PROBLEMS OF QUANTUM FIELD THEORY

II. ELIMINATION OF DIVERGENCES FROM THE SCATTERING MATRIX*)

N. N. Bogoliubov and D. V. Shirkov

CONTENTS

§ 1. Divergences of the scattering matrix of spinor electrodynamics in 2nd and 3rd order . . . . . . . . . . . . . . . . . . . . . . 3
§ 2. General rules for eliminating divergences from the \(S\)-matrix . . . . . . . . . . . . . . . . . . . . . 29
§ 3. Classification of the renormalizability of theories . . . . . . . . . . . . . . . . . . . . . . . . . 48
§ 4. General form of the counterterms of spinor electrodynamics . . . . . . . . . . . . . . . . 58
§ 5. Renormalization of mass and charge in spinor electrodynamics . . . . . . . . . . . . . 78
Cited literature . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91

§ 1. DIVERGENCES OF THE SCATTERING MATRIX OF SPINOR ELECTRODYNAMICS IN 2nd AND 3rd ORDER

The Feynman rules formulated at the end of the preceding article\(^{1}\) for matrix elements make it possible to write expressions for them in the form of integrals over the 4-momenta of intermediate particles. In the simplest cases (for example, in the effects of Compton scattering, pair annihilation, etc.) the indicated integrations are performed trivially with the aid of the vertex \(\delta\)-functions expressing the law of conservation of 4-momentum at the individual vertices of the corresponding diagrams. When there are internal lines in a diagram, corresponding to virtual particles, the number of vertex \(\delta\)-functions proves insufficient for carrying out all the integrations, and the function \(F(p',p)\) in (I.6.26)\(^{**}\) is represented in the form of multiple—

) For the first part of the article, see Uspekhi Fizicheskikh Nauk, Vol. LV, No. 2.
*) The Roman numeral I before the formula number means that the formula from the first part of the authors’ article\(^{1}\) is meant.

... of the integral. In this case it turns out that such integrals in the general case diverge in the region of large momenta. For example, the matrix elements of Compton scattering and pair annihilation diverge in higher orders in \(e^2\) (\(e^4\) and higher). The second-order matrix elements corresponding to the diagrams shown in Fig. 1 also diverge.

Fig. 1.

Let us take the term of the scattering matrix corresponding to the diagram (Fig. 1,a)

\[ -4\pi e^2 : \bar{\psi}(x) \sum_n g^{mn}\gamma^m S^c(x-y)\gamma^n D_0^c(x-y)\psi(y) : . \tag{1.1} \]

Represent it in the form

\[ :\bar{\psi}(x)\Sigma(x-y)\psi(y):, \]

where

\[ \Sigma(x-y)=-4\pi e^2 \sum_n g^{mn}\gamma^m S^c(x-y)\gamma^n D_0^c(x-y). \tag{1.2} \]

By a simple calculation it is not difficult to verify that the Fourier transform \(\Sigma(p)\) of the matrix \(\Sigma(x-y)\),

\[ \Sigma(x-y)=\frac{1}{(2\pi)^4}\int e^{-ip(x-y)}\Sigma(p)\,dp, \]

enters into the matrix element of the \(S\)-matrix in the following way:

\[ F(p',p)\sim \bar{u}(p')\Sigma(p)u(p) \quad (p'=p), \]

where \(\bar{u}\) and \(u\) are spinor amplitudes corresponding to the field functions \(\bar{\psi}\) and \(\psi\). The convergence of the matrix element \(F\) is therefore completely determined by the convergence of the matrix \(\Sigma(p)\). Recalling that in the momentum representation the causal functions have the form (cf. (I.6.15) and (I.6.16))

\[ D_0^c(k)=-\frac{1}{k^2+i\varepsilon}, \tag{1.3} \]

\[ S^c(p)=\frac{m+\hat{p}}{m^2-p^2-i\varepsilon}. \tag{1.4} \]

we find for \(\Sigma(p)\) the expression

\[ \begin{aligned} \Sigma(p) &=-\frac{4\pi e^2}{(2\pi)^4}\sum_n g^{nn}\int dk\,D_0^c(k)\gamma^n S^c(p-k)\gamma^n \\ &=-\frac{e^2}{4\pi^3}\sum_n g^{nn}\int \frac{dk}{k^2+i\varepsilon}\, \gamma^n\frac{\hat p-\hat k+m}{(p-k)^2-m^2+i\varepsilon}\gamma^n . \end{aligned} \tag{1.5} \]

For large \(k\) the integrand decreases as \(k^{-3}\), and therefore the integral

\[ \int d^4 k\cdot k^{-3} \]

diverges, generally speaking, linearly.

Thus we see that the purely formal rules adopted by us for handling products of causal functions in the present case lead to a meaningless result.

In essence, what has appeared here is the circumstance that we have not defined the product of singular functions as an integrable singular function. To solve the problem of defining the coefficients of the chronological product

\[ T\{L(x_1),L(x_2)\} \]

as integrable improper functions, we shall use, similarly to how this was done in (I, § 2)\(^*\), the limiting-transition method. For this purpose let us first consider an auxiliary fictitious case, in which the field operator functions satisfy commutation relations in which the causal \(\Delta^c\)-functions are replaced by \(\operatorname{reg}(\Delta^c)\).

In the expression for \(\Sigma(p)\), instead of (1.3) and (1.4), the functions

\[ \operatorname{reg}\bigl[D_0^c(k)\bigr] = -\frac{1}{k^2+i\varepsilon} -\sum_M C_M\frac{1}{k^2+i\varepsilon-M^2}, \tag{1.6} \]

\[ \operatorname{reg}\bigl[S^c(p)\bigr] = (\hat p+m) \left[ \frac{1}{m^2-p^2-i\varepsilon} + \sum_M C_M\frac{1}{M^2-p^2-i\varepsilon} \right]. \tag{1.7} \]

It then turns out that, for the regularization of \(\Sigma(p)\), one auxiliary mass \(M\) is sufficient. Setting

\[ C_M=-1, \]

we find that for large \(k\), \(\operatorname{reg}\bigl[D_0^c(k)\bigr]\) decreases as \(k^{-4}\), while

\[ \operatorname{reg}\bigl[S^c(p)\bigr] \]

decreases as \(k^{-3}\),

\(^*\) That is, in § 2 of the preceding paper\(^1\).

and therefore the integral

\[ \operatorname{reg}(\Sigma(p))= \frac{e^{2}}{4\pi^{3}}\sum_{n}g^{nn}\int dk\, \frac{M^{2}}{(k^{2}+i\varepsilon)(M^{2}-k^{2}-i\varepsilon)} \times \]

\[ \times\gamma^{n} \frac{(\hat p-\hat k+m)(M^{2}-m^{2})} {[m^{2}-(p-k)^{2}-i\varepsilon]\,[M^{2}-(p-k)^{2}-i\varepsilon]} \gamma^{n} \tag{1.8} \]

for large values of \(k\) converges as

\[ \int^{k}\frac{d^{4}k}{k^{7}}\sim k^{-3}. \]

We shall now investigate the behavior of \(\operatorname{reg}(\Sigma(p))\) as \(M\to\infty\) in the process of removing the regularization. It is convenient to carry out this consideration effectively by computing \(\operatorname{reg}(\Sigma(p))\) in explicit form.

To compute the integral (1.8) we shall use the following auxiliary device.

We shall represent the factors of the denominator of (1.8) in the form

\[ \frac{1}{k^{2}-m^{2}+i\varepsilon} = \frac{1}{i}\int_{0}^{\infty} e^{ia(k^{2}-m^{2}+i\varepsilon)}\,da. \tag{1.9} \]

Then integration with respect to \(k\) will be reduced to taking Gaussian integrals. We therefore establish general rules for evaluating such integrals.

Consider the typical integral

\[ \int_{-\infty}^{\infty} e^{i(at^{2}+bt)}\,dt;\qquad a>0, \]

which we shall always regard as the limit of the expression

\[ \int_{-\infty}^{\infty} e^{i(at^{2}+bt+i\eta t^{2})}\,dt \quad \text{for } \eta>0,\qquad \eta\to0. \tag{1.10} \]

To compute (1.10) we make a change of variables (a rotation of the path of integration in the complex \(t\)-plane)

\[ it^{2}=-\tau^{2};\qquad t=\sqrt{i}\,\tau=\frac{1+i}{\sqrt{2}}\,\tau, \]

then

\[ \int_{-\infty}^{\infty} e^{i(at^{2}+bt)-\eta t^{2}}\,dt = \frac{1+i}{\sqrt{2}} \int_{-\infty}^{\infty} e^{-a\tau^{2}+\frac{i-1}{\sqrt{2}}\,b\tau-i\eta\tau^{2}}\,d\tau. \]

Passing to the new variable

\[ x=\tau+\frac{1-i}{\sqrt{2}}\,\frac{b}{a}, \]

Questions of Quantum Field Theory

In the limit as \(\eta \to 0\), we obtain from this

\[ \int_{-\infty}^{\infty} e^{i(at^2+bt)}\,dt = \frac{1+i}{\sqrt{2}}\,e^{-\frac{i b^2}{4a}} \int_{-\infty}^{\infty} e^{-a x^2}\,dx = \frac{1+i}{\sqrt{2}}\sqrt{\frac{\pi}{a}}\, e^{-\frac{i b^2}{4a}}; \qquad a>0. \tag{1.11} \]

We can now compute the four-dimensional integrals we need, of the type

\[ \int e^{i(ak^2+bk)}\,dk, \]

where \(k^2=(k^0)^2-\mathbf{k}^2;\ bk=b^0k^0-\mathbf{b}\mathbf{k};\ dk=dk^0\,d\mathbf{k}\).

With the aid of (1.11) and of the formula obtained from (1.11) by the operation of complex conjugation, we find:

\[ \int e^{i(ak^2+bk)}\,dk = \int_{-\infty}^{\infty} e^{i(a(k^0)^2-b^0k^0)}\,dk^0 \times \prod_{1\leq \alpha \leq 3} \int_{-\infty}^{\infty} e^{-i(a(k^\alpha)^2-b^\alpha k^\alpha)}\,dk^\alpha = \]

\[ = \frac{1+i}{\sqrt{2}} \left(\frac{1-i}{\sqrt{2}}\right)^3 \frac{\pi^2}{a^2} e^{-\frac{i}{4a}\left((b^0)^2-\mathbf{b}^2\right)}, \]

i.e.

\[ \int e^{i(ak^2+bk)}\,dk = \frac{\pi^2}{i a^2} e^{-\frac{i b^2}{4a}}; \qquad a>0. \tag{1.12} \]

Differentiating this expression once and twice with respect to the components \(b^n\) and \(b^m\), we obtain:

\[ \int e^{i(ak^2+bk)} k^n\,dk = \frac{i b^n}{2a}\, \frac{\pi^2}{a^2} e^{-\frac{i b^2}{4a}}; \qquad a>0, \tag{1.13} \]

and also

\[ \int e^{i(ak^2+bk)} k^m k^n\,dk = \frac{2a g^{mn}-i b^n b^m}{4a^2}\, \frac{\pi^2}{a^2} e^{-\frac{i b^2}{4a}}; \qquad a>0. \tag{1.14} \]

Let us return to the calculation of the integral (1.8). Substituting into it the integral representations of the singular functions of the type (1.9),

\[ \operatorname{reg}\left[D_0^c(k)\right] = i\int_0^\infty e^{i\alpha_1 k^2-\varepsilon \alpha_1} \left(1-e^{-i\alpha_1 M^2}\right)\,d\alpha_1, \tag{1.15} \]

\[ \operatorname{reg}\left[S^c(p)\right] = i(\hat p+m) \int_0^\infty e^{i\alpha_2 p^2-\varepsilon \alpha_2} \left(e^{-i\alpha_2 m^2}-e^{-i\alpha_2 M^2}\right)\,d\alpha_2. \tag{1.16} \]

with the aid of formulas (1.11) and (1.12) we can carry out the integration with respect to \(k\). We obtain thereby, performing the summation over \(n\),

\[ \operatorname{reg}[\Sigma(p)] = \frac{i e^2}{2\pi}\int_0^\infty d\alpha_1 \int_0^\infty d\alpha_2\, \frac{ e^{\, i\frac{\alpha_1\alpha_2 p^2}{\alpha_1+\alpha_2}-\varepsilon(\alpha_1+\alpha_2)} }{(\alpha_1+\alpha_2)^2} \times \]

\[ \times (1-e^{-i\alpha_1 M^2})(e^{-i\alpha_2 m^2}-e^{-i\alpha_2 M^2}) \left(2m-\hat p\,\frac{\alpha_1}{\alpha_1+\alpha_2}\right). \]

Passing to new variables

\[ \alpha_1=\xi\lambda;\qquad \alpha_2=(1-\xi)\lambda, \]

taking account of the Jacobian

\[ \left(\frac{d\alpha_1,d\alpha_2}{d\xi,d\lambda}\right)=\lambda \]

and integrating with respect to \(\lambda\), we find in the limit \(\varepsilon\to 0\)

\[ \operatorname{reg}(\Sigma(p))= \frac{i e^2}{2\pi}\int_0^1 d\xi\,(2m-\hat p\xi)\,I(\xi,M), \]

where

\[ I(\xi,M)= \ln \left| \frac{ \bigl(M^2-\xi p^2\bigr)\cdot \bigl[\xi M^2+(1-\xi)m^2-\xi(1-\xi)p^2\bigr] }{ \bigl(m^2-\xi p^2\bigr)\cdot \bigl[M^2-\xi(1-\xi)p^2\bigr] } \right|, \]

or, after a slight rearrangement of terms:

\[ \operatorname{reg}[\Sigma(p)] = \frac{i e^2}{2\pi}\int_0^1 d\xi\,(2m-\hat p\xi) \ln\left|\xi\,\frac{M^2-\xi p^2}{m^2}\right| + \]

\[ + \frac{i e^2}{2\pi}\int_0^1 d\xi\,(2m-\hat p\xi) \ln\left| \frac{m^2}{m^2-\xi p^2}\cdot \frac{\xi M^2+(1-\xi)m^2-\xi(1-\xi)p^2}{\xi M^2-\xi^2(1-\xi)p^2} \right|. \tag{1.17} \]

The second term in (1.17), as \(M\to\infty\), will converge to a completely definite limit equal to

\[ \Sigma'(p)= \frac{i e^2}{2\pi}\int_0^1 d\xi\,(2m-\hat p\xi) \ln\left|\frac{m^2}{m^2-\xi p^2}\right|. \tag{1.18} \]

The splitting (1.17) has been chosen so that

\[ \Sigma'(0)=0 \quad\text{and}\quad \left.\frac{\partial \Sigma'(0)}{\partial p^n}\right|_{p=0}=0. \tag{1.19} \]

The first term in (1.17), as \(M\to\infty\), diverges logarithmically.

Passing to the configuration representation, we obtain, for sufficiently large \(M\),

\[ \operatorname{reg}[\Sigma(x)] = \frac{i e^{2}}{4\pi} \left\{ \ln\left(\frac{M}{m}\right)^{2} \cdot \left(4m - i\frac{\hat{\partial}}{\partial x}\right) + \left(\frac{i}{2}\frac{\hat{\partial}}{\partial x}-4m\right) \right\}\delta(x) +\Sigma'_M(x), \tag{1.20} \]

\[ \frac{\hat{\partial}}{\partial x}\equiv \sum_k \gamma^k \frac{\partial}{\partial x^k}, \]

where the Fourier transform of the function \(\Sigma'_M(x)\) is represented by the second term of expression (1.17). Repeating the argument (I, § 2), we see that, as \(M\to\infty\), \(\Sigma'_M(x)\) converges in the improper sense to an integrable function

\[ \lim_{M\to\infty}\Sigma'_M(x)=\Sigma'(x), \]

whose Fourier transform is represented by expression (1.18).

As a whole, the function \(\operatorname{reg}(\Sigma(x))\), because of the factor \(\ln\left(\frac{M}{m}\right)^2\), will not converge even in the improper sense.

In view of the fact that the first term of (1.20) vanishes for \(x\ne 0\), we may write

\[ \lim_{M\to\infty}\operatorname{reg}[\Sigma(x)]=\Sigma'(x) \quad \text{for } x\ne 0. \]

We have here carried out the extraction of the divergent part from the singular function \(\Sigma(x)\).

Let us emphasize, however, that the operation of extracting the singularity is not unique.

Indeed, formula (1.20) could, for example, be represented in the form

\[ \operatorname{reg}[\Sigma(x)] = \frac{i e^{2}}{4\pi} \left\{ \ln\left(\frac{M^2}{\mu^2}\right) \left(4m - i\frac{\hat{\partial}}{\partial x}\right) + \left(\frac{i}{2}\frac{\hat{\partial}}{\partial x}-4m\right) \right\}\delta(x) +\Sigma''_M(x), \]

where

\[ \Sigma''_M = \frac{i e^{2}}{4\pi} \ln\left(\frac{\mu^2}{m^2}\right) \left(4m - i\frac{\hat{\partial}}{\partial x}\right)\delta(x) +\Sigma'_M(x), \]

and \(\mu\) is an arbitrary finite mass.

We would obtain for the regular part \(\Sigma_M'\) an expression differing from \(\Sigma_M'\) by terms proportional to \(\delta(x)\) and its first derivative.

A change of the same character in the finite part \(\Sigma_M'\) would also be obtained upon passing to any other method of regularization.

Thus, for example, if \(\Sigma(p)\) is regularized by introducing under the integral (1.5) the cutoff factor\(^{2,3}\)

\[ \frac{M^2}{M^2-k^2}, \]

which is equivalent in our case to regularization of only the photon \(D_0^c\)-function, then the result can be represented in the form

\[ \operatorname{reg}^{F}[\Sigma(x)] = \frac{i e^2}{4\pi} \left\{ \ln\left(\frac{M}{m}\right)^2 \left(4m-i\frac{\hat{\partial}}{\partial x}\right) + \left(\frac{i}{2}\frac{\hat{\partial}}{\partial x}-4m\right) \right\}\delta(x) + \Sigma_F'(x), \]

where the regular function \(\Sigma_F'\) in the momentum representation differs from \(\Sigma'\) by the amount

\[ \frac{i e^2}{4\pi}\int_0^1 d\xi\,(\hat{p}\xi-2m)\ln(1-\xi) = \frac{i e^2}{4\pi}\left(2m-\frac{3}{4}\hat{p}\right). \]

Thus one may note that, when the regularization is removed, it is not \(\operatorname{reg}[\Sigma(p)]\) that converges to a definite limit, but, for example, the expression obtained by subtracting from it the first two terms of the Maclaurin series

\[ \operatorname{reg}[\Sigma(p)]-\operatorname{reg}[\Sigma(0)] - \sum_n \left. \frac{\partial\,\operatorname{reg}[\Sigma(p)]}{\partial p^n} \right|_{p=0} \cdot p^n. \tag{1.21} \]

This expression converges to a limit independent of the method of regularization, since the addition to \(\operatorname{reg}[\Sigma(p)]\) of any polynomial of first degree in \(p\) does not change the “remainder” term (1.21) that has been written.

The general expression for \(\Sigma'(p)\) is obtained by adding to (1.21) an arbitrary polynomial of first degree in \(p\).

From considerations of relativistic covariance this polynomial must have the form

\[ c_1(\hat{p}-m)+c_2\cdot m \]

and, consequently, the general expression for \(\Sigma'(p)\) is obtained in the form

\[ \Sigma'(p)=\frac{ie^2}{2\pi}\left\{\int_0^1 d\xi\,(2m-\hat p\,\xi)\ln\left|\frac{m^2}{m^2-\xi p^2}\right|+ \right. \]

\[ \left. {}+c_1(\hat p-m)+c_2m\right\}. \tag{1.22} \]

Accordingly, in the \(x\)-representation the expression for \(\Sigma'\) is determined up to the term

\[ \frac{ie^2}{2\pi}\left[c_1\left(i\frac{\hat\partial}{\partial x}-m\right)+c_2m\right]\delta(x), \]

which vanishes for \(x\ne0\).

Thus, as should have been expected from general considerations, the arbitrariness in this term of the \(T\)-product appears only in an infinitely small neighborhood of the point \(x=0\).

Let us turn to the second divergent term in \(S_2(x,y)\). The term of the scattering matrix corresponding to the diagram (Fig. 1,b) can be represented in the form

\[ -4\pi e^2:\operatorname{Sp}\{\hat A(x)S^c(x-y)\hat A(y)S^c(y-x)\}:= \]

\[ =\sum_{m,n}:A_m(x)\Pi^{mn}(x-y)A_n(y):, \]

where

\[ \Pi^{mn}(x-y)=-4\pi e^2\operatorname{Sp}\{\gamma^m S^c(x-y)\gamma^n S^c(y-x)\}. \tag{1.23} \]

Passing to the momentum representation

\[ \Pi^{mn}(x-y)=\frac{1}{(2\pi)^4}\int e^{ik(x-y)}\Pi^{mn}(k)\,dk, \]

we find that the integral

\[ \Pi^{mn}(k)=-\frac{e^2}{4\pi^3}\int dp\,\operatorname{Sp}\left\{\gamma^m\frac{\hat p+m}{p^2-m^2+i\varepsilon}\gamma^n\frac{\hat p-\hat k+m}{(p-k)^2-m^2+i\varepsilon}\right\} \tag{1.24} \]

diverges quadratically in the region of large momenta,

\[ \int^p dp\cdot p\sim p^2. \]

For the explicit calculation of \(\Pi^{mn}(k)\) we use the same methods that were applied in calculating \(\Sigma(p)\).

Using the regularized \(S^c\)-functions (1.16), with the aid of formulas (1.12), (1.13), and (1.14), we perform the integration over \(p\).

Computing the loop, we obtain:

\[ \operatorname{reg}\left[\Pi^{mn}(k)\right]= \]

\[ = \frac{e^2}{\pi}\int_0^\infty d\alpha_1 \int_0^\infty d\alpha_2 e^{-\varepsilon(\alpha_1+\alpha_2)} \left(e^{-i\alpha_1 m^2}-e^{-i\alpha_1 M^2}\right)\times \]

\[ \times \left(e^{-i\alpha_2 m^2}-e^{-i\alpha_2 M^2}\right) e^{\frac{i\alpha_1\alpha_2 k^2}{\alpha_1+\alpha_2}} \frac{1}{(\alpha_1+\alpha_2)^2}\times \]

\[ \times \left\{ \frac{i\alpha_1\alpha_2}{(\alpha_1+\alpha_2)^2} \left[2k^n k^m-g^{mn}k^2\right] -g^{mn}\left[\frac{1}{\alpha_1+\alpha_2}+im^2\right] \right\}. \]

Passing to the new variables

\[ \alpha_1=\xi\lambda;\qquad \alpha_2=(1-\xi)\lambda \]

and integrating with respect to \(\lambda\), we find that in the limit of large \(M\)

\[ \operatorname{reg}\left[\Pi^{mn}(k)\right] = \frac{e^2}{2\pi i}g^{mn}(M^2-m^2)+ \]

\[ +\frac{ie^2}{3\pi}\ln\left(\frac{M}{m}\right)^2 \left(k^n k^m-g^{mn}k^2\right)+\Pi^{\prime mn}(k), \tag{1.25} \]

where the regular function \(\Pi^{\prime mn}(k)\) is expressed in the form

\[ \Pi^{\prime mn}(k) = \frac{2ie^2}{\pi}\left(k^n k^m-g^{mn}k^2\right) \int_0^1 d\xi\,(1-\xi)\ln\left| \frac{\xi(1-\xi)m^2}{m^2-\xi(1-\xi)k^2} \right| + \]

\[ +\frac{ie^2}{\pi}g^{mn}\left(m^2-\frac{k^2}{6}\right). \tag{1.26} \]

Passing to the configuration representation, we obtain, for sufficiently large \(M\),

\[ \operatorname{reg}\left[\Pi^{mn}(x)\right] = \frac{e^2}{2\pi i}g^{mn}(M^2-m^2)\delta(x)- \]

\[ -\frac{ie^2}{3\pi}\ln\left(\frac{M}{m}\right)^2 \left\{ g^{nn}\frac{\partial}{\partial x^n} g^{mm}\frac{\partial}{\partial x^m} - g^{mn}\Box \right\}\delta(x) +\Pi_M^{\prime mn}(x). \tag{1.27} \]

In the limit \(M\to\infty\) the term \(\Pi_M^{\prime mn}(x)\) converges in the improper sense to the integrable function \(\Pi^{\prime mn}(x)\),

\[ \lim_{M\to\infty}\Pi_M^{\prime mn}(x)=\Pi^{\prime mn}(x), \]

whose Fourier transform is given by formula (1.26).

The separation of the singularities from \(\Pi^{mn}(x)\) is thus completed. Let us note that, as in the preceding case, the decomposition of \(\operatorname{reg}\left[\Pi^{mn}(x)\right]\) into a singular and a finite part, and consequently also the finite part \(\Pi^{\prime mn}(x)\), is not unique. To \(\Pi^{\prime mn}(k)\) there may be added any expression that is a polynomial

in the components \(k\) of degree no higher than the second, since the singular part is in this case a polynomial in \(k\) of the second degree.

To complete the analysis of the expression \(\Pi^{mn}\), let us formulate one more condition of gradient invariance which \(\Pi^{mn}\) must satisfy.

It is not difficult to see that the function \(\Pi^{mn}(k)\), like the function \(\Sigma(p)\), enters into the matrix elements in the following combination with the potentials:

\[ \sum_{m,n} : A_m(k)\,\Pi^{mn}(k)\,A_n(k) : \tag{1.27a} \]

As is known, the condition of gradient invariance consists in the fact that all physically observable quantities do not change their value under the gradient transformation of the potentials

\[ A_m(x)\to A'_m(x)=A_m(x)+\frac{\partial f(x)}{\partial x^m} \]

or, in the momentum representation,

\[ A_m(k)\to A'_m(k)=A_m(k)+i g^{mn}k_n f(k). \]

Therefore the requirement of invariance of the matrix elements of operators of the type (1.27a)

\[ \sum_{m,n} : A'_m(k)\,\Pi^{mn}(k)\,A'_n(k) : = \sum_{m,n} : A_m(k)\,\Pi^{mn}(k)\,A_n(k) : \]

leads us to the condition

\[ \sum_m g^{rm}k^m \Pi^{mn}(k)=0, \tag{1.28} \]

whence it follows that the function \(\Pi^{mn}\) must have the form

\[ \Pi^{mn}(k)=\bigl(k^m k^n-g^{mn}k^2\bigr)\,\pi(k^2). \tag{1.29} \]

Returning to formulas (1.25) and (1.26), we see that, in view of the divergence of the function \(\Pi^{mn}(k)\), the requirement of gradient invariance (1.28) can be imposed only on the regular part \(\Pi'^{mn}(k)\).

From formula (1.26) it is seen that, in order that \(\Pi'^{mn}\) satisfy condition (1.28), it is necessary to subtract from it, using the above-mentioned arbitrariness, the term

\[ \frac{i e^2}{\pi}\, g^{mn}\left(m^2-\frac{k^2}{6}\right), \]

and also to add the expression

\[ c_3\bigl(k^n k^m-g^{mn}k^2\bigr). \]

In other words, the decomposition (1.27) must be replaced by the following:

\[ \operatorname{reg}\,[\Pi^{mn}(x)] = \frac{e^2}{2\pi i}g^{mn}\left(M^2-3m^2-\frac{\Box}{3}\right)\delta(x) - \]

\[ -\frac{ie^2}{3\pi} \left[ \ln\left(\frac{M}{m}\right)^2-3c_3 \right] \left[ g^{nn}\frac{\partial}{\partial x^n}g^{mm}\frac{\partial}{\partial x^m} - g^{mn}\Box \right]\delta(x) + \Pi_{\mathrm{inv}}^{\prime mn}(x), \]

where the finite part \(\Pi_{\mathrm{inv}}^{\prime mn}(x)\) is invariant with respect to a gradient transformation and, in the momentum representation, has the form
\((1.29)\)

\[ \Pi_{\mathrm{inv}}^{\prime mn}(k)= \]

\[ = \frac{2ie^2}{\pi}\left(k^n k^m-g^{mn}k^2\right) \left[ \int_0^1 d\xi\,\xi(1-\xi) \ln\left| \frac{m^2\xi(1-\xi)}{m^2-\xi(1-\xi)k^2} \right| -\frac{c_3}{2} \right]. \tag{1.30} \]

In a similar way one may study the term of the \(S\)-matrix corresponding to the diagram (Fig. 1, \(b\)). Without going into details, we note only that the corresponding function

\[ R(x_1-x_2) \]

after carrying out the regularization in the momentum representation can be represented in the form

\[ R(k)=R_M'(k)+R_{\mathrm{sing}}(Mk), \]

where, as \(M\to\infty\), the function \(R_M'(k)\) converges to an integrable limit

\[ \lim_{M\to\infty} R_M'(k)=R'(k), \]

while \(R_{\mathrm{sing}}\) tends to a polynomial function of the components of the 4-vector \(k\), diverging as \(M^4\).

We arrive at the conclusion that the second-order term in the scattering matrix

\[ S_2(x_1,x_2)=i^2T[L(x_1),L(x_2)] \]

in the regularized case under consideration \((M<\infty)\) can be represented in the form

\[ S_2(x_1,x_2)=i^2T_M'[L(x_1),L(x_2)]- \]

\[ -i\left\{ a_1^M:\bar{\psi}(x_1)\left(i\frac{\hat{\partial}}{\partial x_1}-m\right) \delta(x_1-x_2)\psi(x_2): - \right. \]

\[ \left. -a_2^M:\bar{\psi}(x_1)\delta(x_1-x_2)\psi(x_2): + a_3^M\sum_{m,n}:A_n(x_1)\times \right. \]

\[ \left. \times \left[ g^{nn}\frac{\partial}{\partial x_1^n}g^{mm}\frac{\partial}{\partial x_1^m} - \Box_{x_1}g^{mn} \right] \delta(x_1-x_2)A_m(x_2): \right\} + \]

\[ +\, a_4^M \sum_{m,n} g^{mn} : A_m(x_1)\,\delta(x_1-x_2)\,A_n(x_2):\;-\; \]

\[ -\, a_5^M \sum_{m,n} g^{mn} : A_m(x_1)\,[\Box_{x_1}\delta(x_1-x_2)]\,A_n(x_2):\;+ \]

\[ +\, R_{\mathrm{sing}}\!\left(M,\frac{\partial}{\partial x}\right)\delta(x_1-x_2)+ \]

\[ +\text{ the same terms with the replacement } x_1 \leftrightarrow x_2 \Big). \tag{1.31} \]

Here, for large \(M\), the constants \(a_1, a_2, a_3, a_4\), and \(a_5\) tend to the expressions

\[ a_1 \sim \frac{e^2}{4\pi}\left[\ln\left(\frac{M}{m}\right)^2-\frac{1}{2}-c_1\right];\qquad a_2 \sim \frac{e^2 m}{4\pi}\left[3\ln\left(\frac{M}{m}\right)^2-\frac{7}{2}+c_0\right]; \]

\[ a_3 \sim \frac{e^2}{3\pi}\left[\ln\left(\frac{M}{m}\right)^2-3c_3\right];\qquad a_4 \sim \frac{e^2}{2\pi}(M^2-3m^2);\qquad a_5 \sim \frac{e^2}{6\pi}, \]

and in the coefficient functions of the expression \(T'_M\) the following must be used in the correspondence rules: for simple lines of Feynman diagrams, the function \(\operatorname{reg}[\Delta^c]\) instead of \(\Delta^c\), and for closed diagrams, the corresponding finite functions \(\Sigma'_M\), \(\Pi'^{mn}_{M,\mathrm{inv}}\), and \(R'_M\).

It is then obvious that all coefficient functions of the expression
\(T'_M[L(x_1),L(x_2)]\) converge to a finite limit when the regularization is removed.

We also see that all divergences in \(S_2(x_1,x_2)\) arise from terms proportional to \(\delta(x_1-x_2)\) and its derivatives, which are nonzero only in an infinitesimal neighborhood of the point \(x_1=x_2\). Precisely in a neighborhood of the point \(x_1=x_2\), as was mentioned in (I, § 4), the \(T\)-product

\[ T[L(x_1),L(x_2)] \]

is not completely defined. Therefore it becomes possible to define

\[ T[L(x_1),L(x_2)] \]

in a neighborhood of the point \(x_1=x_2\) as the limit

\[ \lim_{M\to\infty} T'_M[L(x_1),L(x_2)] = T'[L(x_1),L(x_2)], \]

which will ensure the integrability of \(S_2(x_1,x_2)\).

There is also another, completely equivalent possibility. As was established in (I, § 4), the most general form of \(S_2\) includes an arbitrary quasilocal operator

\[ S_2(x_1,x_2)=i^2 T[L(x_1),L(x_2)] + i\Lambda_2(x_1,x_2). \tag{1.32} \]

Indeed, the indicated terms are proportional to \(e^3\) and do not contain factors corresponding to free electrons and positrons. Therefore, under charge conjugation the corresponding matrix elements change by the factor

Fig. 3.

\[ (-1)^3=-1 \]

and, because of the absence of real electron–positrons, they will describe the same processes. Consequently, they are equal to zero*).

It is therefore necessary to consider only the term in \(S_3\) corresponding to the diagram in Fig. 4.

Fig. 4.

According to the general rules this term has the form

\[ \begin{aligned} &(i\sqrt{4\pi}\,e)^3 : \bar\psi(x_3)A(x_3)\psi(x_3) \bar\psi(x_2)A(x_2)\psi(x_2) \bar\psi(x_1)A(x_1)\psi(x_1):= \\ &\qquad = i\sqrt{4\pi}\,e\sum_n : \bar\psi(x_3)\Gamma^n(x_3,x_1\mid x_2)\psi(x_1)A_n(x_2): ; \end{aligned} \tag{1.36} \]

where the “vertex function” \(\Gamma^n(x,y\mid \xi)\) is defined as follows:

\[ \Gamma^n(x,y\mid \xi) = i\,4\pi e^2 \sum_k g^{kk}\gamma^k S^c(x-\xi)\gamma^n S^c(\xi-y)\gamma^k D_0^c(x-y). \tag{1.37} \]

Passing to the momentum representation,

\[ \Gamma^n(x,y\mid \xi) = \frac{1}{(2\pi)^8} \int e^{ip(y-x)+ik(x-\xi)}\Gamma^n(p,k)\,dp\,dk, \]

*) This assertion is a special case of the well-known Furry theorem (see § 4).

we find that

\[ \Gamma^n(p,k)=i_z\,\frac{e^2}{4\pi^3}\int dq\,D_0^c(p-q)\sum_l \gamma^l g^{ll}S^c(q)\gamma^n S^c(q+k)\gamma^l = \]

\[ = \frac{i e^2}{4\pi^3}\sum_l g^{ll}\int \frac{dq}{(p-q)^2+i\varepsilon}\, \gamma^l\,\frac{\hat q+m}{q^2-m^2+i\varepsilon}\, \gamma^n\,\frac{\hat q+\hat k+m}{(q+k)^2-m^2+i\varepsilon}\,\gamma^l . \tag{1.38} \]

The integral (1.38) diverges logarithmically for large \(q\). To evaluate it we use the regularization procedure adopted by us, (1.15), (1.16). Substituting these expressions for the \(S^c\)- and \(D_0^c\)-functions into (1.38), after some rearrangements of the Dirac matrices we find

\[ \operatorname{reg}[\Gamma^n(p,k)] = \]

\[ = -\frac{e^2}{2\pi^3}\int_0^\infty d\alpha_1\int_0^\infty d\alpha_2\int_0^\infty d\alpha_3\, e^{i\alpha_1 p^2+i\alpha_3 k^2-\varepsilon(\alpha_1+\alpha_2+\alpha_3)} (1-e^{-i\alpha_1 M^2})\times \]

\[ \times (e^{-i\alpha_2 m^2}-e^{-i\alpha_2 M^2}) (e^{-i\alpha_3 m^2}-e^{-i\alpha_3 M^2}) \int dq\,e^{iq^2(\alpha_1+\alpha_2+\alpha_3)+2iq(k\alpha_3-p\alpha_1)}\times \]

\[ \times \left[\gamma^n m^2+(\hat k+\hat q)\gamma^n\hat q -2m(k^n+2q^n)\right]. \]

Carrying out the integration over \(q\) with the aid of formulas (1.12)—(1.14) and passing to new variables

\[ \alpha_1=\lambda\xi_1;\qquad \alpha_2=\lambda(1-\xi_1-\xi_2);\qquad \alpha_3=\lambda\xi_2, \]

we obtain from this, after integrating with respect to \(\lambda\) in the limit of large \(M\),

\[ \operatorname{reg}[\Gamma^n(p,k)] = \]

\[ = -\frac{e^2}{2\pi}\gamma^n\int_0^1 d\xi_1\int_0^{1-\xi_1} d\xi_2 \left\{ \ln\left(\frac{M}{m}\right)^2 +\ln\left| \frac{\xi_1\xi_2(1-\xi_1-\xi_2)} {(1-\xi_1)(1-\xi_2)(\xi_1+\xi_2)} \right| +\right. \]

\[ \left. +\ln\left| \frac{m^2} {m^2(1-\xi_1)-\xi_1(1-\xi_1)p^2-\xi_2(1-\xi_2)k^2-2\xi_1\xi_2(pk)} \right| \right\} - \]

\[ -\frac{e^2}{2\pi}\int_0^1 d\xi_1\int_0^{1-\xi_1} d\xi_2\, \frac{ \gamma^n m^2-2mk^n+\hat k\gamma^n(\hat p\xi_1-\hat k\xi_2) +4m(k^n\xi_2-p^n\xi_1) +(\hat p\xi_1-\hat k\xi_2)\gamma^n(\hat p\xi_1-\hat k\xi_2) } {m^2(1-\xi_1)-\xi_1(1-\xi_1)p^2-\xi_2(1-\xi_2)k^2-2\xi_1\xi_2(pk)}, \]

whence it follows that

\[ \operatorname{reg}[\Gamma^n(p,k)] = -\frac{e^2\gamma^n}{4\pi}\left[\ln\left(\frac{M}{m}\right)^2-\frac12\right] +\Gamma_M^n(p,k), \tag{1.39} \]

where the term \(\Gamma_M^{\prime n}\) in the limit of large \(M\) tends to a finite limit equal to

\[ \begin{aligned} \Gamma^{\prime n}(p,k)= &-\frac{e^2}{2\pi}\gamma^n \left\{ \int_0^1 d\xi_1\int_0^{1-\xi_1} d\xi_2 \ln\left| \frac{m^2(1-\xi_1)} {m^2(1-\xi_1)-\xi_1p^2-\xi_2k^2+(\xi_1p-\xi_2k)^2} \right|-1 \right\} \\ &-\frac{e^2}{2\pi}\int_0^1 d\xi_1\int_0^{1-\xi_1}d\xi_2\, \frac{ \gamma^n m^2-2mk^n+\hat{k}\gamma^n(p\xi_1-k\xi_2) +4m(k^n\xi_2-p^n\xi_1) +(p\xi_1-k\xi_2)\gamma^n(p\xi_1-k\xi_2) }{ m^2(1-\xi_1)-\xi_1p^2-\xi_2k^2+(\xi_1p-\xi_2k)^2 }. \end{aligned} \tag{1.40} \]

The decomposition (1.39) has been chosen so that

\[ \Gamma^{\prime n}(0,0)=0. \tag{1.41} \]

Passing to the coordinate representation, we find that, for sufficiently large \(M\),

\[ \operatorname{reg}\,[\Gamma^n(x,y\mid \xi)] = -\frac{e^2}{4\pi}\gamma^n \left[ \ln\left(\frac{M}{m}\right)^2-\frac12 \right]\delta(x-y)\delta(y-\xi) +\Gamma_M^{\prime n}(x,y\mid \xi), \tag{1.42} \]

where, in the limit \(M\to\infty\), the term \(\Gamma_M^{\prime n}\) tends to an integrable function \(\Gamma^{\prime n}\), which in the momentum representation is determined by formula (1.40). The first term in (1.42), as \(M\to\infty\), diverges logarithmically.

The procedure for extracting the divergence from \(\Gamma^n\) is, as always, not unique. The degree of nonuniqueness is determined by the structure of the singular term.

Fig. 5.

A constant multiplied by the matrix \(\gamma^n\) can therefore be added to expression (1.40). This constant, however, is not arbitrary and is determined by the condition of gradient invariance.

Let us consider this condition as applied to terms of the third order of the scattering matrix. The terms of the \(S\)-matrix entering into \(S_3\) can be

can be divided into two groups. One of them consists of terms containing three electromagnetic-field operators \(A(x_1), A(x_2), A(x_3)\) and no electromagnetic contractions \(D_0^c\); the other contains expressions containing one operator \(A\) and one contraction \(D_0^c\). The terms of the first group correspond to diagrams of the type shown in Fig. 5.

The terms corresponding to the diagrams in Fig. 5, \(a,b,c\), are normal products of terms of the lower-order \(S\)-matrix and therefore are manifestly gauge invariant. The term in Fig. 5, \(g\), is zero by Furry’s theorem and, finally, the gauge invariance of the term in Fig. 5, \(d\) can be established by direct calculation.

The terms of the second group, containing the proper third-order divergences, can in general be written in the form

\[ \sum_m : A_m(x_1)J^m(x_1,x_2,x_3):+ \sum_n : A_n(x_2)J^n(x_2,x_3,x_1):+ \]

\[ +\sum_l : A_l(x_3)J^l(x_3,x_1,x_2): . \]

The requirement of gauge invariance imposes on each of the summands of this expression a condition of the form

\[ \sum_m \int dx_1 \frac{\partial f(x_1)}{\partial x_1^m}\, J^m(x_1,x_2,x_3)=0, \tag{1.43} \]

which, in view of the arbitrariness of the function \(f\), gives

\[ \sum_m \frac{\partial}{\partial x^m}J^m(x,x_1,x_2)=0. \tag{1.44} \]

Let us turn to the structure of the function \(J^m(x,x_1,x_2)\).

This function contains terms corresponding to the four diagrams shown in Fig. 6, and also to four more diagrams differing from those of Fig. 6 by an interchange of the points \(x_1\) and \(x_2\).

Fig. 6.

Let us note, first, that the terms \(J^m\) corresponding to the diagram

of the type of Fig. 6, \(g\), i.e.

\[ J^{m}_{(r)}(x,x_1,x_2)\sim \left\{ \sum_{n,l} \Pi^{mn}(x-x_1)g^{nl}:\bar{\psi}(x_2)\gamma^l\psi(x_2):D^c_0(x_1-x_2) +\right. \]

\[ \left. + \text{a term differing by the interchange of } x_1 \text{ and } x_2 \right\}, \]

after removal of the divergences, by virtue of condition (1.28), automatically give

\[ \sum_m \frac{\partial}{\partial x^m}J^m_{(r)}(x,x_1,x_2)=0. \tag{1.45} \]

Therefore it remains to consider only the terms \(S_3\) corresponding to the first three diagrams of Fig. 6:

\[ i\sum_l g^{ll}\left\{ :\bar{\psi}(x_2)\gamma^l S^c(x_2-x)\gamma^n S^c(x-x_1)\gamma^l\psi(x_1): +\right. \]

\[ \left. +:\bar{\psi}(x_2)\gamma^l S^c(x_2-x_1)\gamma^l S^c(x_1-x)\gamma^n\psi(x): +\right. \]

\[ \left. +:\bar{\psi}(x)\gamma^n S^c(x-x_2)\gamma^l S^c(x_2-x_1)\gamma^l\psi(x_1): \right\}D^c_0(x_1-x_2). \tag{1.46} \]

Formally differentiating them with respect to \(x^n\) and summing over \(n\), taking into account the equations

\[ \left(i\frac{\hat{\partial}}{\partial x}-m\right)\psi(x)=0,\qquad i\sum_n \frac{\partial\bar{\psi}}{\partial x^n}\gamma^n+m\bar{\psi}(x)=0, \]

\[ \left(i\frac{\hat{\partial}}{\partial x}-m\right)S^c(x) =i\sum_n \frac{\partial S^c(x)}{\partial x^n}\gamma^n-mS^c(x)=-\delta(x), \]

we obtain zero.

Thus, the terms of the \(S\)-matrix corresponding to the diagrams of Fig. 6, \(a,b,c\), are indeed gauge invariant in the sum.

The verification of condition (1.44) carried out here was of a purely formal character, since each of the terms in expression (1.46) is in fact divergent. In reality it is necessary to verify the condition of gauge invariance for its finite part. For convenience in the verification it is expedient first to use Feynman’s regularization method, and then to pass to the method of regularization adopted by us.

Regularizing, therefore, only one photon function,

\[ D^c_0(x_1-x_2)\to \operatorname{reg}\,[D^c_0(x_1-x_2)], \]

we obtain the expression

\[ i\sum_l g^{ll}\left\{:\bar\psi(x_2)\gamma^l S^c(x_2-x)\gamma^n S^c(x-x_1)\gamma^l\psi(x_1):+\right. \]

\[ +:\bar\psi(x_2)\gamma^l S^c(x_2-x_1)\gamma^l S^c(x_1-x)\gamma^n\psi(x):+ \]

\[ \left.+:\bar\psi(x)\gamma^n S^c(x-x_2)\gamma^l S^c(x_2-x_1)\gamma^l\psi(x_1):\right\}\operatorname{reg}[D_0^c(x_1-x)], \tag{1.47} \]

which, as a whole, manifestly satisfies condition (1.44), since the factors subject to differentiation

\[ \bar\psi(x),\quad S^c(x-\ldots),\quad S^c(\ldots-x),\quad \psi(x) \]

do not change here.

The singular part of expression (1.47) for \(M<\infty\), taking into account the expansions

\[ \left. \begin{aligned} \operatorname{reg}^{F}[\Sigma(x-y)]&=\frac{a_1^F}{i}\left(i\sum_k\gamma^k\frac{\partial}{\partial x^k}-m\right)\delta(x-y)+ \\ &\qquad +ia_2^F\delta(x-y)+\Sigma_F'(x-y),\\[6pt] \operatorname{reg}^{F}[\Gamma^n(x,y\mid\xi)]&=a_5^F\gamma^n\delta(x-y)\delta(x-\xi)+\Gamma_F^{\prime n}(x,y\mid\xi) \end{aligned} \right\} \tag{1.48} \]

is represented in the form

\[ a_5^F:\bar\psi(x_2)\gamma^n\delta(x_2-x)\delta(x-x_1)\psi(x_1):+ \]

\[ +a_1^F:\bar\psi(x_2)\left[\left(i\widehat{\frac{\partial}{\partial x_2}}-m\right)\delta(x_2-x_1)\right]S^c(x_1-x)\gamma^n\psi(x): \]

\[ -a_2^F:\bar\psi(x_2)\delta(x_2-x_1)S^c(x_1-x)\gamma^n\psi(x):+ \]

\[ +a_1^F:\bar\psi(x)\gamma^n S^c(x-x_2)\left[\left(i\widehat{\frac{\partial}{\partial x_2}}-m\right)\delta(x_2-x_1)\right]\psi(x_1): \]

\[ -a_2^F:\bar\psi(x)\gamma^n S^c(x-x_1)\delta(x_1-x_2)\psi(x_1): . \]

Differentiating this combination with respect to \(x^n\), summing over \(n\), and taking into account the equations for \(\psi\), \(\bar\psi\), and \(S^c\), we arrive at the expression

\[ (a_5^F-a_1^F)\left\{:\bar\psi(x_2)\left[\widehat{\frac{\partial}{\partial x}}\delta(x_2-x)\right]\delta(x-x_1)\psi(x_1):+\right. \]

\[ \left.+:\bar\psi(x_2)\delta(x_2-x)\left[\widehat{\frac{\partial}{\partial x}}\delta(x-x_1)\right]\psi(x_1):\right\}. \]

Thus, in order to ensure the gradient invariance of the sin-

of the singular part of expression (1.48), it is necessary that

\[ a_5^F=a_1^F. \tag{1.49} \]

By virtue of the gradient invariance of the entire expression (1.48) as a whole, the invariance of its finite part is then also ensured. Passing then to the limit \(M\to\infty\) after the renormalization of the effective Lagrangian, we obtain for \(S_3\) a gradient-invariant expression containing no infinities.

The identity of the singular constants (1.49), which ensures gradient invariance, was first established in a somewhat more general form by Ward and is known under the name “Ward identities.”

The Ward identity was obtained by us in regularization according to Feynman. We shall show that it also holds for the method of regularization usually adopted by us. To this end, denoting, for brevity of notation,

\[ \operatorname{reg}^F[\Sigma]=\Sigma_F;\qquad \operatorname{reg}^F[\Gamma^n]=\Gamma_F^n, \]

we represent relations (1.48) in the form

\[ \begin{aligned} \Sigma_F(p)+ia_1^F\hat p-i(a_1^F+a_2^F)m &=\Sigma_F(p)-\Sigma_F(0)-\sum_n \left.\frac{\partial \Sigma_F(p)}{\partial p^n}\right|_{p=0} \cdot p^n+c_0^F+i p c_1^F, \end{aligned} \tag{1.50} \]

\[ \Gamma_F^n(p,k)-a_1^F\gamma^n =\Gamma_F^n(p,k)-\Gamma_F^n(0,0)+i c_2^F\gamma^n, \tag{1.51} \]

whence it follows that

\[ a_1^F\gamma^n = i\left.\frac{\partial \Sigma_F(p)}{\partial p^n}\right|_{p=0}g^{nn} -i c_1^F\gamma^n = \Gamma_F^n(0,0)-i c_2^F\gamma^n. \tag{1.52} \]

By direct calculation it is easy to verify that

\[ i g^{nn}\frac{\partial \Sigma_F(p)}{\partial p^n} =-\Gamma_F^n(p,0). \tag{1.53} \]

Therefore from (1.52) it follows that

\[ c_1^F=c_2^F. \]

Recalling now that, according to (1.21), the combination

\[ \Sigma_F(p)-\Sigma_F(0)-\sum_n \left.\frac{\partial \Sigma_F}{\partial p^n}\right|_{p=0} \cdot p^n \]

does not depend on the method of regularization and, in accordance with (1.19), is equal to expression (1.18), and also, taking into account the analogous-

... considerations for the function \(\Gamma^n\), we arrive at the conclusion that the expressions

\[ \Sigma'(p)+c_0+c_1\hat p \]

and

\[ \Gamma^{\prime n}(p,k)+ic_2\gamma^n \]

satisfy the requirement of gauge invariance under the condition

\[ c_1=c_2. \]

Therefore the singular parts of the functions \(\Sigma(x)\) and \(\Gamma^n(x,y\mid \xi)\), subtracted in our usual method of regularization, may be written in the form

\[ -ia_1\left(i\frac{\hat\partial}{\partial x}-m\right)\delta(x)+ia_2\delta(x), \tag{1.54} \]

\[ -a_1\gamma^n\delta(x-y)\delta(x-\xi), \tag{1.55} \]

where

\[ a_1=\frac{e^2}{4\pi}\left[\ln\left(\frac{M}{m}\right)^2-\frac{1}{2}-C_1\right], \tag{1.56} \]

\[ a_2=\frac{e^2m}{4\pi}\left[3\ln\left(\frac{M}{m}\right)^2-\frac{7}{2}+C_0\right]. \tag{1.57} \]

Thus, Ward’s identity also holds in our method of regularization. Since in the preceding argument only the relations

\[ \Sigma(0)=0;\qquad \left.\frac{\partial\Sigma}{\partial p^n}\right|_{p=0}=0;\qquad \Gamma^n(0,0)=0, \]

were used, it has thereby been shown that Ward’s identity is in fact independent of the method of regularization.

We arrive at the conclusion that the third-order term in the scattering matrix can be represented in the form

\[ \begin{aligned} S_3(x_1,x_2,x_3)= &\,i^3T'_M\bigl[L(x_1),L(x_2),L(x_3)\bigr] -i^2T\bigl[L(x_1),\Lambda_2(x_2,x_3)\bigr] \\ &-i^2T\bigl[L(x_2),\Lambda_2(x_1,x_3)\bigr] -i^2T\bigl[L(x_3),\Lambda_2(x_1,x_2)\bigr] \\ &-i\Lambda_3(x_1,x_2,x_3), \end{aligned} \tag{1.58} \]

where \(T'_M\) is a regular operator function, in whose coefficient functions, for simple lines, the functions \(\operatorname{reg}[\Delta^c]\) must be taken, and for the closed diagrams shown in Fig. 7, —

the functions \(\Pi'_M\) and \(\Sigma'_M\), and for the vertex part of the type shown in Fig. 8, the function \(\Gamma_M^{\prime n}\).

It is then obvious that, when the regularization is removed, all coefficient functions of the operator \(T'_M\) converge to finite limits depending on \(\Delta c\), \(\Pi'\), \(\Sigma'\), and \(\Gamma'\). The terms of expression (1.58) containing the functions

Fig. 7.              Fig. 8.

\[ i\Lambda_2(x_1,x_2)=T[L(x_1),L(x_2)]-T'_M[L(x_1),L(x_2)], \]

take into account the second-order divergences corresponding to diagrams of the type of Fig. 6, while the quasilocal operator

\[ \begin{aligned} \Lambda_3(x_1,x_2,x_3) &=-\sqrt{4\pi}\, e\, a_1 \bigl[:\bar\psi(x_1)\,\hat A(x_2)\psi(x_3): \delta(x_1-x_2) \\ &\quad{}\times \delta(x_1-x_3) +\text{terms differing by permutations of the arguments}\bigr] \end{aligned} \tag{1.59} \]

corresponds to the divergences of the vertex parts in Fig. 3, \(в\).

The terms containing the functions \(\Lambda_2\), after integration over \(x_1,x_2,x_3\), give the following contribution to the \(S\)-matrix:

\[ \frac{1}{2}\int dx_1\,dx_2\,dx_3\, T[L(x_1),\Lambda_2(x_2,x_3)], \]

which is compensated by the terms from \(S_2\) containing the counterterms \(L^{(2)}(x)\),

\[ \frac{i^2}{2!}\int T[L(x_1),L^{(2)}(x_2)]\,dx_1\,dx_2 + \]

\[ +\frac{i^2}{2!}\int T[L(x_2),L^{(2)}(x_1)]\,dx_1\,dx_2 = \]

\[ =-\frac{1}{2}\int dx_1\,dx_2\,dx_3\, T[L(x_1),\Lambda_2(x_2,x_3)]. \]

The last divergent term in (1.58) must be compensated by adding to the interaction Lagrangian a new counter-

term of third order in \(e\)

\[ \int L^{(3)}(x)\,dx = \sqrt{4\pi}\,e a_1 \int dx:\bar{\psi}(x)\hat A(x)\psi(x): = \frac{i}{3!}\int \Lambda_3(x,x_1,x_2)\,dx\,dx_1\,dx_2 . \tag{1.60} \]

Thus, after introducing into the Lagrangian the counterterms \(L^{(2)}(x)\) and \(L^{(3)}(x)\) and passing to the limit \(M\to\infty\), we obtain, for the construction of the second- and third-order terms of the \(S\)-matrix, the integrable expressions

\[ T' [L(x_1),L(x_2)], \]

\[ T' [L(x_1),L(x_2),L(x_3)]. \]

Taking into account the counterterms of third order in \(e\), the interaction Lagrangian takes the form

\[ \begin{aligned} L(x)={}&\sqrt{4\pi}\,e Z_1:\bar{\psi}(x)\hat A(x)\psi(x): \;+\; \\ &+a_1\left\{ \frac{i}{2}\sum_k:\left(\bar{\psi}(x)\gamma^k\frac{\partial\psi}{\partial x^k} -\frac{\partial\bar{\psi}}{\partial x^k}\gamma^k\psi\right): -m:\bar{\psi}\psi: \right\}-\\ &-(a_3-a_5)\sum_{m,n} g^{mm}g^{nn}: \frac{\partial A_n(x)}{\partial x^m} \frac{\partial A_n(x)}{\partial x^m}: -\\ &-a_3:\left(\sum_m g^{mm}\frac{\partial A_m}{\partial x^m}\right)^2 -\delta m:\bar{\psi}(x)\psi(x):\;+\;\\ &+\delta m_\phi\sum_m g^{mm}:A_m(x)A_m(x):\;+\;R, \end{aligned} \tag{1.61} \]

where

\[ Z_1=1+a_1;\qquad \delta m=a_2;\qquad \delta m_\phi=a_4. \]

It will be shown below (§ 4) that the elimination of divergences in spinor electrodynamics in higher orders is carried out by introducing into the Lagrangian counterterms of the same operator structure as the counterterms in (1.61).

Regarding as the basic requirement on the theory the condition of finiteness and gradient invariance of physically observable quantities (in the present case, the matrix elements of the scattering matrix), we shall admit such Lagrangians as ensure the fulfillment of this condition. As we have already seen, for this it proves necessary to allow that \(L(x)\) include both divergent terms \((a_1,a_2,\ldots)\) and gradient-noninvariant terms \((\delta m_\phi,a_5)\). It also turns out that the structure of the counterterms depends on the “region of inclusion” of the interaction, described by the function \(g(x)\).

Up to now we have restricted ourselves to consideration of the case in which the interaction is switched on completely in all space-time and

\[ g(x)=1. \]

It is precisely this case that is important in calculating the matrix elements of scattering processes and mutual transformations of particles, when the actual switching on and switching off of the interaction is referred to the infinitely remote past and future.

It turns out, however, that in those cases in which we are interested in characteristics of systems of particles that are in bound states (energy levels, lifetimes, probabilities of transitions between bound states*), one has to consider the situation in which the interaction is switched on only in a certain part of 4-space and the function \(g(x)\) increases from zero to unity in small regions near the surfaces bounding this part of 4-space. Bearing in mind this latter case, when

\[ g(x)\ne 1, \]

we investigate the structure of the counterterms in the effective Lagrangian \(L(x;g)\), which ensure the finiteness of the operator

\[ S(g)=T\left\{e^{\,i\int L(x;g)g(x)\,dx}\right\}. \]

Let us consider the terms of second order.

The counterterm \(L^{(2)}(x;g)\) is determined from the condition, which is a natural generalization of relation (1.33),

\[ \int L^{(2)}(x;g)g(x)\,dx = \]

\[ =\frac{1}{2i}\int dx_1 dx_2\, g(x_1)g(x_2) \left\{T[L(x_1),L(x_2)]-T'_M[L(x_1),L(x_2)]\right\}. \]

Substituting here the explicit expression for the difference \(T-T'_M\) and integrating by parts, we find:

\[ \begin{aligned} L^{(2)}(x;g)g(x) &=L^{(2)}(x)\cdot g^2(x)- \\ &\quad -a_3\sum_{m,n} g^{mn}g^{nn} \left\{ :A_m(x)A_n(x):\,\frac{\partial g}{\partial x^n}\,\frac{\partial g}{\partial x^m} +\right.\\ &\qquad\left. +2:A_n(x)\frac{\partial A_m}{\partial x^m}:\,\frac{\partial g(x)}{\partial x^n}\,g(x) \right\}-\\ &\quad - (a_3-a_5)\sum_{m,n} \left[ :A_m(x)A_m(x):\,\frac{\partial g}{\partial x^n}\,\frac{\partial g}{\partial x^n} +\right.\\ &\qquad\left. +2:A_m(x)\frac{\partial A_m}{\partial x^n}:\,g(x)\frac{\partial g}{\partial x^n} \right]g^{mm}g^{nn} +\\ &\quad +\text{terms containing }R\text{ and its derivatives}, \tag{1.62} \end{aligned} \]

\[ \text{*) These questions will be dealt with in the next article.} \]

where the operator \(L^{(2)}(x)\) is defined by relation (1.34). It is also not difficult to verify that in third order

\[ L^{(3)}(x;g)=L^{(3)}(x)g^2(x), \]

where \(L^{(3)}(x)\) is defined by relation (1.60).

Thus, in the process of integration by parts, terms appear in \(L(x;g)\) which contain derivatives of the function \(g(x)\) and, in their operator structure, differ from the usual counterterms.

This circumstance will prove to be essential in removing divergences from the Schrödinger equation.

§ 2. GENERAL RULES FOR REMOVING DIVERGENCES FROM THE \(S\)-MATRIX

In the preceding section, using spinor electrodynamics as an example, the construction of integrable expressions for \(S_2\) and \(S_3\) was carried out. We shall now formulate a general method for removing divergences from terms of the \(S\)-matrix of arbitrary order, based on the same principle as the examples considered. First of all, let us replace the true causal functions \(\Delta^c(x)\) by regularized expressions \(\operatorname{reg}[\Delta^c(x)]\) with a certain appropriate number of auxiliary masses \(M_i\). For finite values of these masses, the ordinary \(T\)-product of Lagrangians

\[ T\,[L(x_1),\ldots,L(x_n)] \tag{2.1} \]

is perfectly well defined, and its coefficient functions are continuous. However, as we have just seen in concrete examples, the limiting transition \(M\to\infty\), understood even in an improper sense, cannot be carried out in (2.1).

More precisely, the coefficient functions of the operator (2.1) will converge in an improper sense only in those regions of space-time in which all, without exception, of the arguments \(x_1,x_2,\ldots,x_n\) are distinct from one another. In order to isolate from expression (2.1) the “convergent part,” it is necessary, as in the cases discussed, to apply to it a certain subtractive procedure.

In order to formulate it, it is convenient to start from the general formula (I.4.32), which expresses \(S_n(x_1,\ldots,x_n)\) in terms of the interaction Lagrangian \(L(x)=\Delta_1(x)\) and the quasilocal operators \(\Delta_\nu(\nu\geqslant 2)\):

\[ S_n(x_1,\ldots,x_n)=i^n T[L(x_1),\ldots,L(x_n)]+ \]

\[ +\sum_{\substack{2\leqslant m\leqslant n-1\\ \sum \nu_i=n}} \frac{i^m}{m!}\, P(x_1,\ldots,x_{\nu_1}\mid\ldots\mid\ldots x_n)\times \]

\[ \times T[\Delta_{\nu_1}(x_1,\ldots,x_{\nu_1})\ldots \Delta_{\nu_m}(\ldots,x_n)]+i\Delta_n(x_1,\ldots,x_n). \tag{2.2} \]

As was shown, this expression is the most general

expression satisfying all the conditions imposed on \(S_n\) (symmetry, covariance, unitarity, and causality) for an arbitrary choice of Hermitian covariant quasilocal operators \(\Lambda_\nu\).

If we can choose the quasilocal operators \(\Lambda_\nu\) in such a way that the expressions \(S_n\) prove to be convergent (the convergence of expressions of this kind will always be understood in the improper sense), then their limit as \(M \to \infty\), first, will be an integrable operator function (in the sense of Definition 1, § 2), and, second, will satisfy all the requirements imposed on \(S_n\). Indeed, the covariance condition (I.4.4) is linear in character, and therefore passage to the limit in it is trivial, while the possibility of passing to the limit in the unitarity (I.4.9) and causality (I.4.12) conditions is ensured by theorem (I, § 2) stating that the limit of the ordinary product of two operator functions is equal to the corresponding product of the limits.

Thus, it is sufficient for us to establish a method of constructing, for a given Lagrangian \(L(x)=\Lambda_1(x)\), a chain of quasilocal operators \(\Lambda_\nu\) \((\nu \geqslant 2)\) ensuring the convergence of expression (2.2). For the limiting expressions obtained in this way, all the imposed conditions on \(S_n\), including also the condition of integrability, are automatically fulfilled. To solve this problem it is, evidently, sufficient for us to choose as \(\Lambda_\nu(x_1,\ldots,x_\nu)\) quasilocal operators of the same operator type as (2.1).

In saying here that \(\Lambda_\nu\) belongs to the same operator type as (2.1), we mean that \(\Lambda_\nu\) consists of the same operator terms

\[ \ldots u_\alpha(x_j)\ldots; \]

as (2.1), differing from it only in coefficient functions. From (2.2) it now follows that \(S_n(x_1,\ldots,x_n)\) will be an expression of the same operator type as (2.1).

To simplify the formulas, it is convenient to put

\[ \begin{gathered} i^{1-\nu}\Lambda_\nu(x_1,\ldots,x_\nu)=\Delta_\nu(x_1,\ldots,x_\nu),\\ L(x)=\Lambda_1(x)=\Delta_1(x), \end{gathered} \tag{2.3} \]

then

\[ S_n=i^n T'_n, \tag{2.4} \]

where

\[ \begin{aligned} T'_n={}&T[L(x_1),\ldots,L(x_n)]+\\ &+\sum_{\substack{2\leq m\leq n-1\\ \sum \nu_i=n}} \frac{1}{m!}P(x_1,\ldots,x_{\nu_1}|\ldots|\ldots x_n)\times\\ &\times T[\Delta_{\nu_1}(x_1,\ldots,x_{\nu_1})\cdots \Delta_{\nu_m}(\ldots x_n)] +\Delta_n(x_1,\ldots,x_n). \end{aligned} \tag{2.5} \]

Since \(T'_n\) belongs to the same operator type as in (2.1), we can obviously develop a method for the actual construction of \(T'_n\) by means of a certain transformation of the coefficient functions of the operator \(T\). In doing so it proves convenient to work with graphical representations.

The coefficient function for the \(T\)-product corresponding to some given diagram with \(\nu\) vertices and prescribed internal lines is represented by a product of regularized causal functions of the form

\[ \prod_l \operatorname{reg}\,[\Delta_i^c(x_a-x_b)]. \tag{2.6} \]

Fig. 9.

The coefficient functions of the quasilocal operator \(\Delta\), for the same diagram, are products of \(\delta\)-functions and their derivatives. In the diagram, therefore, the entire set of points \(x_1,\ldots,x_\nu\) appears as a single whole, in view of which, when working with \(\Delta\), it is convenient to introduce the concept of a “generalized node \(G\)” (Fig. 9).

We denote the coefficient function of the operator \(\Delta\) corresponding to the given generalized node \(G\) by

\[ d_G(x_1,\ldots,x_\nu). \]

Let us now consider an expression of the form

\[ T[\Delta_\nu(x_1,\ldots,x_\nu),\,L(x_{\nu+1}),\ldots,L(x_n)]. \tag{2.7} \]

It is clear that the coefficient functions can be obtained from the coefficient functions of the operator

\[ T[L(x_1),\ldots,L(x_\nu),\ldots,L(x_n)] \tag{2.8} \]

by the following procedure.

Consider some diagram corresponding to the operator (2.7). Combining in it the points \(x_1,\ldots,x_\nu\) into the generalized vertex \(G\), we replace the product (2.6), corresponding to lines \(l\) internal with respect to \(G\), by the coefficient function

\[ d_G(x_1,\ldots,x_\nu). \]

We shall formally denote the operation of such a replacement by the symbol

\[ \Delta(G). \]

Then the coefficient functions of the operator (2.7) will be obtained from the coefficient functions of the operator (2.8) by the operation \(\Delta(G)\).

It is also obvious that in the more general case the coefficient functions of the expression

\[ T\bigl[\Delta_{\nu_1}(x_1,\ldots,x_{\nu_1}),\, \Delta_{\nu_2}(x_{\nu_1+1}\ldots x_{\nu_1+\nu_2}),\ldots \Delta_{\nu_m}(\ldots x_n)\bigr] \]

can be obtained from the coefficient functions of the operator (2.8) by the operation

\[ \Delta(G_1)\ldots \Delta(G_m), \]

which consists in combining the points

\[ x_1,\ldots,x_{\nu_1},\ x_{\nu_1+1},\ldots,x_{\nu_1+\nu_2},\ldots,\ldots,x_n \]

into generalized nodes (vertices)

\[ G_1,\ G_2,\ldots,\ G_m, \]

and replacing the parts of the product (2.6) corresponding to the internal lines \(G_1,\ldots,G_m\) by the corresponding \(d_{G_1},\ldots,d_{G_m}\), while keeping unchanged the factors corresponding to the lines connecting different generalized vertices \(G\).

Thus, \(T'_n\) can be obtained from \(T\) by applying the operation

\[ R(G)=1+ \sum_{\substack{2\le m\le n-1\\ G=G_1*G_2*\ldots*G_m}} \frac{1}{m!}\Delta(G_1)\ldots \Delta(G_m)+\Delta(G). \tag{2.9} \]

The sum here is taken over all possible partitions of the set of points \(x_1,\ldots,x_n\) in the diagram \(G\) into generalized nodes \(G_1,G_2,\ldots,G_m\). The symmetrization operator \(P\), which enters formula (2.2), is taken into account here by the fact that under such a partition the points \(x_1,\ldots,x_n\) occur completely symmetrically.

The operation \(R(G)\) has so far been defined purely formally. It will acquire concrete content when rules are established for the actual definition of the function \(d_G\) for the given diagram \(G\). Having specified \(d_G\), we thereby determine \(\Delta_\nu\) and, consequently, \(T'_n\). We shall choose \(d_G\) so that \(T'_n\) proves to be integrable as \(M\to\infty\).

Before proceeding to formulate the method for constructing \(d_G\), let us introduce the notion of connectedness of a diagram. We shall say that a given diagram is disconnected if it can be divided into two parts not connected with each other by lines. In the opposite case we call the diagram connected. A connected diagram will be called “weakly connected” if by removing one line it can be made disconnected, and “strongly connected” if this is impossible.

Let us now note that for a disconnected diagram the coefficient function of the \(T\)-product is represented in the form of a product of two coefficient functions with different arguments. But the produ—

...the product of two functions with different arguments is convergent when each function separately is convergent. It follows from this that, in a disconnected diagram, divergences are automatically removed after they have been removed in its connected parts. Therefore the operator \(\Delta(G)\) for disconnected diagrams should be regarded as equal to zero.

With such a choice of \(\Delta(G)\), the operation \(R(G)\), applied to the coefficient function of the \(T\)-product corresponding to a diagram \(G\) which consists of two disconnected parts \(G_1\) and \(G_2\), decomposes into the product

\[ R(G_1)R(G_2) \]

of two operations acting individually on the coefficient functions of the \(T\)-product corresponding to the diagrams \(G_1\) and \(G_2\).

For weakly connected diagrams we arrive at coefficient functions of the type

\[ K_M(x_1,\ldots,x_k)\cdot \operatorname{reg}\left[\Delta^c(x_a-y_b)\right]\cdot Q_M(y_1,\ldots,y_s). \]

In view of the translational invariance of \(K\) and \(Q\), setting

\[ x-x_a=x'; \qquad y-y_b=y'; \qquad x_a-y_b=\xi, \]

we obtain a product of coefficient functions with independent arguments

\[ K_M(x'_1,\ldots,x'_k)\operatorname{reg}\left[\Delta^c(\xi)\right]\cdot Q_M(y'_1,\ldots,y'_s). \tag{2.10} \]

The whole expression as a whole will again be convergent if \(K\) and \(Q\) are separately convergent. Therefore for weakly connected diagrams one should also set

\[ \Delta(G)=0. \]

Here, as in the preceding case, it is clear that, with the chosen definition of the operator \(\Delta(G)\), the coefficient functions of the operator \(T'_n\) for weakly connected diagrams have the same structure as (2.10).

Thus, in formula (2.9) one may consider decompositions of \(G\) only into generalized vertices \(G_a\) that are strongly connected.

To make the operator \(\Delta(G)\) more concrete, let us introduce also the notion of the index of a diagram. For this purpose we pass to the momentum representation. The coefficient functions of the \(T\)-product in the \(p\)-representation will evidently have the form

\[ J_M(k)=\int \prod_{1<q\le n}\delta\!\left(\Sigma p+k_q\right)\cdot \prod_l\left\{\operatorname{reg}\left[\Delta_l^c(p_l)\right]\cdot dp_l\right\}. \tag{2.11} \]

Here, in the arguments of the \(\delta\)-functions, there stand algebraic sums of the momenta of the internal lines of the diagram converging at the vertex \(q\), to which the external momenta \(k_q\) have been added.

In accordance with the regularization procedure for the \(\Delta^c\)-function adopted by us, we also have

\[ \operatorname{reg}\,[\Delta^c(p)] = Z(p)\left\{\frac{1}{m^2-p^2-i\varepsilon} +\sum_j c_j \frac{1}{M_j^2-m^2-i\varepsilon}\right\}, \]

where \(Z(p)\) is the same polynomial as in the unregularized \(\Delta^c\)-function.

If in (2.11) there were unregularized \(\Delta^c\)-functions, then the whole integral would, generally speaking, turn out to diverge at large momenta. Let us now calculate the total degree of its divergence.

In view of the fact that we are considering connected diagrams, with the help of the \(\delta\)-functions \(4(n-1)\) integrations are removed (one remaining \(\delta\)-function expresses the law of conservation of the total 4-momentum), and we have in all

\[ 4(L-n+1) \]

independent variables of integration, where \(L\) denotes the total number of internal lines.

Just as, when integrating over three-dimensional space, one introduces the radius as the variable of integration, we shall introduce, in carrying out the integration over the \(4(L-n+1)\)-dimensional space, the corresponding “radial” momentum \(P\). Then the product of the independent differentials from

\[ \prod_l dp_l \]

gives the factor

\[ P^{4(L-n+1)}\frac{dp}{P}. \]

Taking into account only the highest powers of \(P\) from the functions \(\Delta^c(p)\), we obtain the factor

\[ P^{\sum_l r_l-2L}=P^{\sum_l (r_l-2)} \]

and therefore, in carrying out the integration over \(P\), the factor multiplying \(dP/P\) at large \(P\) will increase or decrease as

\[ P^{\sum_l (r_l+2)-4(n-1)}. \]

The integral over \(P\) will thus be divergent if

\[ \sum_l (r_l+2)-4(n-1)\ge 0, \]

and convergent if

\[ \sum_l (r_l+2)-4(n-1)<0. \]

The number

\[ \omega(G)=\sum_i (r_i+2)-4(n-1) \tag{2.12} \]

we shall call the index of the diagram \(G\).

Of course, from the convergence of the integral over \(P\) it still does not follow that the whole integral of type (2.11) converges as a whole.

Here a situation may arise similar to that when, in computing the integral

\[ \int_{-\infty}^{\infty} dx \int_{-\infty}^{\infty} dy\, \frac{x}{(y^2+1)^2} \]

the integral with respect to the radial variable \(\rho\)

\[ \rho=\frac{x}{\cos\varphi}=\frac{y}{\sin\varphi}; \]

\[ \int \frac{\rho^3\,d\rho\,\cos^2\varphi\,d\varphi} {(\rho^2\sin^2\varphi+1)^2} \]

converges, but the remaining integration with respect to \(\varphi\), because of singularities at the points \(\varphi=0\), \(\varphi=\pi\), turns out to be divergent.

The index of the diagram \(\omega(G)\) can also be related to the conditional degree of growth with respect to momentum. To compute it, multiply all external momenta and masses by a certain number \(a\), and compute by what factor the integral (2.11) changes, without taking regularization into account and considering only the highest power of \(a\).

It is not hard to see that this factor is precisely equal to

\[ a^{\omega(G)}. \tag{2.13} \]

Thus the index of a diagram is exactly equal to the degree of growth. Let us note that the degree of growth is called conditional because the estimate (2.13) is carried out purely formally, without a careful analysis of the convergence of the integral, and does not take into account the presence of logarithmically divergent factors.

Let us now draw attention to the fact that, when \(G\) is decomposed into a series of generalized vertices

\[ G=G_1 * G_2 * \ldots G_s \]

we shall have

\[ \omega(G)= \sum_{1\le j<s}\omega(G_j)+\sum_i(r_i+2)-4(s-1). \tag{2.14} \]

Let us next consider the coefficient function \(d_{G_j}\). In the momentum representation it has the form

\[ \delta\!\left(\sum p_i\right) Z_{G_j}(\ldots p\ldots), \]

where \(Z_{G_j}(\ldots p\ldots)\) is a certain polynomial in the components \(p\).

As we shall see below, in order to compensate divergences in the \(T\)-product it is sufficient to choose, as \(Z_{G_i}\), a polynomial of degree \(\omega(G_i)\). From identity (2.14) it follows that, with such a choice of \(\mathcal F\), neither the total degree of divergence nor the conditional degree of growth with respect to momentum increases from the application of the operation

\[ \Delta(G_1)\ldots \Delta(G_m). \]

Consequently, the index \(\omega(G)\) is not increased by applying the operation \(R(G)\) as a whole.

As we have already verified in the examples considered above, in analyzing and computing integrals of type (2.11) it is convenient to use the integral representation of causal functions (below we shall call it the “\(\alpha\)-representation”)

\[ \Delta^c(p)=Z(p)i\int_0^\infty e^{i\alpha(p^2-m^2+i\varepsilon)}\,d\alpha, \]

\[ \operatorname{reg}[\Delta^c(p)]=Z(p)i\int_0^\infty e^{i\alpha(p^2-m^2+i\varepsilon)}I(\alpha)\,d\alpha, \]

\[ I(\alpha)=1+\sum_M C_M e^{-i\alpha(M^2-m^2)}. \]

In order to get rid of the factor \(Z(p)\), which violates the purely exponential form of the dependence on \(p\), we shall use the relation

\[ Z(p)=Z(-i\nabla_q)e^{i(pq)}\big|_{q=0}. \]

With the aid of this representation, integration over the internal momenta in (2.11) is reduced to quadratures of Gaussian type

\[ \prod_l Z(-i\nabla_{q_l}) \int e^{\,i\sum_m \alpha_m p_m^2+i\sum_m(p_m q_m)} \prod_{n<q<n}\delta\!\left(\sum p+k_q\right) \prod_l dp_l\big|_{q_l=0} = f(\ldots,k,\ldots,\alpha,\ldots) \tag{2.15} \]

and only integrations over the variables \(\alpha\) remain:

\[ J_M(k)= \delta\!\left(\sum k\right) \int_0^\infty d\alpha_1\ldots \int_0^\infty d\alpha_L\, f(\ldots,k,\ldots,\alpha,\ldots) \times e^{-i\sum_l m_l^2\alpha_l-\varepsilon\sum_l\alpha_l} \prod_{1\le l\le L} I(\alpha_l). \tag{2.16} \]

Carrying out the integration with respect to \(p_l\), we find that

\[ f(\ldots,k,\ldots,\alpha,\ldots) = F(\ldots,k,\ldots,\alpha,\ldots) e^{\,i\sum_{a,b} A_{ab}(\ldots\alpha\ldots)(k_a k_b)}, \tag{2.17} \]

where \(F\) is a polynomial in \(k\) and a rational function of \(\alpha\), possessing nonintegrable poles when some \(\alpha_l\) tend to zero. In view of the fact that the convergence of the integral (2.16) for large \(\alpha\) is ensured by the factors \(e^{-\varepsilon\sum_l \alpha_l}\), the possible divergences of an integral of type (2.11) in the unregularized case are, in this representation, precisely due to the presence of these nonintegrable poles.

To clarify the structure of the singularity, let us pass to new variables

\[ \alpha_j=\lambda \xi_j;\qquad \sum_j \xi_j=1 \]

and, fixing \(\xi_j\ne 0\), compute the order of the pole at the point \(\lambda=0\). Passing for this purpose in (2.15) to new “momenta”

\[ p_l\sqrt{\lambda}=P_l;\qquad q_l\frac{1}{\sqrt{\lambda}}=Q_l;\qquad k_q\sqrt{\lambda}=K_q, \]

we find that

\[ \begin{aligned} f(\ldots,k,\ldots,\alpha,\ldots) &= \lambda^{-2L+2(n-1)} \prod_l \left\{ Z\left(\frac{\nabla Q_l}{i\sqrt{\lambda}}\right) \right\} \times \\ &\quad{}\times \int e^{\,i\sum_m P_m^2+i\sum_m(P_m Q_m)} \prod_{1\le q\le n}\delta(\sum P+K_q)\prod_l dP_l\bigg|_{Q=0}. \end{aligned} \]

Taking into account in \(\prod\{Z(\ldots)\}\) only the highest powers of \(\lambda\), we obtain, for small \(\lambda\),

\[ \begin{aligned} f(\ldots,k,\ldots,\alpha,\ldots) &= \lambda^{-\frac{\omega(G)}{2}-L} F'(\ldots K\ldots) e^{\,i\sum_{a,b} A'_{ab}(K_aK_b)} \\ &= \lambda^{-\frac{\omega(G)}{2}-L} F'(\ldots \sqrt{\lambda}\,k\ldots) e^{\,i\sum_{a,b} A'_{ab}(\xi)\lambda(k_a k_b)} . \end{aligned} \tag{2.18} \]

Thus, the effective order of the pole in \(\lambda\) at \(\lambda=0\), taking into account the value of the determinant

\[ \left| \frac{\partial \alpha_1,\ldots,\partial \alpha_L} {\partial \xi_1,\ldots,\partial \xi_{L-1},\partial \lambda} \right| = \lambda^{L-1}, \]

is indeed determined by the index of the diagram \(\omega(G)\).

Let us now consider the structure of the quadratic form

\[ A_{ab}(\ldots\alpha_j\ldots)=\lambda A_{ab}(\ldots\xi_l\ldots). \]

It is not hard to verify that it does not depend on the polynomials \(Z(-i\nabla_q)\) and is a homogeneous function of degree 1 in the \(\alpha_l\).

Moreover, as we shall now show, it has the positivity property

\[ \sum_{a,b} A_{ab}x_a x_b > 0;\qquad \sum x^2 \ne 0 \tag{2.19} \]

and satisfies the condition

\[ \sum_{a,b} A_{ab}x_a x_b < \sum_l \alpha_l \left(\sum_a |x_a|\right)^2 . \tag{2.20} \]

Relations (2.19) and (2.20) are conveniently checked by induction, since their validity for the simplest diagrams raises no doubt (see the examples analyzed in § 1).

Induction will be carried out in two stages:

a) for a given number of vertices, a new internal line is added;

b) a new vertex is added, together with one line connecting this vertex with one of the old vertices.

Consider the first case. For definiteness we shall assume that an internal line with momentum \(p_0\) has been added between vertices 1 and 2 (Fig. 10).

Fig. 10.

Neglecting factors of the type \(Z(-i\nabla_q)\), which do not affect the structure of the exponent, one may say that in expression (2.15), instead of the factors

\[ \delta\left(\sum_1 p+k_1\right)\delta\left(\sum_2 p+k_2\right) \]

there will appear the factor

\[ e^{i\alpha_0p_0^2} \delta\left(\sum_1 p+k_1+p_0\right) \delta\left(\sum_2 p+k_2-p_0\right) \]

and integration over \(p_0\) will be added. Thus the change in the integral (2.15) can be written in the form

\[ f(k_1,k_2,\ldots|\ldots \alpha \ldots)\to \]

\[ \to \int e^{i\alpha_0p_0^2} f(k_1+p_0,k_2-p_0,\ldots k\ldots|\ldots \alpha \ldots)\,dp_0 = \]

\[ = \int e^{i\alpha_0p_0^2} F(k_1+p_0,k_2-p_0,\ldots k\ldots|\ldots \alpha \ldots)\times \]

\[ \times e^{\,i\sum_{a,b} A_{ab}(k_a+e_a p_0)(k_b+e_b p_0)}\,dp_0, \]

where the symbol \(e_a\) is defined as follows:

\[ e_1=1,\qquad e_2=-1,\qquad e_q=0;\qquad (q>0). \]

Representing the integral obtained in the form (2.17), we note that the function
\(F(k_1+p_0,\ k_2-p_0,\ldots,k,\ldots|\ldots,\alpha,\ldots)\), by virtue of its polynomial character, does not affect the structure of the resulting exponential. Therefore, evaluating the integral

\[ \int e^{i a_0 p_0^2+i\sum A_{ab}(k_a+e_a p_0)(k_b+e_b p_0)}\,dp_0= \]

\[ =\int dp_0 \exp i\left[\left(a_0+\sum A_{ab}e_a e_b\right)^2 p_0^2+ 2\sum A_{ab}e_a(p_0 k_b)+\sum A_{ab}k_a k_b\right], \]

we arrive at an exponential of the form

\[ \exp i\left[\sum A_{ab}k_a k_b- \frac{\left(\sum A_{ab}k_b e_a\right)^2}{a_0+\sum A_{ab}e_a e_b}\right]. \]

Thus, when a new internal line is included in the diagram, the exponent undergoes the transformation

\[ \sum A_{ab}k_a k_b\to \sum A_{ab}k_a k_b- \frac{\left(\sum A_{ab}k_b e_a\right)^2}{a_0+\sum A_{ab}e_a e_b}. \tag{2.21} \]

By virtue of the positivity of the quadratic form (2.19), for

\[ x_a=k_a t_1+e_a t_2 \]

from the condition that its discriminant be non-positive, we find the inequality

\[ \left(\sum A_{ab}k_a k_b\right)\left(\sum A_{ab}e_a e_b\right)> \left(\sum A_{ab}k_b e_a\right)^2, \]

substituting which into (2.20), we obtain:

\[ \sum A_{ab}k_a k_b\to \sum A_{ab}k_a k_b- \frac{\left(\sum A_{ab}k_b e_a\right)^2}{a_0+\sum A_{ab}e_a e_b}> \]

\[ >\sum A_{ab}k_a k_b \frac{a_0}{a_0+\sum A_{ab}e_a e_b}>0. \]

Thus, property (2.19) is not violated when an extra internal line is included.

In view of the fact that, under the transition (2.21), the quadratic form decreases, condition (2.20) is also not violated when an additional internal line is included.

Let us now consider the process of including into the diagram a new vertex and the line connecting it.

For greater definiteness, we shall assume that the new vertex \(O\) is connected by a line with momentum \(p_0\) to vertex \(1\) (Fig. 11).

Fig. 11.

In the integral (2.15) this will cause the following replacement:

\[ \delta\left(\sum p_1+k_1\right)\to e^{ia_0p_0^2}\delta\left(\sum_1 p+k_1+p_0\right)\delta(k_0-p_0) \]

and the appearance of an additional integration over \(p_0\), which can also be represented in the form

\[ f(\ldots k\ldots a\ldots)\to \int dp_0 e^{ia_0p_0^2} f(k_1+p_0,k_2,\ldots,a,\ldots)\delta(p_0-k_0)= \]

\[ = e^{ia_0k_0^2} f(k_1+k_0,k_2,\ldots,a,\ldots). \]

The changes in the exponent are correspondingly equal to

\[ \sum_{a,b} A_{ab} k_a k_b \to a_0 k_0^2+\sum A_{ab}(k_a+e_a k_0)(k_b+e_b k_0), \tag{2.22} \]

where

\[ e_1=1;\qquad e_a=0;\qquad (a\geqslant 2). \]

On the basis of (2.20) we obtain from this

\[ a_0 x_0^2+\sum A_{ab}(x_a+e_a x_0)(x_b+e_b x_0)\leq \]

\[ \leq \alpha_0\sum_a (x_0e_a+x_a)^2+\sum_{1<l}\alpha_l\sum_{l\leq a}(x_a+e_a x_0)^2= \]

\[ =\sum_{0<l}\alpha_l\left|\sum_{l\leq a}(x_a+e_a x_0)\right|^2 =\sum_{0<l}\alpha_l\left(\sum_{0\leq a}|x_a|\right)^2. \]

Thus, condition (2.20) is preserved when a new node with a line is included. Property (2.19) is also preserved, since the quadratic form increases when a node is included.

The proof of properties (2.19) and (2.20) is thereby completed. Let us note that in the proof there was in fact isolated a recipe for constructing the quadratic form, which is contained in formulas (2.21) and (2.22).

Let us now consider the changes that occur in the coefficient functions of the \(T\)-products as a result of the action of the operation \(\Delta(G_j)\) in the integral “\(a\)-representation” (2.15) adopted by us.

Recall that, according to the definition, the operation \(\Delta(G_j)\) in the \(x\)-representation consists in replacing the part of the product

\[ \prod_l \Delta_i^c(x_a-x_b), \]

corresponding to the internal lines of \(G_j\), by the coefficientная

function of the quasilocal operator

\[ d_{G_j}(\ldots x_a \ldots), \]

which in the \(p\)-representation has the form of a polynomial multiplied by a \(\delta\)-function

\[ \delta(\Sigma p)\, Z_{G_j}(\ldots p \ldots). \]

In passing to the integral representation (2.15), in order to bring the polynomial \(Z_G\) to exponential form, we shall use the relation

\[ Z_G(\ldots p \ldots)= Z_G(\ldots -i\nabla_q \ldots)e^{i\Sigma(p_a q_a)}\bigm|_{q=0}, \tag{2.23} \]

which is a natural generalization of the formula for \(Z_l(p)\) from \(\Delta_l^\zeta(p)\). It is then obvious that the quadratic form in the exponent of \(Z_G\), as also of \(Z_l\), will not depend on this.

Thus, as a result of applying the operation \(\Delta(G_j)\), we again obtain an expression of type (2.17), not containing the variables \(\alpha_l\) corresponding to the internal lines of \(G_j\).

The “new” quadratic exponential form is obtained from the “old” one by setting equal to zero all \(\alpha_l\) corresponding to the internal lines of \(G_j\). At the same time, the momenta \(k_a\) corresponding to the vertices of the diagram joined into the generalized vertex \(G_j\) automatically drop out of the quadratic form. It is also obvious that the indicated \(\alpha_l\) will not enter the pre-exponential factor \(F(\ldots k \ldots \alpha \ldots)\).

For the analysis of the singularity we shall now proportionally decrease all \(\alpha_l\), putting

\[ \alpha_l=\xi_l\lambda;\qquad \sum_j \xi_j=1;\qquad \xi_j>0;\ \lambda\to 0, \]

and determine the maximal effective degree of the pole at \(\lambda=0\) in the resulting integral.

Somewhat earlier it was established that the effective degree of the pole automatically turns out to be equal to one half of the conditional degree of growth plus one, and that the conditional degree of growth is not changed under the action of the operation \(\Delta(G_j)\). Consequently, in the present case also the effective degree of the pole is equal to

\[ \frac{\omega(G)}{2}+1. \]

A completely analogous conclusion can be drawn about the invariance of the effective degree of the pole under the action of the operation

\[ \Delta(G_1)\ldots \Delta(G_m). \]

Having established this important property, we proceed to a concrete choice of the operation \(\Delta(G)\). Let us first consider the case when the masses of the particles of all the fields under consideration \(m_l\) are greater than zero.

As we have just found, the effective degree of the pole of the expression

\[ \Delta(G_1)\ldots\Delta(G_m)J(k)=\delta(\Sigma k)I(\ldots k\ldots) \]

in the \(\alpha\)-representation is equal to

\[ \frac{\omega(G)}{2}+1. \]

On the other hand, it follows from formula (2.18) that each differentiation with respect to the components of the momenta \(k_\alpha\) lowers the degree of the pole by \(1/2\). Therefore, taking a partial derivative of order \(N\) with respect to the components \(k_\alpha\),

\[ \frac{\partial^N I(\ldots k\ldots)} {\partial^{N_1}k_1\ldots \partial^{N_m}k_m} \qquad (\Sigma N_i=N), \]

we obtain an expression in which the effective degree of the pole is reduced by \(N/2\) and turns out to be equal to

\[ \frac{\omega(G)-N}{2}+1. \]

Choosing \(N=\omega(G)+1\), we obtain under the integral the factor \(d\lambda\,\lambda^{-1/2}\). Since, however, all our functions were rational functions of the variables \(\alpha\), they must also be rational in \(\lambda\). Therefore the factor \(\lambda^{-1/2}\) is in fact absent.

Let us now note that if from the function

\[ I(\ldots k\ldots) \]

one subtracts the sum of all the first terms of its expansion in a Maclaurin series up to and including the terms of order \(\omega(G)\),

\[ \{I(\ldots k\ldots)\}_{\omega(G)}, \]

then the remainder

\[ I(\ldots k\ldots)-\{I(\ldots k\ldots)\}_{\omega(G)} \tag{2.24} \]

is expressed, by the well-known Schlemihl formula, in the form of an integral of partial derivatives of order \(\omega(G)+1\). Thus, in expression (2.24) the effective degree of the pole at \(\lambda=0\) is equal to zero.

We now define the operation \(\Delta(G)\) as follows:

\[ \Delta(G)=-M(G)\left\{1+\sum_{\substack{2\le m\le n-1\\ G=G_1\cdots G_m}} \frac{1}{m!}\Delta(G_1)\ldots\Delta(G_m)\right\}, \tag{2.25} \]

where the operation \(M(G)\) is defined by the relation

\[ M(G)\bigl[\delta(\Sigma k)F(k)\bigr] = \delta(\Sigma k)\{F(k)\}_{\omega(G)}. \]

On the basis of (2.9) we have:

\[ R(G)J_M(k)=\{1-M(G)\}J_M'(k), \]

where

\[ J'_M(k)=\left(1+\sum_{\substack{2\leq m\leq n-1\\ G=G_1\cdots G_m}} \frac{1}{m!}\,\Delta(G_1)\cdots \Delta(G_m)\right)J^{(k)} \]

and, since in \(J'_M\) the singularity in \(\lambda\) is no higher than in \(J_M\), \(R(G)J_M\) has no singularity at all at the point \(\lambda=0\).

It has thus been proved that the application of the last operation \(\Delta(G)\) eliminates the singularities under the proportional tending of all \(a\) to zero. Since, at the same time, the operations \(\Delta(G_1),\ldots,\Delta(G_m)\) compensate the singularities in the region where only some of the \(a\) tend to zero, it is clear that in \(R(G)J_M\) there are no singularities at all. This last assertion constitutes the content of an important theorem, whose proof, with all the necessary estimates, was recently carried out by Parasiuk \(^{8,9,10}\).

The final result of the theorem states that, after application of the operation \(R(G)\), the function \(J_M(k)\) can be represented in the form

\[ R(G)J_M(k)=\delta(\Sigma k)\int_0^\infty da_1\cdots \int_0^\infty da_L \times \]

\[ \times \prod_{1<l<L}\{I(a_l)\} e^{-i\sum_n a_n m_n^2-\varepsilon\sum_n a_n} f(\ldots,k,\ldots,a,\ldots), \tag{2.26} \]

where the function \(f(\ldots,k,\ldots,a,\ldots)\) is expressed as a sum of terms of the form

\[ F(\ldots,k,\ldots,a,\ldots)e^{\,i\sum A_{ab}(\ldots \alpha\ldots)k'_a k'_b}. \]

Here \(k'_a\) are equal either to \(k_a\) or to zero, and \(F\) satisfies the estimate\(^*\)

\[ \left|F(\ldots,k,\ldots,a,\ldots)\right|\leq \frac{c}{\prod_l\left(a_l^{\,1-\frac{1}{2L}}\right)}, \tag{2.27} \]

\(^*\) The estimate (2.27) cited here, as is always the case in similar situations, is overestimated, but nevertheless it is quite sufficient for our purposes, since it ensures absolute integrability of the function \(F\) in a neighborhood of the points \(a_l=0\). The fact that in the estimate (2.27) it has not been possible to avoid the occurrence of \(a_l\) in the denominator is connected not only with the roughness of the estimating method, but also with the circumstance that, under a nonproportional tending of the various \(a_l\) to zero, when some of them tend faster than others, the character of the growth of the function may change.

where \(L\) is the number of all internal lines of the diagram \(G\), while \(c\) is bounded polynomially in the variables \(k\) and \(\alpha\).

From the estimate (2.27) it follows directly that in formula (2.26) one may pass to the limit \(M \to \infty\) for fixed \(\varepsilon > 0\). Indeed, the factor \(e^{-\varepsilon \sum_l \alpha_l}\) makes the integral (2.26) absolutely convergent for large \(\alpha\), while as \(\alpha \to 0\) the possible singularities, according to (2.27), are integrable.

Thus, in the process of removing the regularization, setting the factors \(I(\alpha)\) equal to unity, we obtain that \(R(G)J_M\) tends, in the ordinary sense, to the expression

\[ R(G)J(k)=\delta(\Sigma k)\int_0^\infty d\alpha_1\ldots \int_0^\infty d\alpha_L e^{-i\sum_l \alpha_l m_l^2-\varepsilon\sum_l \alpha_l} f(\ldots k\ldots \alpha\ldots). \tag{2.28} \]

Let us now consider the analyticity properties of the expression obtained and the possibility of passing to the limit \(\varepsilon \to 0\). Rotating the axes of the variables \(\alpha\) by \(90^\circ\) in the complex plane, i.e. making the change of variables

\[ \alpha_k \to -i\beta_k, \tag{2.29} \]

we write the integral (2.28) in the form

\[ \int_0^\infty d\beta_1\ldots \int_0^\infty d\beta_L e^{-\sum_l \beta_l m_l^2+i\varepsilon\sum_l \beta_l+\sum A_{ab}(\ldots \beta\ldots)k'_a k'_b} \times \]

\[ \times F(\ldots,k,\ldots,\beta\ldots). \tag{2.30} \]

Representing the exponent as follows:

\[ A=\sum A_{ab}(\ldots \beta\ldots)(k_a^0 k_b^0)- \]

\[ -\sum A_{ab}(\ldots \beta\ldots)(\mathbf{k}_a\mathbf{k}_b) -\sum_l \beta_l m_l^2+i\varepsilon\sum_l \beta_l, \]

we find, with the aid of condition (2.20),

\[ A<\sum_l \beta_l\left(\sum_a k_a^0\right)^2-\sum_l \beta_l m_l^2+i\varepsilon\sum_l \beta_l. \]

It is now obvious that, when the condition

\[ \left(\sum_a k_a^0\right)^2<\min m_l^2 \tag{2.31} \]

is fulfilled, the form \(A\) is negative, and the integral (2.30) is absolutely

convergent, and the transformation (2.29) is legitimate. In the expression obtained one can pass to the limit as \(\varepsilon=0\) and obtain a function analytic in the domain defined by the inequality (2.31).

This inequality can be written in a relativistically invariant form

\[ \left|\sum_a (k_a \xi)\right|^2 < \min m^2, \]

where \(\xi\) is a timelike unit vector directed into the future,

\[ \xi^2=1; \qquad \xi^0>0. \]

In the procedure described, we first let \(M\) tend to infinity and then carried out the limiting transition \(\varepsilon \to 0\). However, the rotation operation (2.29) could have been performed here even before passing to the limit \(M \to \infty\). We would then have obtained a linear combination of integrals of type (2.30), including the masses \(M_i\). When the limiting transitions \(M \to \infty\) and \(\varepsilon \to 0\) are performed simultaneously, the terms containing the masses \(M_i\) tend to zero, and we arrive at the very same results.

Thus, in the case considered, the point \(k_a=0\) proves to be regular and Maclaurin series expansions are admissible. This circumstance is due to the fact that, by assumption, all masses \(m_i\) are essentially positive and

\[ \min m_i^2 > 0. \]

In the case when some of the masses are equal to zero, the point \(k_a=0\) may fail to be regular. In such a case, in defining the operation \(\Delta(G)\), the choice of the center of expansion at the point \(k_a=0\) may prove inadmissible, and it must be placed at some point \((k^0,\mathbf{k})\) with purely imaginary time component

\[ k^0=i\omega . \]

Since, however, the selection of any point in momentum space (except the point \(k=0\)) is not invariant from the point of view of four-dimensional rotations, the corresponding polynomial must also be averaged over the sphere

\[ \omega^2+\mathbf{k}^2=\mu^2, \qquad \text{where } \mu^2=\text{const}. \]

With such a choice of the operation \(\Delta(G)\), the conclusions made above concerning the properties of the coefficient functions obtained as a result of the operation \(R(G)\) are evidently preserved, with the difference, however, that analyticity will hold only for points \(k\) with purely imaginary \(k_0\).

Let the momenta \(k\) now be completely arbitrary. So long as the masses have finite, purely imaginary negative additions \(-i\varepsilon\), the integrals contain cutoff factors \(e^{-\varepsilon \sum_l \alpha_l}\), and the functions are

regular. As $\varepsilon$ tends to zero, these functions converge only in an improper sense. The limiting expressions thus obtained, which are the true coefficient functions of the $T'$-products, turn out to be improper and may have singularities in certain regions of the values of their arguments.

However, by virtue of the integrability properties, the operator integrals

$$ \int T'\,[L(x_1)\ldots L(x_n)]\,g(x_1)\ldots g(x_n)\,dx_1\ldots dx_n $$

for sufficiently regular functions $g(x)$ that decrease sufficiently rapidly at infinity turn out to be convergent, and difficulties arise here with the limiting transition $g(x)\to 1$. The corresponding matrix elements of $S(1)$ turn out to be divergent. In these cases one usually speaks of divergences of the type of an infrared catastrophe or resonance denominators.

Singularities of the infrared-catastrophe type arise, as is known, because of the inapplicability of the perturbation-theory method in the description of processes with quanta of very small energy and may be eliminated from the results by the Bloch–Nordsieck method$^{11}$ or by introducing into the photon $D_c$-functions a small constant playing the role of a “photon mass.”

Singularities of the resonance-denominator type arise, for example, in the case when a high-order scattering process, for given values of the momenta, can be reduced to simpler independent processes of lower order.

It should be noted that divergences of these two types also appear in ordinary quantum mechanics in those cases where the use of perturbation theory is illegitimate. Taking this opportunity, let us emphasize that the condition of integrability of the operator functions $S_n(x_1,\ldots,x_n)$ guarantees only the absence of divergences specific to quantum field theory—divergences “at large momenta.”

We formulated above a prescription for constructing integrable coefficient functions for the operators $S_n(x_1,\ldots,x_n)$. Let us now note that the introduction of regularized causal functions

$$ \Delta^c \to \operatorname{reg}[\Delta^c] $$

had, in our reasoning, a purely auxiliary character and was in fact needed by us only in order to make sure that the expressions obtained for $S_n$ satisfy all the conditions imposed on them.

In practice, for example, when computing the indicated coefficient functions, it is quite possible to work with the true $\Delta^c$-functions. Passing then to the $\alpha$-representation, one can apply the operation

\(R(G)\), excluding from the domain of integration over \(\alpha\) only a small region near the point \(\alpha = 0\).

Our prescription for constructing the operation \(\Delta(G)\) contains a certain arbitrariness, consisting in the choice of the center of expansion at the point \(k=0\) (in the case when all \(m_i \ne 0\)) or in fixing the “radius” of the averaging 4-sphere \(\mu\) (in the case when some of the \(m_i=0\)). Therefore the operation \(\Delta(G)\) must be somewhat generalized.

To this end let us consider the system of finite polynomials

\[ Z'_G(\ldots k \ldots) \]

of degree not higher than \(\omega(G)\), such that the expressions

\[ \delta\!\left(\sum k\right) Z'_G(\ldots k \ldots) \]

are momentum representations of the coefficient functions of certain Hermitian covariant quasilocal operators \(\Lambda'(\ldots x \ldots)\). The most general expression for the operation \(\Delta(G)\) can now be obtained by defining it as the sum of the operation introduced earlier (2.25) and the operation of adding the polynomial \(Z'_G\), i.e. in the form

\[ \Delta(G)J_G(\ldots k \ldots) = \delta\!\left(\sum k\right) Z'_G(\ldots k \ldots) - M(G)\left\{1+ \sum_{\substack{2 \le m \le n-1\\ G=G_1*\cdots*G_m}} \frac{1}{m!}\,\Delta(G_1)\ldots \Delta(G_m) \right\}J_G(\ldots k \ldots). \tag{2.32} \]

The corresponding most general expression for the operator \(S_n\) differs from that obtained earlier by the inclusion of Hermitian covariant quasilocal operators \(\Lambda'_n\). But, as we have already established earlier, the addition to \(S_n\) of certain \(\Lambda_n\) can be taken into account by changing the effective interaction Lagrangian. Thus, changing the prescription for constructing integrable coefficient functions is equivalent to adding new finite counterterms to the effective Lagrangian. One may therefore consider that the \(T'\)-product is defined as before, but that there is an arbitrariness in the choice of the Lagrangian. This arbitrariness consists in the possibility of including in \(L(x)\) terms which, according to a formula of type (1.4.35), correspond to quasilocal operators connected with generalized, strongly connected nodes \(G\) of nonnegative index \(\omega(G)\).

Thus, in order to obtain the operator functions \(S_n(x_1,\ldots,x_n)\) we have obtained the following general prescription. Having taken the coefficient function of the ordinary \(T\)-product for some \(G\), one transforms it to the “\(\alpha\)-representation.” Temporarily excluding an infinitely small region near the points \(\alpha_i=0\), one applies the operation

\[ R(G)=1+ \sum_{\substack{2 < m \le n-1\\ G=G_1*\cdots*G_m}} \frac{1}{m!}\,\Delta(G_1)\ldots \Delta(G_m) +\Delta(G), \]

where \(\Delta(G_j)\) contains terms \(Z'_{G_j}\), after which the region near the points \(\alpha_i=0\) is included in the integral. The improper limiting transition \(\varepsilon \to 0\) in the expressions obtained then gives the required coefficient functions of the operators \(S_n\).

§ 3. CLASSIFICATION OF THE RENORMALIZABILITY OF THEORIES

As we established in the preceding section, the change in the effective interaction Lagrangian by the introduction of counterterms into it is due to strongly connected diagrams \(G\) with nonnegative index \(\omega(G)\).

Let us consider the connection between the structure of such diagrams and the structure of the corresponding counterterms in the \(x\)-representation.

If a given strongly connected diagram connects \(n\) vertices and has \(s\) external lines, then the corresponding quasilocal operator will have the form:

\[ : u_{a_1}(x_{i_1}) \ldots u_{a_s}(x_{i_s}) : Z \left( \ldots \frac{\partial}{\partial x} \ldots \right) \delta(x_1-x_2)\ldots \delta(x_{n-1}-x_n), \]

where the degree of the polynomial \(Z\) is equal to the index of the diagram \(\omega(G)\). Then, integrating over all variables \(x_i\), except one, we obtain a counterterm of the Lagrangian

\[ \int dx_2 \ldots dx_n : u_{a_1}(x_{i_1}) \ldots u_{a_s}(x_{i_s}) : Z \left( \ldots \frac{\partial}{\partial x} \ldots \right) \times \]

\[ \times \delta(x-x_2)\delta(x_2-x_3)\ldots \delta(x_{n-1}-x_n). \]

When carrying out the trivial integrations contained here, the derivatives of the \(\delta\)-functions pass to the field operators \(u\), and the result of the integration is represented in the form of a normal product of a certain number of operators of field functions and their derivatives. In this case the total degree of the derivatives proves to be equal to the index of the diagram \(\omega(G)\), and the degree of “linearity” of the entire expression in the operator functions is equal to the number of external lines \(s\).

Therefore, if some given theory (completely determined by the basic “seed” term of the interaction Lagrangian and by the structure of the causal functions) leads to strongly connected diagrams of nonnegative index for which the numbers \(\omega(G)\) and \(s\) turn out to be bounded, then, in order to eliminate all divergences completely, such a theory requires the introduction of counterterms of a finite number of types. By the type of a counterterm we mean here its operator type and the degree of derivatives at each field operator. In the opposite case the number of types of counterterms proves to be infinite.

We shall now analyze the dependence of the index \(\omega(G)\) on the number of external and internal lines of the diagram.

Introduce for this purpose the notion of the index of a vertex, defining it by the equality

\[ \omega_i=\frac{1}{2}\sum_{l_{\mathrm{int}}}(r_l+2)-4, \tag{3.1} \]

where the summation is over all internal lines entering the \(i\)-th vertex. It is not hard to see that the index of a diagram is expressed in terms of the indices of the vertices entering the diagram as follows:

\[ \omega(G)=\sum_{1\leq i\leq n}\omega_i+4, \tag{3.2} \]

since each internal line enters two vertices simultaneously. For a given type of vertices the index \(\omega_i\) assumes its maximum value \(\omega_i^{\max}\) in the case when all lines entering the vertex are internal.

It is now obvious that if

\[ \omega_i^{\max}\leq 0, \tag{3.3} \]

then it follows from (3.2) that

\[ \omega(G)\leq 4. \]

Conversely, if for some types of vertices

\[ \omega_i^{\max}>0, \tag{3.4} \]

then one can always construct a diagram \(G\) containing a sufficient number of vertices of this type so that \(\omega(G)\) becomes greater than any preassigned number. Thus, either the index of a diagram does not exceed four, or it can be made arbitrarily large.

Taking into account that

\[ \omega_i=\omega_i^{\max}-\frac{1}{2}\sum_{l_{\mathrm{ext}}}(r_l+2), \]

where \(l_{\mathrm{ext}}\) are the indices of the external lines entering the given vertex, the dependence of \(\omega(G)\) on the number of external lines can be written in the form

\[ \omega(G)=\sum_i \omega_i^{\max}+4-\frac{1}{2}\sum_{l_{\mathrm{ext}}}(r_l+2), \tag{3.5} \]

where the summation in the last term is over all external lines of the given diagram.

Therefore, in the case (3.3), for a diagram with positive index the number of external lines does not exceed four. In this case both quantities \(\omega(G)\) and \(s\) are bounded by the number 4, the number of types

of the corresponding counterterms turns out to be finite and can be subjected to a detailed classification. In the opposite case (3.4), both sums on the right-hand side of formula (3.5) can be made arbitrarily large for nonnegative \(\omega(G)\). Both characteristics \(\omega(G)\) and \(s\) turn out to be unbounded, and in order to compensate divergences of increasing orders one has to introduce counterterms with an increasing degree of “linearity” and an increasing number of derivatives. It is not possible to obtain a closed expression for the complete effective Lagrangian.

In accordance with the properties indicated, the types of interactions can be divided into two classes\({}^{12}\):

a) interactions of the first kind (all \(\omega_i \leqslant 0\)),

b) interactions of the second kind (some of the \(\omega_i > 0\)).

The corresponding theories are called “renormalizable” and “nonrenormalizable.”

Concerning this definition, only one essential reservation must be made. The point is that in some cases individual vertex factors may compensate one another and thereby lower the effective value.

Consider, for example, the interaction of an ordinary fermion field (spin one half) with a neutral vector meson field of the type

\[ \sum_n :\bar{\psi}(x)\gamma^n\psi(x)u_n(x): . \tag{3.6} \]

In a direct calculation of \(\omega_i\), using the fact that the contraction of the vector field in the \(p\)-representation has the form

\[ i\,\frac{g^{mn}-\dfrac{k^m k^n}{m^2}}{m^2-k^2} \]

and, consequently, \(r_l\) for it is equal to two, we shall obviously obtain

\[ \omega_i = 1 \]

and shall assign the Lagrangian \((r_l)\) to the nonrenormalizable type.

It can be shown, however, that in reality the Lagrangian (3.6) describes an interaction of the first kind. To see this, decompose the vector field, following Stueckelberg\({}^{13}\), into transverse and longitudinal parts

\[ u_n(x)=U_n(x)+\frac{1}{m}\frac{\partial B}{\partial x^n} \]

with the corresponding contractions

\[ \overline{U_n(x)U_m(y)} \sim i g^{mn}\frac{1}{m^2-k^2}, \]

\[ \overline{\frac{\partial B(x)}{\partial x^n}\frac{\partial B(y)}{\partial y^m}} \sim \frac{i k^m k^n}{k^2-m^2}. \]

Let us now recall that the scattering matrix \(S\) depends not on the Lagrangian density \(L(x)\), but on the integral

\[ \int L(x)\,dx \]

as a whole. Substituting into it the expansion that has been made, we have:

\[ \int L(x)\,dx = \]

\[ = \sum_n \int :\bar{\psi}(x)\gamma^n\psi(x)U_n(x):\,dx +\frac{1}{m}\sum_n \int :\bar{\psi}(x)\gamma^n\psi(x)\frac{\partial B}{\partial x^n}: \, dx. \]

Using the conservation law (more precisely, the continuity equation) for the spinor current, which is not violated in the presence of neutral mesons,

\[ \sum_n \frac{\partial}{\partial x^n}:j^n(x): = \sum_n \frac{\partial}{\partial x^n}:\bar{\psi}(x)\gamma^n\psi(x):=0, \]

by integrating the second term by parts we find that it is equal to zero, i.e.

\[ \int L(x)\,dx = \sum_n \int :\bar{\psi}(x)\gamma^n\psi(x)U_n(x):\,dx. \]

The longitudinal field \(\dfrac{\partial B}{\partial x^n}\) therefore in fact drops out of the \(S\)-matrix; the effective values of \(r_l\) and of the vertex index \(\omega_i\) are lowered to zero, and the Lagrangian (3.6) turns out to be renormalizable.

Thus, without considering here in detail these very special cases of compensation of singularities, due by their existence to a certain group of transformations, we shall now give a general classification of the simplest interaction Lagrangians.

When establishing the corresponding indices \(\omega_i^{\max}\), we shall take into account that, in accordance with the general structure of commutation and causal functions, the degree of the polynomial \(r_l\) for a scalar field and a vector field of zero mass (electromagnetic) with scalar coupling is equal to zero; for a spinor field of spin one half it is equal to one; for a vector field (for \(m\ne 0\)) with scalar coupling it is equal to two. In the case of gradient coupling, with the aid of formula (I.5.10) we find that \(r_l\) for scalar and electromagnetic fields is equal to two, and for a vector field to four.

As was noted, the number of external lines in strongly connected diagrams with non-negative index cannot be greater than four. Therefore the maximal degree of linearity of an interaction Lagrangian of the first kind is equal to four. From the formula

\[ \omega_i^{\max}=\frac{1}{2}\sum_l (r_l+2)-4, \tag{3.7} \]

where the summation is taken over all lines emerging from the given vertex, we find that in the case of a Lagrangian containing four operators, all four lines must have index \(r_i=0\), i.e., only quartic products of scalar fields and electromagnetic fields

\[ :\varphi_{a_1}\varphi_{a_2}\varphi_{a_3}\varphi_{a_4}:,\qquad :\varphi_{a_1}\varphi_{a_2}A_kA_k:,\qquad :A_kA_kA_lA_l: \tag{3.8} \]

describe interactions of the first kind \((\omega_i=0)\).

All other quartic interactions, such as, for example, the quartic Fermi interaction of spinor operator functions\(^*\)

\[ (\bar{\psi}_{a_1}O\psi_{a_2})(\bar{\psi}_{a_3}O\psi_{a_4})\qquad (\omega_i=2), \]

used for the description of beta processes, as well as quartic interactions including derivatives and vector field functions, are interactions of the second kind.

Cubic terms in a Lagrangian of the first kind, obviously, may have the following structure:

a) the product of three scalar and electromagnetic functions without derivatives

\[ :\varphi_{a_1}\varphi_{a_2}\varphi_{a_3}:,\qquad :\varphi A_kA_k:\qquad (\omega_i=-1) \tag{3.9} \]

(the other combinations do not form a scalar);

b) the product of three scalar and electromagnetic functions with one first derivative

\[ :\varphi_{a_1}\frac{\partial\varphi_{a_2}}{\partial x^k}A_k:\qquad (\omega_i=0). \tag{3.10} \]

The interaction of charged scalar mesons with the electromagnetic field (scalar electrodynamics) is constructed according to this type;

c) the product of one scalar function, one vector function, and one electromagnetic function

\[ :\varphi_{a_1}\varphi^{k}_{a_2}A_k:; \tag{3.11} \]

d) the product of two spinor functions and one scalar or electromagnetic function

\[ :(\bar{\psi}_1O\psi_2)\,\varphi:,\qquad :(\bar{\psi}_1O^k\psi_2)\,A_k:\qquad (\omega_i=0). \tag{3.12} \]

All other cubic interactions, for example interac—

\(^*\) Here the usual matrix contraction is used

\[ (uOv)=\sum_{\alpha,\beta}u_\alpha O_{\alpha\beta}v_\beta . \]

Interaction of a Spinor Field with a Scalar Field of Gradient Coupling Type

\[ :\left(\bar{\psi}_1 O^k \psi_2\right)\frac{\partial \varphi}{\partial x^k}:, \qquad (\omega_i=1), \]

lead to nonrenormalizable theories*).

The nine Lagrangians listed above, (3.8)—(3.12), exhaust the possible types of interactions of the first kind**, since the quadratic forms given below, satisfying condition (3.3), correspond to vertices into which two lines enter. Such formulas do not describe processes of mutual transformation of particles and therefore represent only possible types of counterterms.

The indicated quadratic forms can be represented by the following combinations:

a) pair products of functions of one and the same field without derivatives

\[ \begin{array}{ll} :\varphi\varphi: & (\omega_i=-2); \\[3pt] :(\bar{\psi}O\psi): & (\omega_i=-1); \end{array} \qquad \begin{array}{ll} :A_k A_k: & (\omega_i=-2); \\[3pt] :\varphi^k\varphi^k: & (\omega_i=0); \end{array} \tag{3.13} \]

b) pair products of functions of one and the same field, including up to two derivatives:

\[ \begin{array}{ll} :\dfrac{\partial\varphi}{\partial x^k}\dfrac{\partial\varphi}{\partial x^k}: & (\omega_i=0); \\[10pt] :\dfrac{\partial A_m}{\partial x^m}\dfrac{\partial A_m}{\partial x^m}: & (\omega_i=0); \end{array} \qquad \begin{array}{ll} :\left(\bar{\psi}O^k\dfrac{\partial\psi}{\partial x^k}\right): & (\omega_i=0); \\[10pt] :\left(\dfrac{\partial A_m}{\partial x^m}\right)^2: & (\omega_i=0). \end{array} \tag{3.14} \]

All the remaining quadratic combinations lead to interactions of the second kind.

The division of Lagrangians into interactions of the first and second kind that we have carried out is, in essence, not sufficiently consistent. It was performed from the point of view of the convergence of individual terms in the expansion of the \(S\)-matrix in powers of the coupling constant. It is quite possible that, after summation of the given series has been carried out, the analytic nature of the functions under study will acquire a different character, which will affect the classification made above \(^{14}\).

There therefore arises the problem—without resorting to perturbation theory—of determining which interactions belong to the first kind and which to the second. In other words, the question arises as to for which Lagrangians of local type it is possible to construct a closed theory. The importance of this problem is due to the fact that, as we shall now

*) Not counting Lagrangian (3.6).
**) In fact, their number is further restricted by charge and gradient invariance (see below).

We shall show that between theories of the 1st and 2nd kind there is an essential physical difference.

For this purpose let us dwell in more detail on the properties of interactions of the second kind. As we have already seen, among the infinite number of types of counterterms arising in such theories there are groups of terms of the same operator type, but with infinitely increasing degrees of derivatives. Such series in powers of derivatives in fact represent expansions of certain nonlocal expressions and therefore in essence constitute nonlocal interactions.

For example, the Lagrangian of the pseudovector meson-nucleon coupling

\[ :\bar{\psi}\gamma^5\gamma^k\psi\,\frac{\partial u}{\partial x^k}: \]

requires the introduction into the effective Lagrangian of an infinite number of counterterms of the form

\[ :\bar{\psi}\gamma^5\gamma^k\psi\, \frac{\partial^n}{\partial (x^\alpha)^n} \left(\frac{\partial u}{\partial x^k}\right):, \]

which in the sum may be regarded as the expansion of the nonlocal expression

\[ \int dy\,\bar{\psi}(x)\gamma^5\gamma^k\psi(x)\, \frac{\partial u(y)}{\partial y^k}\,C(x-y) \]

in a series in powers of derivatives of the function \(u\).

Thus, in the case of interactions of the 2nd kind there occurs an actual disappearance of the localizability of the effective Lagrangian, which begins to depend on the behavior of the field functions not only in an infinitely small neighborhood of the point \(x\). It then turns out that, independently of the smallness of the interaction constant, terms of higher orders become essential at sufficiently large momenta. Indeed, from dimensional considerations it follows that a counterterm containing \(n\) derivatives is proportional to the factor \(l^n\), where \(l\) is a small parameter of the dimension of length (“universal length”), related by a power law to the interaction constant[^15]. In the \(p\)-representation the derivatives \(\partial/\partial x\) are transformed into momenta, and we obtain an expansion in the quantity

\[ pl=\frac{l}{\lambda}, \tag{3.15} \]

where

\[ \lambda=\frac{1}{p} \]

is the Compton wavelength. For sufficiently large \(p\) the Compton wavelength \(\lambda\) becomes comparable with the universal length \(l\), and the expansion parameter (3.15) of the nonlocal Lagrangian in a series

QUESTIONS OF QUANTUM FIELD THEORY

with respect to derivatives ceases to be small. The universal length \(l\) in this case is a characteristic of the physical extension of the particle, and its appearance signals the importance of the influence of the particle’s internal structure.

We arrive at the conclusion that Lagrangians of interactions of the second kind are “fragments” of nonlocalized interactions, represented as though in localized form. For the consistent construction of such theories it is necessary from the very beginning to start from a nonlocal Lagrangian that takes into account the internal structure of elementary particles. It follows from this that there is a profound physical difference between interactions of the first and second kinds. The question becomes essential: “Do all interactions realized in nature belong to the first kind?”\(^{16}\).

Let us emphasize that a sufficiently well-founded answer to this question can be obtained by comparing with experiment the results of calculations not based on perturbation theory.

At the present time it is by no means clear, for example, whether the interactions which we have assigned, on the basis of perturbation theory, to the first kind are in fact interactions of the first kind.

If the answer to the question posed proves to be affirmative, then it should be considered that field theory already possesses methods that make it possible to remove divergences unambiguously and to obtain unambiguous finite results. At present, of course, subtraction techniques are based on perturbation theory, but their application in procedures not resorting to perturbation theory, or resorting to it only partially, apparently causes no particular difficulties\(^{17,18}\).

The presence of infinities in this case is, of course, connected with the assumption of the exact character of elementary particles, and the formal procedure for eliminating infinities is in fact a comparatively small price that has to be paid for abandoning a detailed investigation of the internal structure of elementary particles. The success of the subtraction method in this case indicates that the details of the internal structure of elementary particles do not play a major role in interactions of the first kind.

In the opposite case, if at least some true interactions belong to the second kind, the question of the internal structure of elementary particles and of the consistent description of their interactions by means of nonlocal Lagrangians becomes important.

At the same time, in view of the mutual connectedness of all particles, it will turn out that even if some interactions between them, taken in isolation, belong to the first kind, the complete description of any particles must necessarily be nonlocal.

Since at present there in fact exist no methods for investigating nonlocal interactions in quantum theory

fields, we shall not consider this second possibility at all here. We shall therefore assume, although perhaps this is not the case, that all existing interactions belong to the 1st kind.

Let us now proceed to a more detailed analysis of the possibilities for constructing theories of the 1st kind.

As was recently established, the possible types of terms of the effective interaction Lagrangian are restricted to three fourth-order forms (3.8), six cubic forms (3.9)—(3.12), and eight quadratic forms (3.13)—(3.14).

The most general form of the effective Lagrangian therefore has the form

\[ \begin{aligned} L(x)={}&\sum_{(a)} A_{a_1a_2a_3a_4}:\varphi_{a_1}\varphi_{a_2}\varphi_{a_3}\varphi_{a_4}: \\ &+\sum_{(a,k)} B_{a_1a_2}:\varphi_{a_1}\varphi_{a_2}A_kA_k:g^{kk} +C\sum_{(k,l)} g^{kk}g^{ll}:A_kA_kA_lA_l: \\ &+\sum_{(a)} D_{a_1a_2a_3}:\varphi_{a_1}\varphi_{a_2}\varphi_{a_3}: +\sum_{(a,k)} E_a g^{kk}:\varphi_a A_kA_k: \\ &+\sum_{(a,k)} F_{a_1a_2}:\varphi_{a_1}\frac{\partial\varphi_{a_2}}{\partial x^k}A_k:g^{kk} +\sum_{(a,k)} G_{a_1a_2}:(\bar{\psi}_{a_1}O^k\psi_{a_2})A_k: \\ &+\sum_{(a,k)} H_{a_1a_2}:\varphi_{a_1}\varphi_{a_2}^{k}A_k: +\sum_{(a)} K_{a_1a_2a_3}:(\bar{\psi}_{a_1}O\psi_{a_2})\varphi_{a_3}: \\ &+L\sum_{m,n} g^{mm}g^{nn}:\frac{\partial A_m}{\partial x^m}\frac{\partial A_n}{\partial x^n}: +M\sum_{m,n} g^{mm}g^{nn}:\frac{\partial A_m}{\partial x^n}\frac{\partial A_m}{\partial x^n}: \\ &+\sum_a N_a:(\bar{\psi}_aO\psi_a): +\sum_a P_a:\varphi_a\varphi_a: +Q\sum_k g^{kk}:A_kA_k: \\ &+\sum_{a,k} R_a g^{kk}:\varphi_a^k\varphi_a^k: +\sum_{a,k} W_a:\bar{\psi}_aO^k\frac{\partial\psi_a}{\partial x^k}: \\ &+\sum_{a,k} V_a g^{kk}:\frac{\partial\varphi_a}{\partial x^k}\frac{\partial\varphi_a}{\partial x^k}: . \end{aligned} \tag{3.16} \]

Therefore, in theories of the 1st kind the choice of the interaction Lagrangian reduces to the choice of a finite number of “coupling constants”

\[ A,\ B,\ C,\ldots,W,\ V. \tag{3.17} \]

The number of independent ones among them is reduced as a result of taking into account the requirements of gradient invariance, charge conservation, and inva-

invariance with respect to charge conjugation. In order to fix the theory completely, it is, of course, still necessary to specify the particle masses in the absence of interaction. Thus, any theory of the first kind is completely characterized by a finite set of numbers: the masses of fictitious noninteracting particles and the coupling constants.

In our version of the theory, the basic quantity, in addition to the Lagrangians of free particles, which fix the properties of the noninteracting fields, is the interaction Lagrangian \(L(x)\).

As we have seen, by choosing \(L(x)\) in the usual way, by redefining the \(T\)-product one can obtain integrable expressions for the terms of the \(S\)-matrix:

\[ S_n(x_1,\ldots,x_n)=i^n T' [L(x_1),\ldots,L(x_n)]. \]

Although the prescription for constructing the operator \(T'\) is not unique, the arbitrariness it contains corresponds to a finite change of the coupling constants in the interaction Lagrangian, i.e. to the introduction into the original Lagrangian \(L(x)\) of finite counterterms of the same type as the terms entering expression (3.15). Thus, for complete unambiguity of the calculations it is necessary to specify \(L(x)\) in relation to a fixed prescription for constructing the \(T'\)-product.

On the other hand, as we have seen, an entirely equivalent result is obtained if, instead of redefining the \(T\)-product, one uses the ordinary \(T\)-product (with one or another auxiliary regularization in the intermediate reasoning), but, in place of \(L(x)\), employs a certain effective interaction Lagrangian \(L_{\mathrm{eff}}\), containing, in addition to the original interaction Lagrangian, divergent counterterms of the form (3.15), which compensate the divergences in the ordinary \(T\)-products.

Thus, from the point of view of the \(S\)-matrix, matters stand as though, instead of the original full Lagrangian

\[ L_0(x)+L(x) \]

we have the Lagrangian

\[ L_{\mathrm{full}}=L_0+L_{\mathrm{eff}}. \]

The “masses,” “charges,” etc. entering into it, i.e. the coefficients of the corresponding operator combinations, diverge; however, the observable quantities computed with their aid (including masses, charges, etc.) have finite values.

We thus arrive in this way at the so-called “renormalization” point of view, according to which, in order to obtain finite values for the calculated observable quantities, infinite “bare” masses, charges, etc. are introduced into the Lagrangian. It is then said that the removal of infinities from the theory is achieved by means of

“renormalization” of the fundamental constants. The renormalization factors contain divergent expressions.

This point of view, however, is not pursued sufficiently consistently, since in \(L_{\mathrm{full}}\) one has to introduce such counterterms whose appearance is not reducible to the renormalization of the fundamental quantities (for example, in the case of spinor electrodynamics, explicitly non-gradient-invariant photon-mass terms*) and the term \((\partial A_m/\partial x^m)^2\), quartic terms in meson theories, etc.).

Moreover, when considering the generalized scattering matrix \(S(g)\), as we saw in § 2, the structure of the counterterms changes and turns out to depend on the behavior of the function \(g(x)\). As will be shown below, it is precisely the matrix \(S(g)\) that determines the effective Hamiltonian of the system. One may therefore say that, in regularizing the Schrödinger equation, it is necessary to introduce counterterms different from those required in regularizing the \(S\)-matrix. Finally, the concrete expressions for the divergent coefficients in the counterterms depend on the method of regularization.

In view of all this, we do not adhere to the “renormalization” terminology, regarding the procedure of introducing counterterms as a formal device ensuring the finiteness of the results of calculations.

§ 4. GENERAL FORM OF THE COUNTERTERMS OF SPINOR ELECTRODYNAMICS

As an example of interacting quantized wave fields, let us consider the practically important case of spinor electrodynamics, i.e. the system of interacting vector electromagnetic and fermion spinor fields with the interaction Lagrangian

\[ L(x)=\sqrt{4\pi}\, e:\bar{\psi}(x)\,\hat{A}(x)\psi(x). \tag{4.1} \]

Let us recall that, in accordance with the structure of expression (4.1), in Feynman diagrams at each vertex there occur two fermion lines and one photon line, while the causal functions of the participating fields have the form (cf. (I.5.7) and (I.5.8))

\[ \frac{1}{i}\,\overline{A_m(x)A_n(y)} = g^{mn}D_0^c(x-y) = -\frac{g^{mn}}{(2\pi)^4}\int \frac{dk\,e^{ikx}}{k^2+i\varepsilon}, \tag{4.2} \]

\[ i\,\overline{\psi(x)\bar{\psi}(y)} = S^c(x-y) = \frac{1}{(2\pi)^4}\int dp\,e^{-ipx}\frac{m+\hat{p}}{m^2-p^2 i\varepsilon}. \tag{4.3} \]

Therefore the degree of the polynomial \(P(k)\) in the numerator of the causal function for a photon line is equal to zero, while for a fermion line

*) With a non-gradient-invariant method of regularization, similar to that used in § 1 of the present article.

is equal to unity. The maximal vertex index, determined by the formula

\[ \omega_i^{\max}=-\frac{1}{2}\sum_l (r_l+2)-4, \tag{4.4} \]

turns out to be equal to zero, and the Lagrangian (4.1) therefore belongs to the renormalizable type.

We shall now carry out a classification of divergent diagrams, basing ourselves on formula (3.5). First of all, we note that, because the maximal vertex index \(\omega_i^{\max}\) is equal to zero, the index of a diagram \(\omega(G)\) does not depend on the number of vertices and turns out to depend only on the number and the character of the external lines,

\[ \omega(G)=4-\frac{1}{2}\sum_{l_{\mathrm{nar}}}(r_l+2). \tag{4.5} \]

As was indicated in the preceding paragraph, the maximal number of external lines in divergent diagrams cannot exceed four. It follows from (4.5) that the only diagram of this kind in spinor electrodynamics is the diagram with four external photon lines. The index of this diagram turns out to be equal to zero.

Let us pass to diagrams with three external lines. Because of the continuity property of spinor lines, the number of external spinor lines is always even. It is therefore sufficient to consider diagrams with three external photon lines and diagrams with two external fermion lines and one photon line. The total contributions to matrix elements from diagrams with an odd number of external photon lines, in the absence of external spinor lines, turn out to be equal to zero on the basis of Furry’s theorem, stated below in this paragraph. A diagram with two external spinor lines and one photon line has index \(\omega(G)\) equal to zero.

A diagram with two external spinor lines has index equal to unity, while one with two external photon lines has index equal to two.

Diagrams with one external fermion line are absent because of the continuity principle for fermion lines, while diagrams with one external photon line are forbidden by Furry’s theorem.

This completes the enumeration of divergent diagrams. Before turning to the study of the corresponding quasilocal operators, we shall prove the Furry theorem mentioned above.

First of all, let us note that the general classification of divergent diagrams carried out by us does not take into account the symmetry and invariance properties of the system with respect to various transformations.

The fulfillment of the indicated properties leads to significant restrictions on the possible types of diagrams and, as we shall see below, establishes interrelations between the structure of the regularizing quasi-local operators corresponding to different divergent diagrams. The continuity of spinor lines noted above, for example, is essentially a reflection of the conservation property of the electric charge of fermions.

An important restriction on the possible types of diagrams is imposed by the property of charge invariance, i.e., invariance with respect to a change of sign of the electric charge in processes in whose initial and final states there are no electrically charged fermion particles. Such processes are precisely described by diagrams in which all external lines are photons. A change in the sign of the charge in the charge-invariance transformation consists here, obviously, in changing the sign of the charge of virtual fermions in intermediate states. The motion of the indicated virtual particles in Feynman diagrams is described by closed spinor loops.

Furry’s theorem[^19] consists in the assertion that, for the corresponding diagrams \(G\) containing at least one odd closed spinor loop, the matrix elements cancel mutually. Let us consider such a diagram \(G\), containing an odd closed loop \(L\). It is clear that the matrix element corresponding to this diagram will consist of the sum of two terms, one of which corresponds to the motion of charge in \(L\) clockwise, and the other to the motion of charge in the opposite direction. As we shall now show (the proof is due to Feynman[^20]), the indicated terms differ only by signs and therefore give zero in the sum.

The factor of the matrix element corresponding to the closed loop \(L\), containing \(n\) vertices, according to Feynman’s rules has the form

\[ \operatorname{Sp}\{\gamma S^c(1-2)\gamma S^c(2-3)\ldots \gamma S^c(n-1)\} = \]
\[ = \sum_{(\alpha,\ldots,\nu)} \{\gamma_{\alpha\beta} S^c_{\beta\gamma}(1-2)\gamma_{\gamma\delta} S^c(2-3)\ldots \gamma_{\mu\nu} S^c_{\nu\alpha}(n-1)\}. \tag{4.6} \]

Now let us use the fact that the entire scheme of the spinor field and, in particular, the relation defining the Dirac matrices

\[ \gamma^m\gamma^n+\gamma^n\gamma^m=2g^{mn}, \]

as well as the value of the traces of products of any number of matrices, are invariant under the replacement

\[ \gamma \to -\gamma^{T},\quad \text{i.e.}\quad \gamma_{\alpha\beta}\to -\gamma_{\beta\alpha}. \tag{4.7} \]

As a result of the transformation (4.7), the causal function

\[ S^c_{\beta\gamma}(1-2)=\frac{1}{(2\pi)^4}\int dp\, e^{-ip(1-2)} \frac{(m+\hat p)_{\beta\gamma}}{m^2-p^2-i\varepsilon} \]

takes the form

\[ \frac{1}{(2\pi)^4}\int dp\, e^{-ip(2-1)} \frac{(m+\hat p)_{\gamma\beta}}{m^2-p^2-i\varepsilon} = S^c_{\gamma\beta}(2-1). \]

Therefore, applying the transformation (4.7) to the expression (4.6), we obtain:

\[ \begin{aligned} \operatorname{Sp}\,[\gamma S^c(1-2)\gamma S^c(2-3)\ldots \gamma S^c(n-1)] &=\\ &=(-1)^n \sum_{(\alpha,\ldots,\nu)} \{\gamma_{\beta\alpha}S^c_{\gamma\beta}(2-1)\gamma_{\delta\gamma}S^c_{\varepsilon\delta}(3-2)\ldots \gamma_{\nu\mu}S^c_{\alpha\nu}(1-n)\}\\ &=\\ &=(-1)^n \operatorname{Sp}\,[\gamma S^c(1-n)\gamma\ldots S^c(3-2)\gamma S^c(2-1)]. \end{aligned} \tag{4.8} \]

We see from this that, by virtue of the invariance of the trace with respect to the transformation (4.7), for even \(n\) the factor (4.6) coincides with the multiplier

\[ \operatorname{Sp}\,[\gamma S^c(1-n)\gamma\ldots \gamma S^c(3-2)\gamma S^c(2-1)], \tag{4.9} \]

corresponding to traversal of the contour \(L\) in the opposite direction, while for odd \(n\) it differs from it by a sign.

In this latter case the sum of the matrix elements corresponding to the different directions of traversal of an odd cycle cancels, and Furry’s theorem is proved.

Let us note that the proof of Furry’s theorem given above is formal in character, since here we are dealing with singular products of unregularized causal functions. It is not difficult, however, to see that if \(S^c\) is replaced by \(\operatorname{reg}[S^c]\), the proof remains valid and the sum of the regularized matrix elements gives zero. Therefore, assigning to such diagrams the quasi-local operator equal to zero,

\[ \Lambda_n(x_1,\ldots,x_n)=0, \tag{4.10} \]

we obtain the fulfillment of Furry’s theorem for the complete coefficient functions \(S_n(x_1,\ldots,x_n)\) after the removal of divergences.

We emphasize that the condition (4.10), which ensures the charge invariance of the theory after the removal of divergences, is, generally speaking, not compulsory. Replacing it by some other condition, we could obtain one or another charge-noninvariant theory.

Let us return to the diagrams under consideration. In view of the fact that diagrams having only external photon lines in an odd

number, have an odd number of vertices, they necessarily contain at least one odd spinor cycle, and therefore the matrix elements of diagrams with an odd number of external photon lines always vanish. We used this circumstance above, having forbidden diagrams with three and one external photon lines.

Thus, taking Furry’s theorem into account, we have four types of divergent diagrams in spinor electrodynamics, shown in Fig. 12.

Fig. 12.
a) \(\omega(G)=0\); b) \(\omega(G)=0\); c) \(\omega(G)=2\); d) \(\omega(G)=1\).

The hatched circles in these drawings represent the internal parts of the diagrams, containing an arbitrary\(^*\) number of vertices. Let us recall that the independence of the degree of divergence from the number of vertices in spinor electrodynamics follows from the equality to zero of the maximal vertex index.

Let us now investigate the form of the quasilocal operators corresponding to the divergent diagrams (Fig. 12).

To diagram a), for every even \(n\), beginning with \(n=4\), there corresponds a term in \(\Lambda_n(x_1,\ldots,x_n)\)

\[ A^n \sum_{k,l} g^{kk} g^{ll} : A_k(x_i) A_k(x_j) A_l(x_m) A_l(x_p) : \times \]

\[ \times \delta(x_1-x_2)\delta(x_2-x_3)\ldots\delta(x_{n-1}-x_n). \tag{4.11} \]

The degree of the differential polynomial in this expression is equal to zero in accordance with the value of the index of diagram a).

To diagram b), for every odd \(n\), beginning with \(n=3\), there corresponds the quasilocal operator

\[ B^n \sum_m : \bar{\psi}(x_i)\gamma^m\psi(x_j) A_m(x_k) : \delta(x_1-x_2)\times \]

\[ \times \delta(x_2-x_3)\ldots\delta(x_{n-1}-x_n). \tag{4.12} \]

\(^*\) In a), c), and d) — even; in b) — odd.

To diagram c), for every even \(n\), starting with \(n=2\), there corresponds a term in \(\Lambda_n(x_1,\ldots,x_n)\)

\[ \sum_{(i\ne j,m,k)} :A_m(x_i)\left\{\left(C^n g^{mk}+D^n g^{mm}g^{kk}\frac{\partial}{\partial x_i^m}\frac{\partial}{\partial x_j^k}+\right.\right. \]

\[ \left.\left. +E^n g^{mk}\sum_l g^{ll}\frac{\partial}{\partial x_i^l}\frac{\partial}{\partial x_j^l}\right)\times \right. \]

\[ \left. \times \delta(x_1-x_2)\delta(x_2-x_3)\ldots\delta(x_{n-1}-x_n)\right\}A_k(x_j):. \tag{4.13} \]

The differential polynomial here has degree two in accordance with the index of the corresponding diagram. The polynomial contains no first-degree term because it is impossible to construct an invariant combination with it.

Finally, to diagram d), for every even \(n\), starting with \(n=2\), there corresponds the operator

\[ \sum_{(i\ne j)} :\bar{\psi}(x_i)\left\{\left[F^n+\frac{i}{2}G^n\left(\frac{\widehat{\partial}}{\partial x_i}-\frac{\widehat{\partial}}{\partial x_j}\right)\right]\times \right. \]

\[ \left. \times \delta(x_1-x_2)\delta(x_2-x_3)\ldots\delta(x_{n-1}-x_n)\right\}\psi(x_j):. \tag{4.14} \]

From the expressions given, it is clear that the quasilocal operators \(\Lambda_n\) contain quite many arbitrary constants. We note, however, that up to now we have not taken into account the requirement of gauge invariance.

In order to formulate this requirement, let us first consider an infinitesimal gauge transformation of the potentials of the electromagnetic field

\[ A_m(x)\to A'_m(x)=A_m(x)+\frac{\partial f}{\partial x^m}, \]

where \(f\) is an arbitrary infinitesimal function. Under this transformation the \(n\)-th order term of the scattering matrix \(S(1)\)

\[ \int S_n(x_1,\ldots,x_n)\,dx_1\ldots dx_n \tag{4.15} \]

taking into account the linear dependence of \(S_n\) on the potentials \(A_n\) with different arguments, will acquire the increment

\[ \sum_{m,i}\int \frac{\partial S_n}{\partial A_m(x_i)}\frac{\partial f}{\partial x_i^m}\,dx_1\ldots dx_n. \]

Integrating this expression by parts, we obtain that it trans—

vanishes if identically

\[ \operatorname{div}_{x_i}\frac{\partial S_n(x_1,\ldots,x_n)}{\partial A(x_i)} \equiv \sum_m \frac{\partial}{\partial x_i^m} \frac{\partial S^n(x_1,\ldots,x_n)}{\partial A^m(x_i)} =0. \tag{4.16} \]

It is not hard to see that, when this condition is fulfilled, expression (4.15) will not change under a finite gradient transformation either. This is due to the fact that the coefficients of the higher powers in \(\dfrac{\partial f}{\partial x}\) in the increment of the subintegral expression are expressed in terms of the derivatives with respect to \(A(x_i)\) of (4.16).

Proceeding from the considerations indicated, we shall take condition (4.16) as the condition of gradient invariance of the theory. Below we shall see (cf. the following article) that, in addition to invariance, this condition also ensures the fulfillment of the differential law of conservation of electric current.

Let us now analyze the degree of arbitrariness in the choice of the coefficients \(A^n,\ B^n,\ldots,\ F^n,\ G^n\) in expressions (4.11)—(4.14), which remains after imposing the condition of gradient invariance in the form (4.16).

Consider, for example, the vertex part of the operator function \(S_n(x_1,\ldots,x_n)\),

\[ \sum_{(i\ne j\ne k,\ m)} :\bar{\psi}(x_i)\Gamma^m(x_1,\ldots,x_{k-1},x_{k+1},\ldots,x_n\mid x_k) \times \]

\[ \times \psi(x_j)A_m(x_k): \tag{4.17} \]

In accordance with the condition of gradient invariance, the vertex part must satisfy the condition

\[ \operatorname{div}_{x_k}:\bar{\psi}(x_i)\Gamma^m(x_1,\ldots,x_{k-1},x_{k+1},\ldots,x_n\mid x_k)\psi(x_j)=0. \]

Comparing with (4.12), we see that the function \(\Gamma^m\) is determined up to a term

\[ b^n\gamma^m\hat{\delta}(x_1-x_2)\hat{\delta}(x_2-x_3)\ldots\hat{\delta}(x_{n-1}-x_n), \tag{4.18} \]

where the introduction of \(b^n\) corresponds to a change of the coefficient \(B^n\). However, it is not hard to note that if the vertex part (4.17) satisfies condition (4.16), then with the addition to \(\Gamma^m\) of the term (4.18) it loses this property, since

\[ \operatorname{div}_{x_k}\bar{\psi}(x_i)\gamma\psi(x_j)\hat{\delta}(x_1-x_2)\hat{\delta}(x_2-x_3)\ldots\hat{\delta}(x_{n-1}-x_n)\ne 0 \]

for

\[ k\ne i,\ j. \]

Therefore, if we have chosen the quasilocal operators (4.12) so that the result is gradient-invariant, then this choice is completely

is single-valued, and the indeterminacy is eliminated by the requirement of gradient invariance.

In a completely analogous way one can verify the single-valuedness of the coefficients \(A^n\) in expressions (4.11) and \(C^n\) in expressions (4.13). It is also clear that expressions (4.14), which do not contain the potentials of the electromagnetic field, allow arbitrariness in the choice of the coefficients \(F^n\) and \(G^n\).

Let us consider further the ambiguity of operator expressions of the type

\[ \sum_{(m,k,\ i\ne j)} :A_m(x_i)\Pi^{mk}(x_1,\ldots,x_n)A_k(x_j): \]

with respect to terms of the same structure as the terms with coefficients \(D^n\) and \(E^n\) in expressions (4.13).

Substituting such a term

\[ \sum_{(m,k,\ i\ne j)} :A_m(x_i)\left\{\left(d^n g^{mm}g^{kk}\frac{\partial}{\partial x_i^m}\frac{\partial}{\partial x_j^k}+\right.\right. \]

\[ \left.\left. +\,e^n g^{mk}\sum_l g^{ll}\frac{\partial}{\partial x_i^l}\frac{\partial}{\partial x_j^l}\right) \times \delta(x_1-x_2)\delta(x_2-x_3)\cdots\delta(x_{n-1}-x_n) \right\}A_k(x_j): \]

into condition (4.16), taking into account the differentiability property of the \(\delta\)-functions,

\[ \frac{\partial}{\partial x}\delta(x-y)=-\frac{\partial}{\partial y}\delta(x-y), \]

we find the relation between \(d^n\) and \(e^n\)

\[ d^n+e^n=0, \tag{4.19} \]

which leaves one degree of arbitrariness in the choice of these coefficients.

Thus, for each \(n\) there are in all three undetermined constants \(F^n\), \(G^n\), and, for example, \(D^n\). From the point of view of the counterterms of the Lagrangian, only three numbers turn out to be indeterminate (cf. (1.4.35)):

\[ D=\sum_{n=2}^{\infty}\frac{1}{n!}D^n,\qquad F=\sum_{n=2}^{\infty}\frac{1}{n!}F^n,\qquad G=\sum_{n=2}^{\infty}\frac{1}{n!}G^n. \]

In what follows we shall see that these three numbers enter the results only in two combinations, and the ambiguity is completely eliminated by the choice of the mass and charge of the spinor particle.

We must now show that, by selecting the appropriate counterterms, one can indeed obtain a gradient-invariant theory. As is known to us, the counterterms of the Lagrangian serve to describe one or another subtraction process applied to the theory.

Let us recall that the subtraction procedure we have adopted consists in subtracting from a divergent expression a sufficient number of terms of its Taylor series with expansion center at the point \(p = 0\), and in adding to the result an arbitrary finite polynomial of a specified degree. On the basis of the results just obtained, this arbitrariness reduces to two constants in terms of the type (4.14) and to one constant in terms of the type (4.13).

We shall show that, as a result of such a method of removing divergences, a gradient-invariant theory is obtained.

In the proof we shall use the following variant of regularization by means of auxiliary masses. We shall regularize the photon causal functions (4.2) in the usual way (see (1.6)), while the spinor causal functions we shall regularize not separately, but by replacing their products corresponding to closed cycles,

\[ \operatorname{Sp}\,[\gamma S^{c}(x_{1}-x_{2})\gamma S^{c}(x_{2}-x_{3})\gamma\ldots S^{c}(x_{n-1}-x_{n}) \times \]

\[ \times \gamma S^{c}(x_{n}-x_{1})], \tag{4.20} \]

by expressions

\[ \sum_{M} C_{M}\operatorname{Sp}\,[\gamma S^{c}_{M}(x_{1}-x_{2})\gamma S^{c}_{M}(x_{2}-x_{3})\ldots S^{c}_{M}(x_{n}-x_{1})], \tag{4.21} \]

where \(S^{c}_{M}(x)\) is the causal spinor function with mass \(M\),

\[ S^{c}_{M}(x)=\frac{1}{(2\pi)^{4}}\int dp\,e^{-ipx}\frac{\hat p+M}{M^{2}-p^{2}-i\varepsilon}. \tag{4.22} \]

The indicated method of regularizing the spinor causal functions is one of the variants of Pauli–Villars regularization\(^{21}\).

Let us show, first, that Pauli–Villars regularization removes divergences from the matrix elements corresponding to closed cycles. For this purpose let us consider the result of integrating expression (4.20), written in the momentum representation. We then have

\[ \int dp\, \frac{ \operatorname{Sp}\,[\gamma(\hat p+m)\gamma(\hat p+\hat k_{1}+m)\ldots\gamma(\hat p+\hat k_{n-1}+m)] }{ (m^{2}-p^{2}-i\varepsilon)(m^{2}-(p+k_{1})^{2}-i\varepsilon)\ldots(m^{2}-(p+k_{n-1})^{2}-i\varepsilon) }. \tag{4.23} \]

For large \(p\), the integrand behaves as \(p^{-n}\).

and for \(n\) less than five, the integral diverges as

\[ \int^\infty \frac{p^3\,dp}{p^n} = \int^\infty \frac{dp}{p^{\,n-3}} . \tag{4.24} \]

Let us now note that, for large \(p\), the integrand in (4.23)

\[ \frac{P_n(p)+m^2P_{n-2}(p)+\cdots+m^n} {P_{2n}(p)+m^2P_{2n-2}(p)+\cdots+m^{2n}} \simeq \]

\[ \simeq \left\{ \frac{P_n(p)}{P_{2n}(p)} + m^2\frac{P_n(p)}{P_{2n}(p)} \left( \frac{P_{n-2}(p)}{P_n(p)} - \frac{P_{2n-2}(p)}{P_{2n}(p)} \right) +\cdots \right\} \tag{4.25} \]

(where \(P_i(p)\) is a polynomial of the \(i\)-th degree in the components of \(p\)) can be represented in the form of a power series in \(m^2\), with the degree of growth in \(p\), as \(p\to\infty\), of \(m^2\) to the power \(k\) equal to \(p^{-n-2k}\).

It is therefore clear that if the coefficients \(C_M\) in the sum (4.21) satisfy relations of the type

\[ \sum_M C_M=0;\qquad \sum_M C_M M^2=0;\qquad \ldots\qquad \sum_M C_M M^{2k-2}=0, \tag{4.26} \]

then the integrand for (4.21) as \(p\to\infty\) behaves as

\[ p^{-2k-n}, \]

and the Pauli—Villars procedure regularizes the expressions under consideration together with their first \(2k+n-5\) derivatives with respect to \(x_i\). This property of the Pauli—Villars procedure could also have been established directly by passing to the “\(\alpha\)-representation” of causal functions.

Thus, we shall introduce auxiliary masses only in photon lines and in closed spinor loops. We shall not regularize open spinor loops at all. It is easy to see that it is sufficient to regularize photon functions with one auxiliary mass, and closed spinor loops with two auxiliary masses.

Indeed, the maximal degree of divergence of diagrams with closed spinor loops is equal to two (diagrams of the type of Fig. 12, \(v\)), i.e. \(n=2\), and the introduction of two auxiliary masses reduces the degree by four units. As a result of this procedure the integral, as \(p\to\infty\), begins to converge as

\[ \int^\infty dp\cdot p^{-3}. \]

On the other hand, the maximal degree of divergence of diagrams with internal photon lines is equal to one (diagrams of the type of Fig. 12, \(g\)). The introduction of one auxiliary mass into the photon line reduces the degree of divergence by two units, and the integral

begins to converge as

\[ \int^{\infty} dp\, p^{-2}. \]

Thus there is no need to regularize unclosed spinor cycles.

We see from this that, for finite values of the auxiliary masses \(M_i\), all matrix elements turn out to be convergent. Let us recall, however, that regularization by the method of auxiliary masses is only a technical auxiliary device carried out at an intermediate stage of the reasoning, and that the actual removal of divergences is accomplished by the subtraction procedure.

Therefore, to the regularized expression obtained for the scattering matrix we shall now apply the usual procedure of subtracting the Maclaurin series in the momentum representation, with the simultaneous addition of three arbitrary finite constants. In view of the previously established invariance of the results of such a subtraction with respect to the method of introducing the auxiliary masses, it remains for us only to establish the gradient invariance of the Pauli–Villars regularization and of the subtraction procedure following it.

We shall now prove the gradient invariance of the regularized matrix \(S(1)\) before applying the subtraction procedure to it. In doing so we shall use the circumstance that the totality of \(n\)-th order diagrams with \(k\) \((k>1)\) external photon lines can be obtained from the diagrams of order \((n-1)\) with \(k-1\) external photon lines by the process of inserting an additional \(n\)-th vertex \(\xi\) into any external or internal spinor line. This insertion process (which, for brevity, we shall call the \(\xi\)-process) simultaneously establishes a graphical correspondence between the expressions

\[ \sum' \left. \frac{\partial S_n(x_1,\ldots,x_n)}{\partial A(\xi)} \right|_{\xi=x_i} \quad \text{and} \quad S_{n-1}(x_1,\ldots,x_{n-1}). \tag{4.27} \]

Let us first consider the process of insertion into an internal spinor line. Inserting the \(\xi\)-vertex into the line \((x_1,x_2)\), we obtain the picture shown in Fig. 13.

Fig. 13.

Calculating the divergence with respect to \(\xi\) which interests us in condition (4.16) from the factor

\[ S^{c}(x_2-\xi)\,\gamma\,S^{c}(\xi-x_1), \]

we find, using the equations for \(S^c\),

\[ \operatorname{div}_{\xi}\left[S^c(x_2-\xi)\,\gamma\,S^c(\xi-x_1)\right]= \]

\[ =\frac{1}{i}\,[\delta(x_2-\xi)-\delta(\xi-x_1)]\,S^c(x_2-x_1). \tag{4.28} \]

The process of \(\xi\)-insertion into the internal lines of a complicated fermion cycle may be represented by the diagram shown in Fig. 14.

Computing the divergence of the sum of the terms corresponding to the right-hand side of Fig. 14, with the aid of (4.28), we readily verify

Fig. 14.

that it is proportional to the term corresponding to the diagram in the left-hand part of Fig. 14, with proportionality factor equal to

\[ -\frac{1}{i}\,[\delta(x_n-\xi)-\delta(x_1-\xi)]. \tag{4.29} \]

It follows from this that the indicated divergence vanishes for closed cycles (when \(x_1=x_n\)).

Applying this reasoning to more complicated diagrams consisting of internal photon lines and an arbitrary number of closed and open fermion cycles, we arrive at the relation

\[ \operatorname{div}_{\xi}K^{n+1}(x_1,\ldots,x_n\mid \xi)= \]

\[ =\frac{1}{i}\left[\sum_a \delta(\xi-x_a)-\sum_b \delta(\xi-x_b)\right]K^n(x_1,\ldots,x_n), \tag{4.30} \]

where the summations \(\sum_a\) and \(\sum_b\) extend over all vertices of the diagram into which \((x_a)\) enter and from which \((x_b)\) external fermion lines leave; \(K^n(x_1,\ldots,x_n)\) is the coefficient function of \(n\)-th order, and \(K^{n+1}(x_1,\ldots,x_n\mid \xi)\) is the coefficient function of order \(n+1\) obtained from it by the \(\xi\)-process.

N. N. BOGOLIUBOV AND D. V. SHIRKOV

To complete the proof of the gradient invariance of the regularized matrix \(S(1)\), let us consider also the process of \(\xi\)-insertion into the external (incoming and outgoing) fermion lines, shown in Fig. 15.

Computing the corresponding divergences, we find:

\[ \operatorname{div}_{\xi} S^c(x_1-\xi)\gamma\psi(\xi) = \frac{1}{i}\,\delta(x_1-\xi)\psi(x_1), \tag{4.31} \]

\[ \operatorname{div}_{\xi}\bar{\psi}(\xi)\gamma S^c(\xi-x_n) = i\delta(\xi-x_n)\bar{\psi}(x_n). \tag{4.32} \]

Comparing these expressions with the factor (4.26), we see that the operator functions \(S_n(x_1,\ldots,x_n)\) do indeed satisfy

Fig. 15.

the condition of gradient invariance (4.16), for, as just proved, the divergence of the factors describing closed cycles is equal to zero, while the divergence of the operator corresponding to an open cycle is composed of the divergence of the sum of \(\xi\)-insertions into the coefficient function and of insertions into the external lines, which also vanish, since the factor (4.32) compensates the first term in (4.29), and the factor (4.31) the second term.

Thus the gradient invariance of the matrix \(S(1)\) has been formally proved before the process of regularization. We note, however, that under Pauli–Villars regularization only closed fermion cycles are regularized, and in each of the additional terms in the expressions (4.21), for all causal functions \(S^c\), the mass has the same value.

In view of this, the indicated additional terms after the \(\xi\)-process also have zero divergence, and we arrive at the conclusion that, after applying the Pauli–Villars procedure, the regularized matrix \(S(1)\) also satisfies the condition of gradient invariance.

Let us note here that this property of preserving gradient invariance is an important advantage of Pauli–Villars regularization, which we had in mind when departing from the previously adopted method of introducing auxiliary masses.

We shall now show that the inclusion of quasilocal operators \(\Lambda_n\) does not violate the property of gradient invariance. In the course of the discussion we shall find useful equation (4.30), which with the aid of

by means of the formulas

\[ K^n(x_1,\ldots,x_n)=\int e^{\,i\sum_{i=1}^n p_i x_i}\,K_n(p_1,\ldots,p_n)\,dp_1\ldots dp_n, \]

\[ K^{n+1}(x_1,\ldots,x_n\mid \xi)=\int e^{\,i\sum_{i=1}^n p_i x_i+\pi\xi}\, K_{n+1}(p_1,\ldots,p_n\mid \pi)\,dp_1\ldots dp_n\,d\pi, \]

\[ \delta(\xi-x_a)=(2\pi)^{-4}\int e^{\,i\pi(\xi-x_a)}\,d\pi \]

in the momentum representation takes the form

\[ \pi K_{n+1}(p_1,\ldots,p_n\mid \pi)= \sum_a K_n(p_1,\ldots,p_a+\pi,\ldots,p_n)- \]

\[ -\sum_b K_n(p_1,\ldots,p_b+\pi,\ldots,p_n). \tag{4.33} \]

In view of the translational invariance of the coefficient functions \(K^n\) and \(K^{n+1}\) in the \(x\)-representation, all quantities entering the last relation are proportional to a \(\delta\)-function of the total momentum \(\delta(\sum p+\pi)\).

Fig. 16.

\[ K_{n+1}(p_1,\ldots,p_n\mid \pi)= \delta(\sum p+\pi)\,K'_{n+1}(p_1,\ldots,p_n\mid \pi), \]

\[ K_n(p_1,\ldots,p_i+\pi,\ldots,p_n)= \delta(\sum p+\pi)\,K'_n(p_1,\ldots,p_i+\pi,\ldots,p_n). \]

Separating out this \(\delta\)-function, we have, obviously,

\[ \pi K'_{n+1}(p_1,\ldots,p_n\mid \pi)= \sum_a K'_n(p_1,\ldots,p_a+\pi,\ldots,p_n)- \]

\[ -\sum_b K'_n(p_1,\ldots,p_b+\pi,\ldots,p_n). \tag{4.34} \]

As an illustration, let us note that a diagram of the type of Fig. 12, б can be obtained from Fig. 12, г by the \(\xi\)-process shown in Fig. 16.

The corresponding coefficient functions \(\Sigma\) and \(\Gamma\), entering into the operator expressions

\[ :\bar{\psi}(x_b)\Sigma^{(n)}(x_1,\ldots,x_n)\psi(x_a):, \]

\[ :\bar{\psi}(x_b)\Gamma^{(n+1)}(x_1,\ldots,x_n)\psi(x_a)A(\xi):, \]

according to (4.34), satisfy the relation

\[ \pi\Gamma'_{n+1}(p_1,\ldots,p_n\mid \pi) = i\Sigma'_n(p_1,\ldots,p_a+\pi,\ldots,p_n) - i\Sigma'_n(p_1,\ldots,p_b+\pi,\ldots,p_n). \tag{4.35} \]

Then carrying out the integration over all momenta except the external ones \((\pi,p_a,p_b)\),

\[ \int \Gamma'_{n+1}(p_1,\ldots,p_n\mid \pi)[dp]^{a,b}_n = \Gamma_{n+1}(p_a,p_b\mid \pi), \]

\[ \int \Sigma'_n(p_1,\ldots,p_a+\pi,\ldots,p_n)[dp]^{a,b}_n = \Sigma_n(p_a+\pi,p_b), \]

\[ \int \Sigma'_n(p_1,\ldots,p_b+\pi,\ldots,p_n)[dp]^{a,b}_n = \Sigma_n(p_a,p_b+\pi) \]

(here
\([dp]^{a,b}_n=dp_1\ldots dp_{a-1}dp_{a+1}\ldots dp_{b-1}dp_{b+1}\ldots dp_n\)),
differentiating with respect to \(\pi\) and putting \(\pi=0\), we find:

\[ \Gamma_{n+1}(p_a,p_b\mid 0) = i\frac{\partial\Sigma_n(p_a,p_b)}{\partial p_a} - i\frac{\partial\Sigma(p_a,p_b)}{\partial p_b} \]

or

\[ \Gamma_{n+1}(p,0)=i\frac{\partial\Sigma_n(p)}{\partial p}, \tag{4.36} \]

where it has been set

\[ \left. \begin{aligned} \Gamma_{n+1}(p,0)&=\Gamma_{n+1}(p,-p\mid 0),\\ \Sigma_n(p)&=\Sigma_n(p,-p). \end{aligned} \right\} \tag{4.37} \]

Formula (4.36) establishes a connection between the coefficient function corresponding to the self-energy diagram in the \(n\)-th approximation and the coefficient function of the vertex diagram of the \(n+1\)-st approximation, which occurs both in the unregularized theory (formally) and after Pauli–Villars regularization.

We now turn to the proof of the gauge invariance of the subtraction procedure. In view of the satisfaction of condition (4.16) prior to the subtraction process, it is evidently sufficient for us to verify the gauge invariance of the subtracted quasilocal operators \(\Lambda_n\). If, in addition, it turns out that the subtracted operators corresponding to the diagrams in Figs. 12,a–12,c are separately gauge-invariant, while the operators corresponding to the diagrams in Figs. 12,b and 12,d satisfy relation (4.34), then the gauge-

ent invariance of the subtraction procedure will be established. We shall show that this assertion indeed holds.

Making use of the fact that the condition of gradient invariance of the subtraction procedure in the 2nd and 3rd orders in \(e\) was established by us (see § 1), we shall carry out the proof by induction. Suppose that the quasilocal operators \(\Lambda_n\) satisfy the stated requirements up to some odd

\[ n-1=2\nu-1. \]

Consider the element of the scattering matrix

\[ S_n(x_1,\ldots,x_n) \]

before subtracting from it the quasilocal operator

\[ \Lambda_n(x_1,\ldots,x_n). \]

In accordance with the proved gradient invariance of the \(S\)-matrix before the subtraction process and the assumed gradient invariance of the quasilocal operators of lower orders

\[ \Lambda_\nu(x_1,\ldots,x_n)\qquad \nu=1,\ldots,n-1, \]

the expression \(S_n\) will be gradient-invariant. The specific divergences of \(n\)-th order for even \(n\) have the form shown in Fig. 17.

Fig. 17.

Owing to the gradient invariance of \(S_n\) itself, its parts corresponding to the three indicated diagrams will also be gradient-invariant:

\[ \left. \begin{aligned} &\sum_{\substack{(p,q,r,s)\\(a,b,c,d)}} :A_p(x_a)A_q(x_b)A_r(x_c)A_s(x_d):\,\Box^{pqrs}(x_1,\ldots,x_n);\\ &\sum_{(p,q,a,b)} :A_p(x_a)A_q(x_b):\,\Pi^{pq}(x_1,\ldots,x_n);\\ &\sum_{(a,b)} :\bar{\psi}(x_b)\Sigma(x_1,\ldots,x_n)\psi(x_a): . \end{aligned} \right\} \tag{4.38} \]

The property of gradient invariance of the last expression is obvious, and the four- and two-photon parts give:

\[ \sum_p k_a^p \Box^{pqrs}(k_1,\ldots,k_a,\ldots,k_n)\,g^{pp}=0, \tag{4.39} \]

\[ \sum_p g^{pp}k_a^p\Pi^{pq}(k_1,\ldots,k_a,\ldots,k_n)=0. \tag{4.40} \]

Differentiating (4.39) with respect to \(k_a^p\) and putting \(k_a=0\), we find:

\[ \Box^{pqrs}(k_1,\ldots,k_{a-1},0,k_{a+1},\ldots,k_n)=0. \tag{4.41} \]

Relation (4.41) is very important. Owing only to the logarithmic divergence \((\omega(G)=0)\) of the diagram of Fig. 17,a, the subtraction of the Taylor series reduces to subtracting from \(\Box^{pqrs}\) its value at zero values of the momenta of the external photons.

In consequence of (4.41) this quantity turns out to be equal to zero, and \(\Box^{pqrs}\) itself is convergent. Therefore, in the diagram of Fig. 17,a there is in fact no need to remove divergences.

Let us turn to the two-photon diagram.

Differentiating the left-hand side of relation (4.40) once, twice, and three times with respect to the components \(k_a\), and then putting \(k_a=0\), we find that the first, second, and third coefficients, respectively, of the expansion of \(\Pi^{pq}\) in a Taylor series, multiplied by \(k_a^p\) (summation over \(p\)), vanish. As a result we obtain:

\[ \left. \begin{aligned} \sum_p g^{pp}k_a^p\Pi^{pq}_{(0,k_a)}(k_1,\ldots,k_n)&=0,\\ \sum_p g^{pp}k_a^p\Pi^{pq}_{(1,k_a)}(k_1,\ldots,k_n)&=0,\\ \sum_p g^{pp}k_a^p\Pi^{pq}_{(2,k_a)}(k_1,\ldots,k_n)&=0, \end{aligned} \right\} \tag{4.42} \]

where the symbol \(\Pi_{(i,k)}\) denotes the \(i\)-th term of the Taylor series in the variable \(k\). Expanding, in turn, the expressions obtained in a Taylor series in the variable \(k_b\), up to the second, first, and zeroth order respectively, we obtain in total

\[ \sum_p g^{pp}k_a^p\left(\Pi^{pq}(k_1,\ldots,k_n)\right)^{k_a,k_b}_{3}=0, \tag{4.43} \]

where the symbol \(\{\ldots\}_{i}^{x,y,z,\ldots}\), as usual, denotes the sum of the terms of the Taylor series in the variables \(x,y,z,\ldots\) up to order \((i-1)\) inclusive. But the Taylor series entering into (4.43) is precisely the

coefficient function of the operator \(\Lambda_n\) subtracted from \(S_n\), corresponding to the two-photon diagram. Relation (4.43) is the expression of its gradient invariance. The indicated operator \(\Pi\) is determined up to an arbitrary finite polynomial of second degree in the components \(k=k_a=k_b\). Proceeding from considerations of relativistic covariance, the established uniqueness of the coefficient \(C^n\) in expression (4.13), and relation (4.19), we find that this polynomial, satisfying the requirement of gradient invariance, has the form

\[ d^n\left(g^{pq}k^2-g^{bp}g^{aq}k^b k^a\right). \tag{4.44} \]

The gradient invariance of the operator \(\Lambda_n\) corresponding to the electron self-mass diagram (Fig. 17, \(c\)) is obvious. In accordance with the established arbitrariness of the finite polynomial, this operator is determined up to the expression

\[ f^n m+g^n\hat p. \tag{4.45} \]

The consideration of the terms \(\Lambda_n\) is thus completed.

We pass to the next order \(n+1=2\nu+1\). Because of its oddness, the only diagram from which a divergence must be eliminated is the vertex-type diagram shown in Fig. 18.

Fig. 18.

As was established, we need only verify the fulfillment of relation (4.34) (or of the equivalent relation (4.35)). Owing to the linearity of the coefficient functions of the quasilocal operators \(\Sigma_n^{\prime\,\mathrm{subtr}}\) and \(\Gamma_{n+1}^{\prime\,\mathrm{subtr}}\) with respect to the momenta, relation (4.35) for them takes the form

\[ \Gamma_{n+1}^{\prime\,\mathrm{subtr}}(p_1,\ldots,p_n\mid\pi) = i\,\frac{ \partial \Sigma_n^{\prime\,\mathrm{subtr}}(p_1,\ldots,p_a+\pi,\ldots,p_n) }{\partial p_a} - i\,\frac{ \partial \Sigma_n^{\prime\,\mathrm{subtr}}(p_1,\ldots,p_b+\pi,\ldots,p_n) }{\partial p_b}. \tag{4.46} \]

In view of the fact that, up to arbitrary polynomials, the subtracted parts \(\Gamma\) and \(\Sigma\) are their Maclaurin series of zero and first order,

\[ \Gamma_{n+1}^{\prime\,\mathrm{subtr}}(p_1,\ldots,p_n\mid\pi) = \Gamma_{n+1}'(p_1,\ldots,p_n\mid\pi)\big|_{p_a=p_b=\pi=0} + i b_{n+1}\gamma, \]

\[ \Sigma_n^{\prime\,\mathrm{subtr}}(p_1,\ldots,p_a+\pi,\ldots,p_n) = \left\{\Sigma_n'(p_1,\ldots,p_a+\pi,\ldots,p_n)\right\}_{2}^{\rho_a} + f^n+\frac{g^n}{2}(\hat p_a+\hat\pi), \]

\[ \Sigma_n^{\prime\,\mathrm{subtr}}(p_1,\ldots,p_b+\pi,\ldots,p_n)= \]

\[ =\{\Sigma'_n(p_1,\ldots,p_b+\pi,\ldots,p_n)\}_{2}^{p_b} +f^n+\frac{g^n}{2}\left(\hat p_b+\hat\pi\right), \]

by direct substitution we verify that condition (4.46), taking account of relation (4.36), reduces to the requirement

\[ b_{n+1}=g_n . \tag{4.47} \]

In view of the established uniqueness of the constant \(b_n\), it is also determined by this relation. The subtraction procedure in the \((n+1)\)-st order, when (4.47) is observed, likewise turns out to be gradient-invariant. The induction is thereby completed, and hence the gradient invariance of the matrix \(S(1)\) after elimination of the divergences has been established.

Let us also note that the totality of relations (4.36) and (4.47) establishes, in each order in \(n\), the equality of the constants \(G_n\) and \(B_{n+1}\) in the operators (4.12) and (4.14), i.e. constitutes the complete Ward identity.

We shall now write out the counterterms of the Lagrangian regularizing the matrix \(S(1)\).

Substituting the expressions (4.12), (4.13), and (4.14) into formula (I.4.35), after integration by parts and summation over \(\nu\), we obtain the complete interaction Lagrangian in the form

\[ \begin{aligned} L(x)=L(x;1)&=\sqrt{4\pi}\,eZ_1:\bar\psi(x)\hat A(x)\psi(x):+ \\ &\quad+\delta m:\bar\psi(x)\psi(x): +(Z_2-1)\left\{\frac{i}{2}\sum_k:\left(\bar\psi(x)\gamma^k\frac{\partial\psi}{\partial x^k}\right.\right.\\ &\quad\left.\left.-\frac{\partial\bar\psi}{\partial x^k}\gamma^k\psi(x)\right):-m:\bar\psi\psi:\right\}\\ &\quad-(Z_3-1)\left\{\frac{1}{2}\sum_{m,n}g^{mm}g^{nn}: \frac{\partial A_n(x)}{\partial x^m}\times\right.\\ &\quad\left.\times\frac{\partial A_n(x)}{\partial x^m}: -\frac{1}{2}:\left(\sum_n g^{nn}\frac{\partial A_n}{\partial x^n}\right)^2:\right\}. \end{aligned} \tag{4.48} \]

where the notations used are

\[ \left. \begin{aligned} Z_1&=1+\sum_{\nu=1}^{\infty}\frac{B_{2\nu+1}}{2\nu!}, & Z_2&=1+\sum_{\nu=1}^{\infty}\frac{G_{2\nu}}{2\nu!}, \\[6pt] Z_3&=1+\sum_{\nu=1}^{\infty}\frac{D_{2\nu}}{2\nu!}, & \delta m&=\sum_{\nu=1}^{\infty}\frac{F_{2\nu}+G_{2\nu}}{2\nu!}. \end{aligned} \right\} \tag{4.49} \]

In view of the previously established relation

\[ B_{2\nu+1}=G_{2\nu}, \]

Ward’s identity in the present notation takes the form

\[ Z_1=Z_2. \tag{4.50} \]

The constants \(Z_1, Z_2, Z_3, \delta m\), defined by the relations (4.49), depend on the auxiliary masses \(M_i\), and when these masses are taken to infinity the coefficients in their expansions diverge logarithmically. However, the matrix elements of the matrix

\[ S(1)=T\left\{e^{\,i\int L(x;1)\,dx}\right\} \]

tend, in the limit \(M_i\to\infty\), to finite values.

In view of the gradient invariance of the Pauli–Villars procedure, the Lagrangian (4.48), in contrast to the Lagrangian (1.61), obtained with the aid of a gradient-noninvariant regularization, contains no photon-mass terms \(\delta m_\phi\) and \(a_5\), and is therefore explicitly gradient-invariant.

We see from this that the form of the counterterms depends essentially on the method of regularization. The matrix \(S(1)\), after the limiting passage \(M_i\to\infty\), of course does not depend on it.

Substituting the expression for the quasilocal operators into formula (I.4.31), we obtain the counterterms regularizing the matrix in the form

\[ \begin{aligned} L(x;g)g(x)={}&\sqrt{4\pi}\,e\,g(x)Z_1(g):\bar\psi(x)\hat A(x)\psi(x):+\\ &+\delta m(g):\bar\psi(x)\psi(x):+(Z_2(g)-1)\left\{\frac{i}{2}\sum_k:\left(\bar\psi(x)\gamma^k\frac{\partial\psi}{\partial x^k}\right.\right.\\ &\left.\left.-\frac{\partial\bar\psi}{\partial x^k}\gamma^k\psi(x)\right):-m:\bar\psi(x)\psi(x):\right\}-(Z_3(g)-1)\times\\ &\times\left\{\frac{1}{2}\sum_{m,n}g^{mm}g^{nn}:\frac{\partial A_m(x)}{\partial x^n}\frac{\partial A_m(x)}{\partial x^n}:-\frac{1}{2}:\left(\sum_n g^{nn}\frac{\partial A_n}{\partial x^n}\right)^2:\right\}+\\ &+\frac{Z_3(g)-1}{g^2(x)}\cdot\frac{1}{2}\sum_{m,n}g^{mm}g^{nn}:\left\{A_m(x)A_n(x)\frac{\partial g}{\partial x^m}\frac{\partial g}{\partial x^n}+\right.\\ &\left.+2\frac{\partial A_n(x)}{\partial x^m}A_n(x)g(x)\frac{\partial g}{\partial x^m}+A_n(x)A_m(x)\frac{\partial g}{\partial x^n}\frac{\partial g}{\partial x^m}-\right.\\ &\left.-2A_n(x)\frac{\partial A_m(x)}{\partial x^m}\frac{\partial g}{\partial x^n}g(x)\right\}: . \end{aligned} \tag{4.51} \]

where

\[ \begin{aligned} Z_1(g)&=1+\sum_{\nu=1}^{\infty}\frac{[g(x)]^{2\nu}B_{2\nu+1}}{2\nu!};\\ Z_2(g)&=1+\sum_{\nu=1}^{\infty}\frac{[g(x)]^{2\nu}C_{2\nu}}{2\nu!};\\ Z_3(g)&=1+\sum_{\nu=1}^{\infty}\frac{[g(x)]^{2\nu}D_{2\nu}}{2\nu!};\\ \delta m(g)&=\sum_{\nu=1}^{\infty}\frac{[g(x)]^{2\nu}(F_{2\nu}+G_{2\nu})}{2\nu!}. \end{aligned} \tag{4.52} \]

Expression (4.51) is already gradient-invariant; the matrix \(S(g)\) is also gradient-invariant.

§ 5. RENORMALIZATION OF MASS AND CHARGE IN SPINOR ELECTRODYNAMICS

We now proceed to a more detailed examination of the scattering matrix \(S(1)\). In the preceding section its gradient invariance was established. Let us note, preliminarily, that the property of gradient invariance permits one to add to the electromagnetic-potential pairing

\[ \overline{A_m(k)}A_n(k'), \]

used in the process of reducing the terms of the matrix \(S(1)\) to normal form, an expression of the type

\[ k^m k'^n f(k^2)\,\delta(k+k'), \]

where \(f(k^2)\) is an arbitrary function of \(k^2\). In other words, the matrix elements of the scattering matrix do not change their values when

\[ i\,\overline{A_m(k)}A_n(k') \equiv i\langle T(A_m(k)A_n(k'))\rangle_0 = g^{mn}\frac{1}{k^2}\,\delta(k-k') \]

is replaced by the expression*)

\[ \left(g^{mn}\frac{1}{k^2}+k^m k^n f(k^2)\right)\delta(k+k'). \tag{5.1} \]

For the proof, let us consider the gradient transformation

\[ A_n \to A'_n=A_n+g^{mn}k^n F(k^2)(k\cdot A(k)). \]

*) First noted by Landau and collaborators\(^{22}\).

Defining the chronological pairing of the new operators \(A'\), we find:

\[ i\,\overline{A'_n(k)A'_m(k')} = \]

\[ = \frac{1}{k^2}\left[g^{mn} + g^{nn}k^n g^{mm}k^m\left(2F(k^2)+k^2F(k^2)F(k^2)\right)\right]\delta(k+k'). \]

In view of the gradient invariance of \(S(1)\), its matrix elements will not depend on the function

\[ f(k^2)=2F(k^2)+k^2F(k^2)F(k^2), \]

and the required assertion is proved. Let us further note that, putting

\[ F(k^2)=-\frac{1}{k^2}, \]

we obtain an expression for the pairing of the electromagnetic field

\[ i\,\overline{A'_n(k)A'_m(k)} =\frac{1}{k^2}\left(g^{mn}-\frac{g^{mm}g^{nn}k^m k^n}{k^2}\right)\delta(k+k'), \tag{5.2} \]

possessing the property of transversality

\[ \sum_n k^n\,\overline{A'_n(k)A'_m(k')}=0. \tag{5.3} \]

It will be established below that the pairing (5.2) plays a special role in the theory of renormalizations.

We shall now show that the introduction of the five counterterms (4.48) into the interaction Lagrangian is equivalent to a certain renormalization of the mass and charge of the free fermion*). For this purpose, the complete effective interaction Lagrangian, with the term \(\partial m:\bar\psi\psi:\) omitted, is

\[ L'(x)=Z_1\sqrt{4\pi}\,e:\bar\psi(x)\hat A(x)\psi(x):+ \]

\[ +(Z_2-1)\left\{\frac{i}{2}\sum_n:\left(\bar\psi(x)\gamma^n\frac{\partial}{\partial x^n} -\frac{\partial\bar\psi}{\partial x^n}\gamma^n\psi(x)\right):-m:\bar\psi(x)\psi(x):\right\} \]

\[ -(Z_3-1)\left\{\frac{1}{2}\sum_{m,n}g^{mm}g^{nn}:\frac{\partial A_m(x)}{\partial x^n}\times \right. \]

\[ \left. \times\frac{\partial A_m(x)}{\partial x^n}: -\frac{1}{2}:\left(\sum_m g^{mm}\frac{\partial A_m}{\partial x^m}\right)^2:\right\}. \tag{5.4} \]

* We emphasize that the reasoning given below is not entirely consistent in character. Strictly speaking, it applies only to the study of the influence on Green’s functions of finite additions to the interaction Lagrangian having the same operator structure as (5.5). Accordingly, the multiplicative relations obtained below have real meaning only for finite renormalizations, and are purely formal when divergent counterterms are considered.

we write in the momentum representation

\[ L'(p)=Z_1\sqrt{4\pi}\,e:\bar\psi \hat A\psi: +(Z_2-1):\bar\psi(p)\times \]

\[ \times(\hat p-m)\psi(-p): -\frac{Z_3-1}{2}\sum_{m,n}:A_m(p)\bigl(g^{mn}p^2-p^m p^n\bigr)A_n(-p): \tag{5.5} \]

To the Lagrangian (5.5), unlike the case of the ordinary “bare” interaction Lagrangian (4.1), there will correspond, in Feynman diagrams, vertices of three types. The term \((Z_2-1)\) will correspond to vertices at which two fermion lines meet and which, for brevity, we shall call \(Z_2\)-vertices. The term \((Z_3-1)\) will correspond to \(Z_3\)-vertices, joining two photon lines. Finally, the term \(Z_1\) will correspond to a vertex of the ordinary type (a \(Z_1\)-vertex).

Let us now consider the structure of the “propagation factors” corresponding to the internal lines of the new, more complicated diagrams. We begin with internal fermion lines. Assigning the factor

\[ (Z_2-1)(\hat p-m) \]

to a \(Z_2\)-vertex, and the factor \(\sqrt{4\pi}\,e\,Z_1\) to a \(Z_1\)-vertex, we find that to simple internal fermion lines there will correspond the contraction\(^*\)

\[ \overline{\psi(-p)\psi(p)}=\frac{i}{\hat p-m} \]

independently of the type of vertices which this internal line connects. Let us now calculate the “complete propagation factor” corresponding to the motion of a particle between two \(Z_1\)-vertices. It is obvious that it will be represented by a sum of factors corresponding to open fermion chains having their beginning and end at vertices of type \(Z_1\) and containing an arbitrary number of vertices of type \(Z_2\). Computing these factors successively, taking into account the appearance of the factor \(i\) in each extra order of the \(S\)-matrix and, owing to cancellation of the factorial \(n!\) in the denominator upon passing to the momentum representation, we find:

\[ \frac{i}{\hat p-m}\, i(Z_2-1)(\hat p-m)\frac{i}{\hat p-m} =-(Z_2-1)\frac{i}{\hat p-m}, \]

so that to the insertion of each extra \(Z_2\)-vertex there corresponds the appearance of the multiplier \(Y_2=1-Z_2\).

Therefore, summing the factors corresponding to the various numbers of \(Z_2\)-vertices, from zero to infinity, we find:

\[ \frac{i}{\hat p-m}(1+Y_2+Y_2^2+\cdots) = \frac{i}{\hat p-m}\frac{1}{1-Y_2} = \frac{1}{Z_2}\frac{i}{\hat p-m}. \]

\[ {}^*\text{ For brevity, in this equality we omit the }\delta\text{-function of the total momentum.} \]

We arrive at the conclusion that the appearance in the interaction Lagrangian of the counterterm \((Z_2-1)\), from the point of view of internal fermion lines, is equivalent to the following renormalization of the fermion causal function:

\[ S^c(p)\to S'{}^c(p)=Z_2^{-1}S^c(p). \tag{5.7} \]

By an entirely analogous procedure we verify that taking account of the counterterm \((Z_2-1)\) in external fermion lines leads to the appearance of the factor \(Z_2^{-1}\) at the operators \(\bar\psi\) and \(\psi\) corresponding to the free ends of the line:

\[ \psi(p)\to Z_2^{-1}\psi(p);\qquad \bar\psi(p)\to Z_2^{-1}\bar\psi(p). \tag{5.8} \]

Let us turn to internal photon lines. We shall associate with the vertex \(Z_3\) the factor

\[ \frac{1-Z_3}{2}\left(g^{mn}p^2-p^m p^n\right) = -\frac{Y_3}{2}\left(g^{mn}p^2-p^n p^m\right). \]

Then any internal photon line will still correspond to the contraction

\[ \overline{A_m(p)A_n(p)}=\frac{-i}{p^2}g^{mn}. \]

Let us now consider the structure of the factors corresponding to the motion of a photon between two \(Z_1\)-vertices. Considering successive insertions of \(Z_3\)-vertices, taking into account the identity

\[ \sum_{l,k}\left(g^{ml}p^2-p^m p^l\right) \frac{g^{lk}}{p^2} \left(g^{kn}p^2-p^k p^n\right) = g^{mn}p^2-p^m p^n, \]

we obtain a sum of terms of the form

\[ -\frac{i}{p^2}g^{mn}; \sum_{k,l}\frac{i}{p^2}g^{mk} \left(\frac{Y_3}{2}\right) \left(g^{kl}p^2-p^k p^l\right)2i\cdot\frac{i}{p^2}g^{ln} = \]

\[ = -Y_3\frac{i}{p^2} \left(g^{mn}-\frac{p_m p_n}{p^2}\right); \quad -Y_3^2\frac{i}{p^2} \left(g^{mn}-\frac{p_m p_n}{p^2}\right); \ldots, \]

where

\[ p_m=g_{mn}p^n. \]

Here the factor 2 in the multiplier referring to the vertex \(Z_3\) is due to the two different orders of contraction of the operators \(A(p)\) entering the counterterm \(Z_3\). Performing summation over all possible numbers of \(Z_3\)-vertices, we arrive at the expression

\[ -\frac{1}{Z_3}\frac{i}{p^2}g^{mn} - \frac{Z_3-1}{Z_3}\frac{i}{p^2}\cdot\frac{p_m p_n}{p^2} = \]

\[ = -\frac{1}{Z_3}\frac{i}{p^2} \left(g^{mn}-\frac{p_m p_n}{p^2}\right) -\frac{i}{p^2}\cdot\frac{p_m p_n}{p^2}. \]

Thus, the counterterm \((Z_3-1)\) does not lead to a simple renormalization of the photon Green function. In addition to the renormalized bare function, there appears a not completely renormalized “longitudinal” supplement. In the usual consideration of the scattering matrix this term is neglected, using the fact noted above that the matrix elements of the \(S\)-matrix are independent of terms of this type. Therefore the fact of “incomplete renormalization” of the photon function proves to be inessential here.

The situation changes when one passes to the general theory of Green functions that are sums of diagrams composed only of internal lines, since in this case the basic apparatus is formulated without any reference whatsoever to matrix elements and the Lorentz condition.

In this case the longitudinal term can no longer be neglected; however, the difficulty that arises can be bypassed if, as the zeroth approximation for the photon Green function, one chooses the transverse pairing (5.2). Carrying out in this case the summation of the contributions from vertices of type \(Z_3\), one easily verifies that taking account of the counterterm \(Z_3-1\) leads already to a simple multiplicative renormalization of the photon function:

\[ D^{tr}_{mn}(k)=\frac{1}{ik^2}\left(g^{mn}-\frac{k_m k_n}{k^2}\right)\to \frac{1}{Z_3}\frac{1}{ik^2}\left(g^{mn}-\frac{k_m k_n}{k^2}\right)= \]

\[ =\frac{1}{Z_3}D^{tr}_{mn}(k). \tag{5.9} \]

Considering next the process of inserting \(Z_3\)-vertices into the external photon lines of diagrams, we find, taking into account the weakened Lorentz condition imposed on the admissible states, that counterterms quadratic in \(A(p)\) lead to the transformation

\[ A_m(p)\to Z_3^{-1}A_m(p). \tag{5.10} \]

Summarizing the results of the preceding reasoning on the basis of (5.7), (5.8), (5.9), and (5.10), we arrive at the conclusion that the passage from the “bare” interaction Lagrangian (5.6) to expression (5.5) is equivalent to the following transformation of the propagation factors and free operators in the \(v\)-matrix:

\[ S^c(p)\to Z_2^{-1}S^c(p);\qquad D^{tr}(p)\to Z_3^{-1}D^{tr}(p); \]

\[ \bar\psi(p)\to Z_2^{-1}\bar\psi(p);\qquad \psi(p)\to Z_2^{-1}\psi(p);\qquad A(p)\to Z_3^{-1}A(p) \tag{5.11} \]

and of the fermion charge

\[ e\to Z_1 e. \tag{5.12} \]

In order to pass to the usual causal functions, it is sufficient to renormalize the operators \(\bar\psi,\ \psi\), and \(A\).

Assuming

\[ \bar{\psi}'=Z_2^{1/2}\bar{\psi};\qquad \psi'=Z_2^{1/2}\psi;\qquad A'=Z_3^{1/2}A, \tag{5.13} \]

we obtain from (5.11) the usual causal functions for internal lines.

If we take into account the renormalization (5.13) in the operators of the amplitudes of states, then we find that the operators \(\psi'\) and \(A'\), without any additional factors, will now correspond to the external photon and spinor lines.

Introducing into \(L\) the term \(\delta m\bar{\psi}\psi\), we arrive at the expression

\[ L=Z_1\sqrt{4\pi}\,e:\bar{\psi}\hat{A}\psi:-\delta m:\bar{\psi}\psi: =\sqrt{4\pi}e':\bar{\psi}'\hat{A}'\psi':-\delta m':\bar{\psi}'\psi':, \tag{5.14} \]

where, taking into account the Ward identity (4.50),

\[ e'=eZ_1Z_2^{-1}Z_3^{-1/2}=eZ_3^{-1/2};\qquad \delta m'=\delta m Z_2^{-1}. \tag{5.15} \]

Let us note here that, in contrast to all the other quadratic counterterms, the expression \(\delta m\bar{\psi}\psi\) cannot be reduced to some renormalization or replacement of causal functions and potentials, and therefore must necessarily be left in the interaction Lagrangian. Indeed, it is not difficult to see that the inclusion of vertices of the type \(\delta m'\) in internal fermion lines leads to a new fermion mass

\[ \frac{1}{\hat{p}-m}\to \frac{1}{\hat{p}-m'};\qquad m'=m+\delta m. \]

However, the inclusion of \(\delta m'\)-vertices in external fermion lines leads to the expression

\[ \psi(p)\to \frac{\hat{p}-m}{\hat{p}-m'}\psi(p), \]

which gives zero when the matrix element is calculated. Thus, from the point of view of the matrix elements of the \(S\)-matrix, the introduction of the counterterm \(\delta m\bar{\psi}\psi\) cannot be consistently described by a change of the fermion mass.

On the basis of formula (5.14) we arrive at the conclusion that, in the process of eliminating divergences, we have two arbitrary constants \(e'\) and \(\delta m'\), which can be determined from the condition that the quantities \(e\) and \(m\) coincide with the experimental values of the charge and mass of the electron.

We shall now analyze in somewhat greater detail the elements of arbitrariness contained in the subtractive formalism we have used.

Let us first consider the strongly connected diagrams, subject to the subtraction procedure, with two external spinor lines (the fermion self-energy diagrams) shown in Fig. 19. Denote the sum of all such diagrams (more precisely, the sum of the corresponding coefficient functions) by \(\Sigma(p)\). Then the complete fermion propagator factor can be represented by a sum of terms of the type depicted in Fig. 20 and containing any number of elements \(\Sigma\). Carrying out the indicated summation, we find:

Fig. 19.

\[ \frac{1}{\hat p-m} + \frac{1}{\hat p-m}\,\Sigma\,\frac{1}{\hat p-m} + \frac{1}{\hat p-m}\,\Sigma\,\frac{1}{\hat p-m}\,\Sigma\,\frac{1}{\hat p-m} +\cdots = \]

\[ = \frac{1}{\hat p-m} \left[ 1-\frac{\Sigma(p)}{\hat p-m} \right]^{-1}. \]

the complete fermion propagator factor in the form

\[ G(p)=i(\hat p-m-\Sigma(p))^{-1}, \tag{5.16} \]

Similarly, summing the contribution to the photon lines from the photon self-energy parts \(\Pi(k)\),

\[ \Pi^{mn}(p)=\pi(p)\left(g^{mn}-\frac{p^m p^n}{p^2}\right), \]

we arrive at the complete photon propagator factor

\[ \mathfrak{G}_{mn}(p)=\frac{i}{p^2-\pi(p)} \left(g^{mn}-\frac{p^m p^n}{p^2}\right). \tag{5.17} \]

Considering diagrams of vertex type, i.e. diagrams with two external fermion lines and one external photon line,

Fig. 20.

we introduce their sum \(\Gamma(p,k)\), which we shall represent in the form

\[ e\Gamma^n(p,k)=e\gamma^n+\Lambda^n(p,k). \tag{5.18} \]

Let us now note that relations of the type (5.16), (5.17), and (5.18) are valid both before the subtraction procedure and after it. We shall now establish the form of the transformation of the quantities \(G\), \(\mathfrak{G}\), and \(\Gamma\) upon add-

...in the Lagrangian of the counterterms (5.5), i.e., of all counterterms except \(\delta m\bar\psi\psi\).

As was established earlier, the factors of the internal lines of the diagram and the vertex factors \(e\) undergo transformations of the form (5.11), (5.12). Assigning the renormalization factors to the vertices, we obtain the transformation of the fermion charge (5.15) at all internal vertices of the proper-energy parts \(\Sigma\) and \(\Pi\), except for the pair of vertices in each case into which the external lines enter. Therefore in \(\Sigma(p)\) two factors \(\sqrt{Z_2}\) will be missing, and in \(\Pi(p)\) two factors \(\sqrt{Z_3}\). Multiplying, we obtain the transformation law for \(\Sigma\) and \(\Pi\):

\[ \Sigma_{\mathrm{unren}}(p)=\Sigma(p,e)\to Z_2\Sigma_{\mathrm{ren}}(p)=Z_2\Sigma(p,e'), \tag{5.19} \]

\[ \Pi_{\mathrm{unren}}(k)=\Pi(k,e)\to Z_3\Pi_{\mathrm{ren}}(k)=Z_3\Pi(k,e'). \tag{5.20} \]

Passing to the transformations \(G\) and \(\mathfrak{G}\), we note that in the summation process leading to expressions (5.16) and (5.17) it is necessary only to replace

\[ \Sigma(p,e)\to Z_2\Sigma(p,e');\qquad \pi(k,e)\to Z_3\pi(k,e'); \]

\[ \frac{1}{\hat p-m}\to Z_2^{-1}\frac{1}{\hat p-m};\qquad \frac{1}{k^2}\to Z_3^{+1}\frac{1}{k^2}, \]

which gives

\[ G(p,e)\sim(\hat p-m-\Sigma(p,e))^{-1}\to Z_2^{-1}G(p,e'), \tag{5.21} \]

\[ \mathfrak{G}(k,e)\sim(k^2-\pi(k,e))^{-1}\to Z_3^{-1}\mathfrak{G}(k,e'). \tag{5.22} \]

By analogous reasoning it is easy to see that as a result of the subtraction procedure, instead of

\[ e\Gamma^n(p,k,e)=e\gamma^n+\Lambda^n(p,k,e) \]

we obtain the expression

\[ e\gamma^n+Z_1^{-1}Z_2Z_3^{1/2}\Lambda^n(p,k,e')= \]

\[ =Z_1^{-1}Z_2Z_3^{1/2}\bigl(e'\gamma^n+\Lambda^n(p,k,e')\bigr) =Z_1^{-1}Z_2Z_3^{1/2}e'\Gamma^n(p,k,e') \tag{5.23} \]

or, taking into account the Ward identity,

\[ e\Gamma^n(p,k,e)\to Z_3^{1/2}e'\Gamma^n(p,k,e'). \tag{5.24} \]

Let us also note that, with the aid of the factors \(G\), \(\mathfrak{G}\), and \(\Gamma\), the calculation of any process of arbitrarily high order can be carried out by means of the so-called “skeleton” Feynman diagrams. The collection of skeleton diagrams is obtained from the collection of ...

of all connected diagrams by eliminating from them diagrams including the elements shown in Fig. 21. Therefore skeletal diagrams contain neither self-energy parts nor vertex parts. But, in computing the corresponding coefficient functions, we must use not the ordinary propagation factors \(S^c\) and \(D^c\), but the complete propagation factors \(G\) and \(\mathfrak G\), and to the vertices of a skeletal diagram assign not \(\gamma^n\), but \(\Gamma^n\). It is then obvious that the factors \(G\) and \(\mathfrak G\) are complete influence functions, i.e. complete Green functions including all radiative corrections. We emphasize that with such a computational scheme, working with the renormalized functions \(G\), \(\mathfrak G\), and \(\Gamma\), we shall no longer have to resort to the subtraction procedure, since, as was shown earlier, subtraction need be applied only to elements of diagrams of the type shown in Fig. 21.

Fig. 21.

Let us now turn to specifying the subtraction procedure and to the final removal of the arbitrariness in the subtracted polynomials. In the general definition of the subtraction operation (§ 2) we agreed to subtract from divergent expressions (in the present case \(\Sigma\), \(\Pi\), and \(\Lambda\)) the first terms of the expansion at the zero point. For practical purposes, however, it proves more convenient to take the center of expansion of the proper self-energy part of the fermion at the point \(\hat p = m\). We shall therefore put

\[ \Sigma_{\mathrm{ren}}(\hat p) = \Sigma(\hat p) - \left. \Sigma(\hat p) \right|_{\hat p=m} - \left. \frac{\partial \Sigma(\hat p)}{\partial \hat p} \right|_{\hat p=m} (\hat p-m) + A(\hat p-m). \tag{5.25} \]

Here we have used the circumstance that \(\Sigma\) depends on \(p\) only through \(\hat p\) and \(p^2=(\hat p)^2\). With this method of subtraction

\[ \left. \Sigma_{\mathrm{ren}}(\hat p) \right|_{\hat p=m} = 0 \tag{5.26} \]

and the mass \(m\) turns out to be equal to the experimental mass of the fermion.

This can be noticed directly either from the fact that the pole of the complete Green’s function

\[ G_{\mathrm{ren}}(p)=\frac{i}{\hat p-m-\Sigma_{\mathrm{ren}}(p)} \]

turns out to coincide with the pole of the function

\[ S^c(p)=\frac{i}{\hat p-m}, \]

or from the fact that radiative corrections to the external fermion lines give a zero contribution by virtue of property (5.26).

Let us next consider the subtraction process for the vertex part \(\Gamma^n(p,k)\). As was shown in the preceding paragraph, the requirement of gradient invariance implies that the subtraction procedure for \(\Lambda\) is completely determined by the subtraction for \(\Sigma\)*). From (5.25) we obtain directly:

\[ \Lambda^n_{\mathrm{ren}}(p,k)=\Lambda^n(p,k)-\Lambda^n_{(1)}+A\gamma^n, \]

\[ e\Gamma^n_{\mathrm{ren}}(p,k)=e(1+A)\gamma^n+\Lambda^n(p,k)-\Lambda^n_{(1)}, \tag{5.27} \]

where

\[ \Lambda^n(p,0)=i\,\frac{\partial \Sigma(\hat p)}{\partial p^n}\,g^{nn}, \]

\[ \Lambda^n_{\mathrm{ren}}(p,0)=ig^{nn}\,\frac{\partial \Sigma_{\mathrm{ren}}(\hat p)}{\partial p^n} \tag{5.28} \]

and, consequently,

\[ \Lambda^n_{(1)}=i\,\frac{\partial \Sigma(\hat p)}{\partial \hat p}\bigg|_{\hat p=m}\cdot \gamma^n . \]

Proceeding to the concrete determination of the constant \(A\), let us note, on the one hand, that by the manner in which this constant is introduced its value does not affect the magnitude of the fermion mass \(m\). Moreover, by virtue of the Ward identity the value of \(A\) does not affect the magnitude of the fermion charge \(e\). To see this, let us observe that the exclusion of \(A\) from expressions (5.25) and (5.27) can be carried out by means of a finite renormalization of the field operators and of the charge \(e\). However, by virtue of the Ward identity the corresponding finite constants \(z_1\) and \(z_2\) will be equal to each other and the value of \(e\) will not change (cf. (5.15)).

\[ \text{*) Since the expansion of }\Lambda\text{ in }e\text{ begins with a term of order }e^3, \]
an arbitrary constant is determined by the form of \(\Sigma\) in the preceding order in \(e\). For the term \(e\gamma^n\) such a correspondence does not exist.

Therefore it may prove convenient to determine \(A\) from the requirement that the radiative corrections to the external fermion lines be equal to zero:

\[ \left[ \frac{1}{\hat p-m}\,\Sigma_{\mathrm{ren}}(\hat p) \right]_{\hat p=m} =0. \tag{5.29} \]

In view of the fact that the sum of the first three terms on the right-hand side of (5.25) has at the point \(\hat p=m\) a zero of order higher than the first, the indicated condition gives:

\[ A=0. \tag{5.30} \]

It remains for us to determine also the subtraction operation for the proper energy part of the photon. Placing, as usual, the center of the expansion at the point \(k=0\), we have:

\[ \Pi_{\mathrm{ren}}^{mn}(k) = \Pi^{mn}(k)-\Pi^{mn}(0) -\sum_l \left. \frac{\partial \Pi^{mn}}{\partial k^l} \right|_{k=0} k^l - \frac{1}{2}\sum_{l,s} \left. \frac{\partial^2 \Pi^{mn}}{\partial k^l \partial k^s} \right|_{k=0} \cdot k^l k^s + C\left(g^{mn}k^2-k^m k^n\right). \tag{5.31} \]

The finite constant \(C\) cannot be directly determined from a requirement analogous to the requirement of conservation of the fermion mass, since the term

\[ C\left(g^{mn}k^2-k^m k^n\right) \tag{5.32} \]

does not change the pole of the function \(\mathfrak{G}\) (at \(k=0\)). The expression (5.30), however, can be removed from the interaction Lagrangian by simultaneously carrying out a finite renormalization of the charge by the factor

\[ (1+C)^{-1/2}. \tag{5.33} \]

We thus arrive at the possibility of determining the arbitrary finite constant \(C\) from the condition that \(e\) coincide with the experimental value of the fermion charge. Assuming that the experimental value of the charge is determined in the act of scattering, by a fermion, of a photon of zero energy, we obtain that such a process is described by the expression

\[ e\bar\psi(p)\Gamma_{\mathrm{ren}}^{\,n}(p,0)\psi(p). \tag{5.34} \]

When computing (5.34), the contribution to the difference entering into \(\Gamma_{\mathrm{ren}}^{\,n}\)

\[ \Lambda^n(p,0)-\Lambda_{(1)}^n = ig^{nn}\frac{\partial \Sigma(\hat p)}{\partial p^n} - i\gamma^n \left. \frac{\partial \Sigma(\hat p)}{\partial \hat p} \right|_{\hat p=m} \tag{5.34a} \]

turns out to be equal to zero. To verify this, note that \(\Sigma(\hat p)\) can be represented in the form

\[ i\Sigma(\hat p)=(\hat p-m)f(p^2)+m\varphi(p^2). \]

Calculating (5.34a), we find:

\[ \Lambda^n(p,0)-\Lambda^n_{(1)} = (\hat p-m)2p^n\frac{\partial f}{\partial p^2} + 2p^n\frac{\partial\varphi}{\partial p^2}m + \]

\[ +\gamma^n[f(p^2)-f(m^2)]-2\hat p\gamma^n m\frac{\partial\varphi}{\partial p^2}. \]

Defining the matrix element

\[ \bar\psi(p)\bigl(\Lambda^n(p,0)-\Lambda^n_{(1)}\bigr)\psi(p) \]

with account taken of the field equations

\[ \bar\psi(p)(\hat p-m)=0;\qquad (\hat p-m)\psi(p)=0 \]

and using the transformation

\[ 2(p^n-m\gamma^n)=(\hat p-m)\gamma^n+\gamma^n(\hat p-m), \]

we convince ourselves that it is equal to zero.

Taking the renormalization (5.33) into account, we finally obtain

\[ e\bar\psi(p)\Gamma^n_{\mathrm{ren}}(p,0)\psi(p) = \frac{e}{\sqrt{1+C}}\,\bar\psi(p)\gamma^n\psi(p), \]

whence it follows that the condition for \(e\) to coincide with the experimental value of the charge has the form

\[ C=0. \tag{5.35} \]

It is not difficult to verify that condition (5.35) also ensures the absence of radiative corrections in the external photon lines. We can now formulate the following prescription for a subtraction procedure ensuring coincidence of the “seed” constants \(m\) and \(e\) with the experimental values of the fermion mass and charge.

To eliminate the infinities from the proper energy parts of the fermion \(\Sigma(p)\) and the photon \(\Pi(k)\), the first terms of the Taylor polynomials with centers of expansion about the points \(\hat p=m\) and \(k=0\), respectively, are subtracted from them,

\[ \Sigma_{\mathrm{ren}}(\hat p) = \Sigma(\hat p)-\Sigma(m) - \left. \frac{\partial\Sigma(\hat p)}{\partial \hat p} \right|_{\hat p=m} \cdot(\hat p-m), \tag{5.36} \]

\[ \Pi_{\mathrm{ren}}(k)= \]

\[ = \Pi(k)-\Pi(0) - \left. \sum_n \frac{\partial\Pi}{\partial k^n} \right|_{k=0} k^n - \left. \frac{1}{2}\sum_{n,m}\frac{\partial^2\Pi}{\partial k^n\partial k^m} \right|_{k=0} k^n k^m. \tag{5.37} \]

In this case the subtracted part \(\Lambda_1\) of the vertex function

\[ \Lambda_{\mathrm{ren}}^{\,n}(p,k)=\Lambda^{n}(p,k)-\Lambda_{(1)}^{\,n} \tag{5.38} \]

has the form

\[ \Lambda_{(1)}^{\,n}= i\,\left.\frac{\partial \widehat{\Sigma}(p)}{\partial \widehat{p}}\right|_{\widehat{p}=m}\cdot \gamma^{n}. \tag{5.39} \]

Let us finally give the explicit expressions for the functions \(\Sigma\), \(\Pi\), and \(\Lambda\) in the first nonvanishing approximation, satisfying the subtracted conditions.

The corresponding expression for the self-energy part of the fermion \(\Sigma\) can be obtained by substituting (1.22) into formula (5.36). In doing so, when calculating the derivative of (1.22) with respect to \(\widehat{p}\) at the point \(\widehat{p}=m\), we arrive at a logarithmically divergent expression of the form

\[ \left. \int_{0}^{1}\frac{dx}{m^{2}-xp^{2}} \right|_{m^{2}\to p^{2}} \sim \left. \ln(p^{2}-m^{2}) \right|_{p^{2}\to m^{2}}, \]

which is a manifestation of the infrared catastrophe. To eliminate the indicated divergence, as is known, one introduces, in the intermediate stages of the argument, a vanishingly small photon mass \(\lambda_{0}\). Modifying the corresponding calculations of § 1, we obtain, instead of (1.22),

\[ \Sigma^{(2)}(p)=\frac{i e^{2}}{2\pi}\left\{ c_{1}(\widehat{p}-m)+c_{2}m+ \int_{0}^{1} dx\,(2m-\widehat{p}x)\, \ln \frac{m^{2}(1-x)} {(m^{2}-xp^{2})(1-x)+x\lambda_{0}^{2}} \right\}. \tag{5.40} \]

Determining here the constants \(c_{1}\) and \(c_{2}\) from the conditions

\[ \Sigma^{(2)}(m)=0;\qquad \left. \frac{\partial \Sigma^{(2)}(p)}{\partial \widehat{p}} \right|_{\widehat{p}=m} =0 \tag{5.41} \]

and calculating the integral with respect to \(x\), we arrive at the expression

\[ \Sigma^{(2)}(p)= \frac{e^{2}}{2\pi i} \left\{ \frac{p^{2}-m^{2}}{p^{2}}\ln(p^{2}-m^{2}) \left[ \frac{p^{2}+m^{2}}{2p^{2}}\,\widehat{p}-2m \right] + \frac{\widehat{p}}{2}\,\frac{p^{2}-m^{2}}{p^{2}} + (\widehat{p}-m)\left(1+\ln\frac{\lambda_{0}^{2}}{m^{2}}\right) \right\}. \tag{5.42} \]

Proceeding in an analogous manner, we construct the quantity \(\Pi^{(2)}(k)\), determining the constant \(c_{3}\) in formula (1.30) in accordance with (5.29),

i.e., from the condition that the second derivatives vanish,

\[ \left.\frac{\partial^2 \Pi^{(2)mn}(k)}{\partial k^p \partial k^e}\right|_{k=0}. \]

In this way we obtain

\[ \Pi^{(2)mn}(k)=\frac{2e^2}{\pi i}(k^n k^m-g^{mn}k^2)\int_0^1 dx\cdot x(1-x)\ln\left|\frac{m^2-x(1-x)k^2}{m^2}\right|. \tag{5.43} \]

Finally, the expression for the vertex part

\[ \sqrt{4\pi e}\,\Gamma^{(3)n}(p,k)=\sqrt{4\pi e}\left(\gamma^n+\widetilde{\Lambda}^{(3)n}(p,k)\right) \]

will be obtained from condition (5.28)

\[ \widetilde{\Lambda}^{(3)n}(p,0)=i g^{mn}\frac{\partial \Sigma^{(2)}(p)}{\partial p^n}, \tag{5.44} \]

which gives the following expression:

\[ \begin{aligned} \widetilde{\Lambda}^{(3)n}(p,k) &=\frac{e^2\gamma^n}{2\pi}\left\{ \int_0^1 dx\int_0^{1-x} dy\, \ln\left|\frac{(1-x)m^2}{(1-x)m^2-xp^2-yk^2+(xp-yk)^2}\right| +\ln\frac{\lambda_0^2}{m^2}-\frac{3}{4} \right\} \\ &\quad+\frac{e^2}{2\pi}\int_0^1 dx\int_0^{1-x} dy \times \\ &\quad\times \frac{ \gamma^n m^2-2k^n m^2-\hat{k}\gamma^n(x\hat{p}-y\hat{k})+4m(k^n y-p^n x)+(\hat{p}x-\hat{k}y)\gamma^n(\hat{p}x-\hat{k}y) }{ (1-x)m^2-xp^2-yk^2+(xp-yk)^2 }. \end{aligned} \tag{5.45} \]

The quantities \(\Sigma^{(2)}\), \(\Pi^{(2)}\), and \(\Lambda^{(3)}\), expressed by relations (5.42), (5.43), and (5.45), make it possible to compute directly the radiative corrections of the lowest orders.

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