Difficulties in the Theory of Radiation

M. A. Markov

§ 1. Introduction

The fundamental difficulties of field theory have recently been attracting ever greater attention from authoritative physicists.

In classical electrodynamics these questions are of a very abstract interest, since in essence there are no experimental facts whose explanation could be expected from their resolution. On the contrary, the specific, yet concrete, difficulties of nuclear physics and cosmic-ray physics leave (for the time being, at any rate) hope of linking their fate precisely with the solution of the general problems of field theory and, in general, with the fate of the further development of the fundamental foundations of modern theoretical physics.

It is known that at present quantum electrodynamics, as a mathematically rigorous theory, does not exist. Quantum theory not only gives no explanation for the existence in nature of particles with different masses and other properties, and gives no concrete values for intrinsic magnetic moments—the theory contains divergent expressions closely connected with its mathematical apparatus. Strictly speaking, the basic equation of the theory—the Dirac equation—represents (in the presence of a field) the trivial relation

\[ \frac{1}{c}\frac{\partial \psi}{\partial t}=\infty, \tag{0} \]

for its right-hand side contains divergent expressions.

In recent years some new ideas have appeared in the literature, and new points of view have been put forward concerning the further paths of development of theoretical physics. These new ideas do not have the character of completed theoretical constructions, but constitute that “atmosphere of search” in which the birth of a genuine theory is to be expected—one very likely radically different from those now being discussed.

The imperfect character of these constructions makes a critical attitude toward them natural; but excessive skepticism would nevertheless be inappropriate and, perhaps, an obstacle to the development of science.

The ideas in question attract attention also because they are associated with the names of Dirac\(^1\), Heisenberg\(^2\), Pauli\(^3\), Heitler\(^4\), and other authors of modern quantum physics.

In classical electrodynamics the difficulties are inherent in the representation of the electron as a point charge. The self-energy of such a charge is infinite:

\[ \frac{e^2}{r}\xrightarrow[r\to 0]{}\infty . \tag{1} \]

In momentum space, the expression corresponding to (1) is obtained in the form of a divergent integral:

\[ e^2\int_0^\infty dk \tag{2} \]

(\(k\) is the wave number, with the dimension of inverse length).

Quantum theory not only has not solved the classical difficulties of a point source of the field, but has even multiplied them. In many cases it leads to divergent integrals of the more general form

\[ \int_0^\infty f(k)\,dk \to \infty, \tag{3} \]

where \(f(k)\) is either an even or an odd power of \(k\), while Dirac’s theory of the vacuum leads, for example, to a logarithmic divergence.

Integrals with an even power of \(k\),

\[ f(k)=k^{2n},\qquad n=0,1,2 \tag{4} \]

in the cases considered are connected with the classical difficulty of the point electron. This is either the integral \(\int_0^\infty dk\), connected with the self electrostatic energy, or

\[ \int_0^\infty k^2\,dk \sim \frac{1}{r^3} \tag{5} \]

—an integral connected with the energy of a magnetic dipole.

Integrals with an odd power of \(k\):

\[ f(k)=k^{2n+1},\qquad n=0,1,2 \tag{6} \]

in the cases considered are of a purely quantum character. The appearance of odd divergent integrals is connected with a certain characteristic feature of field quantization. Namely, the field is regarded as a set of linear harmonic oscillators, and to each of them, as to a mechanical system, quantum theory is applied.

Quantum theory gives for the energy of an oscillator the discrete value:

\[ W=\left(N+\frac{1}{2}\right)h\nu \tag{7} \]

(\(\nu\) is the characteristic frequency of the oscillator, \(N\) is an integer).
For \(N=0\) we obtain

\[ W=\frac{1}{2}h\nu, \tag{8} \]

i.e. each oscillator possesses a certain minimal, so-called “zero-point energy,” which it does not radiate under any circumstances. Therefore the electromagnetic field, as a system of oscillators of all frequencies \(\nu\), must possess a certain zero-point energy, namely the sum of the zero-point energies of all the oscillators of the field. This circumstance leads to the integral:

\[ \int_0^\infty k^3\,dk. \tag{9} \]

Integrals of this type (an odd power of \(k\)) are encountered in calculations of many physical effects in higher approximations of perturbation theory, thereby creating uncertainty in the result. They appear in attempts to calculate quantum-theoretically the natural width of a spectral line, the Compton effect of higher order (scattering of several photons by one incident photon). A divergence of type (6)

\[ \int_0^\infty k\,dk \tag{9'} \]

is also obtained for the so-called “transverse” proper mass of the electron,\(^6\) which owes its origin to the presence in the theory of the same zero states of the oscillators of the field, but this term appears only in the presence of the electron.

If expression (2) is obtained by the method of perturbation theory from the term of the Hamiltonian function of the electron describing the scalar potential \(e\varphi\), then the transverse proper energy owes its origin to the term \(\frac{e^2 A^2}{2m}\), where \(\overline{A}\) is the vector potential. In classical theory, for an electron at rest this term is absent. In the quantum domain, quadratic expressions for the field cannot vanish, since there are zero-point oscillations of the field oscillators.

Despite the different nature of the divergences, what is common to all cases is the circumstance that if in any of these integrals one limits the upper bound by some \(k_{\max}\), then the expression becomes convergent. To limit oneself to some \(k_{\max}\) means

to restrict oneself to some wave of minimum length

\[ \lambda_{\min}=\frac{1}{k_{\max}}, \tag{10} \]

i.e. to introduce into the theory some minimum distance, which, in particular, would play the role of the “radius” of the electron.

Several years ago it was considered, in essence, universally recognized that the modern theory is valid only down to lengths of the order of the “electron radius” (\(r\sim 10^{-13}\) cm) and that the region of smaller wavelengths is the region of future theories (see Dirac’s report at the Leningrad Conference on the Atomic Nucleus, 1932, or Heitler’s book The Quantum Theory of Radiation).

However, apparently, one may say that the well-known progress of physical ideas in recent years consisted (and perhaps still consists) in the gradual overcoming of this point of view—in the extension of the limits of applicability of quantum theory in its modern form—the Hamiltonian method, the description by means of the \(\psi\)-function.

The study of cosmic rays (radiation in the braking of an electron by the Coulomb field of the nucleus, the creation of “pairs” by very hard photons) played a very important role here. In all the observed—let us say, for definiteness, electrodynamic—effects the theory proves valid for very large wave numbers, i.e. for small wavelengths.

In the first attempts to understand certain peculiarities of the phenomena of cosmic rays (we have in mind those of them which are now, with the discovery of particles of mass \(\sim 100\) electron masses—mesons—easily explained, for example, the great penetrating power of the rays), ideas of the limited applicability of the modern theory to these processes were not progressive, since in the “explanation” only the negative assertion of “inapplicability” was used.

But a serious obstacle to the extension of the limits of applicability of modern electrodynamics is presented by the ideas of the electromagnetic origin of the proper mass of the electron, at least in its classical form (Maxwell’s equations). Indeed, if one assumes that the mass of the electron is of electromagnetic origin, then the observed value of the electron’s proper mass unambiguously selects the value of the upper limit of integration in expression (2). This limit corresponds to the distance

\[ r_0 \sim \frac{e^2}{mc^2}, \]

i.e. to the “classical radius of the electron,” as this combination of universal constants is often called.

Recently, in order to overcome these difficulties, a number of new ideas have been advanced, often very unexpected and ingenious. One of them belongs to Wentzel\(^7\) and Dirac\(^1\). This is the so-called “limiting \(\lambda\)-process,” which turns integrals of type (2) into zero, to which

corresponds to renouncing the electromagnetic “origin” of the electron mass. Another idea belongs to Bopp and Podolsky, who obtain a finite value of the integral, likewise without restricting the limits of applicability of the methods of modern physics. Roughly speaking, the latter authors achieve this by adding to the Coulomb field \(e/r\) a certain additional field \(e^{-\alpha r}/r\), which is subtracted from the first, giving the potential

\[ \varphi=\frac{e}{r}-\frac{e^{-\alpha r}}{r}. \tag{11} \]

As \(r \to 0\), expression (11) remains finite.

On the other hand, there are attempts to restrict the limits of quantum theory, linking the overcoming of the difficulties of modern physics with the construction of a fundamentally new theory. This point of view has recently been developed by Heisenberg[^2]; it presupposes, in a certain domain of phenomena, the abandonment of the Hamiltonian method and, in certain limits, of the wave function. This point of view is connected with a fundamental revision of the basic physical concepts. These ideas are very attractive; they have, and will have, many supporters. The reason for this lies in the fact that, as the history of the development of physics teaches, fundamental difficulties of old theories were usually overcome by way of a fundamental change in the theories themselves, by way, in particular, of a critical revision of the applicability of old concepts in a new domain of phenomena.

This point of view is readily assimilated by analogy and has become customary in recent decades: classical mechanics and the special theory of relativity, the special theory of relativity and the general theory of relativity, classical mechanics and quantum theory. But to what extent it will prove fruitful in solving the next concrete problems confronting physics—the problem of nuclear forces (or, more narrowly, the deuteron problem, \(\beta\)-decay), cosmic rays (the birth of mesons, passage through the atmosphere and their disappearance)—is difficult to assert.

Whether everything here is taken from the Hamiltonian method, from the \(\psi\)-function, or whether more modest attempts to continue work on these problems by modernizing somewhat the Hamiltonian function and the mathematical apparatus of space-time description in general also deserve attention.

Let us begin a more detailed consideration of the state of the question with attempts to introduce a “critical length” into the mathematical apparatus of the modern theory—that is, with attempts to regard the charge as non-pointlike.

§ 2. NON-POINT SOURCE OF THE FIELD

The difficulties of field theory under discussion disappear when one considers a source of the field with a charge distributed in space. An insurmountable objection to such a program is the requirement of relativistic invariance of the equations of motion. One can

to prove rigorously that such a (i.e. extended) electron cannot be described by means of a wave function, i.e. that the corresponding Dirac equation has no solutions.

The relativistically invariant equations of motion of modern electrodynamics refer to a point source of the field. Mathematically, this circumstance is expressed by the fact that on the right-hand side of d’Alembert’s equation for, for example, the scalar potential, the charge density is written as a function which vanishes everywhere except at the point where the electron is located, where its value becomes infinitely large—the so-called \(\delta\)-function:

\[ \frac{1}{c^2}\frac{\partial^2\varphi}{\partial t^2}-\nabla^2\varphi = 4\pi e\,\delta(\bar r-\bar r_s). \tag{12} \]

Here \(\bar r\) is any point of space (the field coordinate), \(\bar r_s\) is the coordinate of the electron.

The first elementary attempt to introduce an extended electron into the theory would consist in replacing the \(\delta\)-function on the right-hand side of equation (12) by some regular density function \(\rho(\bar r-\bar r_s)^8\).

Thus, the equation for the field of a charge at rest will be written as

\[ \nabla^2\varphi=-4\pi\rho(\bar r-\bar r_s). \tag{13} \]

For illustration we might choose the function \(\rho\), for example, in the form

\[ \rho = \frac{e}{\pi^2}\, \frac{r_0}{\left\{r_0^2+|\bar r-\bar r_s|^2\right\}^2}, \tag{14} \]

\(|\bar r-\bar r_s|\) is the absolute value of the distance of any point of space from the center of the particle, \(r_0\) is a constant with the dimension of length (“the radius of the electron”).

It is easy to see that

\[ \int \rho\,dv = \frac{e}{\pi^2}\int \frac{r_0}{\left\{r_0^2+r^2\right\}^2}\, 4\pi r^2\,dr = e. \]

Let us examine in more detail the objections which such an attempt encounters from the standpoint of the requirements of relativistic invariance.

It is easy to verify that expression (14) can be rewritten in the form of the integral

\[ 4\pi\rho = \frac{1}{2\pi^2} \int \cos \bar k(\bar r-\bar r_s)\,e^{-r_0 k}\,dk_x\,dk_y\,dk_z, \tag{15} \]

where \(\bar k\) is the wave vector.

On the other hand, it is easy to see that expression (14), as \(r_0\) tends to zero, is a \(\delta\)-function. For \(r_0=0\) it vanishes everywhere except at the point \(r=r_s\), where it tends to infinity as \(\frac{1}{r^4}\).

Thus, we easily obtain an integral representation of the \(\delta\)-function by setting \(r_0\) equal to zero in expression (15):

\[ \delta(\bar r-\bar r_s)=\frac{1}{(2\pi)^3}\int \cos \bar k(\bar r-\bar r_s)\,dk_x\,dk_y\,dk_z . \tag{16} \]

We see that expression (15) differs from expression (16) by the presence of the factor \(e^{-r_0 k}\). We shall call this factor the “cutoff factor,” or “form factor,” since one or another choice of this factor gives one or another distribution of charge density, one or another “form” of the electron. This form factor is a function of \(k\), rapidly decreasing with increasing value of the wave number (not more slowly than \(k^{-3}\), since \(dk_x\,dk_y\,dk_z\) grows as \(k^3\)); otherwise the factor is for the time being arbitrary. Conversely, one or another choice of the function for the charge density in (13) leads to one or another form factor in expression (15). In what follows we shall denote the form factor by the function \(B(k)\).

Keeping in mind questions of relativistic invariance, let us give expression (16) a relativistically invariant form. We make the first step in this direction:

\[ 4\pi e\,\delta(\bar r-\bar r_s) = \frac{e}{2\pi^2} \int \cos\{\bar k(\bar r-\bar r_s)-c|k|(t-t_s)\}\,dk_x\,dk_y\,dk_z \tag{17} \]

\[ \text{for } t=t_s. \]

Here, for space-time symmetry, different times have formally been written for the field and the particle, \(t\) and \(t_s\), by analogy with the different designations of the coordinates of the field \(\bar r\) and of the coordinates of the particle \(\bar r_s\).

It is easy to see that the expression for the \(\delta\)-function can be obtained as the time derivative of a certain relativistic invariant:

\[ 4\pi e\,\delta(\bar r-\bar r_s)= \]

\[ = -\frac{e}{2\pi^2}\frac{\partial}{\partial t} \int \sin\{\bar k(\bar r-\bar r_s)-c|k|(t-t_s)\} \frac{dk_x\,dk_y\,dk_z}{|k|} \tag{18} \]

\[ \text{for } t=t_s. \]

The argument of \(\sin\) is the scalar product of two four-dimensional vectors (i.e. an invariant), and \(\dfrac{dk_x\,dk_y\,dk_z}{|k|}\), as is known, is also an invariant.

The entire integral is an invariant playing an important role in modern quantum field theory, and is called the four-dimensional \(\delta\)-function\({}^{6}\):

\[ D(X-X_s)= \frac{1}{(2\pi)^3} \int \sin\{\bar k(\bar r-\bar r_s)-c|k|(t-t_s)\} \frac{dk_x\,dk_y\,dk_z}{|k|}. \tag{19} \]

Expression (15) is correspondingly rewritten as

\[ 4\pi\rho = -\frac{1}{2\pi^2} \left[ \frac{\partial}{\partial t} \int \sin\{\bar k(\bar r-\bar r_s)-c|k|(t-t_s)\} \times \right. \]

\[ \left. \times \frac{dk_x\,dk_y\,dk_z}{|k|}\cdot e^{-r_0|k|} \right] \tag{20} \]

\[ \text{for } t=t_s. \]

In order that the differentiated expression in (20) be invariant, it is necessary that the form factor be an invariant.

The form factor in the form \(e^{-r_0 k}\) is not relativistically invariant. In order to satisfy the requirement of relativistic invariance, let us introduce, instead of \(r_0\), a certain four-dimensional vector \(\lambda\):

\[ \lambda_0,\ \bar{\lambda} \tag{21} \]

(\(\lambda_0\) is its time component, and \(\bar{\lambda}\) its spatial component).

Let this degree of freedom characterize the electron, and for an electron at rest

\[ \lambda_0 \ne 0;\qquad \bar{\lambda}=0; \tag{22} \]

then for an electron at rest the damping factor still has the form \(e^{-\lambda_0 k}\), and the charge density of an electron at rest still has the form (15).

In general, the relativistically invariant form of the given factor is written as follows:

\[ B(k)=e^{-\sum_{\mu=1}^{4}\lambda_\mu k_\mu}, \tag{23} \]

where \(\sum_{\mu=1}^{4}\lambda_\mu k_\mu\) is the scalar product of two four-dimensional vectors: the vector \(\lambda\) and the wave vector \(k\).

Thus, the charge density in the relativistically invariant theory of an extended electron can be written as

\[ 4\pi\rho = -\frac{e}{2\pi^2} \left[ \frac{\partial}{\partial t} \int \sin\{ \bar{k}(\bar{r}-\bar{r}_s)-c|k|(t-t_s)\} \,B(k)\times \right. \]

\[ \left. {}\times \frac{dk_x\,dk_y\,dk_z}{|k|} \right]. \tag{24} \]

This expression transforms under Lorentz transformations as the time component of a vector (the time derivative of an invariant), i.e. precisely as the charge density must transform.

Sometimes in the literature one encounters the assertion that charge density cannot be introduced as a purely spatial function, since in another coordinate system a time dependence will appear^8 (the time dimensions of the electron?). But the concept of the temporal extent of the electron is so foreign to the whole structure of modern physics that the image of such an electron has not been seriously discussed.

Of course, the image of an electron extended in time may be the subject of special discussion, but in the present case we are dealing with a certain misunderstanding. Expression (24) refers, by definition, always to \(t=t_s\), and therefore time always drops out of it, i.e. by definition, after transformation to another coordinate system, the section \(t'=t'_s\) is taken.

The point is that d’Alembert’s equation in any coordinate system, by its very meaning, contains the quantities of the field and of the particles taken at one and the same instant of time \((t'=t'_s)\).

A rigorous mathematical proof of this assertion can be obtained with the aid of the many-time formalism, developed by Dirac–Fock–Podolsky\(^9\), where the right-hand side of d’Alembert’s equation (the charge density in the present case) is obtained only in passing to one common time for all particles and fields \((t' = t_s)\), while the external form of the expression for the charge density is obtained precisely in the form (24).

It should be noted that, in setting out the question, we have so far not only failed to discover the promised contradiction between an extended charge and the requirement of relativism, but have also given, to some extent, an invariant formulation of the problem. Namely, the introduction of an extended charge in a relativistically invariant manner reduces to the introduction of a relativistically invariant cut-off factor (form factor) \(B(k)\). Various authors (Vataghin\(^ {10}\), Heisenberg, Pauli, Serber\(^ {11}\), and others) have attempted to construct the theory in just this way.

Let us consider more rigorously the admissibility of introducing such form factors into the mathematical apparatus of the modern theory and, anticipating somewhat, let us say that such a theory proves, generally speaking, to be internally contradictory\(^ {12}\).

§ 3. RELATIVISTIC FORM FACTORS

Up to now we have considered only the field equations (d’Alembert’s equation), i.e. the law of motion for the field, and have not at all touched upon the law of motion of the particles (Dirac’s equation). But only the totality of the laws of motion of the particles and the field forms the electrodynamical system of equations.

The equation of motion of a system of electrons is written:

\[ i\hbar \frac{\partial \psi}{\partial t}=H\psi; \tag{25} \]

\(H\) is the Hamiltonian function. In expanded form, for \(s\) electrons it is written:

\[ H=\sum_s H_s=\sum_s e_s\varphi_s-\bar{\alpha}_s(\bar{P}_s-e_s\bar{A}_s)-\beta_s m_s, \tag{26} \]

\[ s=1,\,2,\ldots,n, \]

where \(s\) is the number of the electron, \(e_s\) is its charge, \(m_s\) its mass, \(\bar{\alpha}_s\) and \(\beta_s\) are Dirac matrices, \(\varphi\) is the scalar potential, \(\bar{A}_s\) is the vector potential, and \(\bar{P}_s\) is the momentum operator of the \(s\)-th particle.

If the Hamiltonian function of the problem is known, then, by the general rule, it is easy to find the change in time of any physical quantity \(C(rpa)\).

\[ \frac{\partial c}{\partial t}=[HC], \tag{27} \]

where \([\,]\) denotes the operation called the “Poisson bracket.”

Above we were speaking of the Hamiltonian function for particles (in what follows we shall denote it by \(H_p\)). The Hamiltonian function for the field \((H_F)\) in the absence of particles is given in the form of the expression:

\[ H_F=\frac{1}{8\pi}\int (E^2+H^2)\,dv . \tag{28} \]

Now, by the function \(H\) we shall in what follows understand the total function for the field and the particles:

\[ H=H_p+H_F . \tag{29} \]

Having such a function at our disposal, we can find the law of variation in time of any physical quantity referring both to a particle and to the field. For this it is only necessary, according to the general rule, to compute the corresponding Poisson brackets (27).

Thus, equation (25) with the Hamiltonian function (29) is obliged to give us a complete description of electrodynamical problems. We can, for example, compute the change of the electric field with time \(\dfrac{\partial E}{\partial t}\) or of the magnetic field \(\dfrac{\partial H}{\partial t}\), or the second derivative of the scalar potential

\[ \frac{1}{c^2}\frac{\partial^2\varphi}{\partial t^2}. \tag{30} \]

In the right-hand side of expression (30) there must inevitably appear terms completing this expression to the full form of d’Alembert’s equation (12), while in the intermediate calculations the \(\delta\)-function, as is easily verified, is obtained in the form (18). It should be noted that such an external form of the \(\delta\)-function is obtained if one uses potentials expanded in a Fourier series or integral, i.e. if the potentials \(\varphi\) and \(\bar A\) are represented in the form:

\[ \left. \begin{aligned} \varphi&=\sum_\nu \varphi_\nu^+ e^{\,i(ck_\nu t-\bar k_\nu \bar r)} +\varphi_\nu^- e^{-i(ck_\nu t-\bar k_\nu \bar r)},\\ \bar A&=\sum_\nu \bar A_\nu^+ e^{\,i(ck_\nu t-\bar k_\nu \bar r)} +\bar A_\nu^- e^{-i(ck_\nu t-\bar k_\nu \bar r)}. \end{aligned} \right\} \tag{31} \]

In order that in expression (30) one obtain not a \(\delta\)-function but some regular function describing some distribution of the electron charge density in space (for example, function (14)), it is necessary to multiply each term of series (31) by the form factor \(B(k)\).

In the case (14) such a factor is

\[ B(k)=e^{-\sum_\mu \lambda_\mu k_\mu^2}. \]

Thus, substituting into the Hamiltonian function the expression for the potentials in the form

\[ \varphi=\sum_{\nu}\varphi_{\nu}^{+}e^{i(ck_{\nu}t-\bar{k}_{\nu}\bar{r})}B(k) +\varphi_{\nu}^{-}e^{-i(ck_{\nu}t-\bar{k}_{\nu}\bar{r})}B(k_{\nu}), \tag{32} \]

\[ \bar{A}=\sum_{\nu}\bar{A}_{\nu}^{+}e^{i(ck_{\nu}t-\bar{k}_{\nu}\bar{r})}B(k_{\nu}) +\bar{A}_{\nu}^{-}e^{-i(ck_{\nu}t-\bar{k}_{\nu}\bar{r})}B(k_{\nu}) \tag{33} \]

instead of the ordinary series (31), we obtain, in Hamiltonian form, a relativistically invariant theory of a point charge, free of the difficulty with divergences.

The introduction into the theory of an extended electron is completely equivalent to the procedure considered for introducing a form factor into the Hamiltonian function. However, one can indicate such form factors whose presence in the Hamiltonian function gives a substantial difference from the ordinary theory, but nevertheless, in the final analysis, preserves the image of a point charge. Such cut-off factors we shall consider further in connection with the Wentzel–Dirac \(\lambda\)-process.

Attempts to resolve the difficulties of electrodynamics by means of relativistically invariant form factors have appeared repeatedly in the literature. One may point to the attempt of Batalin \(^{10}\), then of Heisenberg, later of Schrödinger \(^{11}\), Heitler, and others. These attempts differ essentially in the form of the factor \(B(k)\).

Despite its outwardly relativistically invariant form, the theory of a point charge (of a relativistically invariant form factor) proves unsatisfactory. It can be shown that the corresponding system of equations has no solutions, i.e. that there exists no \(\psi\)-function describing an extended charge. This strictly mathematical assertion of the unsuitability of the theory of the relativistic form factor has the following simple physical meaning. The introduction of a cut-off factor turns out to be completely equivalent to considering a classical “rigid” electron, i.e. such an electron along which the signal propagates with a speed greater than the critical one, which is incompatible with the requirement of the theory of relativity.

In order to show physically clearly and at the same time mathematically rigorously this defect of the theory, let us consider two electrons \(S\) and \(S'\), situated at \(\bar{r}_{s}\) and \(\bar{r}_{s'}\).

The corresponding Hamiltonian functions are written as:

\[ H_s=e_s\varphi_s-\alpha_s(\bar{P}_s-e_s\bar{A}_s)-\beta_s m_s, \tag{34} \]

\[ H_{s'}=e_{s'}\varphi_{s'}-\alpha_{s'}(\bar{P}_{s'}-e_{s'}\bar{A}_{s'})-\beta_{s'}m_{s'}. \tag{35} \]

By the general rule (27) we can compute the derivative of the “energy” of the second electron with respect to the time of the first,

\[ \frac{\partial H_{s'}}{\partial t_s}=[H_s H_{s'}]. \tag{36} \]

In the ordinary theory (without the form factor) the result of the calculation is as follows:

\[ \frac{\partial H_{s'}}{\partial t_s} = e_s e_{s'} \left(1-\frac{\bar a_s \bar a_{s'}}{c^2}\right) \int \sin\{ck(t_s-t_{s'})-\bar k(\bar r_s-\bar r_{s'})\} \frac{dk_x dk_y dk_z}{k} = e_s e_{s'} \left(1-\frac{\bar a_s \bar a_{s'}}{c^2}\right) D(x_s-x_{s'}). \tag{37} \]

The integral on the right-hand side of expression (37) is the already familiar four-dimensional \(\delta\)-function; as is known, it vanishes for those values of the argument for which the condition

\[ (t_s-t_{s'})^2 c^2 < (\bar r_s-\bar r_{s'})^2 \tag{38} \]

is satisfied.

The physical meaning of this condition is evident: it is the condition of the finiteness of the velocity of propagation of a light signal. If the instants of time \(t_s\) and \(t_{s'}\) are chosen so that, during the interval \(t_s-t_{s'}\), an electromagnetic disturbance issuing from the point \(r_s\) (the position of one electron) and propagating with velocity \(c\) cannot reach the point \(r_{s'}\) (the position of another electron), then expression (37) vanishes, and the electrons are mutually independent.

In the case of a cutoff factor the situation is essentially different. Now, instead of expression (37), we obtain:

\[ \frac{\partial H_{s'}}{\partial t_s} = [H_s H_{s'}] = \]

\[ = e_s e_{s'} \left(1-\frac{\bar a_s \bar a_{s'}}{c^2}\right) \int \sin\{ck(t_s-t_{s'})-\bar k(\bar r_s-\bar r_{s'})\} \frac{dk_x dk_y dk_z}{k} B_s B_{s'}. \tag{39} \]

For a cutoff factor taken in the form (23), in the case of two charges at rest, calculation of the integral (39) leads to the expression:

\[ \int = \frac{\lambda_0 c(t_s-t_{s'})} {\left\{\lambda_0^2+\left[|\bar r_s-\bar r_{s'}|+c(t_s-t_{s'})\right]^2\right\} \left\{\lambda_0^2+\left[|\bar r_s-\bar r_{s'}|-c(t_s-t_{s'})\right]^2\right\}}. \tag{40} \]

Even under the condition

\[ c(t_s-t_{s'}) \ll |\bar r_s-\bar r_{s'}|, \tag{41} \]

when an electromagnetic disturbance issuing from the point \(r_s\) manifestly cannot reach the point \(r_{s'}\), expression (40) does not vanish:

\[ \int \sim \frac{\lambda_0 c(t-t_{s'})} {\left\{\lambda_0^2+|\bar r_s-\bar r_{s'}|^2\right\}^2}. \tag{42} \]

This means: no matter how far the second electron is from the first, nevertheless, after an arbitrarily small time it experiences as a whole the influence of the first electron, i.e. the requirements of relativity are violated.

In a more rigorous investigation of the question by the method of the many-time formalism of electrodynamics developed by Dirac–Fock–Podolsky (which, because of its complexity, we shall not touch upon here), setting expression (37) equal to zero represents the necessary solvability conditions for the system of differential equations of electrodynamics. In the theory of the cutoff factor these conditions (the so-called Bloch conditions \(^{13}\)) are not satisfied, i.e., the corresponding system of equations has no solution; consequently, there is no \(\psi\)-function describing an extended electron.

More detailed investigations show that the Lorentz normalization condition

\[ \frac{1}{c}\frac{\partial \varphi}{dt}+\operatorname{div}\overline{A}=\eta(xt) \]

also ceases to be compatible with the equation of motion of the particle, and the situation is not remedied by the addition of an arbitrary function.

We see that the attempt to introduce into the modern mathematical apparatus of the theory a critical length \((\lambda_0)\) leads to serious conflicts with such fundamental concepts as the \(\psi\)-function or, more precisely and cautiously, with the description of a phenomenon by means of a \(\psi\)-function specified in space-time, i.e., to a contradiction with Hamilton’s method.

Thus, if hopes for resolving the fundamental difficulties of the modern field theory are nevertheless to be connected with further attempts at the program of a “critical length,” then it is necessary to change, in a fundamental way, the basic concepts of quantum theory, at least in those domains where this critical length has essential significance.

In other words, the critical length \(\lambda\) would limit the bounds of applicability of quantum theory by analogy with the way in which the constant \(c\) limits the bounds of the classical theory, or the critical velocity \(\hbar\) limits the applicability of Newtonian mechanics.

Heisenberg has recently been attempting to carry out such a revision of quantum-mechanical concepts, replacing Hamilton’s method of quantum theory by a certain new mathematical apparatus.

§ 4. HEISENBERG’S IDEAS

In seeking a new theory, Heisenberg \(^{2}\) proceeds from the “principle of observability” (which at one time helped him to discover quantum mechanics and to comprehend its physical content).

“Only quantities observable in principle should enter the theory,”—using this criterion, Heisenberg attempts, among the concepts of the theory now in existence, to select those which must be preserved as necessary elements of the future, as yet hypothetical, theory.

Since, in the presence of a fundamental length, the space-time description loses its meaning, it remains to turn to

to an energy description or, more generally, to a description of physical phenomena mainly from the point of view of conservation laws.

According to Heisenberg, the following must be observable:

1. Discrete eigenvalues of the energies of stationary closed systems.

2. In stationary collision processes (for example, an incident plane wave and a scattered spherical one)—the asymptotic behavior of the wave function at infinity.

Indeed, in the overwhelming majority of cases physics is interested in the so-called “cross section” of one process or another. The “process,” in essence, is usually reduced to the results of the interaction of a plane wave with some scatterer. In the language of description from the point of view of conservation laws, the problem is formulated as follows: from infinity, some particle with energy-momentum vector \(\vec P\) “falls” upon the scatterer. It is required to predict the result of the interaction by that moment of time when the colliding systems can again be regarded as free or, strictly speaking, when the scattered particles have gone off to infinity.

Indeed, in studying collisions the experimenter deals only with free particles. He observes those particles which existed before the collision and those free particles which resulted after the collision. Scattering here must be understood in the broad sense of the word—it may lead to the production of any number of new particles in accordance with the conservation laws.

It is natural to regard this problem as a stationary problem: there is incident a flux of identical particles not interacting with one another (a plane wave), and one seeks at infinity the stationary flux of scattered particles.

Between the first and second classes of observable quantities Heisenberg sees a closer connection than appears at first glance. The construction of the mathematical apparatus of the theory must be connected with finding such an operator \(S\) which would transform the state with prescribed momenta \(P'_1\) and \(P'_2\) of the colliding particles into a state with momenta \(P''_1\) and \(P''_2\) of the particles after the collision.

In the general case (i.e., in the case of many particles participating in the collision) such an operator (matrix) is schematically represented in the form:

\(S:\)

	\(P'_1P'_2\)	\(P'_1P'_2P'_3\)	\(P'_1P'_2P'_3P'_4\)
\(P'_1P'_2\)
\(P'_1P'_2P'_3\)
\(P'_1P'_2P'_3P'_4\)

DIFFICULTIES OF THE THEORY OF RADIATION

In explicit form, the prototype of such an operator can be obtained, as Heisenberg indicated, from the first approximation of the quantum theory of scattering.

The asymptotic form of the function describing the scattered wave (for example, the flux of charged particles scattered by a Coulomb center) is written in the form[^14]:

\[ \frac{e^{ikr}}{r}\cdot f(\theta), \]

where

\[ f(\theta)=\frac{1}{2ik}\sum_{n=0}^{\infty}(2n+1)\left[e^{2i\eta_n}-1\right]P_n(\cos\theta) \]

\[ k=mv/\hbar. \]

We see, therefore, that in order to obtain the picture of the scattering phenomenon that interests us, it is sufficient to know the quantity \(S\)

\[ S\sim e^{2i\eta}. \]

Thus, in order to obtain the asymptotic expression of the scattered wave there is no need for the Hamiltonian method itself. From Heisenberg’s point of view, the Hamiltonian method gives an “overly detailed” description of the phenomenon—such a detailed description does not correspond to the fundamental possibilities of observation, and therefore the Hamiltonian method is inadequate to physical reality. An adequate description of physical reality is achieved with the aid of knowledge of \(\eta\).

In the simple case, \(\eta\) is related to the interaction function \(v(r)\) in the following way:

\[ \eta_n=-\frac{\pi}{2}\frac{8\pi^2m}{h^2}\int_{0}^{\infty}v(r)\left[J_{n+1/2}(kr)\right]^2 r\,dr. \]

In the general case, the solution of the scattering problem is connected with specifying the function \(\eta\).

Those difficulties with divergences discussed above are trivially absent in Heisenberg’s scheme, since, by definition, transitions are considered between states with one and the same energy.

It is very significant that Heisenberg’s scheme, as he has shown, is relativistically invariant. Although Heisenberg’s sufficiently concrete indication of the possible role of \(\eta\) in the apparatus of the future theory is the most essential achievement of the program, nevertheless Heisenberg’s scheme is not internally complete, since in the theory there are no grounds or rules for one or another choice of the operator \(\eta\).

Moreover, strictly speaking, the mathematical apparatus of the theory proposed by Heisenberg is not adequate to his physical program. Heisenberg’s basic idea consists in the fact that the introduction of certain

“critical length” into a future theory will inevitably limit the space-time description of physical phenomena.

In the mathematical apparatus proposed by Heisenberg there is no critical length. There are no new fundamental limitations on the possibilities of measurement that follow from it. Hence the question arises whether Heisenberg’s apparatus is not a certain element of another method of, again, “detailed” space-time description. It is precisely this circumstance, it seems to us, that is clarified in Stueckelberg’s recent works. But in that case one must be more cautious about the proof of the relativistic invariance of Heisenberg’s scheme. We saw, in the example of attempts to solve the problem with the aid of relativistically invariant form factors, that formal well-being in satisfying the usual invariance requirements of the mathematical apparatus must be supplemented by the requirement that there be no connection outside the light cone.

Here certain kinds of $\eta$-functions, for example those which Heisenberg associates with nonlocal interaction, apparently do not satisfy this requirement. In any case, as yet there are no rigorous grounds for believing that the matrix $\eta$ gives greater freedom for introducing a critical length into the theory than Hamilton’s method. Moreover, within Heisenberg’s mathematical scheme there is no internal necessity for introducing a critical length: the difficulties of field theory under discussion (divergences) are absent, by definition, in Heisenberg’s apparatus also for point interactions.

It is possible that Heisenberg’s proposal should, in the end, be regarded as a new method of “detailed” space-time description of point interactions. This description may indeed be free of the difficulty with divergences, but the internal consistency of the method for various classes of $\eta$ has not yet been sufficiently investigated.

Among the concrete difficulties of Heisenberg’s theory one should include, for example, the problem of the natural width of spectral lines. This problem does not find its immediate solution by means of the apparatus proposed by Heisenberg (transitions between states strictly with one and the same energy).

In connection with the last remarks, the attempt of Heitler and Peng^4 to replace the wave equation of quantum theory by the computational scheme of perturbation theory deserves attention. From a certain point of view, the attempts of Heitler–Peng represent a realization of Heisenberg’s special case.

§ 5. COMPUTATIONAL SCHEME OF PERTURBATION THEORY

In its mathematical meaning, “perturbation theory” should give approximate expressions for the solutions of the Dirac (or Proca–Kemmer) equations. But since the electrodynamic Dirac equation essentially has no solutions (0),

then in fact at the present time we are compelled to use the mathematical apparatus of perturbation theory as an independent computational scheme, “independent” of the Dirac equation.

Therefore the idea is natural of legitimizing this circumstance—in other words, of defining in a rigorous way the physical concepts contained in this “computational scheme,” of defining operations on them in an internally consistent manner, and thus obtaining a consistent computational scheme for various electrodynamic processes, free of the known difficulties of electrodynamics 4,12.

The physical concepts of the scheme under consideration are a free electron with inertial mass \(m\) and free photons. All electrodynamic processes reduce to the emission and absorption of photons by the electron. Each electrodynamic effect under consideration is realized by a chain of processes of emission and absorption of photons. Each such “chain” \((H_{AB})\) carries the system from a state \(A\) to a state \(B\). The rules for constructing matrix elements and cross sections can be defined directly.

Assuming that the processes are realized by means of a “chain” of minimal length, we exclude divergences, since they are connected with an inadmissible lengthening of the chain. Divergences arise in connection with those matrix elements which characterize the emission and then the absorption of one and the same photon.

Usually such a computational scheme has a relativistically non-invariant external form. But, using the relativistically invariant form of perturbation theory given by Stueckelberg 15, one can convince oneself of the relativistically invariant content of such a computational scheme.

Understanding by \(u(N^j)\) the wave function of radiation, where \(N^j\) denotes some distribution of photons,

\[ N^j=(N^j_1\ldots N^j_k). \]

Understanding by \(\psi^j(x)\) the wave function of the electron for a given state of radiation \((j)\), we can write the wave function of the electrodynamic system:

\[ \psi=\sum \psi^j(x)\,u(N^j). \]

Here \(x\) is a point of space-time \((x,y,z,t)\).

Writing the Dirac equation in the form:

\[ \left[\frac{1}{i}\left(\gamma\cdot\frac{\partial}{\partial x}\right)+\frac{mc}{\hbar}+\vartheta(x)\right]\psi(x)=0, \]

where \(\vartheta(x)\) is the quantized field) and expanding \(\psi^j(x)\) in a four-dimensional Fourier series or integral*)

\[ \psi^j(x)=\int e^{i(l,x)} A^j(l)\,dl_1\,dl_2\,dl_3\,dl_4 \tag{a} \]

) \(\gamma\) are the Dirac matrices, \(m\) is the electron mass, \(\hbar\) is Planck’s constant.
*) And not in a three-dimensional series, as is done in the usual theory of perturbations.

(where \((l,x)\) is the scalar product of two four-dimensional vectors), for the amplitudes \(A^j(l)\) we shall obtain certain algebraic equations, and the quantities \(A^j(l)\) do not depend on space-time and therefore are invariants.

It is easy to see that the zeroth approximation for \(A^j(l)\) satisfies the system of algebraic equations

\[ H(l)A^j(l)=0; \]

here \(H(l)\) denotes the expression

\[ H(l)=(l,\gamma)+\frac{mc}{\hbar}. \]

The first and subsequent approximations are easily obtained by the method of successive approximations.

The first approximation is interpreted physically as the result of a single emission or absorption of a quantum by the electron. The subsequent approximations determine processes involving the emission and absorption by the electron of many photons. This first approximation is, as it were, the elementary link of any chain of transitions with the emission and absorption of any number of photons.

So far we have outlined schematically the usual, but relativistically invariant, perturbation theory. It is now possible to take a further formal step: to abstract from the Dirac equation and postulate the link of the chain as a fundamental relation describing the emission or absorption of a photon by an electron, having the character of a law of nature. And all the more complicated effects that interest the physicist—chains—are composed successively of these elementary links.

Here the general requirement is the realization of the given transition by a minimal number of links. The element of the chain (emission or absorption) is written mathematically in invariant form:

\[ A^j(l)=L_k^{j0}A^0(l-k). \]

\(L\) is an invariant operator taking the given state \(A^0(l-k)\) into the state \(A^j(l)\) with the emission or absorption of a quantum. Without going into details, one may give the explicit form of the operator according to Stueckelberg:

\[ L_k^{j0}=-\frac{K(l)}{R(l)}\,v_{k,j0}(\sigma^k,\gamma), \]

where \(v_{k,j0}\) is the result of the action of the radiation operator on the given wave function of the radiation, and

\[ K(l)=-(l,\gamma)+\frac{mc}{\hbar} \]

and

\[ R(l)=K(l)H(l) \]

(\(\sigma^k\) is the polarization vector of the photon).

Expressions for more complicated transitions are obtained by successive application of the operator \(L\) and by summing all possible cases in the given process by which the given final state is realized.

Knowing the invariant amplitudes \(A^j(l)\), we can construct the cross section, using in the intermediate calculations relation (a), or subsequently finding more direct methods for constructing cross sections from the symbols \(A^j(l)\), since the cross section is also invariant and does not depend on the space-time variables.

The scheme under discussion gives a general method for finding the operator with whose aid collision problems are solved*). In this sense the scheme has much in common with Heisenberg’s scheme. On the other hand, the scheme under discussion is a space-time description, since by means of the Fourier transformation (a) one can write the \(\psi\)-function as a function of \(x,t\).

Here, just as in the general Heisenberg scheme, there are no mathematical circumstances of the apparatus itself that organically forbid the transition from the description in energy-momentum space to coordinate space. But then there are no grounds for asserting a priori that there do not exist differential equations whose solutions give the \(\psi\) which we can obtain as a result of the transformation. Moreover, we may think that there exist such Hamiltonian operators, describing point interactions, which give exactly the same result as the mathematical schemes under discussion.

One such Hamiltonian operator has recently been discovered by Dirac (the theory of the so-called “limiting \(\lambda\)-process”).

§ 6. THE WENTZEL–DIRAC \(\lambda\)-PROCESS

Attempts to regard the charge as non-pointlike, as was shown, are completely equivalent to attempts to introduce an invariant form factor into the theory; but the latter attempts have a more general character. As a rule, form factors turn divergent integrals into finite expressions different from zero. For example, with the aid of the factor (23), expression (2) for a charge at rest takes the form:

\[ e^2 \int_0^\infty e^{-2\lambda_0 k}\,dk=\frac{e^2}{2\lambda_0}. \tag{43} \]

But one can indicate such exceptional cases in which the form factor turns the divergent expression to zero.

*) We have set forth the ideas of Heitler and Peng in a simplified form, in the form in which they were discussed by the author somewhat earlier (23). Heitler and Peng introduced into this scheme an allowance for damping, which we have not touched upon here.

Consider, for example, the factors:

\[ B(k)=+\sqrt{\cos(k_\mu \lambda_\mu)}. \tag{44} \]

In this case the potential (32) is rewritten as

\[ \varphi=\sum_\gamma\left\{\varphi_\gamma^{+}e^{i(ck_\gamma t-\bar{k}_\gamma \bar{r})} +\varphi_\gamma^{-}e^{-i(ck_\gamma t-\bar{k}_\gamma \bar{r})}\right\} \sqrt{\cos(k_\mu \lambda_\mu)}. \tag{45} \]

Then, with the aid of the second approximation, for the intrinsic electrostatic energy of a charge at rest \((\lambda_0\ne 0;\ \bar{\lambda}=0)\), instead of (2) we obtain the expression

\[ e^2\int_0^\infty \cos k\lambda_0\,dk. \tag{46} \]

This integral has no definite meaning, but it may be regarded as the limit of the following integral:

\[ e^2\int \cos k\lambda_0\,dk = \lim_{\alpha\to0} e^2\int e^{-\alpha k}\cos k\lambda_0\,dk = e^2\,\frac{\alpha}{\alpha^2+\lambda_0^2}=0. \tag{47} \]

It is essential that the expression vanishes independently of the magnitude of \(\lambda_0\); \(\lambda_0\) may be arbitrarily small.

Since \(\lambda_0\ne 0\) leads to the same difficulties for the relativistically invariant form factors, the general consideration of which was given at the beginning of the article, an unexpected property of the present form factor proves to be the independence of expression (47) from the magnitude \(\lambda_0\). Thus the quantity \(\lambda_0\) may, in the final result, be made to tend to zero. Hence the name “limiting \(\lambda\)-process.”

This property of the form factor ensures the relativistic invariance of the theory. Here a new fundamental circumstance in physics draws attention to itself: a theory is being discussed in which the intrinsic electrostatic energy of the electron is equal to zero. Before discussing in more detail other consequences of this new theory, let us carefully consider the last remark.

§ 7. THE IDEA OF THE ELECTROMAGNETIC INTRINSIC MASS OF THE ELECTRON

Since the relation between energy and mass was established, the idea of the electromagnetic mass of the elementary electric charge has seemed attractive. In its final and consistent form this idea was to lead to the construction of mechanics from field theory, i.e., to the derivation of the equations of motion of a charge from field theory.

Despite the half-century age of this circle of ideas, it must be noted that at the present time there are no real programs

DIFFICULTIES OF THE THEORY OF RADIATION

of constructing mechanics from field theory, programs lying within the circle of ideas of the quantum physics of elementary particles.

Here, over the course of the last decades, rather new difficulties than hopes for success have emerged1. It has become clear that particles possess spin, and so forth.

On the other hand, if we turn to the actual content of the equations describing the modern theory of the interaction of elementary particles with a field (of any particles with any field), then these equations inevitably lead to the appearance of expressions which, in essence, ought naturally to be interpreted as the masses or energies of the proper field of the given particle.

The appearance of such expressions in the modern mathematical apparatus of the theory indicates that the question under discussion is already closely connected with the apparatus itself, that to a certain extent it is here posed concretely—and, moreover, posed in an unsatisfactory manner.

In an unsatisfactory manner because, first, these expressions diverge; second, even if it were possible to make these expressions finite, it would be difficult to interpret them as particle masses, since the latter (i.e. the masses of the particles) enter the existing equations from the very beginning as purely mechanical inertial masses of particles free from field and charge:

\[ i\hbar \frac{\partial \psi}{\partial t}=(\bar{\alpha}\bar{p}+\beta m)\psi . \]

In other words, we wish to emphasize the thought that, within the framework of the existing theory, these expressions are in essence “superfluous” formations from the point of view of the apparatus of the theory itself. This “superfluous” character of the expressions under discussion is curiously emphasized in all concrete calculations of actually observed effects. Here the ignoring of these “superfluous” terms (despite their infinite absolute value) does not lead to any misunderstandings. All finite effects of the interaction of the electron, for example, with radiation, calculated according to the modern theory, agree excellently with experimental data. And, conversely, there is as yet not a single experimental effect which would produce a disagreement with the theory such that the cause of this disagreement could be sought in the illegitimate deletion of these infinite terms from the equations.

The impression is created that, if it proved possible to propose such a theory of the interaction of particles with radiation in which the terms with the proper energy of the particle’s field, and all similar divergent expressions, were naturally absent, while otherwise for all observed effects this theory gave the correct expressions, then such a theory would in any case deserve attention and discussion.

It is precisely this problem that Dirac set himself in his latest works on quantum electrodynamics.

§ 8. EVEN AND ODD DIVERGENCES

Calculating the transverse self-energy of the electron, and using the factor (44), we obtain, instead of (9′), the integral

\[ \underset{a\to 0}{\int_{0}^{\infty}} k\,dk\,e^{-ak}\cos k\lambda_0 = \left[\frac{a^2-\lambda_0^2}{(a^2+\lambda_0^2)^2}\right]_{a\to 0} = -\frac{1}{\lambda_0^2}. \tag{48} \]

We see an essential difference between the last case and case (47): there the integral turns to zero independently of the numerical value of \(\lambda_0\); here the integral is finite, but different from zero, and in the limiting case \(\lambda_0=0\) it still diverges. If, together with Dirac, one considers only the limiting value of the quantities (as \(\lambda_0\to 0\), i.e., the “limiting \(\lambda\)-process”), then, as we see, the factor (44) solves the problem in case (2) (the electrostatic self-energy of the electron), but leaves unchanged the difficulty with the transverse energy (9′).

It is easy to verify in general form that, with the aid of the factor (44), all even divergences vanish:

\[ \int_{0}^{\infty} k^{2n}\,dk. \]

But the difficulties of the odd divergences

\[ \int_{0}^{\infty} k^{2n+1}\,dk \]

are preserved in the \(\lambda\)-process.

If one takes the factor \(B(k)\) in the form of a sine,

\[ B(k)=\sqrt{\sin\left(\sum_{\mu=1}^{4} k_\mu \lambda_\mu\right)}, \tag{49} \]

then in the case of the electrostatic energy we obtain the integral

\[ \underset{a\to 0}{\int_{0}^{\infty}} e^{-ak}\sin \lambda_0 k\,dk = \left[\frac{\lambda_0}{a^2+\lambda_0^2}\right]_{a\to 0} = \frac{1}{\lambda_0}, \tag{50} \]

and for the transverse self-energy the expression

\[ \underset{a\to 0}{\int_{0}^{\infty}} e^{-ak}\sin k\lambda_0\,dk=0. \tag{51} \]

It can be verified in the general case that the sine factor gives a result “opposite” to the cosine factor, namely: in the latter case, for finite \(\lambda_0\), the odd divergences turn into zero, while the even ones, for finite \(\lambda_0\), are finite; but in the limiting transition \((\lambda_0=0)\) the even divergences remain, i.e. this factor gives zero for the transverse proper energy and infinity for the electrostatic one. But for the limiting transition the cosine factor is of interest, since at \(\lambda_0=0\), \(\cos \lambda_0 k=1\), and all ordinary finite effects remain unchanged (\(\cos \lambda_0 k\) enters as a multiplier).

With the sine factor such a limiting transition is impossible, since

\[ \sin k\lambda_0=0 \]

for

\[ \lambda_0=0. \]

Therefore the limiting \(\lambda\)-process of Dirac proves effective only against even divergences.

§ 9. COULOMB’S LAW

The form factor, taken in the form (44), changes the expression for Coulomb’s law in a curious way. Let us consider two electrons at rest at the points \(\overline r_1\) and \(\overline r_2\), with charges \(e_1\) and \(e_2\).

In the usual way, with the aid of the second approximation of perturbation theory, we obtain the expression for the interaction energy between two charges:

\[ W=-\frac{e_1e_2}{|\overline r_1-\overline r_2|}\cdot \frac{2}{\pi}\int_0^\infty \frac{\sin k|\overline r_1-\overline r_2|}{k}\,dk = -\frac{e_1e_2}{|\overline r_1-\overline r_2|}. \tag{52} \]

Dirac’s theory leads to the appearance in the integral (52) of the characteristic cosine factor

\[ \begin{aligned} W&=-\frac{e_1e_2}{|\overline r_1-\overline r_2|}\cdot \frac{2}{\pi}\int_0^\infty \cos k\lambda_0\, \frac{\sin k|\overline r_1-\overline r_2|}{k}\,dk \\ &=-\frac{e_1e_2}{|\overline r_1-\overline r_2|}\cdot \frac{1}{\pi}\cdot\frac{2}{2} \left\{ \int_0^\infty \left[ \sin\bigl(|\overline r_1-\overline r_2|+\lambda_0\bigr)k + \sin\bigl(|\overline r_1-\overline r_2|-\lambda_0\bigr)k \right]\frac{dk}{k} \right\}. \tag{52′} \end{aligned} \]

The first integral is equal to:

\[ \int_0^\infty \frac{\sin k\bigl(|\overline r_1-\overline r_2|+\lambda_0\bigr)}{k}\,dk = \frac{\pi}{2}. \]

The second integral has two values depending on whether
\(|\bar r_1-\bar r_2|>\lambda_0\) or \(|\bar r_1-\bar r_2|<\lambda_0\):

\[ \int_0^\infty \frac{\sin k\left(|\bar r_1-\bar r_2|-\lambda_0\right)}{k}\,dk = \begin{cases} \dfrac{\pi}{2}, & \text{if } |\bar r_1-\bar r_2|>\lambda_0,\\[6pt] -\dfrac{\pi}{2}, & \text{if } |\bar r_1-\bar r_2|<\lambda_0. \end{cases} \tag{53} \]

In the first case we have

\[ W_{|\bar r_1-\bar r_2|>\lambda_0} = -\frac{e_1e_2}{|\bar r_1-\bar r_2|}\cdot \frac{1}{\pi} \left(\frac{\pi}{2}+\frac{\pi}{2}\right) = -\frac{e_1e_2}{|\bar r_1-\bar r_2|}. \tag{54} \]

In the second case

\[ W_{|\bar r_1-\bar r_2|<\lambda_0} = -\frac{e_1e_2}{|\bar r_1-\bar r_2|}\cdot \frac{1}{\pi} \left(\frac{\pi}{2}-\frac{\pi}{2}\right) = 0. \tag{55} \]

In other words, if the distance between two charges is greater than \(\lambda_0\), then Coulomb’s law remains in its former form; if the distance between two charges is less than \(\lambda_0\), then the interaction between the charges is completely absent. In the limiting case there remains only one such point, namely the coordinate of the electron itself.

The potential function vanishes at zero and at infinity; therefore, figuratively speaking, the classical work of “assembling” a charge from infinity into a point (or into the region \(\lambda_0\)) is equal to zero, and the existence of a charge does not require external forces holding together the “elements” of a charge placed in the region \(\lambda_0\).

We easily obtain Poisson’s equation by replacing, in expression (15), written for the charge density, the factor \(e^{-b k}\) by \(\cos\lambda_0 k\). Carrying out integration over the angles and over the wave numbers \((k)\), we obtain the right-hand side of the new Poisson equation, which is now written as

\[ \nabla^2\varphi = \frac{e}{r}\frac{\partial}{\partial r} \left\{ \delta(r+\lambda_0)+\delta(r-\lambda_0) \right\}, \tag{55'} \]

where

\[ r=|(\bar r-\bar r_s)|. \]

Here \(\delta(r\pm\lambda_0)\) is understood not as a spatial, but as a linear integral,

\[ \delta(r\pm\lambda_0) = \frac{1}{\pi}\int_0^\infty \cos k(r\pm\lambda_0)\,dk. \]

For \(\lambda_0=0\) we obtain

\[ \nabla^2\varphi = \frac{e}{r}\delta'(r) = -4\pi e\delta(r), \]

i.e. the ordinary theory.

§ 10. THE \(\lambda\)-PROCESS IN DIRAC FORM

In the preceding sections we set forth Dirac’s new theory as a special case of the theory of the relativistically invariant form factor. In Dirac’s original works the theory is given as a generalization of the classical Poisson brackets.

If the quantities \(B(qp)\) and \(C(qp)\) are functions of canonically conjugate variables, for example, the momentum \(p\) and the coordinate \(q\), then by the Poisson bracket \([\,]\) is meant the following simple mathematical operation:

\[ [BC]=\frac{\partial B}{\partial p}\frac{\partial C}{\partial q} -\frac{\partial B}{\partial q}\frac{\partial C}{\partial p}. \tag{55} \]

Consequently, the Poisson bracket for \(p\) and \(q\) themselves is

\[ [pq]=1. \tag{57} \]

If \(p\) and \(q\) are the momentum and coordinate of a linear harmonic oscillator, then the energy of such an oscillator is written as

\[ \frac{1}{2}(p^2+\nu^2 q^2)=W. \tag{58} \]

\(ck_\nu=\nu\) is the frequency of the oscillator.

We introduce new variables

\[ p=\sqrt{2\nu}\,A^+;\qquad q=\frac{1}{\sqrt{2\nu}}\,A. \tag{59} \]

Then the expression for the energy of the oscillator takes the form:

\[ W=(A^{+2}+A^2)\nu; \]

for \(A\) and \(A^+\), as is easily verified, one also has

\[ [A^+A]=1. \tag{60} \]

Using the expressions for the potential in the form (31), one can express the energy of the field through the amplitudes \(A^+\) and \(A\),

\[ H_F=\sum_\nu\{A_\nu^{+2}(k_\nu)+A_\nu^2(k_\nu)\}\,Ck_\nu =\sum_\nu W_\nu, \tag{61} \]

i.e. the energy of the electromagnetic field may be regarded as the sum of the energies \((W_\nu)\) of linear harmonic oscillators.

Dirac’s idea consists in proposing to change the Poisson bracket (60), replacing on the right-hand side of the relation \(1\) by \(\cos(k_\mu^\nu \lambda_\mu)\)

\[ (A_\nu^+A_\nu)=\cos(k_\mu^\nu\lambda_\mu). \tag{62} \]

With the help of the Poisson brackets (60) and (62), written for the amplitudes, one can compute, on the basis of the rules (56), the Poisson bracket for the potentials themselves (the series (31)).

Carrying out the corresponding calculations, we in the ordinary case (i.e. in the case of (60)) obtain

\[ [A(x)A(x')] = D(x-x'), \tag{63} \]

where \(A(x)\) is taken at the point \(x\) (at the point \(\bar r\) at the moment \(t\)), and \(A(x')\) is taken at the point \(x'\) (at the point \(\bar r'\) at another moment \(t'\)). \(D(x-x')\) is a four-dimensional \(\delta\)-function.

If we compute the Poisson bracket for the potentials taken at the world points \(x\) and \(x'\), using for the amplitude condition (62), and not (60), then we obtain

\[ [A(x)A(x')] = \frac{1}{2}\{D(x-x'+\lambda)+D(x-x'-\lambda)\}. \tag{64} \]

On the right stand four-dimensional \(\delta\)-functions which differ from the \(\delta\)-function (63) in that they are nonzero not on the light cone, but on cones somewhat displaced relative to the light cone. The quantity \(\lambda\) characterizes this displacement.

Conversely, taking condition (64) as the basis, and expanding the potentials in a Fourier series, one can obtain condition (62).

Dirac expounds the theory starting from condition (64). Strictly speaking, here we are dealing with another way of introducing the same relativistically cutting-off factor. We could have expounded the general theory of the cutting-off factor (§ 3) by means of changing the Poisson brackets according to the type of (62).

Namely, putting instead of (60) the condition

\[ [A^+(k)A(k)] = B^+(k)B(k). \tag{65} \]

In the case of the factor \(B^+=B=\sqrt{\cos k_\mu\lambda_\mu}\) we obtain condition (62). In the case of the factor \(B^+=B=e^{-\sum k_\mu\lambda_\mu}\) we have

\[ [A^+(k)A(k)] = e^{-2\sum k_\mu\lambda_\mu}. \]

In the case of the factor \(B^+=B=\sqrt{\sin\sum k_\mu\lambda_\mu}\) we obtain

\[ [A^+(k)A(k)] = \sin\sum k_\mu\lambda_\mu. \]

But for the Poisson bracket of the potentials themselves we do not in general always have a simple analytical expression.

Here, using the sine factor (49), we obtain for the potentials themselves the Poisson bracket in the form

\[ [A^+(x)A(x')] = \frac{1}{2}\{\Delta(x-x'+\lambda)+\Delta(x-x'-\lambda)\}. \tag{66} \]

By \(\Delta(x-x')\) we understand the second four-dimensional invariant function

\[ \Delta(x-x')=\frac{1}{(2\pi)^3}\int \cos\{\bar k(\bar r-\bar r')-c(k)(t-t')\}\frac{dk_x\,dk_y\,dk_z}{k}, \tag{19'} \]

which differs from function (19) in that under the integral there stands a cosine instead of a sine[^17].

This factor cannot be used for the limiting transition, as we have already indicated, since for \(\lambda=0\)

\[ [A^{+}(x)A(x')]=0. \]

The expression does not pass over into the classical one.
The Poisson bracket for the potentials themselves in the case (23) is given by the expression:

\[ [A^{+}(x)A(x')] = \frac{1}{2}\,[D(x-x'+i\lambda)+D(x-x'-i\lambda)] + \frac{i}{2}\,[\Delta(x-x'+i\lambda)-\Delta(x-x'-i\lambda)]. \tag{67} \]

However, there are certain differences in these ways of introducing cutoff factors.

With the aid of a Poisson bracket of type (64) or (62), a difference from the ordinary theory is already established for the vacuum. Thus the four-vector \(\lambda_{0},\vec{\lambda}\), if we assume it to be nonzero, must have some physical meaning precisely for the vacuum.

On the other hand, the condition \(\dfrac{\lambda_{0}}{|\vec{\lambda}|}>1\) (timelike) is imposed on this vector. This means that in some coordinate system (the physical meaning of which is not clear) \(\vec{\lambda}=0\) and \(\lambda_{0}\ne0\).

If, however, one introduces the cutoff factor, as we did above, only into the Hamiltonian function written for a particle interacting with fields, it has meaning only in the presence of particles.

But since in Dirac’s theory this vector nevertheless tends to zero in the final result, the difference is immaterial.

§ 11. THE \(\lambda\)-PROCESS IN WENTZEL’S PRESENTATION

The basic properties of the limiting \(\lambda\)-process were set forth in Wentzel’s works about ten years ago, but interest in it arose in connection with the new mathematical formulation (62), (64), given in recent years by Dirac. The form is very convenient for generalizing the theory to other, non-electromagnetic, fields.

In Wentzel’s works the process has the following simple content.

The equations of motion of particles (the Dirac, Proca, etc., equations) contain the field potentials. It is essential that the values of these potentials be taken at the given instant at the point where the particles are located:

\[ A(t,r)\to A(t_s,r_s), \]

i.e.

\[ t\to t_s;\qquad r\to r_s. \]

Here \(t_s\) and \(\bar r_s\) are the time and coordinates of the particle, \(\bar r\) is any point of space (i.e. of the field), which may be taken at any instant of time \(t\), in general \(t \ne t_s\).

As Wentzel showed, this coordination of field and particle has the following ambiguous character: if we refer the potential (the expression appearing in the Hamiltonian function written for the charge) to some event displaced relative to the particle, i.e. put

\[ t=t_s+\Delta t \quad \text{and} \quad r=r_s+\Delta r, \]

and let \(\Delta t\) and \(\Delta r\) tend to zero only at the final result of the calculations, then we have two possibilities for the tending of \(\Delta t\) and \(\Delta r\) to zero:

I.

\[ \lim \frac{\Delta t c}{\Delta r}<1. \tag{68} \]

This means that the displacement has the character of a spatial vector, and

II.

\[ \lim \frac{\Delta t c}{\Delta r}>1. \tag{69} \]

This means that the displacement has the character of a temporal vector. In the first case the result of the final calculations differs in no way from the usual one; in the second case the situation is different.

We can let \(t\) tend to \(t_s\), starting from the “past” or from the “future”; therefore Wentzel takes the arithmetic mean of these limits. Figuratively speaking, the limiting value of the “past” and the “future” is represented equally in the “present” as a result of the action on the charge of both the retarded and the advanced field.

Wentzel’s ideas are, in essence, a proposal to regard the point electron as the limit of an electron extended in time.

In order to clarify the meaning of the last statement, let us first consider the image of Wentzel’s electron extended in space, i.e. let us first consider an example of a form factor with a purely spatial vector \(\bar\lambda \ne 0,\ \lambda_0=0\):

\[ [A^{+}A]=\cos \bar k \bar\lambda = B^{+}(k)B(k). \tag{70} \]

We know that for a point charge Poisson’s equation is written in the form:

\[ \nabla^2 \varphi = -4\pi e\,\delta(\bar r-\bar r_s). \]

Calculations show that, introducing the form factor (65) into the brackets of Poisson, in the general case we obtain for the density the expression:

\[ \rho=\frac{e}{8\pi^3}\int \cos \bar k(\bar r-\bar r_s)\,B^{+}(k)B(k)\,dk_x\,dk_y\,dk_z \tag{71} \]

or, taking into account the particular form of factor (70), we have:

\[ \rho=\frac{1}{(2\pi)^3}\int \cos \bar{k}\bar{R}\cos \bar{k}\bar{\lambda}\, dk_x\,dk_y\,dk_z, \tag{72} \]

where by \(\bar{R}\) is denoted the difference \(\bar{R}=\bar{r}-\bar{r}_s\).

We can rewrite expression (72) as the half-sum of two terms

\[ \rho=\frac{1}{(2\pi)^3}\frac{1}{2}\left\{\int \cos \bar{k}(\bar{R}+\bar{\lambda})\, dk_x\,dk_y\,dk_z+ \int \cos \bar{k}(\bar{R}-\bar{\lambda})\, dk_x\,dk_y\,dk_z\right\}, \]

which, on the basis of (16), is written as the half-sum of two \(\delta\)-functions

\[ \rho=\frac{1}{2}\{\delta(\bar{R}+\bar{\lambda})+\delta(\bar{R}-\bar{\lambda})\} \tag{73} \]

and Poisson’s equation

\[ \nabla^2\varphi=-4\pi e\frac{1}{2}\{\delta(\bar{R}+\bar{\lambda})+\delta(\bar{R}-\bar{\lambda})\}. \tag{74} \]

Since both \(\delta\)-functions refer to one and the same charge, we have a simplified model of an extended electron, when the electron charge is distributed at two points. The electron “rod,” a “hollow” linear electron.

Calculating the electron’s own electrostatic energy, in the case of factor (70) we obtain the expression

\[ W\sim \frac{e^2}{\lambda}. \tag{75} \]

This expression tends to infinity as \(\lambda\) tends to zero. Therefore, in a \(\lambda\)-process (the tendency, in the final result, \(\lambda\to 0\)), factor (70) with the spatial vector \(\lambda\) proves to be useless.

In Dirac’s case the factor \(B=\cos k\lambda_0\), where \(\lambda_0=ct_0\) is a time vector. In the present case a coordinate system is chosen in which the spatial part of this vector vanishes.

The charge density \(\rho\) in this case is written:

\[ \rho=\frac{1}{(2\pi)^3}\int \cos \bar{k}(\bar{r}-\bar{r}_s)\cos k\lambda_0\, dk_x\,dk_y\,dk_z. \tag{76} \]

We could, as in the case of (72), carry out the integration and would obtain expression (55), which does not have a “transparent” form. We shall proceed somewhat differently and rewrite (76) in the form (17), introducing, by analogy with \(r\) and \(r_s\), the times \(t\) and \(t_s\).

Denoting \(\lambda_0=c\tau_0\) and transforming the product of cosines into their half-sum, we obtain

\[ \rho=\frac{1}{(2\pi)^3}\cdot\frac{1}{2} \int\left[\cos\{\bar k(\bar r-\bar r_s)-kc(t-t_s-\tau)\}+\right. \]

\[ \left. +\cos\{\bar k(\bar r-\bar r_s)-kc(t-t_s+\tau_0)\}\right]\,dk_x\,dk_y\,dk_z= \]

for \(t=t_s\)

\[ =\frac{1}{(2\pi)^3}\cdot\frac{1}{2} \int\left[\cos\{\bar k(\bar r-\bar r_s)+kc\tau_0\}+\right. \]

\[ \left. +\cos\{\bar k(\bar r-\bar r_s)-kc\tau_0\}\right]\,dk_x\,dk_y\,dk_z . \tag{77} \]

The last expression shows that, owing to the presence of the time vector \(\lambda_0=c\tau_0\), the charge is split, as it were, into two instants of time, \(+\tau_0\) and \(-\tau_0\), somewhat shifted into the “past” and the “future” with respect to the time to which the field \(t\) and the center of gravity of the electron \(t_s\) \((t=t_s)\) refer.

Thus, the connection of Dirac’s \(\lambda\)-process with Wenzel’s ideas is readily established. If a charge extended in space (the case \(\lambda_0\ne 0,\ \lambda_0=0\)) means that the motion of the center of gravity of the electron is influenced by the value of the field existing at a given instant at different points of space (in the case considered above, for example, an electron whose center of gravity is situated at \(r_s\) is acted upon by fields located at the points \(r_s+\lambda\) and \(r_s-\lambda\)), and the electron, as a rigid “rod,” in space experiences as a whole the influence of the fields, then in the case of the time vector \(\lambda_0\), as the preceding consideration shows, the electron is acted upon by the field that existed at the given point a time

\[ \frac{\lambda_0}{c}=\tau_0 \]

“ago,” and by the field that will exist at the given point

\[ \frac{\lambda_0}{c}=\tau_0 \]

“after the given instant” to which this action refers. The electron—point-like in space—experiences the influence of a “past” instant, as if it were, speaking by analogy with the spatial case, “rigid” in time.

Of course, “rigidity” both in space and in time essentially means transmission of the action with instantaneous velocity from the periphery to the center of the electron. Since in the final result \(\lambda_0\) tends to zero, this last remark causes no difficulty.

All that has been said about the \(\lambda\)-process may be summarized as follows: before Wenzel and Dirac, the point electron was regarded as the limit of a charge extended in space. Wenzel and Dirac regard the point electron as the limit of a charge extended symmetrically in time. It then turns out that, in the second case, the classical difficulties of a point charge are absent.

§ 12. THE LORENTZ EQUATION

(the back-reaction of the field on the electron)

Taking into account the back-reaction of the field on the electron, in the simple classical case we obtain the well-known equation

\[ m\ddot R(t)=-\frac{1}{3}\frac{e^2}{ac^2}\ddot R+\frac{2}{3}\frac{e^2}{c^3}\dddot R+\ldots+F\ldots, \tag{78} \]

where \(m\) is the mechanical mass of the electron, \(a\) is the “electron radius,” and \(F\) is the external force. In the coefficient multiplying the higher derivatives there enters the radius of the particle \(a\). Therefore, for a point electron \((a\to 0)\) the equation contains derivatives no higher than the third order. For a point electron, treated in the usual way \((a\to 0)\), the electromagnetic mass of the electron \(\dfrac{e^2}{ca^2}\to\infty\). In Dirac’s theory the electron is pointlike, but owing to the peculiarities of the theory the electron’s proper electromagnetic mass turns into zero.

Thus, in Dirac’s theory equation (78) has the form

\[ m\ddot R(t)=\frac{2}{3}\frac{e^2}{c^3}\dddot R+F. \tag{79} \]

Beginning the construction of his new theory, Dirac proceeds from equation (79), or more precisely, makes an attempt to justify it by considering, alongside the retarded potentials, the advanced potentials*). But, taking equation (79) as given, Dirac\(^{18}\) arrives at sharply unusual conclusions concerning the behavior of an electron in an electromagnetic field: the electron may experience the influence of an electromagnetic impulse before this impulse reaches the electron (i.e., reaches the center of gravity). This would mean that the electron is in some sense extended, and that a signal along it propagates with a velocity greater than the critical one.

The \(\lambda\)-process of Wentzel–Dirac also leads to equation (79); one might therefore think that the \(\lambda\)-process likewise endows the electron with these special qualities, as did the first version of the theory of the classical electron given by Dirac in 1938. A more careful consideration, however, shows that this is not so\(^{25}\). One may be convinced of this\(^{25}\) by retaining the vector in explicit form in deriving equation (78).

In the \(\lambda\)-theory of Wentzel–Dirac there are no exceptions to the fulfillment of the requirements of relativistic invariance. A general proof of this circumstance is given by the Poisson bracket (64). For finite \(\lambda\ne 0\) we would indeed have the case of transmission of action with a velocity greater than the speed of light (the connection would be effected not only on the light cone), but since \(\lambda\) tends, in the final analysis, to zero,

*) The first version of the theory of 1938.

then the right-hand side of (64) in the limit does not differ from the ordinary four-dimensional \(\delta\)-function, which attests to the fulfillment of the requirement of the critical signal velocity.

The original system of equations is likewise symmetric with respect to the sign of time \({}^{25}\).

It is true that, from the fundamental equations of the theory:

\[ m\ddot{\bar R} = \int \rho(\bar r-\bar R)\,\bar E\,dv + \frac{1}{c}\int \rho(\bar r-\bar R)\,[\dot{\bar R}\bar H]\,dv, \]

\[ \bar E = -\frac{1}{c}\frac{\partial \bar A}{\partial t} - \operatorname{grad}\varphi; \qquad \bar H=\operatorname{rot}\bar A, \tag{80} \]

\[ \nabla^{2}\varphi - \frac{1}{c^{2}}\frac{\partial^{2}\varphi}{\partial t^{2}} = 4\pi\rho(\bar r-\bar R), \]

\[ \nabla^{2}\bar A - \frac{1}{c^{2}}\frac{\partial^{2}\bar A}{\partial t^{2}} = -4\pi\rho(\bar r-\bar R) \]

we can, eliminating the field, obtain equation (79). Here \(\rho\) is the density function. We write (80) in general form, since the case of the \(\lambda\)-theory differs only by a special form of the charge-density function (76).

Since the original equations (80) require, for their solution, the specification at the initial instant of the quantities and of their first derivatives, these initial conditions must also be sufficient for the solution of the equivalent equation (78); therefore, in solving equation (78), one cannot arbitrarily prescribe (as Dirac does in his 1938 work) the initial or final values of \(\ddot R\), or even of higher derivatives.

In classical electrodynamics, the question of initial conditions in the transition from the system of equations (80) to (78) was considered in detail by Belousov \({}^{19}\). Of course, one may a priori take equation (77) or (78) as given, without any connection with the system (80), but such a theory has no relation to the Dirac–Wentzel theory.

§ 13. ODD DIVERGENCE

To overcome the odd divergence, Dirac \({}^{1}\) proposes to introduce a certain new auxiliary field, the quantization of which gives photons with negative energy and a negative zero-point energy of empty space.

If the energy of the ordinary electromagnetic field is

\[ H_{+} = \sum \frac{1}{2}\left(A_i^{+}A_i+A_iA_i^{+}\right)h\nu_i = \sum\left(N_{i+}+\frac{1}{2}\right)h\nu_i, \]

then the energy of the new field will be written in the form:

\[ H_{-} = \sum \frac{1}{2}\left(a_i^{+}a_i+a_ia_i^{+}\right)h\nu = -\sum\left(N_{i-}+\frac{1}{2}\right)h\nu_i, \]

i.e. the sum

\[ H_{+}+H_{-} = \sum\left(N_{i+}-N_{i-}\right)h\nu_i \]

does not possess the zero-point energy of empty space.

Since the odd divergences are connected with the presence of zero oscillations of the vacuum, then along the path proposed by Dirac one may hope to avoid the difficulties of odd divergences.

It turns out that the operators \(a^+\) and \(a\) exist, but they possess the property:

\[ [a^+a] = -1, \]

whereas the usual amplitude operators \(A^+\) and \(A\) satisfy the relation:

\[ [A^+A] = 1. \]

The odd integrals are now taken over the limits from \(-\infty\) to \(+\infty\) and vanish (for example,

\[ \int_{-\infty}^{+\infty} k\,dk = 0 \]

). But, unfortunately, the consistent carrying out of this second idea of Dirac requires many new assumptions. The difficulties here consist in restricting this new auxiliary field to its auxiliary functions and obtaining the old expressions for the emission and absorption of quanta (spontaneous radiation, Einstein coefficients). In all other respects, the influence of this field on physical processes must be excluded. Such a problem is solved by Dirac in no ordinary way. Dirac introduces into consideration a certain “imaginary world,” a “mathematical world,” in which all concepts and laws of interaction of particles with positive and negative quanta are defined, and then a connection is established between this mathematical world and the actually observed phenomena of the physical world.

The fundamentally different methodological conceptions of Dirac and Heisenberg attract attention. Heisenberg attempts, by considering only observable quantities, to restrict the variants of theoretical constructions and, in this way, to find a new theory. Dirac, by bringing into consideration a mathematical scheme of the “mathematical,” “imaginary” world, richer in formal possibilities, attempts to resolve the difficulties of the contemporary theory by using this additional freedom of mathematical constructions.

If the \(\lambda\)-process in cases of classical difficulties does not lead to any new difficulties, then this cannot yet be said of Dirac’s second idea.

On the other hand, the odd difficulty is a purely quantum difficulty; it may have its cause in our inability to quantize such systems.

One may think that the difficulties of nonclassical divergences are ultimately connected with the unsuccessful analogy of considering the field as an aggregate of quantum oscillators, and empty space as a “solid body.”

§ 7. Non-electromagnetic Fields

All the attempts considered are easily generalized to the case of non-electromagnetic fields.

The \(\lambda\)-process for the case of a meson field was considered by Pauli, Jauch\(^{20}\), and others. The characteristic results here are the following. In those cases where the quanta of the field have no rest mass of their own (photon, neutrino), the \(\lambda\)-process turns the divergent integrals of the former theory into zero. In those cases, however, where the field quanta possess a nonzero proper mass, the divergent expressions of the ordinary theory are reduced by means of the \(\lambda\)-process to finite expressions different from zero.

For the self-energy of the source of the field one obtains a finite expression of order:

\[ \sim g\mu, \]

where \(\mu\) is the proper mass of the field quantum, and \(g\) is the coupling constant. For existing theories of nuclear interactions this quantity is too small in comparison with the proper mass of the proton or neutron. On the other hand, the anomalous magnetic moment given by the \(\lambda\)-process for the proton and neutron is not correct in sign. This last failure of the \(\lambda\)-process may, of course, be attributed to the imperfection of the theory of nuclear forces, but nevertheless this circumstance at present does not testify in favor of the \(\lambda\)-theory.

True, here there is the tempting possibility, in principle, of regarding the proper mass of elementary particles as equal to one universal constant, identical with the mass of the electron (since, according to the \(\lambda\)-theory, the electromagnetic field adds nothing to this mass), and of ascribing the differences in the inertial masses of elementary particles to their fields, which, according to the \(\lambda\)-theory, in known cases give certain additional finite quantities. But, unfortunately, in existing theories (weak couplings) these additions are small\(^{21}\).

It must be especially emphasized that both Heisenberg’s attempts and the \(\lambda\)-theory overcome difficulties common to all fields of divergences. The theory of nuclear fields has its own specific difficulties—the anomalous increase with energy of cross sections for many effects, the absence of stationary states of systems of nuclear particles for some fields, etc. The \(\lambda\)-process does not solve these difficulties and offers no hope for their solution in the future, since mathematically the \(\lambda\)-theory is quite complete.

Of course, it is entirely possible that some future theory will resolve all these difficulties in a unified way, but one must also think that in a unified way only difficulties common to all field theories are to be resolved.

But here, as we see, the \(\lambda\)-process gives a solution to the question only in the classical cases—quantum divergences require new changes in the theory.

Such a change is partially achieved by Dirac by introducing a new auxiliary field, but it is difficult to escape the intuitive feeling of primitiveness in our method of quantizing systems with an infinite number of degrees of freedom. Perhaps it is more expedient to seek other ways of quantizing such systems than to introduce new fields whose whole purpose consists in the formal correction of the failures of secondary quantization.

Moreover, there is a danger that Dirac’s second proposal leads to certain new difficulties^22.

Finally, there remains the important question of the logarithmic divergence for the proper electrostatic energy of the source when the Dirac vacuum is taken into account—here we have an indication either of the viciousness of the method of the \(\lambda\)-process, or of the untenability of the usual accounting for the influence of the “filled background.”

In conclusion, one may add that for \(\lambda\) different from zero all odd divergences still disappear, while the even ones remain finite and do not require the introduction of negative probabilities, which also does away with the logarithmic divergence of the electron’s proper energy. Finally, this would give a natural “cutoff” to nuclear forces, etc. But in the present case the vector \(\lambda\) must be given a different physical meaning: either it must be connected with the field, or introduced as a new degree of freedom for the particle; but, most importantly, we nevertheless obtain a theory which in the region \(\lambda\) contradicts relativity. It is easy to see that a signal in this region (the size of the particle) propagates instantaneously.

Of course, rejection of relativity “in the small” is possible in principle (Dirac^18), but then the value and advantage of one or another proposed mathematical apparatus is very low: a simple classical particle—a solid sphere—fully corresponds to this conception.

Rejection of relativity in the “small” essentially gives an infinite variety of possibilities—here, too, some limiting point of view is needed.

The requirements of relativistic invariance at present so reduce the possibilities of mathematical speculation that they make rational in theoretical physics “mathematical creation” as a scientific method of research. True, if the electron is regarded as pointlike, but in an indeterminate way localized in the region \(\lambda\), then a connection not on the light cone could be interpreted not as a contradiction with relativity, but as the result of an essentially imprecise localization of charge^23. But such an assertion is purely verbal in character—so long as there is no adequate mathematical apparatus, for Hamilton’s method, as we see, gives no information about the region \(\lambda\) (in this region there are no solutions of the equation of motion).

In reviewing the various attempts to construct new theories of radiation, we have confined ourselves to linear theories and equations of the second order.

We have dealt with the $\lambda$-theory somewhat more fully because it is so far the only relativistically invariant theory that has a mathematically complete form and, physically, does not lead to new complications. Moreover, in its actual results in electrodynamics (the “throwing away” of infinities), it is very close to Heisenberg’s actual scheme and to the ideas of Heitler–Peng. The latter circumstance gave Heitler grounds[^24] to regard Dirac’s $\lambda$-theory as a rigorous exposition of his ideas—but we now know that, in the case of the meson field, the theories give fundamentally different results.

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1. Reference number as printed on the page. ↩