does not possess the properties of \(\Delta(x-x')\)

*) The extension of both particles present in the interaction is written, as is known, by the more general expression:

\[ H(x)=g\int \psi^{+}(x')U(x)\psi(x'')F(x'x x'')\,dx'\,dx''. \]

**) As is known, \(D(x-x')\) in electrodynamics, for example, is related to the charge density in the following way:

\[ \rho(x-x')=e\delta(x-x')=e\frac{\partial}{\partial t'}D(x-x')\quad \text{for } t=t' \]

tend to zero on a space-like surface, i.e. in the case when

\[ (\mathbf{x}-\mathbf{x}_1)^2>c^2(t-t')^2. \tag{9} \]

Consequently, upon introducing the form factor \(F(xx'')\) in (7), the system of equations (4) becomes, generally speaking, inconsistent, while equation (3) becomes contradictory. The indicated inconsistency of the equations is a mathematical consequence of the possibility—admitted by the introduction of the form factor—of an infinite velocity of signal propagation.

Upon a more concrete examination of the question it is more convenient to introduce form factors in momentum space, and not in coordinate space. Then the scalar field \(U_F(x)\), acting on an extended particle, will be written in the form

\[ \int U(x'')F(xx'')\,dx''=U_F(x)= \]

\[ =\sum_k f(k)\,U(k)e^{ik,x}+f^{+}(k)\,U^{+}(k)e^{-ik,x}, \tag{10} \]

\(f(k)\) must be a function decreasing with the growth of \(k\) in such a way that the integrals, divergent in the theory of point interactions, would now be represented by convergent expressions.

The function \(f(k)\) must be invariant with respect to the full group of Lorentz transformations. The latter requirement will be fulfilled if \(f(k)\) is a function of an invariant containing the four-dimensional vector \(k\). The only such invariant is the four-dimensional product of the vector \(k\) with some four-dimensional vector \(l\). The vector \(l\) must necessarily refer to the characteristic of the particle (if \(l\ne k\)); otherwise a privileged coordinate system arises*).

If no new internal degrees of freedom are introduced, then such a vector \(l\) could be a vector proportional to the wave number of the particle**). Consequently, \(f(k)\) may be written in the form

\[ f\bigl(r_0^2(k,l)\bigr); \tag{11} \]

where \(r_0\) is a constant having the dimension of length. Assuming, for simplicity, the proper mass of the quantum of the scalar field equal to zero, we obtain for the bracket (8) the expression in explicit form:

\[ [U_F(x);\; U_F(x')]\simeq \]

\[ \simeq \int \sin\{k(\mathbf{x}-\mathbf{x}')-ck(t-t')\}\,f^2(k)\,\frac{dk_x\,dk_y\,dk_z}{k}; \tag{12} \]

*) For example, in empty space, a coordinate system in which \(l_4\ne 0,\ \mathbf{l}=0\), if the vector \(l\) is time-like.

**) In the general case the operator is \(l_x\sim i\dfrac{\partial}{\partial x}\).

when \(f(k)=1\) the expression on the right-hand side of (12) becomes the well-known Dirac \(D\)-function; \(f(k)\) “smears” the Dirac \(D\)-function.

Let, for example, \(f(kl)=e^{-k_1 l r_0^2}\). Then it is easy to see that the right-hand side of (12) will contain, besides the usual one, a second Dirac function \(D^{(1)}\), which does not vanish outside the light cone; namely, it will have the form

\[ \simeq \frac{1}{2}\{D(x-x'-i2b)+D(x-x'+i2b)\}+ \]

\[ +\frac{i}{2}\{D^{(1)}(x-x'-i2b)-D^{(1)}(x-x'+i2b)\}, \tag{13} \]

\[ b=r_0^2 l. \]

In a similar way one may consider all the known attempts to introduce a form factor within the framework of the Hamilton equation; moreover, they all prove to be, to the same extent, internally contradictory.

b) The extension of particles within the framework of the \(S\)-matrix method

The idea of abandoning the rigid restrictions of the Hamilton method and replacing it by a certain “computational scheme” arose immediately after the general difficulties of introducing extended particles within the framework of the Hamilton equation\(^2\) (1940) had become clear. But decisive successes in this direction have not been achieved up to the present.

The existing proposals, briefly speaking, amount to introducing a form factor into the \(S\)-matrix obtained from the ordinary theory of point interactions, and considering the resulting \(S\)-matrix independently of the initial equations.

In the case of point interactions the \(S\)-matrix is written, as is known, in the following form:

\[ S=1+\sum_{n=1}^{\infty} S_n, \]

where

\[ S_n=(-i)^n \int_{-\infty}^{+\infty}\cdots\int_{-\infty}^{+\infty} H(x_1)\,\theta^+(\sigma_1\sigma_2)\,H(x_2)\,\theta^+(\sigma_2\sigma_3)\cdots \]

\[ \cdots H(x_n)\,dx_1\cdots dx_n \tag{14} \]

and

\[ \theta^+(\sigma_1\sigma_2)= \begin{cases} 1, & \text{if } \sigma_1 \text{ is after } \sigma_2,\\ 0, & \text{if } \sigma_1 \text{ is before } \sigma_2. \end{cases} \tag{15} \]

The invariance of the properties of \(\theta^+\) (15) is closely connected with the finite velocity of signal propagation and, ultimately, with the principle of causality.

It is essential to emphasize that the properties (15) have meaning in all coordinate systems because, and only because, on a space-like surface precisely condition (5) is satisfied:

\[ [H(x_1)H(x_2)]=0. \tag{5′} \]

In the case of nonlocal interactions (when there is a form factor \(F\) in \(H(x_1)\) and \(H(x_2)\)), relation (5′) is not satisfied on a space-like surface; in this case the invariant meaning of the assertions “before” and “after” is lost, the ordering in time begins to depend on the coordinate system, and the \(S\)-matrix in the form (14) loses its meaning*). \(S\)-matrices in a form different from (14) have so far not been analyzed objectively. It is still unclear whether the possibilities of the \(S\)-matrix method for introducing an extension of particles are indeed broader than the possibilities of the Hamiltonian method. The point is that in the \(S\)-matrix method the general requirements that such a mathematical apparatus must satisfy have not yet been formulated with sufficient clarity. It should be borne in mind that also for the Hamiltonian method a consistent relativistic formulation has been found only recently, in the Tomonaga–Schwinger equations. Great caution is required in questions of relativistic invariance. The simultaneous formalism, for example, or even the many-time Dirac–Fock–Podolsky equations for a single particle in a scalar field, do not formally contradict the introduction of a relativistically invariant form factor. But a consistent consideration of this question within the framework of the Tomonaga–Schwinger equation leads here to internal contradictions.

Recently, attempts have been intensively developed to introduce a form factor into the equations of motion of the Heisenberg representation. The most consistent attempt is given in work \(^{3}\). But a priori it is obvious that the difficulties of representing the interaction, the cause of which is physically clear, cannot disappear under another, equivalent mathematical formulation. There are attempts to localize deviations from the requirements of relativity theory to a “small region” \(^{4,5}\). In principle this can be done by using a rapidly decreasing form factor, or even a form factor with discontinuous functions: \(F(x-x')=0\), if \(|\mathbf{x}-\mathbf{x}'|>r_0\). This can be achieved most simply by introducing, for example, such a form factor, which leads to a shifted argument of a \(\Delta\)-function of the same type as occurs in the \(\lambda\)-process:

\[ [U(x)U(x')]=\frac{1}{2}\{D(x-x'+\lambda)+D(x-x'-\lambda)\}. \]

But here \(\lambda\) is not made to tend to zero, but is proportional, for example, to the electron four-momentum \((\lambda\sim p)\).

*) If the interval \(x_1-x_2\) is time-like, then the sign of \(t_1-t_2\) is preserved in all coordinate systems. If \(x_1-x_2\) is a space-like interval, then \(t_1-t_2\) can change sign under transformations. But for point interactions in this case (14) vanishes owing to the fulfillment of (5′).

Under these conditions the Tomonaga–Schwinger equation (3) is not satisfied, but the infinite system of Tomonaga equations (4) is compatible for all points satisfying the condition

\[ |t-t' \pm \lambda_4|c < |\mathbf{X}-\mathbf{X}' \pm \lambda|. \tag{a} \]

But it is clear that, even for \(t=t'\), in the region

\[ |\lambda_4|c > |\mathbf{X}-\mathbf{X}' \pm \lambda| \tag{b} \]

the system of equations (4) is already incompatible. Of course, in this region \(\lambda\) one may abandon the Hamiltonian method and the finiteness of signal propagation, but the latter purely negative assertion must be supplemented by some positive content: either by some generalization of Lorentz transformations, or by some particular features of a new theory, “limiting” the accuracy of a macroscopic verification of the velocity of signal propagation, so that the latter circumstance would not be connected with the level of experimental possibilities. In quantum theory there is, for example, no unambiguous connection between the past and the future (in the sense of classical mechanics), not because the principle of causality is invalid, but because the coordinate and momentum do not simultaneously characterize exactly the state of a particle. Likewise here, if the Lorentz transformation is preserved in its former form, it is necessary in any case that the accuracy of carrying them out should be limited by some suitable moments, for example by the atomism of charge², etc., so that a greater accuracy would not correspond to the nature of the phenomenon.

The most important thing, however, is that in the region (b) there is no compatibility of the equations, i.e. there is no mathematical apparatus in the region that is most interesting and essential for the present question. True, in the region (b) there is the trivial solution \(\psi=0\), but it must naturally be “joined” with the \(\psi\)-function in the region (c); the difficulties arising here have not yet been overcome.

Form factors that decrease rapidly with distance lead to the result that, in the macroscopic region, for macroscopic distances \(\mathbf{X}-\mathbf{X}'\), the integrability conditions (5) are fulfilled only approximately. But, strictly speaking, the equations remain incompatible, and even the “approximate” lawfulness of using the system of equations in the case where the bracket (5) differs little from zero has not been mathematically investigated.

In other words, at present there exists no consistent mathematical apparatus adequate to the physical considerations set forth above. The last remark applies equally to the \(S\)-matrix method, since an infinite interval for physics means macroscopic distances.

In view of the fact that in the writing of (11) the form factor \(f(r_0^2(k,l))\) depends on \(l \sim \dfrac{\partial}{\partial x}\), where \(x\) is the coordinate of the source of the field, the same pecu-

...features of the interaction of the field with the source, one may introduce additional commutation relations between the amplitudes of the field and the coordinate of the source, for example \(^{2,4}\),

\[ U(k)x_\nu - x_\nu U(k) = -ir_{\nu k}U(k); \]

whence

\[ [x_\nu [x_\nu U(k)]] = (ir_{\nu k})^2 U(k). \]

For the case \(\sum (ir_\nu)^2 = \lambda^2\) this theory was investigated in detail by Yukawa \(^{6}\), but, as is known, here as well the integrability conditions of equations (4) are not satisfied (see the Appendix).

Finally, one may point out one more general consideration of an experimental character against all kinds of form factors*). Indeed, the purpose of introducing form factors is to eliminate divergences. Suppose we have a divergent integral of the form

\[ \int k^2\,dk \sim k^3,\qquad k \to \infty. \]

For the convergence of such integrals, form factors are required which decrease faster than \(\dfrac{1}{k^{3/2}}\). However, in the presence of such form factors, the interaction with the field will rapidly decrease with increasing energy of the colliding particles (see § 3). As a consequence of this circumstance, perturbation theory at high energies, for example for meson fields, should become more and more applicable, and the cross sections themselves should tend to zero**).

This conclusion is in clear contradiction with the data on cosmic rays, with the weak penetrating power of primary radiation of extremely high energy.

*) More precisely, against all form factors that preserve the Hermiticity of the Hamiltonian function.

**) For example, in the scattering of protons by protons (pseudoscalar meson field), the characteristic factor \(f_1^2 f_2^2\) appears in the terms of the differential cross section:

\[ \frac{d\sigma}{d\Omega} = \frac{1}{4\pi} \left(\frac{g^2}{\hbar c}\right)^2 \frac{k^4}{\varepsilon^2} \left\{ \left[ \frac{\sin^2 \dfrac{\theta}{2}} {4k^2 \sin^2 \dfrac{\theta}{2}+\varkappa^2} \right] \right\} f_1^2 f_2^2+\ldots, \]

where \(f_1^2\) and \(f_2^2\) are functions decreasing with increasing argument (the energy of the colliding particles):

\[ f_1=f_1[a^2(\omega_q E-\mathbf{q}\mathbf{p})];\qquad f_2=f_2[a^2(\omega_q E+\mathbf{q}\mathbf{p})]. \]

Here \(\mathbf{p}\) is the momentum of the particles in the center-of-inertia system before the collision, \(\mathbf{q}=(\mathbf{p}\mp \mathbf{p}')\), \(\mathbf{p}'\) is the momentum after the collision, \(\omega_q=\sqrt{\mu^2c^2+q^2}\), and \(\mu\) is the meson mass.

At high energies \(\omega_q \sim |q|\) and \(E \sim p\). If \(f_1(0)\sim 1\), then in every...

§ 2. DYNAMICALLY DEFORMABLE FORM FACTOR

The question naturally arises: does there exist such a class of form factors that would lead to a signal propagation velocity \((v)\) along an extended particle less than or equal to the velocity of light:

\[ v \leq c. \tag{16} \]

There is an answer to this question, but, as follows from what comes later, it entails a quite different interpretation of the concept of an elementary particle in comparison with the usual one.

Form factors characterizing such models of distributed charges must change under the influence of external forces*). In accordance with this idea, corresponding “equations of motion” must be written for the form factors themselves. This conclusion follows logically from an analysis of the character of the failures of all theories with nondeformable form factors.

In constructing a theory with a dynamically deformable form factor one may proceed in various ways**).

Making the preceding considerations more concrete, let us discuss one particular possibility, which is connected with the following remark:

To write equations of motion for a form factor (for some new function \(F(x)\)) means, in essence, generally speaking, to introduce into consideration a new field \(F\).

Modern theory has at its disposal a large number of different fields, and it is therefore natural to try not to introduce new ones, but to use the existing fields as such “mutual form factors.”

Apparently, the actual situation in modern field theory is partly prepared for such a formulation of the question, if all interacting fields are considered in their universal interrelation with one another.

in which case \(f_2 = f_2(2\omega_q E_q)\), i.e. the cross sections fall with increasing energy of the colliding particles and with increasing energy transferred in the collisions. Taking higher approximations into account (the factorial growth of the number of chains) should not alter this circumstance. Indeed, if the growth in the number of chains proves essential for real processes, then it is essential also for the calculation of the self-mass. In other words, the form factor must then be chosen from the very beginning so as to take this circumstance into account, i.e. with a stronger dependence on energy–momentum.

*) If the ideal image of an absolutely rigid body was a physical model for all preceding attempts to represent an extended particle by means of a form factor that does not change under the action of external forces, then the image of a charged “cloud” or “liquid drop” might serve as a model of such an extended particle, a signal along which would propagate with velocity \(v < c\).

**) It is possible, for example, formally to regard a particle as a “liquid drop” or “cloud” [7]. For the present only one thing is clear: within the framework of differential equations there are certain possibilities which have not yet been investigated.

M. A. MARKOV

“Mutual Form Factors”

Let us consider, as an example, the proper electromagnetic mass of the proton. Within the framework of the existing theory this problem is formulated as follows: the equation of motion of a “free” proton is written in the form

\[ \frac{\hbar}{i}\frac{\partial}{\partial t}\psi(x_p)+(\alpha p+\beta m_0 c^2)\psi(x_p)=0. \tag{17} \]

The electromagnetic field will give an additional increment to the proton mass, \(\Delta m_{\mathrm{e.m.}}\), due to the electromagnetic field. Consequently, the original equation (17) must contain not the experimental mass of the proton, but a certain “initial” hypothetical mass \((m_0)\), such that

\[ m_0=m_{\mathrm{exp}}-\Delta m_{\mathrm{e.m.}}. \]

From the modern point of view, every elementary particle interacts (directly or indirectly) with all fields. Consequently, each of these fields contributes its share to its proper mass. The mass of the “free” proton \(m_0\) is that mass which is due to the resultant effect of all fields except the given one (i.e., except the electromagnetic one).

If this mass \(m_0\) is also of field origin, then it is distributed over the space in which all the other fields are excited.

The electric charge of the hypothetical (initial) proton (17), to which the electromagnetic field is then “connected,” is determined by other electrically charged fields surrounding the proton. This charge cloud has a complex structure; it is determined by charged \(\pi\)-mesons, other kinds of charged mesons, and finally by the charged \(\beta\)-field.

Thus before us there arises the image of a truly extended “bare” proton (i.e., a proton before the “connection” of the electromagnetic field), that is, of the proton which ordinarily, in essence phenomenologically, we try to describe (very roughly) by the free equation (17).

It is very natural and physically attractive to use the function \(F(x)\), characterizing this charge cloud, as a form factor for the interaction of the electromagnetic field with the proton*).

The great complexity of “elementary” particles is gradually being realized by us, but, apparently, the moment is coming when decisive conclusions must be drawn from this circumstance.

The idea of the extendedness of particles in its usual aspect is too classical; in essence it presupposes parts of a whole, endowed—

\[ \text{*)} \]

*) Hence it is clear that the idea of “mutual form factors” has no relation to the ideas of so-called “realistic regularizations,” within the framework of which the pointlike nature of particles is preserved, while divergences are absorbed by the features of various fields.

...having the same physical nature as the whole. This circumstance is usually concealed in the idea of a dynamically nondeformable form factor, but in the aspect of a dynamically deformable form factor there inevitably arises the question of the “structure of the drop,” of the structure of the “cloud.” It is desirable to answer this question not in the sense of mechanical divisibility, but in the spirit of contemporary theories, which establish the closest connections between different fields.

From the preceding analysis it follows that equation (17) must be regarded, in essence, as a phenomenological equation of motion of the center of gravity of a “complex system.”

If one follows this path consistently, then the proton coordinate \(x_p\) should be regarded as the value of the coordinate averaged over the form factor

\[ x_p=\int xF(x)\,dx . \tag{18} \]

To illustrate a possible theory, let us simplify the problem by assuming that the distribution of electric-charge density around the proton is determined entirely by the charged meson field \(\varphi\). In other words,

\[ x_p=\int x\rho(x)\,dx, \tag{19} \]

where \(\rho(x)\) represents the meson density. In the case of scalar mesons

\[ \rho(x)\simeq ie\left(\dot{\varphi}^{*}\varphi-\varphi\dot{\varphi}^{*}\right). \]

The interaction of the proton (more precisely, of the meson cloud) with the electromagnetic field will now be written in the form

\[ H'=-e\int \Phi_0(x^\mu)\rho(x_p,x^\mu)\,dx^\mu+ \]

\[ +ea\int \Phi(x^\mu)\rho(x_p,x^\mu)\,dx^\mu, \tag{20} \]

and the whole equation will be rewritten as follows:

\[ \left\{\frac{\hbar}{i}\frac{\partial}{\partial t}+\alpha p+\beta m_0c^2+H'\right\}\Psi(x_p)=0, \tag{21} \]

where

\[ x_p=\int x\rho(x)\,dx . \]

The equations for the electromagnetic field will be written in the form

\[ \begin{aligned} \square \Phi &= ae\rho(x),\\ \square \Phi_0 &= e\rho(x). \end{aligned} \tag{22} \]

But subsequently we must write the equations of motion for the meson field. For a “free” meson field we have:

\[ \Box \varphi(x^\mu)=0, \tag{23} \]

where, in view of the preceding discussion, it is natural to assume that

\[ x^\mu=\int x f(x)\,dx \tag{24} \]

(\(f\) is a form factor characterizing the dimensions of the \(\pi\)-meson).

At present we do not know what field mainly determines the dimensions of the \(\pi\)-meson. It is not excluded that such a field could be, for example, the nucleon field

\[ f(x)=\psi^*(x)\psi(x). \tag{25} \]

The presence of the \(\pi\)-meson in the state of a nucleon–antinucleon pair*) illustrates the possible character of such representations.

In order to write a closed system of equations, it is necessary to write the equation of the meson field (23) with a right-hand side, since only from this equation can one determine the density \(\rho(x^\mu)\) of the meson field in the presence of a real nucleon.

In the theory of point interactions, on the right-hand side of equation (23) there stands a \(\delta\)-function or derivatives of it.

In the case of a scalar meson field and a point interaction of the meson and nucleon fields, we would have

\[ \Box \varphi = g\,\delta(r_\mu-r'_\mu) \tag{26} \]

and the known static solution**)

\[ \varphi \sim g^2\frac{e^{-kr^\mu}}{r^\mu}, \tag{27} \]

or rather, its analogue in the case of an extended source, could be used in (19) and (21) as the form factor for the interaction of the proton with the electromagnetic field.

Naturally, for the interaction of the meson field with nucleons we must also introduce a form factor representing the dimensions of the nucleon in relation to the meson field.

Recently a large number of different meson fields have been discovered, but as yet, with respect to these fields in relation to the \(\pi\)-meson field, there is no

*) A particular case here is the nucleon–antinucleon model of the meson proposed by Fermi–Yang\({}^8\), or the nucleon–\(\mu\)-meson model of Wentsel\({}^9\).

**) In the general case, of course, one must take the nonstatic solution.

clarity*); therefore, concrete statements about the form factor in the interaction of $\pi$-mesons with nucleons can hardly be sufficiently definite at the present time.

It is clear from physical considerations that correctly written equations with dynamically deformable form factors must be compatible with, and not contradict, the theory of relativity.

The existence of charged macroscopic bodies vividly illustrates this proposition.

This does not mean that equations of the form (21), written for the motion of the center of gravity of a particle, are correct in this sense. Consistent equations in this sense must be written for the motion of the “cloud”; in particular, they may be of the type of kinetic equations.

The preceding considerations are perhaps not so much an attempt to illustrate the character of a possible theory as to emphasize the phenomenological, very rough features of the existing theory of elementary particles. In this connection we shall make one more remark.

In the modern theory, a fundamental role is played by various kinds of four-dimensional $\Delta$-functions. For a meson field, for example,

\[ [\psi(x_\mu)\varphi(x'_\mu)] \simeq \Delta(x_\mu - x'_\mu). \tag{28} \]

Relations of this kind must be regarded from a new point of view as coarsened, phenomenologized relations, coarsened to the same extent as the equations

\[ \square \varphi(x_\mu)=0 \tag{29} \]

for a “free” meson field. Indeed, on the basis of (23), the characteristic properties of commutation brackets and $\Delta$-functions refer to “certain averaged” coordinates: $\bar{x}_\mu$ and $\bar{x}'_\mu$.

In other words, relations of type (28) are valid when the dimensions of the cloud of the $\psi$-field (over which, by the averaging (25), $\bar{x}_\mu$ and $\bar{x}'_\mu$ are obtained) are small in comparison with the distance $x_\mu - x'_\mu$.

Strictly speaking, different fields (for example, $\varphi$ and $\psi$) also should not commute in small regions. Consequently, in the right-hand side of (28) there should stand some complicated function of the Green functions or $\Delta$-functions of all the fields. Gradually we arrive at the idea—

*) Apparently, most new types of mesons decay into $\pi$-mesons, i.e., there must exist a close connection between these fields and $\pi$-mesons. How far this connection goes at the present time is not clear. For the time being, such a possibility is also not excluded: owing to the strong interaction of $\pi$-mesons with one another, they combine temporarily into mesons of larger mass, which would be very natural in the aspect of the Fermi–Yang hypothesis of the nucleon–antinucleon structure of the $\pi$-meson. In other words, parallel to matter consisting of nucleons, one may suppose a short-lived formation of nucleons and antinucleons, which manifests itself in the form of heavy mesons.

to the notion according to which all fields make their contribution to the image of any elementary particle, and different particles represent certain states of this common field. At present it is not clear to what extent this property of the noncommutativity of different fields can be formulated explicitly and used for constructing a consistent theory (i.e., on another basis we return to the idea of nonlocalizable fields), formulated with the aid of new permutation relations.

It is essential to emphasize that the aggregate system of equations describing the mutual relations between fields makes it possible, in principle, to eliminate all fields except the given one, or all fields except two given fields interacting with one another.

Indeed, from the second equation (22) one can determine, for example, \(\Phi_0\):

\[ \Phi_0(x)=e\int \rho(x')G(xx')\,dx', \tag{30} \]

where \(G(xx')\) is the corresponding Green’s function. \(\Phi_0\) in the form (30) may be substituted into (20), for example,

\[ -e\int \Phi_0(x_\mu)\rho(x_p x^\mu)\,dx^\mu = \]

\[ = -e\iint \rho(x')\rho(x_p x^\mu)G(x^\mu x')\,dx'\,dx^\mu . \tag{31} \]

If one recalls that

\[ \rho(x)=ie(\varphi^{**}\varphi-\varphi\varphi^{**}), \]

where \(\varphi(x)\) is the meson field, then expression (31) already contains, under the integral sign, the fourth power of the meson field. Nonlinear interactions of this kind, and nonlinear equations in general (when the meson field is eliminated through the nucleon field), must be characteristic for a consistent treatment of one or a limited number of interacting fields.

Eliminating, for example, all fields except the nucleon ones, and taking the mass of the initial “bare” nucleon \(m_0=0\), we should in principle obtain the experimental values of the nucleon masses and the equation of motion of nucleons free from external fields.

The same should also hold for all other fields. Nonlinear equations of this kind may also serve as the starting point for searches for the apparatus of a new theory.

Thus, if the idea of mutual form factors corresponds to reality, then before us lies only the beginning of a very long path toward the creation of a consistent theory of the interaction of fields, the number of which increases rapidly.

We see that the attempt to consider consistently the idea of “mutual form factors” grows into a very complex system of rela-

related equations. The experience of the development of science shows that in those cases when a problem is complicated in an essential way, it is practically solved by other, more adequate methods, even if approximate ones, while a strict, consistent formulation of the problem remains only an ideal case of a correctly formulated problem*).

Let us consider separate cases of approximate approaches to the strict problem.

§ 3. PHENOMENA IN THE REGION OF APPROXIMATE APPLICABILITY OF THE UNDEFORMED FORM FACTOR

In the region of relatively small energy actions on an elementary particle, one may neglect the deformation of the form factor and, taking into account the dimensions of elementary particles by means of some simple “smearing,” make qualitative estimates of the characteristic influence of particle sizes in various effects.

The hypothesis of a deformable form factor provides a justification for the legitimacy of such estimates within certain energy limits. The expediency of such estimates is dictated by the present state of experiment. As was emphasized above, in experiment at the present time there appear precisely such lengths as lie on the scale of possible dimensions of elementary particles.

The influence of dimensions on the character of effects is very specific. Let us consider several simple examples.

a) Scattering of \(\pi^+\)-mesons by a proton

For simplicity of consideration, let us choose the factor (11) in the form**)

\[ f=be^{-r_0^2(k,l)}, \]

where \(k\) and \(l\) are, respectively, the wave numbers of the meson and the proton,

\[ b=e^{\,r_0^2 \frac{mc}{\hbar}\frac{Mc}{\hbar}} \]

is a normalizing coefficient which ensures that \(f(k)\) is equal to unity at small meson momenta. In the center-of-inertia system \(\mathbf{k}=-\mathbf{l}\); consequently, in the scattering of a positive meson in the first act of interaction there will enter the factor

\[ f_1=be^{-r_0^2\left\{\left(\frac{m^2c^2}{\hbar^2}+k^2\right)^{1/2}\left(\frac{M^2c^2}{\hbar^2}+k^2\right)^{1/2}+\mathbf{k}\mathbf{k}'\right\}} . \tag{32} \]

(The proton with momentum \(\hbar \mathbf{l}=-\mathbf{k}\hbar\) emits a meson with momentum \(\hbar \mathbf{k}'\).)

) For example, the classical problem of many interacting bodies.
*) If one does not confine oneself to purely illustrative purposes, then it is advisable to choose \(f((k,l))\), where \(n\) is such that the factor is also suitable for the case of negative energies. \(k\) and \(l\) should be successively taken from the corresponding Feynman diagrams.

In the second act of interaction there will enter the factor

\[ f_2=b e^{-r_0^2\left\{\left(\frac{m^2c^2}{\hbar^2}+k^2\right)^{1/2} \left(\frac{M^2c^2}{\hbar^2}+(\mathbf{k}'+\mathbf{k})^2\right)^{1/2} +(\mathbf{k}'+\mathbf{k})\mathbf{k}\right\}}, \tag{33} \]

(the nucleon with momentum \(\hbar(1-\mathbf{k}')=-\hbar(\mathbf{k}+\mathbf{k}')\) absorbs the initial \(\pi^+\)-meson with momentum \(\hbar\mathbf{k}\)). The differential scattering cross section of positive mesons on a proton will acquire the additional multiplier

\[ \left(\frac{d\sigma^+}{d\Omega}\right)_f = \frac{d\sigma^+}{d\Omega}(f_1^2 f_2^2)^+ . \tag{34} \]

Choosing, for simplicity of notation,

\[ r_0^2=\alpha\frac{\hbar}{mc}\frac{\hbar}{Mc};\qquad \frac{M}{m}=\varkappa\sim 6,\quad \alpha\sim 1, \tag{35} \]

we obtain

\[ \left. \begin{aligned} f_1&= e^{-\alpha\left\{(1+k'^2)^{1/2}\left(1+\frac{k^2}{\varkappa^2}\right)^{1/2} +\frac{\mathbf{k}\mathbf{k}'}{\varkappa}-1\right\}},\\[4pt] f_2&= e^{-\alpha\left\{(1+k^2)^{1/2} \left(\frac{(\mathbf{k}'+\mathbf{k})^2}{\varkappa^2}\right)^{1/2} +\frac{(\mathbf{k}'+\mathbf{k})\mathbf{k}}{\varkappa}-1\right\}}, \end{aligned} \right\} \tag{36} \]

where now \(\mathbf{k}\) and \(\mathbf{k}'\) are the momenta of the initial and final mesons in energy units \(mc^2\). Considering meson kinetic energies \(\sim 1\) (i.e. \(mc^2\)), we may neglect \(\frac{k^2}{\varkappa^2}\) and \(\frac{(\mathbf{k}'+\mathbf{k})^2}{\varkappa^2}\) in comparison with unity.

Noting that in the c.m. system \(|\mathbf{k}|=|\mathbf{k}'|\), \(E=\sqrt{1+k^2}\), we obtain:

\[ (f_1^2 f_2^2)^+ = e^{-4\alpha\left\{E-1+\frac{E^2-1}{2\varkappa}(1+2\cos\theta)\right\}} . \tag{37} \]

For small \(k\), \(f_1, f_2 \simeq 1\), and the cross section (37) coincides with the cross section obtained for a point interaction. For a pseudoscalar meson with pseudovector coupling, the scattering cross section grows with the energy of the incident meson. Taking into account the dimensions of the nucleon \((f_1^2 f_2^2)\) leads to a decrease of the scattering cross section at high energies of the incident meson. The energy dependence of the total cross section must in this case contain a characteristic maximum.

For the differential cross section, the finite size of the nucleon should lead to a characteristic increase of the scattering of \(\pi^+\)-mesons backward and, in general, through large angles*). The momenta \(\mathbf{k}\) and \(\mathbf{k}'\), in absolute va-

*) In the scattering of \(\pi^-\)-mesons on protons, in the first act of interaction (in contrast to the scattering of \(\pi^+\)-mesons) absorption of a \(\pi^-\)-mes-

are equal to one another. If \(\mathbf{k}\) and \(\mathbf{k}'\) approximately coincide in direction (small scattering angles), then in the exponent \(kk'/\varkappa \sim k^2/\varkappa\), i.e., terms of the form appearing in (37), will diminish scattering through small angles.

If \(\mathbf{k}'\sim -\mathbf{k}\) (backward scattering), then the corresponding terms enter into (37) with a minus sign, and the factors suppress the cross section less strongly. The ratio of the corresponding cross sections has the form

\[ \left(\frac{d\sigma^+}{d\Omega}\right)_{f;\,\vartheta=0} \bigg/ \left(\frac{d\sigma^+}{d\Omega}\right)_{f;\,\vartheta=\pi} = \left[ \left(\frac{d\sigma^+}{d\Omega}\right)_{\vartheta=0} \bigg/ \left(\frac{d\sigma^+}{d\Omega}\right)_{\vartheta=\pi} \right] \times e^{-\frac{4\alpha\left(E^2-1\right)}{\varkappa}\,(\cos\vartheta-\cos\pi)} . \tag{38} \]

Strong scattering through large angles is a characteristic feature of the experimental data on the scattering of \(\pi^+\)-mesons*) on protons; smaller cross sections and relative isotropy are characteristic of the scattering \(\pi^-\to\pi^-\).

b) Production of charged mesons by photons

In this effect it is necessary to consider both the interaction of \(\pi\)-mesons with nucleons and the interaction, for example, of mesons with photons.

If the dimensions of the meson are determined mainly by the cloud of nucleon–antinucleon pairs, then the charge cloud of the \(\pi\)-meson has dimensions \(\sim \hbar/Mc\), and with respect to photons of energy \(\sim 400\) MeV may, in a rough approximation, be regarded as pointlike.

In the case of a pseudoscalar meson field with pseudovector interaction, the principal contribution to the cross section is given by the interaction \(H_{\mathrm{eg}}\), which, among other things, ensures the gradient invariance of the equations

\[ H_{\mathrm{eg}}\sim \gamma A\gamma_5\varphi . \tag{39} \]

zone. This circumstance leads to another expression for the form factor:

\[ \left(f_1^2 f_2^2\right)^- = e^{-4\alpha\left\{E-1+\frac{E^2-1}{\varkappa}\right\}} . \]

For the latter, the absence of angular dependence and a stronger decrease of \(\pi^-\) scattering in accordance with experiment are characteristic.

*) It is essential that

\[ \frac{\left(f_1^2 f_2^2\right)^+} {\left(f_1^2 f_2^2\right)^-} \simeq e^{2\alpha} \quad \text{for } \vartheta\sim\pi, \]

if \(\alpha\sim 1\), \(e^{2\alpha}\sim 7\), i.e., in agreement with experiment, scattering through large angles of \(\pi^+\)-mesons is much greater than scattering of \(\pi^-\)-mesons.

In this case, without an intermediate state there takes place the process of absorption of a photon and emission of a real meson with momentum \(k\). Consequently, in the same approximation as in the case of meson scattering, the differential cross section for the production of charged mesons on a nucleon will have the form (in the laboratory system)

\[ \left(\frac{d\sigma}{d\Omega}\right)_f = \frac{d\sigma}{d\Omega} e^{-2\alpha\{E_\mu-1\}} = \frac{d\sigma}{d\Omega} f_1^{2}, \tag{40} \]

where \((E_\mu-1)\) is the momentum of the produced meson, and \(\dfrac{d\sigma}{d\Omega}\) is the cross section obtained without applying the form factor.

The characteristic features of the influence of the form factor on the process under consideration are as follows:

1. The form factor (40) decreases the number of produced mesons with large momenta (at small angles).

2. The maximum of the cross section as a function of the \(\gamma\)-quantum energy is flatter and occurs at higher energies than in the case of meson scattering.

c) Photoproduction of neutral mesons

As calculations show, the observed cross sections for the photoproduction of neutral mesons in the region of small energies can be obtained if the effect is interpreted as the production of charged mesons and then their scattering on the nucleon with charge exchange into neutral ones. In the case of photoproduction of a neutral meson, the influence of the form factor reduces to the appearance of a sharper maximum in the dependence of the total cross section on the \(\gamma\)-quantum energy.

It is known that the experimental data concerning the production of charged mesons at small angles (with respect to the direction of the momentum of the \(\gamma\)-quantum) are characterized by a decrease of the cross section (as the angle decreases), which is in agreement with the preceding remarks. It is also known that all calculations (for weak and strong coupling) lead to an increase of the cross section at small angles.

If the dimensions of the \(\pi\)-mesons with respect to the electromagnetic field also lie in the region of the length \(\hbar/mc\), then the preceding consideration of photoproduction effects becomes somewhat more complicated.

The discussed extension of mesons with respect to the electromagnetic field can substantially change the usual interpretation of such a phenomenon as the absorption of slow negative mesons by protons. The process under consideration leads either to the appearance of a slow neutral \(\pi^0\)-meson, or of a single \(\gamma\)-quantum.

From a comparison of the two latter cross sections one can determine the ratio of the constants \(e^2/\hbar c : g_0^2/\hbar c\); it is known that from this, even for pseudovector coupling of the pseudoscalar neutral meson field, one obtains

turn out to be too large constants.

\[ \left(\frac{g_0^2}{\hbar c}\sim 10\right). \]

The introduction of a form factor weakens the interaction with the electromagnetic field, which reduces the effective value of the fine-structure constant \((e^2/\hbar c)\), thereby imitating a large value of the constant \(g_0^2/\hbar c\), if, at small momenta of the neutral meson (large wavelength), the dimensions of the nucleon are not substantial in comparison with the meson field. The absorption of slow \(\pi^-\)-mesons by a proton, however, gives ambiguous evidence concerning the electromagnetic dimensions of the meson in the region of lengths \(\hbar/mc\). Small constants \(g_0^2/\hbar c\) can also be obtained, as is known, in the case of an admixture of pseudoscalar coupling with the constant

\[ \frac{f_0^2}{\hbar c}\sim 1. \]

All the examples cited show that, in principle, the dimensions of elementary particles can already at the present time be an object of experimental investigation.

Unfortunately, all the preceding estimates relate to comparison with perturbation theory. True, when the form factor is taken into account, the applicability of perturbation theory is improved, but nevertheless, in the case of meson fields it is desirable to make the preceding comparisons with more exact solutions of the existing equations, for example taking into account radiation reaction, or with a more general solution obtained by Tamm’s method. Thus, an answer to the question we have posed can be given only by a more thorough quantitative consideration of the problem.

It is necessary to note one more specific feature of the developing ideas about elementary particles. These ideas allow for an entirely different character of the interaction of particles and fields in comparison with that which we are accustomed to deal with in the modern theory of elementary particles. One of the fundamental propositions of the existing theory is expressed in the fact that a process, for example the absorption of field quanta by free particles, does not occur in the first order of perturbation theory. The conservation laws of energy and momentum and the “elementarity of the particle” (the absence of excited degrees of freedom) make such a process impossible. The preceding treatment of an elementary particle as a very complicated system admits, in principle, interactions leading to real short-lived states of the “compound-nucleus” type.

Fermi’s well-known attempt to interpret the multiple production of particles in a single act may be regarded as an example precisely of such a new kind of interaction.

In Fermi’s theory, contrary to weak-coupling theory, the total cross section does not decrease with energy. Fermi regards his theory as a limiting case of a theory with a large coupling parameter. Such an interpretation is not clear without further qualification, since in the known calculated cases of interaction with a large coupling parameter (“strong coupling”), because of the large reaction of the radiation, the cross sections of the effects rapidly die out with increasing energy and, as is known, are small in comparison with the cross sections obtained in theories with weak coupling. In its time, interest in theories with strong coupling arose, as is known, in connection with the necessity of explaining the observed small scattering cross section of mesons in cosmic rays (subsequently these particles turned out to be \(\pi\)-mesons).

\[ * \qquad * \qquad * \]

At present, a more trivial solution of the question under discussion is also not excluded.

It may be that the dynamically deformable form factor should be regarded as a phenomenological account of all higher-order corrections of the modern theory in the case when all fields are considered in mutual coupling. This question can hardly be clarified without a substantial rationalization of the methods of solving the existing field equations.

In the case of electrodynamics, the striking fact is that taking account of vacuum polarization reduces the divergence of the self-energy to a logarithmic one. It is possible that a more exact solution of the existing equations of electrodynamics may eliminate the divergences altogether.

It is known, for example, that the (as yet imperfect) attempts to obtain a renormalized equation for two interacting fields lead to the appearance of a certain form factor. Naturally, this form factor must not be rigid; it is an “envelope” in the spirit of the idea of mutual form factors. Here a connection is suggested between the new ideas in renormalizations and deformable form factors.

APPENDIX

To the equation

\[ [x_\nu [x^\nu U]]-\lambda^2 U=0 \tag{I} \]

one may compare the equation

\[ [p_\nu [p^\nu U]]-(imc)^2U=0; \tag{II} \]

equation (II) is, as is easily verified, another-

... the usual form of the equation for a scalar field:

\[ \left(\frac{\partial^2}{\partial x_\mu \partial x^\mu}-\varkappa^2\right)U(x)=0, \]

where

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

Yukawa showed\(^6\) that equations (I) and (II) can be written in the nontrivial form:

\[ \left(\frac{\partial^2}{\partial X_\mu \partial X^\mu}-\varkappa^2\right)U(X_\mu,r_\mu)=0 \tag{II′} \]

and

\[ (r_\mu r^\mu-\lambda^2)U(X_\mu,r_\mu)=0, \tag{I′} \]

where

\[ X_\mu=\frac12(x'_\mu+x''_\mu); \]

\[ r_\mu=x'_\mu-x''_\mu. \]

Here the coordinate \(X_\mu\) may be interpreted as the coordinate of the particle, the coordinate of its “center of gravity,” while the relative coordinate \(r_\mu\) may be connected with the “dimensions” of the particle.

It is true, however, that the new vector \(r_\mu\) cannot so simply, without further qualification, be connected with the dimensions of the particle. Indeed, suppose we define it as a spatial or temporal vector. This means, for example, that there exists such a privileged coordinate system (in empty space) in which \(r_4=0\), and \(r_i\ne0\) (i.e. \(r^2=\lambda^2\)). Since this coordinate system is not physically distinguished in any way in the system of equations (I) and (II), we arrive at a privileged coordinate system in empty space, i.e. we arrive at a contradiction with relativity. The situation can be corrected by an additional condition linking the given coordinate system with the particle itself. In Yukawa’s view, the role of such a condition must be fulfilled by the additional equation introduced by him:

\[ r_\mu \frac{\partial}{\partial X_\mu}U(X_\mu,r_\mu)=0 \tag{III} \]

or *)

\[ k_\mu r^\mu=0. \tag{III′} \]

*) There exists a more general condition \(k_\mu r^\mu+b=0\), which in fact was used already in that paper,\(^2\) where the very idea of nonlocality was proposed.

Now the joint solution of equations \((\mathrm{I}')\), \((\mathrm{II}')\), and \((\mathrm{III}')\) is written in the form

\[ U(X_\mu r_\mu)= \]

\[ =\int\cdots\int (dk)^4 \varphi(k_\mu r_\mu)e^{ik_\mu X^\mu}\delta(k_\mu k^\mu+\varkappa^2)\delta(r_\mu r^\mu-\lambda^2)\delta(k_\mu r^\mu); \]

for a particle for which \(R_4=-\varkappa;\ \mathbf{k}=0\), the wave function takes the form

\[ \varphi(x_1 r_\mu)\delta(r_\mu r^\mu-\lambda^2)\delta(\varkappa k_4)e^{-i\varkappa X_1}, \]

where, consequently, on the basis of \(r_4=0\),

\[ r_1^2+r_2^2+r_3^2=\lambda^2. \]

The latter circumstance is interpreted by Yukawa in such a way that, in the coordinate system associated with the particle, the particle is a sphere of radius \(\lambda\). Unfortunately, this interpretation does not quite correspond to the actual content of the given theory. The point is that the condition \(r_4=0\) does not uniquely determine the coordinate system associated with the particle \((\mathbf{k}=0)\). Indeed, the requirement \(r_4=0\) determines a certain coordinate system, distinguished by the fact that in it all \(\mathbf{k}\) are perpendicular to \(\mathbf{r}\). The latter circumstance follows directly from the condition \(r_\mu k^\mu=0\) for \(r_4=0\).

It is easy to compute the Poisson bracket of \([U(X_\mu r_\mu); U(X'_\mu r_\mu)]\). Since the given commutator is an invariant, it is most simply computed in the coordinate system in which \(r_4=0\). Here, instead of the ordinary four-dimensional \(\Delta\)-function, one obtains the expression

\[ \Delta'=-\frac{2\pi i}{\lambda}\int_0^\infty \frac{k\,dk}{\sqrt{k^2-\varkappa^2}}\, J_0(kR\sin\theta)\sin\sqrt{k^2+\varkappa^2}\,X^4, \tag{IV} \]

where

\[ \mathbf{R}\mathbf{r}=rR\cos\theta. \]

Expression \((\mathrm{IV})\) is curious in that, for \(\mathbf{r}\) perpendicular to the radius-vector \(\mathbf{R}=X-X'\), the Poisson bracket under consideration is different from zero and does not depend on the magnitude \(|R|\). In other words, in contradiction to the theory of relativity, the two points \(X\) and \(X'\), under the given conditions, are always connected by a signal with superluminal velocity.

It is essential to note yet another feature of the nonlocal-field variant under consideration. Yukawa attempted to attach meaning to the extension of the particle itself, i.e. of a free particle, apart from any relation to the field interacting with it.

Subsequently it became clear11 that, by means of certain transformations, a nonlocal free field can be converted into a local one. Only in the case of interacting nonlocal fields is it impossible to carry out such a transformation. Thus, we return again to the idea of a nonlocal interaction, as it was initially formulated2.

References

1. G. Wataghin, Zeits. f. Phys. 88, 92 (1934), and others*).
2. M. Markov, ZhETF 10, 1311 (1940).
3. C. Bloch, Kgl. Danske. Videnskab. Selsk. kab., Mat.-fys. Medd. 24, No. 1 (1950).
4. D. Blokhintsev, ZhETF 18, 566 (1948).
5. M. Markov, ZhETF 21, 11 (1951).
6. H. Yukawa, Phys. Rev. 77, 219 (1949).
7. A. Vlasov, Theory of Many Particles. Gostekhizdat, 1950.
8. E. Fermi and C. Jang, Phys. Rev. 76, 1739 (1949).
9. G. Wentzel, Phys. Rev. 79, 710 (1950).
10. Yorô Ôno and Masao Sugwara, Progr. of Theor. Phys. 6, No. 2, 182 (1951).
11. Hara and Shimazu, Progr. of Theor. Phys. 5, 1055 (1950).

*) At present there is an extensive literature on this question.