ELIMINATION OF DIVERGENCES IN QUANTUM ELECTRODYNAMICS

A. Akhiezer and R. Polovin

1. INTRODUCTION

1. While explaining and predicting an enormous range of phenomena, quantum electrodynamics nevertheless leads to a whole series of characteristic difficulties—infinite self-energies of the electron and photon, and divergent expressions for higher approximations of perturbation theory. These divergences arise when the interaction of the electron and photon with the zero-point oscillations of the electromagnetic and electron-positron fields is taken into account.

The new relativistically invariant form of perturbation theory developed in recent years has led to considerable progress in quantum electrodynamics. It has made it possible to classify the divergences occurring in the theory and has led to the conclusion that, independently of the order of approximation of perturbation theory, there exist only three basic types of divergences, associated with the self-energies of the electron and photon in the second approximation of perturbation theory and with the scattering of an electron in an external field accompanied by the emission and subsequent absorption of a virtual photon.

In sufficiently high approximations of perturbation theory, very complicated infinite expressions may appear, but it is extremely important that these divergences can always be represented in the form of combinations of the three simplest types of divergences indicated above.

The possibility of classifying divergences is a very important result of the theory, but the new stage in the development of quantum electrodynamics is connected not only with the establishment of a classification of divergences, but also with the establishment of general rules that make it possible to eliminate these divergences in an unambiguous manner.

The formulation of these rules—the so-called regularization rules—is based on the new idea of renormalizing the mass and charge of the electron.

This idea consists in the following. The interaction of the electron with the zero-point oscillations of the electromagnetic and electron-positron fields*) leads to a divergent addition to the energy of the electron. This addition to the energy is equivalent to a certain addition to the mass of the electron, which is called the electromagnetic mass of the electron.

The process of regularization consists in our not taking this addition into account, assuming that for a free electron it is already included in its mass. Thus we are, as it were, carrying out a renormalization of the mass of the electron.

The energy of interaction of an electron bound in an atom with the zero-point oscillations of the electromagnetic and electron-positron fields also proves to be infinite, but it can no longer be discarded. On the contrary, it turns out that the difference between the interaction energies of the bound and the free electron with the zero-point oscillations of the electromagnetic and electron-positron fields is a finite quantity. This difference, which can be defined unambiguously, is the radiative shift of the energy levels of the electron in the atom.

The situation is analogous with the charge of the electron. Owing to the interaction of the charge with the zero-point oscillations of the electron-positron and electromagnetic fields**) it acquires an infinite addition. The process of regularization consists in our not taking this addition into account, assuming that it is inseparable from the charge itself. Thus we are, as it were, carrying out a renormalization of the charge of the electron.

The most important result of the theory is that the elimination of all the divergences encountered in quantum electrodynamics is achieved, essentially speaking, by the renormalization of two constants—the mass and the charge of the electron.

The establishment of regularization rules made possible the calculation of higher approximations of perturbation theory—the so-called radiative corrections. In this way an explanation was found for the experimentally observed radiative shift of the levels of the hydrogen atom and the anomalous magnetic moment of the electron. In these effects the interaction of the electromagnetic field with the “vacuum” of the electron-positron field and the interaction of electrons with the “vacuum” of the electromagnetic field are manifested in a remarkable way.

The radiative shift of atomic levels, the additional magnetic moment of the electron, and the radiative corrections to electron scattering show that the “vacuum” possesses physical properties and that it cannot be regarded as a metaphysical “empty” space—

*) In the first approximation one should take into account only the interaction of the electron with the zero-point oscillations of the electromagnetic field.

**) In the first approximation one should take into account only the interaction of the charge with the zero-point oscillations of the electron-positron field.

space. The development of ideas about the physical vacuum is an essential result of modern quantum electrodynamics.

In effects connected with the interaction of the electron-positron and electromagnetic fields with the “vacuum,” Lenin’s well-known thesis on the inexhaustibility of the properties of the electron has found new confirmation.

The rules of regularization cannot yet be rigorously derived from the fundamental equations describing the electromagnetic and electron-positron fields. However, one may say that the apparatus of modern quantum electrodynamics makes it possible to single out unambiguously, from divergent expressions, finite values having physical meaning.

In the present article the ideas and methods for eliminating divergences in quantum electrodynamics are set forth. Since the probabilities of various processes are most naturally determined with the aid of the scattering matrix, we shall further present the methods for eliminating divergences from the scattering matrix.

1. Let us first recall certain general properties of the scattering matrix*).

The scattering matrix \(S\) can be expanded in a series in powers of the electron charge \(e\):

\[ S=\sum_n S^{(n)}; \]

the \(n\)-th term in this expansion, proportional to \(e^n\), describes all processes of \(n\)-th order. The matrix element \(S^{(n)}\) corresponding to some process \(i \to f\) can be represented in the form

\[ S^{(n)}_{i\to f}=\sum_n M^{(n)}_{i\to f}, \tag{1.1} \]

where the separate terms in the sum differ from one another by the arrangement of the emission and absorption operators of particles. These terms can, as is known, be represented graphically, and the graphs corresponding to \(M^{(n)}_{i\to f}\) are topologically equivalent and differ from one another only in the order of arrangement of the electron and photon lines.

In order to determine the quantity \(M^{(n)}_{i\to f}\) corresponding to some graph, one must remember the following rules\(^3\).

1) To each external electron line there corresponds the spinor of one of the types \(u,\ \bar u,\ v,\ \bar v\), where \(u\) and \(\bar u\) correspond to the absorption and emission of an electron, and \(\bar v\) and \(v\) to the absorption and emission of a positron. The four-momentum of the positron is equal to the four-momentum determining the spinor \(v\), taken with the opposite sign (\(v\) corresponds to a plane wave with negative frequency).

*) See review \(^1\).

2) To each external photon line representing a photon with frequency \(\omega\) and polarization \(\hat e\), there corresponds the operator \(\dfrac{\hat e}{\sqrt{2\omega}}\) *). To each external photon line representing the external electromagnetic field \(A_\mu(x)\), there corresponds the operator \(\dfrac{1}{(2\pi)^4}\hat a(q)\), where

\[ \hat a(q)=\int \hat A(x)e^{-iqx}\,d^4x. \]

3) To each internal electron line there corresponds the operator

\[ \frac{\hat p-m}{p^2+m^2}, \]

where \(p\) is the four-dimensional momentum associated with the line, and \(m\) is the electron mass. (To an internal electron line there corresponds the function

\[ \frac{1}{2}S^F=-\frac{i}{(2\pi)^4}\frac{\hat p-m}{p^2+m^2}. \]

)

4) To each internal photon line with momentum \(k\) there corresponds the factor \(\dfrac{1}{k^2}\), and to its ends the operators \(\gamma_\nu\). (To an internal photon line there corresponds the function

\[ \frac{1}{2}D^F=-\frac{i}{(2\pi)^4}\frac{1}{k^2}. \]

)

5) To each vertex of the graph there corresponds \(\delta(\sum p)\), containing the momenta of all lines meeting at this vertex (the momenta at the two ends of an internal line should be taken with opposite signs).

6) All operators (acting on spinor indices) are arranged from right to left in the same order in which they occur when moving along the electron line.

7) If the graph contains a closed electron loop, then the expression \(M_{i\to f}^{(n)}\) includes the trace (Spur) of the product of the matrices

\[ \frac{\hat p-m}{p^2+m^2} \]

and \(\gamma_\nu\), belonging to the separate lines of the loop and its vertices.

*) If \(a_\mu(\mathbf a, ia_0)\) is a certain four-dimensional vector, then \(\hat a=a_\mu\gamma_\mu\), where \(\gamma_\mu\) are the Dirac matrices satisfying the conditions \(\gamma_\mu\gamma_\nu+\gamma_\nu\gamma_\mu=2\delta_{\mu\nu}\). By the scalar product of two vectors \(a\) and \(b\) we mean the quantity \(ab=\mathbf a\mathbf b-a_0b_0\). The speed of light \(c\) and Planck’s constant \(\hbar\) are everywhere taken equal to unity.

Matrix elements of graphs containing one or several closed electron loops with an odd number of electron lines are equal to zero (Furry’s theorem).

8) Integration is carried out over the four-dimensional momenta of internal lines representing virtual*) particles, and summation over the four-dimensional polarizations of virtual photons.

9) The numerical factor standing before the general expression for \(M_{i\to f}^{(n)}\) is equal to \(\pm \xi e^n(-i)^F(2\pi)^{4(n-F)}\), where \(\xi=\dfrac{r}{n!}\), \(F\) is the total number of internal lines, and \(r\) is the number of similar terms corresponding to \(M_{i\to f}^{(n)}\) in \(S^{(n)}\) (the sign should be found in accordance with the rule for permuting electron-positron operators).

As an illustration of these rules, let us determine the matrix element for scattering of an electron in an external field with emission and absorption of a virtual photon. This process is represented by graph 2 in Fig. 1. In the general formula (1.1) for \(S_{i\to f}^{(n)}\), in this case there enters only one term \(M_{i\to f}^{(n)}\), which, according to the rules set forth, can be represented in the form

Fig. 1.

\[ M_{i\to f}^{(n)} = (-i)^3 e^3 \bar{u}_2 \left( \int \gamma_\nu \frac{i\left(\hat{p}_2-\hat{k}\right)-m} {(p_2-k)^2+m^2} \frac{\hat{a}(q)}{(2\pi)^4} \times \right. \]

\[ \left. \times \frac{i\left(\hat{p}_1-\hat{k}\right)-m} {(p_1-k)^2+m^2} \gamma_\nu \frac{d^4 k}{k^2} \right) u_1, \tag{1.2} \]

where \(u_1\) and \(u_2\) are spinor amplitudes corresponding to the four-dimensional momenta \(p_1\) and \(p_2\) of the electron before and after scattering, \(q=p_2-p_1\). The in-

*) The momentum of a virtual electron, unlike that of a real electron, does not satisfy the relation \(p^2+m^2=0\), and the momentum of a virtual photon, unlike that of a real photon, does not satisfy the relation \(p^2=0\).

entering the expression for \(M^{(3)}_{i\to f}\), three four-dimensional \(\delta\)-functions we have eliminated by integrating over \(q\) and over the momenta of the internal electron lines, replacing here \(q\) by \(p_2-p_1\) and the momenta of the internal electron lines by \(p_1-k\) and \(p_2-k\). For this reason the integration in (1.2) is carried out only over the four-dimensional momentum of the virtual photon \(k\). In (1.2) there is also summation over the polarization states of the virtual photon, i.e., over the four values of the index \(\nu\).

1. It can be shown that in the general case \(M^{(n)}_{i\to f}\) is determined by the following integral in momentum space:

\[ M^{(n)}_{i\to f} = \pm e^n(-i)^F(2\pi)^{4(n-F)} \int d^4p_1\ldots d^4p_{F_e} \int d^4k_1\ldots d^4k_{F_p} \times \]

\[ \times \int d^4q_1\ldots d^4q_s \sum_{\nu}\prod_1^n \delta(\Sigma_p)\, O\left\{ \prod \left( u(p_i)\bar v(p_i)-\frac{\hat e_i}{\sqrt{2\omega_i}} \right) \right. \times \]

\[ \left. \times \prod \left( \bar u(p_f)v(p_f)-\frac{\hat e_f}{\sqrt{2\omega_f}} \right) \prod_1^s\frac{\hat a(q)}{(2\pi)^4} \prod_1^{F_e} \left( \frac{i\hat p-m}{p^2+m^2} \right) \times \]

\[ \times \prod_1^{F_p} \gamma_\nu \frac{1}{k^2}\gamma_\nu, \tag{1.3} \]

where the integration is performed over \(4F_e\) variables \(p_1,p_2,\ldots,p_{F_e}\), arising from factors of the type \(\frac{1}{2}SF\), over \(4F_p\) variables \(k_1,k_2,\ldots,k_{F_p}\), arising from factors of the type \(\frac{1}{2}DF\), and also over \(4S\) variables \(q_1,\ldots,q_s\), arising from the expansion of the external potentials in a Fourier integral. One may say that the integration is carried out over the four-dimensional momenta of the virtual particles (\(F_e\) and \(F_p\) are the numbers of internal electron and photon lines, i.e., the numbers of virtual electrons and photons; \(F_e+F_p=F\)), and also over the variables \(q\). The summation is performed over the four values of the indices \(\nu\), denoting the different polarizations of the virtual photons, with each virtual photon corresponding to a definite index \(\nu\), taking the values \(\nu=1,2,3,4\).

The separate factors in (1.3) have the following meaning:

\[ \prod \left( u(p_i)\bar v(p_i)-\frac{\hat e_i}{\sqrt{2\omega_i}} \right) \]

denotes the product of the spinors \(u\), \(\bar v\) and the operators \(\dfrac{\hat e_i}{\sqrt{2\omega_i}}\), describing electrons, positrons, and photons in the initial state. \(i\) (\(p_i\) is the four-dimensional momentum of an electron or positron; \(e_i\) is the unit polarization vector of a photon with momentum

\(k_i,\ \hat e=e_\mu\gamma_\mu);\ \Pi\left(\bar u(p_f)v(p_f)\dfrac{\hat e_f}{\sqrt{2\omega_f}}\right)\) denotes the analogous product for the final state \(f\).

The factors
\[ \frac{i\hat p-m}{p^2+m^2}\,(\hat p=-p_\mu\gamma_\mu) \quad\text{and}\quad \gamma_\mu\frac{1}{k^2}\gamma_\mu \]
arise from contractions of electron and photon operators, respectively; the number of the former is equal to \(F_e\), and of the latter to \(F_p\).

The numerical factor \(\xi=\dfrac{r}{n!}\) takes into account the presence of \(r\) equivalent normal \((N)\) products of the type of interest to us in the expansion of \(S^{(n)}\) into \(N\)-products.*) All of them, as is easy to show, have the same sign and therefore make the same contribution to \(M_{i\to f}^{(n)}\).

The individual terms in (1.3) may have different signs. The sign \(\pm\) appearing on the right-hand side of (1.3) can be simply determined in each concrete case in accordance with the rule for permuting electron operators.

Finally, the symbol \(O\) serves to denote a definite order in the arrangement of the operators in (1.3), namely: the operators (acting on spinor indices) must be arranged, counting from right to left, in the sequence in which they occur when one moves along the electron line of the graph.

2. STRUCTURE OF GRAPHS

1. The momentum representation of the scattering matrix makes it possible to carry out a general investigation of the properties of its matrix elements. Turning to the consideration of this question, we first make several general remarks on the structure of the graphs representing matrix elements.

If a graph contains several parts not connected with one another, then it evidently represents the same number of mutually unrelated scattering processes. The matrix element corresponding to such a graph then decomposes into a series of separate factors, which are matrix elements corresponding to the individual scattering processes. It is clear that it is sufficient to restrict oneself to the study of those matrix elements and graphs which do not decompose into separate, mutually unconnected parts.

*) In an \(N\)-product the absorption operators stand on the right, and the emission operators of particles on the left.

In many cases a graph may contain one or several parts that are connected with the remaining parts of the graph by only two homogeneous (i.e., electron or photon) lines. Such parts we shall call self-energy parts*).

A part of a graph connected with the remaining parts by only two electron lines will be called an electron self-energy part.

A part of a graph connected with the remaining parts by only two electron lines will be called an electron self-energy part.

A part of a graph connected with the remaining parts by only two photon lines will be called a photon self-energy part.

If in a graph a part can be singled out which is connected with the remaining parts by only two electron lines and one photon line, then such a part will be called a vertex, or corner, part.

Schematically these parts of graphs are shown in Fig. 2, in which they are denoted respectively by \(W_e\), \(W_p\), \(V\) (the remaining parts of the graph are denoted by the squares \(A\), \(B\), \(C\)). The simplest structures \(W_e\), \(W_p\), and \(V\) are presented in Fig. 3.

Fig. 2.

Fig. 3.

We shall distinguish two groups of parts of graphs corresponding to self-energy parts, vertex parts, and other internal parts of the graph: reducible and irreducible parts.

We shall call irreducible such a part that cannot be divided into parts connected with one another by only a single line and that does not contain within itself self-energy parts and vertex parts. In the opposite case we call

*) This name is connected with the fact that self-energy parts describe the interaction of an electron with the zero-point oscillations of the electromagnetic field and the interaction of a photon with the zero-point oscillations of the electron-positron field; see below.

part of the self-energy, the vertex part, or another internal part of the graph is reducible.

Figure 4 shows parts of the electron self-energy of the second and fourth orders.

Graph 1 is irreducible; the remaining graphs are reducible. Of these, graph 2 separates into two parts of the self-energy which are connected to one another by a single electron line;

Fig. 4.

graph 3 contains a part of the electron self-energy \(W'_e\); graph 4 contains a part of the photon self-energy \(W'_p\); graph 5 contains the vertex part \(V_a\) at the vertex \(a\) and the vertex part \(V_b\) at the vertex \(b\) (the internal photon line \(k_a\) and the internal electron lines \(p_a\) and \(p\) belong to \(V_a\); the photon line \(k_b\) and the electron lines \(p_b\) and \(p_1\) are external with respect to \(V_a\); analogously, the internal photon line \(k_b\) and the internal electron lines \(p_b\) and \(p\) belong to \(V_b\); the photon line \(k_a\) and the electron lines \(p_a\) and \(p_2\) are external with respect to \(V_b\)).

It is easy to verify that graph 1 is the only irreducible graph of the electron self-energy.

Figure 5 shows parts of the photon self-energy of the second and fourth orders.

Graph 1 is irreducible; the remaining graphs are reducible. It is easy to verify that graph 1 shown in Fig. 5 is the only irreducible graph of the photon self-energy.

1. The expediency of separating out parts of the self-energy and vertex parts and studying them separately is connected with the fact that, in the general expression (1.3) for \(M_{r \to s}^{(n)}\), separate factors correspond to the parts of the self-energy and to the vertex parts, and these factors do not depend on the structure of the remaining parts of the graph (\(\tau\), i.e., on the structure of the squares \(A, B, C\) in Fig. 2), and therefore, if they are calculated in advance, they can be used in determining the matrix elements of many processes.

In this case the graphs may be replaced by certain “equivalent” skeleton graphs which do not contain parts of the self-

own energy and vertex parts, but with modified expressions for the connections of the operators and other quantities relating to the lines and vertices of the graph from which the self-energy parts and vertex parts have been removed.

The skeleton graphs are shown in Fig. 6.

An electron line connecting \(A\) and \(B\), from which a part of the electron self-energy has been removed, now corresponds not to \(\dfrac{1}{2}S^F\),

Fig. 5.

but to some other operator \(\dfrac{1}{2}\delta S^F\), depending on the momentum \(p\) of the electron line and on the particular form of the part of the electron self-energy \(W_e\). If \(W_e\) is the irreducible part of the electron self-energy shown in Fig. 3, then \(\dfrac{1}{2}\delta S^F(p)\), according to (1.3), can be represented in the form

Fig. 6.

\[ \frac{1}{2}\,\delta S^F(p) = \frac{e^2}{(2\pi)^8}\, \frac{i\hat p-m}{p^2+m^2} \left( \int \gamma_\nu\, \frac{i(\hat p-\hat k)-m}{(p-k)^2+m^2}\, \gamma_\nu\, \frac{d^4k}{k^2} \right) \frac{i\hat p-m}{p^2+m^2}. \tag{2.1} \]

Thus,*)

\[ \frac{1}{2}\,\delta S^F(p) = \frac{1}{2}S^F(p)\,\Sigma(W_e,p)\,\frac{1}{2}S^F(p), \tag{2.2} \]

*) Recall that

\[ \frac{1}{2}S^F(p) = -\frac{i}{(2\pi)^4}\, \frac{i\hat p-m}{p^2+m^2}. \]

ELIMINATION OF DIVERGENCES IN QUANTUM ELECTRODYNAMICS

where

\[ \Sigma(W_e,p)=e^2\Sigma^{(2)}(W_e,p) =-e^2\int \gamma_\nu \frac{i(\hat p-\hat k)-m}{(p-k)^2+m^2}\gamma_\nu \frac{d^4k}{k^2}. \tag{2.3} \]

This formula is valid for the irreducible part of the electron self-energy, but \(\frac12\delta S^F\) can always be represented in the form (2.3), where \(\Sigma(W_e,p)\) will be determined by a formula more complicated than (2.3), depending on the concrete structure of the electron self-energy \(W_e\).

Let us note the circumstance that the integral (2.3) diverges linearly in the region of large momenta \(k\).

If a part of the graph \(A\) is absent, i.e., the electron line connecting \(W_e\) with \(A\) represents a free electron or positron, then the electron line with the removed part of the electron self-energy will correspond not to the spinor \(u(p)\), but to the modified spinor \(\delta u(p)\), having the form

\[ \delta u(p)=\frac12 S^F(p)\Sigma(W_e,p)u(p), \tag{2.4} \]

where \(\Sigma(W_e,p)\) is the same operator that enters into (2.2). In an analogous manner, the spinor \(\bar u(p)\) must be replaced in the skeleton graph by the spinor

\[ \delta \bar u(p)=\bar u(p)\Sigma(W_e,p)\frac12 S^F(p). \tag{2.4'} \]

The same formulas are valid for the spinors \(v\) and \(\bar v\).

The photon line connecting \(A\) and \(B\) in the skeleton graph corresponds not to \(\frac12 D^F\), but to some other function \(\frac12\delta D^F\), depending on the momentum \(k\) of the photon line and on the concrete form of the part of the photon self-energy \(W_p\).

If \(W_p\) is the irreducible part of the photon self-energy shown in Fig. 3, then \(\frac12\delta D^F\) may, according to (1.3), be represented in the form

\[ \frac12\delta D^F(k) =-\frac12\frac{e^2}{(2\pi)^8}\frac{1}{k^2} \int \operatorname{Spur}\left\{ \gamma_\nu \frac{i\hat p-m}{p^2+m^2} \gamma_\nu \frac{i(\hat p+\hat k)-m}{(p+k)^2+m^2} \right\} \frac{d^4p}{k^2}. \]

Thus\(^*\),

\[ \frac12\delta D^F(k) =\frac12 D^F(k)\Pi(W_p,k)\frac12 D^F(k), \tag{2.5} \]

\[ {}^*\text{Recall that}\qquad \frac12 D^F(p)=-\frac{i}{(2\pi)^4}\frac{1}{p^2}. \]

where

\[ \Pi(W_p,k)=e^2\Pi^{(2)}(W_p,k)= \]

\[ =\frac{1}{2}e^2\int \operatorname{Spur} \left\{ \gamma_\nu \frac{i\hat p-m}{p^2+m^2} \gamma_\nu \frac{i(\hat p+\hat k)+m}{(p+k)^2+m^2} \right\}\,d^4p. \tag{2.6} \]

This formula is valid for the irreducible part of the photon self-energy, but \(\frac{1}{2}\delta D^F(k)\) can always be represented in the form (2.5), where \(\Pi(W_p,k)\) will be determined by a more complicated formula than (2.6), depending on the concrete structure of the part of the photon self-energy \(W_p\).

Let us note the circumstance that the integral (2.6) diverges quadratically in the region of large momenta \(p\).

If the part of graph \(A\) is absent, i.e. the photon line joining \(W_p\) with \(A\) represents a free photon or an external electromagnetic field, then to the photon line with the removed part of the photon self-energy there will correspond not the operator \(\hat A(k)\), but the modified operator \(\delta\hat A(k)\), of the form

\[ \delta\hat A(k)=\hat A(k)\Pi(W_p,k)\frac{1}{2}D^F(p), \tag{2.7} \]

where \(\Pi\) is the same quantity that enters into (2.5).

Finally, if the graph contains a vertex part, then to the point \(1\) of the skeletal graph (see graph 3 in Fig. 6) there will correspond not the matrix \(\gamma_\mu\), but some other operator \(\Lambda_\mu\), depending on the form of the vertex part \(V\) and on the momenta of the electron lines \(p_1\) and \(p_2\) approaching point \(1\).

In the case of the irreducible vertex part of third order shown in Fig. 3, the operator \(\Lambda_\mu\), according to (1.3), can be represented in the form

\[ \Lambda_\mu(V,p_1,p_2)=e^3\Lambda_\mu^{(3)}(V,p_1,p_2)= \]

\[ =\frac{ie^3}{(2\pi)^4}\int \left\{ \gamma_\nu \frac{i(\hat p_2-\hat k)-m}{(p_2-k)^2+m^2} \gamma_\mu \frac{i(\hat p_1-\hat k)-m}{(p_1-k)^2+m^2} \gamma_\nu \frac{d^4k}{k^2} \right\}. \tag{2.8} \]

We see that \(\Lambda_\mu\) is expressed by an integral which diverges logarithmically in the region of large \(k\).

Thus, if a graph contains parts of the self-energy and vertex parts, then its matrix element can be obtained from the matrix element of the skeletal graph, if in the correspond-

to make, in the latter places, the substitution:

\[ \left. \begin{gathered} -\frac{1}{2}SF \longrightarrow -\frac{1}{2}\delta SF,\quad \frac{1}{2}DF \longrightarrow \frac{1}{2}\delta DF,\\ u,v \longrightarrow \delta u,\delta v;\quad \bar u,\bar v \longrightarrow \delta\bar u,\delta\bar v,\\ A_\mu \longrightarrow \delta A_\mu,\quad \gamma_\mu \longrightarrow \Lambda_\mu . \end{gathered} \right\} \tag{2.9} \]

1. Naturally, the study of the internal, inserted parts of a graph should not be confined to only the parts of the self-energy and the vertex parts just considered; it is also necessary to study those parts of the graph which are connected with the remaining parts by a larger number of lines than in the cases considered. In doing so, however, one should bear in mind that if a graph contains a certain part connected with the remaining parts by only an odd number of photon lines, then the corresponding matrix element is equal to zero (Furry’s theorem).

In analyzing the singularities of the scattering matrix, besides the self-energy parts and vertex parts, those parts of the graph which are connected with the remaining parts by only four photon lines are essential. The simplest form of such a part of a graph is shown in Fig. 7.

We shall call it a part for the scattering of a photon by a photon, since, if the parts \(A,B,C,D\) are absent, this graph represents the scattering of a photon by a photon.

Fig. 7.

The self-energy parts and vertex parts lead to divergent matrix elements, since the operators \(\delta SF\), \(\Lambda_\mu\), \(\delta\hat A_\mu\), as well as the spinors \(\delta u\) and \(\delta\bar u\) and the function \(\delta DF\), are expressed by integrals which diverge in the region of large momenta of the virtual particles.*)

Our principal further task consists in eliminating these divergences arising upon integration over the region of large momenta of the virtual particles, and in separating from the divergent matrix elements finite quantities having physical meaning.

Let us note that in a number of questions a divergence arises in the region of small momenta. This divergence, known under the name of the infrared catastrophe, is connected with the inapplicability of perturbation theory and can be eliminated comparatively simply. Therefore, below we consider only divergences in the region of large momenta of the virtual particles.

* ) Let us recall that the virtual particles connected with the internal lines of the graphs, and therefore the time component of their four-dimensional “momentum,” are not connected with the spatial part.

3. CLASSIFICATION OF DIVERGENCES

1. Before proceeding to the solution of the problem posed, we shall clarify the character of the divergences that arise. Let us note that, for a sufficiently complicated structure of the self-energy parts and of the vertex parts, divergent factors will appear in the matrix element which, generally speaking, can lead to infinities of arbitrarily high order. If, for example, the self-energy part shown in graph 1 of Fig. 4 leads to some infinity, which we shall conventionally regard as an infinity of the first order, then the self-energy part shown in graph 2 of Fig. 4 will obviously lead to an infinity of the second order.

It follows from this that one must first of all study the divergences which arise from irreducible internal parts of the graph (i.e., irreducible parts of the self-energy and vertex parts). We shall now show that irreducible internal parts of graphs lead to only four kinds of divergences of matrix elements.

Let \(n\) denote the number of vertices of an irreducible internal part of the graph, \(F\) the number of its internal lines, and \(N\) the number of external lines of this part connecting it with the remaining parts of the graph (i.e., with the squares \(A, B, C\) in Fig. 2). Let \(N_e\) denote, in addition, the number of external electron lines, and \(N_p\) the number of external photon lines, \(N=N_e+N_p\). Consider the part of the general matrix element associated with the variables pertaining to the internal part of the graph under consideration. This part of the matrix element is evidently an integral over \(4F\) variables—the momenta of the internal lines of the irreducible internal part of the graph. The momenta of the \(N\) external lines will enter this integral as constant parameters.

Since the integrand contains \(n\) four-dimensional \(\delta\)-functions, not all of the \(4F\) integration variables will be independent. We can evidently eliminate only \(n-1\) of these \(\delta\)-functions by integrating over \(4(n-1)\) variables, but we must retain one four-dimensional \(\delta\)-function, which will express the conservation law for the four-dimensional momentum for the part of the graph under consideration. This conservation law expresses the relation between the momenta of the lines connecting the self-energy parts \(W_e, W_p\) and the vertex part \(V\) with the remaining parts of the graph. Therefore one may assert that the number of independent integration variables will be \(4(F-n+1)\).

The integrand is evidently a rational function; moreover, according to (1.3), the degree of the numerator will be \(F_e\), and that of the denominator \(2F\) (\(F_e\) is the number of internal electron lines).

Since our graph is irreducible, the integrand does not split into separate factors containing variables not connected with one another, and the convergence of the integral in the region of large momenta is determined by the difference of the degrees of the numerator and denominator or, more precisely, by the quantity

\[ K=2F-F_e-4(F-n+1). \]

If \(K\geqslant 1\), then the integral converges; for \(K=0\) the integral diverges logarithmically; for \(K=-1\) linearly; for \(K=-2\) quadratically, etc.

Since two electron lines and one photon line pass through each vertex of the graph, it is evident that

\[ 2F_e+N_e=2n,\qquad 2F_p+N_p=n, \]

whence it follows that

\[ K=\frac{3}{2}N_e+N_p-4. \tag{3.1} \]

1. Relation (3.1) makes it possible to enumerate all possible cases of divergence of irreducible internal parts of graphs. There are, evidently, seven types of divergences arising for the following values of the numbers \(N_e\) and \(N_p\):

1) \(N_e=2,\quad N_p=0,\quad K=-1\), — linear divergence;
2) \(N_e=2,\quad N_p=1,\quad K=0\), — logarithmic divergence;
3) \(N_e=0,\quad N_p=1,\quad K=-3\), — cubic divergence;
4) \(N_e=0,\quad N_p=2,\quad K=-2\), — quadratic divergence;
5) \(N_e=0,\quad N_p=3,\quad K=-1\), — linear divergence;
6) \(N_e=0,\quad N_p=4,\quad K=0\), — logarithmic divergence;
7) \(N_e=0,\quad N_p=0,\quad K=-4\), — divergence of the fourth order.

In Fig. 8 the simplest irreducible graphs corresponding to these divergences are shown.

Let us recall that there exists only one irreducible graph for the self-energy of the electron and one irreducible graph for the self-energy of the photon; there may be arbitrarily many irreducible graphs of the vertex part and of photon–photon scattering (in Fig. 8 the simplest graphs of these types are shown).

Using Furry’s theorem, we can at once exclude cases 3) and 5) from consideration, since in these cases the internal parts of the graphs are joined to the remaining parts by an odd number of photon lines, and the matrix elements corresponding to the graphs are equal to zero.

Furthermore, case 7) evidently has no physical meaning, since here the internal part of the graph is not connected at all with its remaining parts. Graph 7 evidently describes the vacuum—vacuum transition in the absence of external fields in the second approximation,

and the matrix element corresponding to it determines the second-order correction to the amplitude of this transition. On the other hand, it is obvious that the probability of the vacuum—vacuum transition in the absence

1

\[ N_e = 2,\quad N_p = 0 \]
— linear divergence

2

\[ N_e = 2,\quad N_p = 1 \]
— logarithmic divergence

3

\[ N_e = 0,\quad N_p = 1 \]
— cubic divergence

4

\[ N_e = 0,\quad N_p = 2 \]
— quadratic divergence

5

\[ N_e = 0,\quad N_p = 3 \]
— linear divergence

6

\[ N_e = 0,\quad N_p = 4 \]
— logarithmic divergence

7

\[ N_e = 0,\quad N_p = 0 \]
— divergence of 4th order

Fig. 8.

of fields is equal to unity, and therefore all corrections to it should be regarded as equal to zero.

As for the scattering of a photon by a photon, then, as can be shown by direct calculation, at least in the lowest (fourth) approximation the sum of the matrix elements corresponding to the various diagrams of photon scattering by a photon (they dif-

…are determined by the order in which the photon lines are arranged), contains no divergence. Therefore one may assert that in reality there exist only three types of divergences, associated with the irreducible parts of the proper energy of the electron and photon and with the irreducible vertex parts. We shall now turn to the investigation of these divergences.

4. ELIMINATION OF DIVERGENCES FROM THE SCATTERING MATRIX

1. We shall now pose the following question: is it possible to eliminate divergences from the scattering matrix and to single out, in a unique manner, from the divergent matrix elements finite expressions having physical meaning.

Let us note preliminarily that the processes represented by parts of the proper energy lead to infinite proper energies of the free electron and photon.*) To verify this, let us consider the interaction of a free electron with the zero oscillations of the electromagnetic field, i.e. with the vacuum of the electromagnetic field. This process is evidently represented by graph 1 of Fig. 3, i.e. by a part of the proper energy of the electron. The difference from the general case shown in Fig. 2 consists in the fact that here the parts \(A\) and \(B\) are absent and the momentum \(p\) is the momentum of a real electron, satisfying the relation \(p^2 + m^2 = 0\).

Using the general rules for writing matrix elements, we obtain the following expression for the matrix element corresponding to the process under consideration:

\[ S^{(2)}_{p\to p} = -e^2 \bar u(p) \left( \int \gamma_\nu \frac{i\hat p' - m}{p'^2 + m^2} \gamma_\nu \frac{1}{k^2} \times \right. \]

\[ \left. \times \delta(p-k-p')\delta(p-k-p')\,d^4p'\,d^4k\,u(p) \right). \tag{4.1} \]

Eliminating one \(\delta\)-function by integration with respect to \(p'\), we replace the second \(\delta\)-function by an integral over four-dimensional space:

\[ \delta(q)=\frac{1}{(2\pi)^4}\int e^{iqx}\,d^4x,\qquad q=p-k-p'. \]

Since \(q=0\), the domain of integration must here be regarded as bounded. Assuming that the spatial volume is equal to unity and that the interval of variation of time, i.e. the time of interaction of the electron with the electromagnetic field, is equal to \(\Delta t\), we obtain the fol—

*) For this reason these parts of graphs are called parts of the proper energy.

the following expression for the matrix element \(S^{(2)}_{p\to p}\):

\[ S^{(2)}_{p\to p} = -\frac{\Delta t e^2}{(2\pi)^4}\, \bar u(p) \left( \int \gamma_\nu \frac{i(\hat p-\hat k)-m}{(p-k)^2+m^2} \gamma_\nu \frac{d^4k}{k^2} \right) u(p). \]

On the other hand, we may consider that the interaction of the electron with the zero-point oscillations of the electromagnetic field leads to a change in the energy of the electron. If this change is equal to \(\Delta\varepsilon\), then during the time \(\Delta t\) the electron wave function changes by \((e^{-i\Delta\varepsilon\Delta t}-1)u(p)\). In order to find the matrix element for the transition of the electron from the initial state with energy \(\varepsilon\), when the electron has not yet interacted with the field, to the final state with energy \(\varepsilon+\Delta\varepsilon\), this expression must be multiplied by \(u^*(p)\) (we recall that the spinors \(u(p)\) and \(u^*(p)\) are related by the normalization condition \(u^*u=1\)). Assuming that \(\Delta\varepsilon\Delta t\ll 1\), we find that the matrix element is

\[ S^{(2)}_{p\to p}=-i\Delta t\Delta\varepsilon . \]

Comparison of this expression with that obtained above shows that the “proper” energy of the electron is equal to *)

\[ \Delta\varepsilon = -\frac{i e^2}{(2\pi)^4}\, \bar u(p) \left( \int \gamma_\nu \frac{i(\hat p-\hat k)-m}{(p-k)^2+m^2} \gamma_\nu \frac{d^4k}{k^2} \right) u(p). \]

This expression is easily reduced to an equivalent change in the electron mass \(\Delta m\):

\[ \varepsilon\Delta\varepsilon=m\Delta m. \]

Noting that from the Dirac equation there follows the relation

\[ \varepsilon \bar u(p)u(p)=m u^*(p)u(p)=m, \]

we obtain the following expression for the change in mass \(\Delta m\):

\[ \Delta m = -\frac{i e^2}{(2\pi)^4}\, \frac{1}{\bar u u}\, \bar u \left( \int \gamma_\nu \frac{i(\hat p-\hat k)-m}{(p-k)^2+m^2} \gamma_\nu \frac{d^4k}{k^2} \right) u. \tag{4.2} \]

This mass may be called the electromagnetic mass of the electron. We see that expression (4.2) diverges linearly in the region of large momenta of the virtual photon, which is in accordance with the results obtained above on the classification of divergences **).

Let us now consider the interaction of a photon with zero-point oscillations, i.e. with the vacuum of the electron-positron field. This pro-

*) Here \(e\) is measured in Heaviside units.

**) By first carrying out the integration over the angles, one can show that this expression in fact diverges logarithmically.

process is depicted by graph 2 in Fig. 3, i.e. by a part of the photon self-energy. The difference from the general case shown in Fig. 2 consists in the fact that here the parts \(A\) and \(B\) are absent and the momentum \(k\) is the momentum of a real photon, satisfying the relation \(k^{2}=0\).

Using the general rules for writing matrix elements and repeating the preceding arguments, we obtain the following expression for the self-energy of a photon of frequency \(\omega\), due to its interaction with the vacuum of the electron-positron field

\[ \Delta \varepsilon_{p}=\frac{e^{2}}{2\omega}\int \operatorname{Spur}\left\{\gamma_{\nu}\frac{i\hat p-m}{p^{2}+m^{2}}\,\gamma_{\nu}\frac{i(\hat p-\hat k)-m}{(p-k)^{2}+m^{2}}\right\}\frac{d^{4}p}{(2\pi)^{4}} . \tag{4.3} \]

This expression diverges quadratically in the region of large momenta of the virtual electron, which is in agreement with the results obtained above on the classification of divergences.

1. Thus, the parts of the self-energy do indeed lead to infinite self-energies of the free electron and photon.

The presence of these infinite energies is an essential defect of the theory. However, the circumstance that the parts of the graphs which we have called parts of the self-energy are connected with the infinite self-energies of the free electron and photon makes it possible to establish a general and unambiguous procedure for extracting from divergent matrix elements, containing irreducible parts of the self-energy, finite expressions having physical meaning. This procedure is based on the fact that the self-energies of the free electron and photon are infinite only formally in the existing theory; in reality they are equal to zero. In other words, when we are dealing with a free electron, its experimentally determined mass \(m\) already includes, as a constituent part (if such exists at all), the electromagnetic mass. Therefore we need not take into account the change in the electron mass \(\Delta m\), and we must impose the physical requirement that this change be equal to zero, i.e.

\[ \Delta \varepsilon = 0. \tag{4.4} \]

In an analogous way, the photon is a particle whose mass and self-energy are equal to zero. Therefore we need not take into account the self-energy of the photon, and we must impose the physical requirement that it be equal to zero:

\[ \Delta \varepsilon_{p}=0. \tag{4.5} \]

From this follow important consequences concerning the divergent quantities \(\Sigma(W_{e},p)\) and \(\Pi(W_{p},k)\), introduced in (2.3) and (2.6).

Let us first consider the irreducible part of the self-energy of the electron, shown in graph 1 of Fig. 9. Here \(A\) denotes some process as a result of which a free electron with momentum \(p\) appears, but before this electron finally appears as a free particle there takes place an interaction of the electron with the vacuum of the electromagnetic field, which

Fig. 9.

is represented by the part of the self-energy \(W_e\). It is clear that graph 1 of Fig. 9 describes a second-order correction to the scattering process shown in graph 2 of Fig. 9, as a result of which the electron immediately appears as free.

Guided by the general rules for writing matrix elements, we may, according to (2.9), not take into account the part of the self-energy \(W_e\) in graph 1 of Fig. 9, but instead must replace, in the skeleton graph, the spinor \(u(p)\) by the spinor \(\delta u(p)\)

\[ \delta u(p)= -\frac{1}{2}\, S^F(p)\Sigma(W_e,p)u(p). \]

On the other hand, according to the considerations set forth above, we should not take the process 1 into account at all, since the electromagnetic mass of the electron is already included in its total mass \(m\). Therefore the spinor \(\delta u(p)\) must physically be equal to zero, and since the operator \(S^F(p)\) for a free electron \((p^2+m^2=0)\) has, obviously, at \(\hat p=im\) a pole of the first order, \(\Sigma(W_e,p)\) must physically have a zero of the second order at \(\hat p=im\).

Let us now consider the operator \(\Sigma(W_e,p)\), where the momentum \(p\) of the virtual electron does not, in general, satisfy the relation \(p^2+m^2=0\). Such an operator appears, as we know, if a part of the electron self-energy connects two parts of a graph (see Fig. 2). Above we saw (see formula (2.3)) that the operator \(\Sigma(W_e,p)\) has the form of an integral of a certain rational function \(R(\hat t,\hat p)\)

\[ \Sigma(W_e,p)=\int R(\hat t,\hat p)\,d^4t, \]

where the matrix vector \(\hat p\) enters into \(R\) not independently, but in the form of a sum with the vector \(\hat t\) (\(t\) denotes the momentum of the virtual particle—

photon—with respect to which the integration is carried out). This integral diverges linearly in the region of large \(|t|\).

We shall show that \(\Sigma(W_e,p)\) can be represented in the form

\[ \Sigma(W_e,p)=\Sigma_1+\Sigma_0(\hat p-im)+\Sigma_R(W_e,p), \tag{4.6} \]

where \(\Sigma_1\) and \(\Sigma_0\) are divergent constants independent of \(\hat p\), and \(\Sigma_R=(\hat p-im)s(\hat p)\) is an operator containing no divergences, with \(s(\hat p)\) vanishing for \(\hat p=im\).

For this purpose let us represent \(R(\hat t,\hat p)\) in the form

\[ R(\hat t,\hat p)=R(\hat t,0)+p_\alpha\left(\frac{\partial R}{\partial p_\alpha}\right)_{p=0}+R_c(\hat t,\hat p). \]

We then obtain the following expression for \(\Sigma(W_e,p)\):

\[ \Sigma(W_e,p)=A' + B'_\alpha p'_\alpha+\Sigma_c(W_e,p),\qquad \Sigma_c=\int R_c(\hat t,\hat p)\,d^4t, \tag{4.6'} \]

where \(A'\) and \(B'_\alpha\) are divergent operators independent of \(p\), and \(\Sigma_c(W_e,p)\) is an operator containing no divergences. Indeed, \(R_c\) has, evidently, the form

\[ R_c(\hat t,\hat p)=R(\hat t,\hat p)-R(\hat t,0)-p_\alpha\left(\frac{\partial R}{\partial p_\alpha}\right)_{p=0} = \]

\[ =\frac{1}{2}p_\alpha p_\beta \left(\frac{\partial^2 R}{\partial p_\alpha \partial p_\beta}\right)_{p=\xi\ne 0}, \]

where the second derivative is taken at some point \(p=\xi\ne 0\). Since \(\hat p\) enters into \(R\) in the form of a sum with the vector \(\hat t\), it may be asserted that \(\Sigma_c\) is determined by the integral of the second derivative of \(R\) with respect to \(\hat t\). Hence it follows that for large \(|t|\) the integrand \(\Sigma_c\) will contain an additional factor of order \(|t|^{-2}\) in comparison with the integrand \(\Sigma\), and since \(\Sigma\) diverges linearly, \(\Sigma_c\) will converge absolutely.

From invariance considerations it follows that the operator \(\Sigma_c\) must have the form

\[ \Sigma_c(W_e,p)=s_1(W_e,p^2)+s_2(W_e,p^2)\hat p, \tag{4.6''} \]

where \(s_1\) and \(s_2\) are certain functions of \(p^2\). From these same considerations

it follows that the divergent operator \(B'_\alpha\) has the form

\[ B'_\alpha=\gamma_\alpha B', \]

where \(B'\) is a certain divergent constant.

Expression (4.6″) can evidently be rewritten in the form

\[ \Sigma_c(W_e,p)=\Sigma_c(W_e,0)+b(\hat p-im)+\Sigma_R(W_e,p), \]

where

\[ \Sigma_R(W_e,p)=(\hat p-im)s(\hat p) \]

vanishes for \(\hat p=im\). Substituting this expression in (4.6′) and introducing the notation

\[ A'+\Sigma_c(W_e,0)=\Sigma_1,\qquad B'+b=\Sigma_0, \]

we obtain relation (4.6), which is the expansion of the operator \(\Sigma(W_e,p)\) in a series in powers of \(\hat p-im\).

We have explained above that the divergent operator \(\Sigma(W_e,p)\) must physically have a zero of second order at \(\hat p=im\). This condition is satisfied by the operator \(\Sigma_R(W_e,p)\), but not by the divergent operators \(\Sigma_1\) and \(\Sigma_0(\hat p-im)\) (the first of these operators diverges, obviously, linearly, and the second logarithmically).

We shall now adopt as a postulate that it is the operator \(\Sigma_R(W_e,p)\), and not the operator \(\Sigma(W_e,p)\), that has physical meaning. In other words, in (4.6) we shall discard the divergent terms \(\Sigma_1\) and \(\Sigma_0(\hat p-im)\), and in all matrix elements containing the linearly divergent operator \(\Sigma(W_e,p)\) we shall replace \(\Sigma\) by \(\Sigma_R\).

The convergent operator \(\Sigma_R(W_e,p)\), by which the linearly divergent operator \(\Sigma(W_e,p)\) is to be replaced in matrix elements, will be called the regularized operator \(\Sigma\).

1. Let us now consider the irreducible part of the photon self-energy represented by diagram 3 in Fig. 9. Here \(A\) denotes a certain process as a result of which a photon with momentum \(k\) arises, but before this photon finally appears as a free particle, the interaction of the photon with the zero-point oscillations of the electron-positron field takes place, which is depicted by the part of the photon self-energy \(W_p\). It is clear that diagram 3 of Fig. 9 describes a second-order correction to the scattering process depicted in diagram 4 of Fig. 9, as a result of which a free photon is emitted at once. According to (2.9), we may disregard the part of the self-energy \(W_p\), but instead we must, in the skeleton diagram, replace the operator \(\hat A(k)\) by the operator

\[ \delta\hat A(k)=\hat A(k)\Pi(W_p,k)\frac{1}{2}D^F(k). \]

On the other hand, guided by the considerations set forth above, we should not take process 3 into account at all. Therefore

the operator \(\delta \hat A(k)\) must physically be equal to zero, and since the function \(D^F(k)\) has a pole of second order at \(k=0\), the function \(\Pi(W_p,k)\) must physically have a zero of third order at \(k=0\).

Let us now turn to the function \(\Pi(W_p,k)\), where the momentum of the virtual photon \(k\), generally speaking, does not satisfy the relation \(k^2=0\). Such a function, as we know, diverges quadratically and appears if part of the photon self-energy connects two parts of the graph \(A\) and \(B\) (see Fig. 2).

According to (2.6), the function \(\Pi(W_p,k)\) has the form of an integral of some rational function \(Q(\hat t,\hat k)\)

\[ \Pi(W_p,k)=\int Q(\hat t,\hat k)\,d^4t, \]

where the matrix vector \(\hat k\) enters into \(Q\) not independently, but in the form of a sum with the vector \(\hat t\). This integral, as we know, diverges quadratically in the region of large \(|t|\).

Proceeding in the same way as in the derivation of (4.6), it is easy to show that the function \(\Pi(W_p,k)\) can be represented in the form

\[ \Pi(W_p,k)=\Pi_2+\Pi_0 k^2+\Pi_R(W_p,k), \tag{4.7} \]

where \(\Pi_2\) and \(\Pi_0\) are divergent constants independent of \(k\), and \(\Pi_R(W_p,k)=k^2P(W_p,k)\) is a function not containing divergences, with \(P(W_p,k)\) tending to zero at \(k=0\).

Above we explained that the divergent expression \(\Pi(W_p,k)\) must physically have a zero of third order at \(k=0\). This condition is satisfied by \(\Pi_R(W_p,k)\), but not by the first two terms on the right-hand side of (4.7) (of these, the first term diverges quadratically, and the second logarithmically).

We shall adopt as a postulate that the quantity \(\Pi_R(W_p,k)\), and not \(\Pi(W_p,k)\), has physical meaning. In other words, we shall discard in (4.7) the divergent terms \(\Pi_2\) and \(\Pi_0 k^2\), and in all matrix elements containing the quadratically divergent expression \(\Pi(W_p,k)\), we shall replace \(\Pi(W_p,k)\) by the finite function \(\Pi_R(W_p,k)\). We shall call the function \(\Pi_R(W_p,k)\) the regularized function \(\Pi(W_p,k)\).

1. We have shown how matrix elements containing irreducible parts of the self-energy can be freed from divergences.

We shall now show that an unambiguous removal of divergences can also be carried out in those cases when the graph contains an irreducible vertex part.

If the graph contains an irreducible vertex part \(V\) (see Fig. 2), then at the vertex 1 of the skeleton graph (not containing

of this vertex part, see graph 3 in Fig. 6) the operator \(\gamma_\mu\) must be replaced by the operator

\[ \Lambda_\mu=\Lambda_\mu(V,\ p_1,\ p_2,\ k). \]

As we saw above, this operator diverges logarithmically in the region of large momenta of the virtual particles. To eliminate this divergence it is enough to note that if \(p_1=p_2=p_0\), where \(p_0\) is the momentum of a free electron, then no operator \(\Lambda_\mu\) at all should be assigned to vertex 1, since in this case the graph does not represent any real scattering process. In other words, the operator \(\Lambda_\mu(V,\ p_0,\ p_0,\ 0)\) (if \(p_1=p_2\), then \(k=0\)) should be regarded as a physical zero, and it may be subtracted from the operator \(\Lambda_\mu(V,\ p_1,\ p_2,\ k)\). It is easy to see that the difference of these operators

\[ \Lambda_{\mu R}(V,\ p_1,\ p_2,\ k)=\Lambda_\mu(V,\ p_1,\ p_2,\ k)-\Lambda_\mu(V,\ p_0,\ p_0,\ 0) \tag{4.8} \]

contains no divergences.

We shall adopt as a postulate that it is not the operator \(\Lambda_\mu\) that has physical meaning, but the operator \(\Lambda_{\mu R}\), by which we must replace the operator \(\Lambda_\mu\) in divergent matrix elements. We shall call the operator \(\Lambda_{\mu R}\) the regularized operator \(\Lambda_\mu\).

1. Thus, we see that the three existing types of divergences, connected with the irreducible parts of the self-energy, the vertex parts, and the parts of scattering of a photon by a photon, can be eliminated in an unambiguous way, proceeding from simple physical considerations*).

The elimination of divergences is carried out by one general method, which consists in the following. If \(M(p,k)\) is a divergent part of a matrix element connected with irreducible internal parts of the graph (i.e. parts of the self-energy, vertex parts, and parts of scattering of a photon by a photon), and \(p\) and \(k\) are the momenta of the electron and photon lines connecting the part of the graph under consideration with the remaining parts, then, in order to eliminate the divergence in \(M(p,k)\), one should subtract from \(M(p,k)\) several first terms of the expansion of \(M(p,k)\) in a series in powers of \(\hat p-im\) or \(k\). The number of terms to be subtracted must be the minimum necessary to ensure convergence of the remainder \(M_R(p,k)\). This remainder is the regularized, physically meaningful value of the matrix element \(M(p,k)\). As for \(p\) and \(k\), these may be both the momenta of internal and the momenta of external lines of the graph.

*) This method of eliminating divergences belongs to Dyson\(^{4,8}\).

In practice, in solving concrete problems, one may in the divergent integral defining \(M(p,k)\),

\[ M(p,k)=\int R(\hat p,\hat k,\hat t)\,dt \]

first carry out the integration over some finite invariant region*). The resulting expression should be regularized by subtracting from it the necessary number of terms in the expansion in a power series, and then passing to the limit of an infinitely large region of integration \(N\to\infty\)**).

Since regularization consists in subtracting from a divergent expression several first terms of the expansion of this expression in a power series, it follows from this that the operation of regularization is additive, i.e., if a divergent expression is a sum of several terms, then they may be regularized separately. In doing so one should bear in mind that if a certain term is finite, this still does not mean that it coincides with the regularized expression.

Regularization may be carried out in several steps, immediately discarding those terms which must disappear in the finite result.

Up to now we have considered divergences connected with irreducible internal parts of graphs. However, the regularization method set forth is sufficient for eliminating divergences connected with reducible parts of the proper energy, of arbitrarily complicated reducible parts, and of the vertex parts for photon scattering by a photon.

Consider, for example, a graph which contains the reducible vertex part \(V_{11}\), depicted in Fig. 10. The composition of \(V_{11}\) includes the irreducible parts of the proper energy \(W'_e\), \(W''_e\), \(W_p\), and the irreducible vertex part \(V_3\). It is clear that the part of the matrix element associated with \(V_{11}\) will diverge not only when integrating over the variables \(p,k\), which pertain to the graph as a whole, but also when integrating over the variables \(p'\), \(p''\), \(p'''\), which pertain

*) A finite invariant region may be defined by means of the inequalities

\[ t^2\leq N,\qquad \frac{|ts|^2}{|s^2|}<N, \]

where \(s\) is some arbitrary timelike vector, and \(N\) is a positive scalar which, after regularization, is made to tend to infinity.

**) In integrating over the momenta of virtual particles one may also introduce, under the integral sign, various “cutoff” factors which ensure convergence of the integral for large \(|t|\) (cf. in this connection the method of regularization by means of auxiliary masses\({}^{6,12}\)).

to each of the inserted parts, i.e. \(W'_e\), \(W''_e\), \(W_p\), \(V_3\), if the remaining variables are kept fixed.

The elimination of divergences in this case can be carried out successively according to the rules given above, beginning

Fig. 10.

with the internal parts \(W'_e\), \(W''_e\), \(W_p\), \(V_3\), and ending with the part \(V_{11}\) as a whole.

In an analogous way, beginning with the internal inserted parts and ending with the whole part as a whole, the divergences associated with reducible parts of the self-energy can be eliminated.

5. RENORMALIZATIONS OF MASS AND CHARGE

1. We now turn to an explanation of the physical ideas which underlie the regularization method set forth above.

In studying the divergence associated with a part of the photon self-energy, we showed that it leads to an infinite photon self-energy and, on this basis, regarded as a physical zero the divergent operator \(\delta \hat A(k)\) corresponding to the dotted line in graph 3 of Fig. 9. However, if the dotted line in graph 3 of Fig. 9 represents not a photon but a certain prescribed external electromagnetic field, then the divergent operator \(\delta \hat A(k)\) can no longer be regarded as a physical zero. In this case, from \(\delta \hat A(k)\), according to the rules just given, one can extract a finite expression having physical meaning and describing the polarization of the vacuum.

It is known that the d’Alembert operator, applied to the potential \(A_\mu^{(e)}(x)\) describing the prescribed external field, determines the current \(I_\mu^{(e)}(x)\) that creates this field. Therefore, \(-\Box\,\delta A_\mu(x)\) represents a correction to the external current \(I_\mu^{(e)}(x)\), caused by the interaction of this current with the zero oscillations of the electron-positron-

of the field. It can be shown that

\[ -\Box \delta A_\mu(x)=a_2 I_\mu^{(e)}(x)+I_\mu^{(p)}(x), \]

where \(a_2\) is a quadratically divergent constant, and \(I_\mu^{(p)}(x)\) is a finite quantity. If the divergence is eliminated from \(\delta A_\mu(x)\), then the regularized value \(\delta A_{\mu R}(x)\) will satisfy the relation

\[ -\Box \delta A_{\mu R}(x)=I_\mu^{(p)}(x). \tag{5.1} \]

Therefore one may say that regularization eliminates, in the correction to the external current caused by the interaction of this current with the zero-point oscillations of the electron–positron field, the infinite term \(a_2 I_\mu^{(e)}(x)\), proportional to the initial external current \(I_\mu^{(e)}(x)\).

In particular, if we have some charge \(e\) (for example, the charge of the electron), then, owing to the interaction of this charge with the zero-point oscillations of the electron–positron field, it acquires an infinite addition proportional to \(e\). The process of regularization consists in our not taking this addition into account, considering that it has no physical meaning. One may say that the addition to the charge of the electron is inseparable from its charge itself, and that the process of regularization, essentially speaking, reduces to a renormalization of the charge: the sum of the charge of a hypothetical electron not interacting with the zero-point oscillations of the electron–positron field, and the infinite addition to the charge given by the modern theory and caused by interaction with these oscillations, is in fact finite and represents the total experimentally observed charge of the electron.

1. An analogous situation occurs in the elimination of divergences connected with parts of the electron self-energy. We saw above that these divergences lead to an infinite electromagnetic mass of the electron, i.e. to a divergent addition to the mass of the electron caused by the interaction of the electron with the zero-point oscillations of the electromagnetic field. The method of regularization set forth consists in our not taking this addition into account, considering that it is inseparable from the total mass of the electron. One may therefore say that the process of regularization reduces to a renormalization of the electron mass: the sum of the mass of the “bare” electron, i.e. of a hypothetical electron not interacting with the zero-point oscillations of the electromagnetic field, and the infinite electromagnetic mass of the electron given by the theory and caused by interaction with these oscillations, is in fact finite and represents the experimentally observed mass of the electron.

Thus, the physical idea underlying the method of regularization consists, in essence, in the renormalization of the constants \(m\) and \(e^{9,7}\).

1. We shall now try to formulate more rigorously the idea of renormalizing the mass and charge of the electron. Let us first recall

(see item 4), that quantum electrodynamics contains essentially five infinite constants, connected with the irreducible graphs of the proper energy of the electron and photon, the irreducible vertex parts, and the irreducible parts of photon scattering by a photon. These constants \(\Sigma_1, \Sigma_0, \Pi_2, \Pi_0, L_0\) enter into the operators \(\Sigma(W_e,p)\), \(\Pi(W_p,k)\), and \(\Lambda_\mu\) in the following way:

\[ \left. \begin{aligned} \Sigma(W_e,p)&=\Sigma_1+\left(\hat p-im\right)\Sigma_0+\Sigma_R(W_e,p),\\ \Pi(W_p,k)&=\Pi_2+k^2\Pi_0+\Pi_R(W_p,k),\\ \Lambda_\mu(V,p_1,p_2,k)&=L_0\gamma_\mu+\Lambda_{\mu R}(V,p_1,p_2,k). \end{aligned} \right\} \tag{5.2} \]

Above we saw that the constants \(\Sigma_0, \Pi_0, L_0\) diverge logarithmically, \(\Sigma_1\) linearly, and \(\Pi_2\) quadratically. As for the divergent constant \(\Pi_2\), it does not depend on the photon momentum and can therefore simply be discarded on grounds of gradient invariance. It may therefore be considered that four divergent constants actually exist: \(\Sigma_1, \Sigma_0, \Pi_0, L_0\).

We shall now show that two of these constants, namely the linearly divergent constant \(\Sigma_1\) and the logarithmically divergent constant \(\Pi_0\), can be eliminated if one slightly changes the form of the interaction energy between the electron-positron and electromagnetic fields\(^{10}\).

As is known, one usually starts from the following expression for the density of the interaction energy between the fields:

\[ V(x)=-j_\mu(x)A_\mu(x). \tag{5.3} \]

Let us add to this expression two terms, \(-\delta m\,\bar\psi\psi\) and \(-\delta f F_{\mu\nu}^2\), where \(F_{\mu\nu}\) is the electromagnetic-field tensor, and \(\delta m\) and \(\delta f\) are certain two constants (divergent, but not depending on \(\psi\) and \(F_{\mu\nu}\)), i.e. we shall assume that the density of the interaction energy between the fields is determined by the formula

\[ V^*(x)=-j_\mu(x)A_\mu(x)-\delta m\,\bar\psi(x)\psi(x)-\frac14\,\delta f F_{\mu\nu}^2(x). \tag{5.4} \]

Substitution of this expression, instead of \(V(x)\), into the general formula for the scattering matrix \(S=P e^{-i\int V\,d^4x}\)*) will lead to a change in the form of \(S\), as well as of the operators \(\Sigma(W_e,p)\) and \(\Pi(W_p,k)\). It is clear that the second term in (5.4) will influence the electron transitions, and since it does not contain the electron momentum, by choosing the constant \(\delta m\) appropriately we can eliminate the divergent term \(\Sigma_1\) in the expression for \(\Sigma(W_e,p)\). Analogously, the third term in (5.4) will influence the photon transitions, and since it is proportional to the square of the field tensor,

\(\rule{3cm}{0.4pt}\)

*) \(P\) denotes the chronological operator.

i.e., is proportional to the square of the photon momentum, then by a corresponding choice of the constant \(\delta f\) we can eliminate the divergent term \(\Pi_0 k^2\) in the expression for \(\Pi(W_p^\gamma, k)\).

Thus, adding to the expression for the interaction-energy density of the fields two terms, \(-\delta m \bar\psi \psi\) and \(-\dfrac{1}{4}\delta f F_{\mu\nu}^2\), makes it possible to eliminate from the theory two divergent constants, \(\Sigma_1\) and \(\Pi_0\). On the other hand, this addition physically means a renormalization of the electron mass and a renormalization of the electromagnetic field. Indeed, the Lagrangian function of the free electron-positron field contains the term \(-m\bar\psi\psi\); therefore the addition of the term \(-\delta m\bar\psi\psi\) may be interpreted as a renormalization of the electron mass. The Lagrangian function of the free electromagnetic field is equal to \(-\dfrac{1}{4}F_{\mu\nu}F_{\mu\nu}\); therefore the addition of the term \(-\dfrac{1}{4}\delta f F_{\mu\nu}^2\) means a renormalization of the electromagnetic field, i.e., a transition from the field \(F_{\mu\nu}\) to the field \(F_{\mu\nu}^{*}\), equal to

\[ F_{\mu\nu}^{*}=F_{\mu\nu}(1+\delta f)^{1/2}. \]

This field renormalization may also be interpreted as a renormalization of the electron charge, i.e., a transition from the charge \(e\) to the charge \(e^{*}\):

\[ e^{*}=e(1+\delta f)^{-1/2}. \]

We see, therefore, that the elimination of the divergent constants \(\Sigma_1\) and \(\Pi_0\) is achieved, essentially speaking, by a renormalization of the electron mass and charge. After the renormalization of the electron mass and charge, two divergent constants, \(\Sigma_0'\) and \(L_0\), still remain. It can be shown that in matrix elements they lead to expressions that always cancel out\(^5\).

1. The methods of regularization described eliminate divergences not at once in the entire scattering matrix, but in its individual matrix elements or, in other words, in the individual terms of the expansion of the scattering matrix in a series in powers of the electron charge. The question arises whether this power series will converge after regularization of its individual terms. If it converged, then one could consider its sum to represent the true scattering matrix. There are, however, considerations indicating the divergence of the expansion of the scattering matrix in a series in powers of \(e\) after regularization of the individual terms of the expansion\(^6\). The terms of this expansion at first decrease and then, in all probability, begin to grow without bound, the growth beginning when the number of terms reaches \(\sim 137\). The first few terms of the expansion usually used by us in estimating specific physical effects give only an asymptotic representation of the scattering matrix, which, as comparison of the theory with experiment shows, is, however, a sufficiently good approximation.

6. RADIATIVE CORRECTIONS TO ELECTRON SCATTERING. VACUUM POLARIZATION

1. Let us consider, as an example, the elimination of divergences from the matrix elements that determine vacuum polarization and radiative corrections to electron scattering in an external field. These effects appear in the third approximation of perturbation theory, to which we shall restrict ourselves. The diagrams depicting the effects of interest to us are given in Fig. 1. Using the results obtained, one may at once exclude diagrams 4 and 5 from consideration, since they contain parts of the self-energy “on the path” of a free electron.

We shall call diagram 2 the diagram of the radiative corrections proper, and diagram 3 the diagram of vacuum polarization. The corresponding matrix elements, according to (1.3), are equal to

\[ M_1^{(3)}=(-i)^3 e^3 \overline{u}_2 \left( \int \gamma_\mu \frac{i(\hat p_2-\hat k)-m}{(p_2-k)^2+m^2} \frac{\hat a(q)}{(2\pi)^4} \times \right. \]

\[ \left. \times \frac{i(\hat p_1-\hat k)-m}{(p_1-k)^2+m^2} \gamma_\mu \frac{d^4k}{k^2} \right)u_1, \tag{6.1} \]

\[ M_2^{(3)}=-(-i)^3 e^3 \overline{u}_2 \left( \gamma_\mu \frac{1}{q^2} \int \operatorname{Spur} \left\{ \frac{i(\hat p+\hat q)-m}{(p+q)^2+m^2} \times \right. \]

\[ \left. \times \frac{\hat a(q)}{(2\pi)^4} \frac{i\hat p-m}{p^2+m^2} \gamma_\mu \right\} d^4p \right)u_1, \tag{6.2} \]

where \(a_\mu(q)\) is the Fourier component of the external potential \(A_\mu^{(e)}(x)\):

\[ a_\mu(q)=\int A_\mu^{(e)}(x)e^{-iqx}d^4x \]

and \(u_1\) and \(u_2\) are the spinor amplitudes corresponding to electron states with momenta \(p_1\) and \(p_2\); \(q=p_2-p_1\). The extra minus sign in (6.2), as compared with (6.1), arises because diagram 3 has a closed electron line. For the same reason, (6.2) contains the trace (Spur) of matrices belonging to this closed line.

The matrix element \(M_1^{(3)}\) may be represented in the form

\[ M_1^{(3)}=\frac{i e^3}{(2\pi)^4}\overline{u}_2 \mathfrak{A} u_1, \tag{6.3} \]

where

\[ \mathfrak{A}= \left[ 4m^2\hat a(q)+2q^2\hat a(q) \right]I + \]

\[ + i\left[ 4ma_\sigma(q) + 2i\left( \gamma_\sigma \hat a(q)\hat q - \hat q \hat a(q)\gamma_\sigma \right) \right]I_\sigma + 2\gamma_\sigma \hat a(q)\gamma_\tau I_{\sigma\tau}. \]

and

\[ \left. \begin{aligned} I&=\int R^{-1}\,d^4k,\\ I_\sigma&=\int k_\sigma R^{-1}\,d^4k,\\ I_{\sigma\tau}&=\int k_\sigma k_\tau R^{-1}\,d^4k,\\ R&=(k^2-2p_1k)(k^2-2p_2k)k^2 \end{aligned} \right\} \tag{6.4} \]

(\(pk\) is the scalar product of the four-dimensional vectors \(p\) and \(k\)).

Of these integrals, in the region of large \(|k|\) only the third integral diverges, and logarithmically, in agreement with the results obtained above on the classification of divergences (graph 1 represents the irreducible vertex part). Therefore it is necessary to regularize the integral \(I_{\sigma\tau}\), subtracting from it the value of \(I_{\sigma\tau}\) at \(q=0\):

\[ I_{\sigma\tau R}=I_{\sigma\tau}-I_{\sigma\tau\,q=0}. \]

Let us note that the integral \(I\) diverges for small \(|k|\). This divergence is called the infrared catastrophe and is connected with the inapplicability of perturbation theory in the region of small \(|k|\).

Here we shall restrict ourselves to considering the interaction of an electron with photons whose frequency exceeds a certain minimum value \(\omega_{\min}\). In practice, when integrating over photon momenta, it is more convenient to use a somewhat different condition, equivalent to the condition \(\omega>\omega_{\min}\), namely, without imposing restrictions on the frequency of the photons, to assume that the photon possesses some small mass \(\lambda\), different from zero. It can be shown\(^7\) that, if the electron momentum is small in comparison with \(m\), then \(\lambda\) is related to \(\omega_{\min}\) by the relation

\[ \ln\lambda=\ln 2\omega_{\min}+\frac{5}{6}. \tag{6.5} \]

As a result of straightforward but lengthy calculations, the following expression is obtained for \(\mathfrak{A}\):

\[ \mathfrak{A}=4\pi^2 i\,\hat a(q) \left\{ 2\theta\,\operatorname{ctg}2\theta\left(\ln\frac{m}{\lambda}-1\right) -2\operatorname{ctg}2\theta\int_0^\theta \xi\,\operatorname{tg}\xi\,d\xi -\frac{\theta}{2}\operatorname{tg}\theta \right\} + \]

\[ +\frac{\pi^2}{2m}\left[\hat a(q)\hat q-\hat q\hat a(q)\right]\frac{2\theta}{\sin 2\theta} + \]

\[ +\left(\frac14 A_0(N)-\frac38+\frac12\ln m\right)\cdot 4\pi^2 i\,\hat a(q), \tag{6.6} \]

where \(4m^2+q^2=4m^2\cos^2\theta,\ q=p_2-p_1\), and \(A_0(N)\) is a logarithmically divergent constant.

We must now regularize this expression. Since \(\mathfrak A\) diverges logarithmically, as was shown above, in order to regularize \(\mathfrak A\) it is necessary to subtract from \(\mathfrak A\) the value of \(\mathfrak A\) at \(q=0\), i.e. at \(\theta=0\) (see (4.8); in the present case \(p_1\) and \(p_2\) are the momenta of a free electron).

The regularized expression \(\mathfrak A\) has the following form:

\[ \mathfrak A_R = 4\pi^2 i\,\hat a(q) \left[ \left( \frac{2\theta}{\tg 2\theta}-1 \right) \left( \ln \frac{m}{\lambda}-1 \right) - \frac{2}{\tg 2\theta} \int_0^\theta \xi \tg \xi\, d\xi - \frac{\theta}{2}\tg\theta \right] + \frac{\pi^2}{2m} \left[ \hat a(q)\hat q-\hat q\hat a(q) \right] \frac{2\theta}{\sin 2\theta}. \tag{6.7} \]

Substituting (6.7) into (6.3), we obtain the matrix element corresponding to graph 2 of Fig. 1:

\[ M_1^{(3)} = i\,\frac{e^3}{(2\pi)^4}\,\bar u_2 \mathfrak A_R u_1, \tag{6.8} \]

where here the interaction of the electron with long-wave photons is not taken into account.

1. The matrix element \(M_2^{(3)}\), describing the polarization of the electron-positron vacuum (graph 3 of Fig. 1), is determined by formula (6.2). Introducing the notation

\[ T_{\mu\nu} = \int \operatorname{Spur} \left\{ \frac{i(\hat p+\hat q)-m}{(p+q)^2+m^2}\, \gamma_\nu\, \frac{i\hat p-m}{p^2+m^2}\, \gamma_\mu \right\} d^4p, \tag{6.9} \]

we rewrite \(M_2^{(3)}\) in the form

\[ M_2^{(3)} = - i\,\frac{e^3}{(2\pi)^4}\, \bar u_2 \left( \gamma_\mu \frac{a_\nu(q)}{q^2} T_{\mu\nu} \right) u_1. \tag{6.10} \]

The tensor \(T_{\mu\nu}\) has a simple physical meaning. Indeed, replacing graph 3 of Fig. 1 by a skeleton graph containing no photon self-energy part, we must replace the external potential \(A_\mu^{(e)}(x)\) by some new potential \(\delta A_\mu(x)\), which in first-order perturbation theory leads to the matrix element \(M_2^{(3)}\):

\[ M_2^{(3)} = e\bar u_2\gamma_\mu\,\delta a_\mu(q)u_1, \qquad \delta a_\mu(q) = \int \delta A_\mu(x)e^{-iqx}d^4x. \]

Comparison of this formula with (6.10) shows that

\[ \delta a_\mu(q)=-i\,\frac{e^2}{(2\pi)^4}\,\frac{T_{\mu\nu}}{q^2}\,a_\nu(q). \tag{6.11} \]

The potential \(\delta A_\mu(x)\) should be regarded as an addition to the “assigned” external potential \(A_\mu^{(e)}(x)\), caused by the polarization of the electron-positron vacuum. Applying the operator \(-\Box\) to \(\delta A_\mu(x)\), we find an addition to the external current \(I_\mu^{(e)}(x)\), caused by the interaction of \(I_\mu^{(e)}(x)\) with the zero oscillations of the electron-positron field. Denoting the \(q\)-component of the Fourier transform of this addition by \(\delta I_\mu(q)\), we obtain, according to (6.11),

\[ \delta I_\mu(q)=-i\,\frac{e^2}{(2\pi)^4}\,T_{\mu\nu}a_\nu(q). \tag{6.12} \]

It can be shown that \(T_{\mu\nu}\) is determined by the following formula\(^3\):

\[ T_{\mu\nu}=4\pi^2 i\,(q_\mu q_\nu-\delta_{\mu\nu}q^2)\, \frac{4m^2-2q^2}{-3q^2}\,(1-\vartheta\operatorname{ctg}\vartheta). \tag{6.13} \]

This expression must be regularized, i.e. the first terms of the expansion of \(T_{\mu\nu}\) in powers of \(q_s\) must be subtracted from \(T_{\mu\nu}\). Since \(T_{\mu\nu}\) contains the factor \(q_\mu q_\nu-\delta_{\mu\nu}q^2\), which is quadratic with respect to \(q_s\), the regularization reduces to subtracting from the factor

\[ \frac{4m^2-2q^2}{-3q^2}(1-\vartheta\operatorname{ctg}\vartheta) \]

the free term. It is easy to verify that this free term is equal to \(\frac{1}{9}\). Therefore the final regularized expression for the tensor \(T_{\mu\nu}\) has the form\(^3\):

\[ T_{\mu\nu R}=4\pi^2 i\,(q_\mu q_\nu-\delta_{\mu\nu}q^2) \left[ \frac{4m^2-2q^2}{-3q^2}(1-\vartheta\operatorname{ctg}\vartheta)-\frac{1}{9} \right]. \tag{6.14} \]

If \(T_{\mu\nu R}\) is multiplied by \(-i\,\dfrac{e^2}{(2\pi)^4}a_\nu(q)\), then, according to (6.12), we obtain the addition \(I_\mu^{(p)}(q)\) to the Fourier component of the original “external” current \(I_\mu^{(e)}(q)\), caused by its interaction with the zero oscillations of the electron-positron field:

\[ I_\mu^{(p)}(q)=\frac{e^2}{(2\pi)^4}(q_\mu q_\nu-\delta_{\mu\nu}q^2) \left[ \frac{4m^2-2q^2}{-3q^2}(1-\vartheta\operatorname{ctg}\vartheta)-\frac{1}{9} \right]a_\nu(q). \tag{6.15} \]

It is easy to see that this addition satisfies the charge-conservation law

\[ q_\mu I_\mu^{(p)}(q)=0. \]

Noting that \(q_\mu a_\mu(q)=0\) and \(q^2 a_\mu(q)=I_\mu^{(e)}(q)\), we rewrite \(I_\mu^{(p)}(q)\) in the form

\[ I_\mu^{(p)}(q) = -\frac{e^2}{(2\pi)^2} \left[ \frac{4m^2-2q^2}{-3q^2}(1-\theta\operatorname{ctg}\theta)-\frac{1}{9} \right] I_\mu^{(e)}(q). \tag{6.16} \]

Expanding \(T_{\mu\nu R}\) in a series in powers of \(q_\sigma\) and retaining terms of fourth order, we obtain:

\[ T_{\mu\nu R} = -\frac{4\pi^2 i}{15m^2}(q_\mu q_\nu-\delta_{\mu\nu}q^2)q^2. \tag{6.17} \]

In this case the addition to the Fourier component of the current due to polarization of the electron-positron vacuum has the form\({}^{2}\)

\[ I_\mu^{(p)}(q)=\frac{e^2}{60\pi^2m^2}\,q^2 I_\mu^{(e)}(q). \tag{6.18} \]

From the Fourier components one can easily pass to the coordinate and time functions themselves\({}^{2}\):

\[ I_\mu^{(p)}(x) = -\frac{e^2}{60\pi^2m^2}\,\Box I_\mu^{(e)}(x). \tag{6.19} \]

If, in the expansion of \(T_{\mu\nu R}\) in powers of \(q_\sigma\), terms of sixth order are retained, then instead of (6.19) we obtain\({}^{2}\):

\[ I_\mu^{(p)}(x) = -\frac{e^2}{60\pi^2m^2}\,\Box I_\mu^{(e)}(x) - \frac{e^2}{680\pi^2} \left(\frac{1}{m^2}\Box\right)^2 I_\mu^{(e)}(x). \tag{6.20} \]

Substituting formula (6.14), which determines \(T_{\mu\nu R}\), into (6.10), we obtain the following expression for the matrix element \(M_2^{(3)}\):

\[ M_2^{(3)} = -i\frac{e^3}{(2\pi)^4}\, \bar u_2 \left( \gamma_\mu\frac{a_\nu(q)}{q^2} - T_{\mu\nu R} \right) u_1 . \tag{6.21} \]

3. To determine the probability of the processes represented by the diagrams in Fig. 1, one must, obviously, find the total matrix element, equal to \(S_{i\to f}^{(3)}=M_1^{(3)}+M_2^{(3)}\). Using formulas (6.8), (6.21), (6.7), and (6.14), we represent the total matrix element in the form

\[ S_{i\to f}^{(3)}=\bar u_2 S^{(3)}(q)u_1, \tag{6.22} \]

where

\[ S^{(3)}(q)= i\,\frac{e^3}{(2\pi)^4} \left[\mathfrak{X}_R-\gamma_\mu \frac{a_\nu(q)}{q^2}\,T_{\mu\nu R}\right]. \]

The expansion of \(S^{(3)}(q)\) in powers of \(q_\sigma\) up to and including terms of third order has the following form:

\[ S^{(3)}(q)=-\frac{e^3}{(4\pi)^2} \left\{ \frac{4q^2}{3m^2}\,\hat a(q) \left(\ln\frac{m}{\lambda}-\frac{3}{8}-\frac{1}{5}\right) + \frac{i}{2m}\left[\hat q\,\hat a(q)-\hat a(q)\hat q\right] \right\}. \tag{6.23} \]

If to \(S^{(3)}_{i\to f}\) we add the matrix element
\(S^{(1)}_{i\to f}=\bar u_2 e a(q)u_1\), which determines electron scattering in the first approximation, then we find the resulting matrix element \(S^{(1)+(3)}_{i\to f}\), which takes into account the interaction of the electron with the zero-point oscillations of the field up to terms of order \(e^4\):

\[ S^{(1)+(3)}_{i\to f} = \bar u_2\{\gamma e a(q)+ i\beta[ e\varphi(q)+\delta U(q)]\}u_1, \tag{6.24} \]

where

\[ \delta U(q)=\frac{1}{i}\,\beta S^{(3)}(q). \]

Expression (6.24) shows that the quantity \(\delta U(q)\) may be interpreted as the Fourier component of an addition to the potential energy of the electron, caused by the interaction of the electron with the zero-point oscillations of the field. We shall call this addition the effective potential energy determining the interaction of the electron with the zero-point oscillations of the field.

The effective potential energy as a function of the coordinates may be represented in the form

\[ \delta U(x)= -\frac{e^3}{(4\pi)^2}\frac{1}{m}\,(\beta\sigma H-i\beta\alpha E) + \]

\[ +\frac{e^3}{(4\pi)^2}\frac{4}{3m^2} \left(\ln\frac{m}{\lambda}-\frac{3}{8}-\frac{1}{5}\right) (\Box\varphi-\alpha\Box A), \tag{6.25} \]

where \(E\) and \(H\) are the electric and magnetic fields, \(\varphi\) and \(A\) are the scalar and vector potentials, and \(\sigma_i\) are the spin matrices related to \(\gamma_i\) by the relations

\[ \sigma_1=\frac{1}{i}\gamma_2\gamma_3,\qquad \sigma_2=\frac{1}{i}\gamma_3\gamma_1,\qquad \sigma_3=\frac{1}{i}\gamma_1\gamma_2. \]

From expression (6.25) for \(\delta U\) it follows that an electron situated in a constant magnetic field, owing to its interaction with the zero-point oscillations of the electromagnetic field, acquires an additional energy equal to \(-\dfrac{1}{2\pi}\dfrac{1}{137}\dfrac{e}{2m}\beta\mathfrak{H}\). On the other hand, the energy of an electron in a magnetic field is equal to

\[ U_m=-\mu\beta\mathfrak{H}, \]

where \(\mu\) is the magnetic moment of the electron. Therefore one may say that, owing to the interaction of the electron with the zero-point oscillations of the electromagnetic field, it acquires an additional magnetic moment equal to

\[ \Delta\mu=\frac{\alpha}{2\pi}\mu_0, \tag{6.26} \]

where \(\mu_0=\dfrac{e\hbar}{2mc}\) is the Bohr magneton and \(\alpha=\dfrac{e^2}{\hbar c}=\dfrac{1}{137}\) is the fine-structure constant. Thus, to terms of order \(e^4\), the magnetic moment of the electron is equal to

\[ \mu=\mu_0\left(1+\frac{\alpha}{2\pi}\right). \]

1. The expression for \(\delta U\) also makes it possible to determine the radiative corrections to the scattering of an electron in the Coulomb field of a nucleus. We shall give here only the results of the calculations \(^{2,11}\).

The cross section of purely elastic scattering, with allowance for radiative corrections and the second Born approximation, is determined by the following formula*):

\[ ds_s= \]

\[ =\left(\frac{Z\alpha}{2mv^2\sin^2\dfrac{\vartheta}{2}}\right)^2 (1-v^2) \left(1-v^2\sin^2\frac{\vartheta}{2}\right) \{1-\chi+\delta_B\}\,do, \tag{6.27} \]

where

\[ \delta_B=\pi\alpha vZ\sin\frac{\vartheta}{2} \left(1-\sin\frac{\vartheta}{2}\right) \left(1-v^2\sin^2\frac{\vartheta}{2}\right)^{-1}, \]

\[ \chi=\frac{\alpha}{\pi} \left[ 2(1-\Phi\,\operatorname{cth}\Phi) \left(1-\frac{1}{3}\operatorname{cth}^2\Phi\right) -\frac{2}{9} +\Phi\,\operatorname{th}\Phi +\right. \]

\[ \left. +2(1-2\Phi\,\operatorname{cth}2\Phi) \left(1+\ln\frac{\lambda}{m}\right) +\right. \]

\[ \left. +(1-v^2)\frac{\operatorname{ch}2\Phi}{\sin\dfrac{\vartheta}{2}} \int_{\cos\dfrac{\vartheta}{2}}^{1} \frac{\zeta\ln(1-v^2\zeta^2)\,d\zeta} {(1-v^2\zeta^2)\sqrt{\zeta^2-\cos^2\dfrac{\vartheta}{2}}} +\right. \]

\[ \left. +2\Phi\,\operatorname{cth}2\Phi\ln\frac{1}{1-v^2} +\frac{v^2\sin^2\dfrac{\vartheta}{2}} {1-v^2\sin^2\dfrac{\vartheta}{2}} -\frac{2\Phi}{\operatorname{sh}2\Phi} \right]. \]

*) \(\delta_B\) takes into account the second Born approximation.

and \(\Phi\) is related to the scattering angle \(\vartheta\) by the relation

\[ \operatorname{sh}\Phi = -\frac{v}{\sqrt{1-v^{2}}}\,\sin\frac{\vartheta}{2}. \]

(\(v\) is the velocity of the electron).

This expression contains the “mass” of the photon \(\lambda\), which can be eliminated if, alongside purely elastic scattering, one also considers inelastic scattering of the electron with emission of a photon whose energy does not exceed the value \(\Delta\varepsilon\) \((\Delta\varepsilon \ll m)\). Let us denote the cross section of this scattering by \(ds'\). The photon “mass” does not enter the total scattering cross section \(ds_{\Delta\varepsilon}=ds_s+ds'\), which alone has physical meaning.

The total scattering cross section \(ds_{\Delta\varepsilon}\) with an energy loss not exceeding \(\Delta\varepsilon\) can be represented in the form \(^{2,11}\)

\[ ds_{\Delta\varepsilon} = \left( -\frac{Z\alpha}{2mv^{2}\sin^{2}\dfrac{\vartheta}{2}} \right)^{2} (1-v^{2}) \left( 1-v^{2}\sin^{2}\frac{\vartheta}{2} \right) (1+\delta_{B}-\delta_{R}), \]

where \(\delta_{B}\) and \(\delta_{R}\) take into account the second Born approximation and the radiative corrections. The latter quantity is equal to \(^{11}\)

\[ \delta_{R} = \frac{\alpha}{\pi} \left\{ 2(1-2\Phi\,\operatorname{cth}2\Phi) \left(1+\ln\frac{2\Delta\varepsilon}{m}\right) + \Phi\,\operatorname{th}\Phi +\right. \]

\[ \left. +\,2(1-\Phi\,\operatorname{cth}\Phi) \left(1-\frac{1}{3}\operatorname{cth}^{2}\Phi\right) -\frac{2}{9} +\frac{1}{v}\ln\frac{1-v}{1+v} +\right. \]

\[ \left. +\,2\Phi\,\operatorname{cth}2\Phi \ln\frac{1}{1-v^{2}} + \frac{ v^{2}\sin^{2}\dfrac{\vartheta}{2} }{ 1-v^{2}\sin^{2}\dfrac{\vartheta}{2} } -\frac{2\Phi}{\operatorname{sh}2\Phi} +\right. \]

\[ \left. + \frac{(1-v^{2})\,\operatorname{ch}2\Phi}{v\sin\dfrac{\vartheta}{2}} \int_{\cos\dfrac{\vartheta}{2}}^{1} \left[ \ln\frac{1+v\zeta}{1-v\zeta} - \ln\frac{1-v\zeta}{1+v\zeta} \right] \frac{d\zeta}{\sqrt{\zeta^{2}-\cos^{2}\dfrac{\vartheta}{2}}} \right\}. \tag{6.28} \]

Let us emphasize that these formulas can be used only for sufficiently small values of \(\delta_{B}\) and \(\delta_{R}\). As for the second Born approximation, it gives correct results for values of \(Z\) smaller than approximately 15; therefore formula (6.23) is valid for sufficiently light nuclei. Below we give the values of \(\delta_{R}\) for various values of the electron energy \(\varepsilon\) and of the energy loss \(\Delta\varepsilon\). \(^{11}\)

Values of \(\delta_R\) in percent

\(\varepsilon\) (MeV)	2.5	2.5	2.5	4.0	4.0	4.0	9.5	9.5	9.5
\(\vartheta\) (degrees)	45	90	135	45	90	135	45	90	135
\(\Delta\varepsilon = 10\) keV	4.8	7.4	8.7	6.9	9.9	11.3	12.4	15.9	17.5
\(\Delta\varepsilon = 25\) keV	3.9	6.0	7.1	5.7	8.1	9.3	10.5	13.5	14.8
\(\Delta\varepsilon = 50\) keV	3.2	5.0	5.9	4.7	6.8	7.9	9.0	11.7	12.8
\(\Delta\varepsilon = 100\) keV	2.5	3.9	4.7	3.8	5.5	6.4	7.6	9.9	10.8

Let us note that formula (6.28) for \(\delta_R\) becomes inapplicable as \(\Delta\varepsilon \to 0\). Since, strictly speaking, as \(\Delta\varepsilon \to 0\) the scattering cross section must tend to zero, we obtain the correct behavior of the scattering cross section for \(\Delta\varepsilon \to 0\) if we replace \(1-\delta_R+\delta_B\) by \(e^{-\delta_R+\delta_B}\) (the further terms in the expansion of \(e^{-\delta_R+\delta_B}\) in powers of \(-\delta_R+\delta_B\) then describe higher-order effects).

References

1. V. B. Berestetskii, UFN 46, 231 (1952).
2. J. Schwinger, Phys. Rev. 74, 1439 (1948); 75, 651 (1949); 75, 1912 (1949).
3. R. Feynman, Phys. Rev. 76, 749, 769 (1949).
4. F. Dyson, Phys. Rev. 75, 486 (1949).
5. F. Dyson, Phys. Rev. 83, 608 (1951).
6. F. Dyson, Phys. Rev. 85, 631 (1952).
7. Shift of the Levels of Atomic Electrons. Collection of articles, Moscow, 1950.
8. Problems of Modern Physics, third series, issue 11, IL, Moscow, 1951.
9. H. Bethe, Phys. Rev. 72, 339 (1947).
10. S. Gupta, Proc. Phys. Soc. 64, 426 (1951).
11. L. Elton and H. Robertson, Proc. Phys. Soc. 65, 145 (1952).
12. W. Pauli and F. Villars, Rev. Mod. Phys. 21, 434 (1949).