Physics

A. D. SUKHANOV

ON THE QUESTION OF “SURFACE” DIVERGENCES

(Presented by Academician N. N. Bogolyubov on 29 III 1962)

1. The most serious shortcoming of the usual scheme of the Hamiltonian formalism \(^{(1)}\) is that, on its basis, even in perturbation theory, it is not possible to construct a finite Dyson matrix \(S(\sigma,-\infty)\), owing to the appearance, in addition to “ultraviolet” divergences, of special “surface” divergences \(^{(2)}\).

In this connection it was proposed \(^{(3,4)}\) to turn to the construction of the scheme of the Hamiltonian formalism within the framework of Bogolyubov’s method \(^{(5)}\). Along this path it proved possible to obtain the Tomonaga—Schwinger equation and a local interaction Hamiltonian of the expected form (for fixed regularization masses); however, when the boson self-energy diagram was considered, the problem of “surface” divergences again arose \(^{(4,5)}\).

In order to analyze more fully the mathematical and physical nature of this problem, let us consider it by the example of the usual Hamiltonian

\[ H(\tau)=\lim_{\substack{g\to\theta_\tau\\ x^0=\theta(\tau-x^0)}}\int\left\{-L(x)- \]

\[ -\sum_{n=1}^{\infty}\frac{1}{n!}\int \Lambda_{n+1}(x,x_1\ldots x_n)\,g(x_1)\ldots g(x_n)\,dx_1\ldots dx_n \right\}g'(x)\,dx \tag{1} \]

for the theory with \(L(x)=e:\varphi^4(x):\). Here, for simplicity, we shall restrict ourselves to considering, in second order, only that part of the quasilocal operator \(\Lambda_2(x,y)\) corresponding to the boson self-energy diagram, which contains derivatives of the \(\delta\)-function.

1. In this case

\[ \widetilde{H}_2(\tau)= \]

\[ =\frac{1}{2}\frac{e^2}{2!}\,B_2 \lim_{g\to\theta_\tau} \int dx\,dy\,g'(\tau-x^0)\,g(\tau-y^0):\varphi(x)\varphi(y): \left(\frac{\partial^2}{\partial x^2}+\frac{\partial^2}{\partial y^2}\right)\delta(x-y), \tag{2} \]

where \(B_2\) is a logarithmically divergent constant \(^{(4)}\).

Following the procedure proposed in \(^{(4)}\), we transfer the derivatives from \(\delta(x-y)\) and integrate with respect to \(y\). Then, after some transformations, we obtain

\[ \widetilde{H}_2(\tau)= \frac{1}{2}\frac{e^2}{2!}B_2 \lim_{g\to\theta_\tau}\int dx \left\{ :\varphi\frac{\partial^2\varphi}{\partial x^2}:\,[g^2(\tau-x^0)]' - :\frac{\partial\varphi^2}{\partial x^0}:\,[g'(\tau-x^0)]^2 \right\}. \tag{3} \]

In the first term in the curly brackets we apply the mean-value theorem and take the fields outside the integral sign with respect to \(x^0\), after which we take the integral with respect to \(x^0\) and pass to the limit \(g\to\theta_\tau\). In the present case this passage raises no doubts, so that

\[ \widetilde{H}_2(\tau)= -\int d\bar{x}\,\widetilde{L}_2(\bar{x},\tau;1) -\frac{1}{2}\frac{e^2}{2!}B_2 \lim_{g\to\theta_\tau}\int dx: \frac{\partial\varphi^2}{\partial x^0}: [g'(\tau-x^0)]^2, \tag{4} \]

where \(\widetilde{L}_2(\bar{x},\tau;1)\) is the corresponding part of the effective Lagrangian \(^{(5)}\) \(L(x;1)\). However, the attempt to pass to the limit in the second term leads to the appearance of a “surface” divergence. Indeed, if we directly

substitute \(g=\theta(\tau-x^0)\) under the integral sign, we obtain a product of \(\delta\)-functions of one and the same argument, i.e., a nonintegrable expression of the type indicated in \((5)\).

1. The physical cause of the appearance of such expressions is, in principle, the same as the cause of the appearance of “ultraviolet” divergences, namely, the illegitimate use of the mathematical concept of a point in quantum field theory for the description of physical processes. At the same time, if “ultraviolet” divergences arise as a consequence of the actual indefiniteness of chronological products and their products at points where the arguments coincide, then the cause of the appearance of “surface” divergences lies in the actual indefiniteness of the expressions that arise in the vicinity of the surface on which the interaction is switched off, i.e., at the point \(x^0=\tau\).

From the mathematical point of view, the appearance of such divergences in Bogoliubov’s method \((5)\), when passing directly to the limit \(g\to\theta_\tau\), is also understandable, since the quasilocal operators \(\Lambda_n\) introduced in \((5)\) are, even for fixed regularizing masses, generalized functions that are integrable only on the class of sufficiently smooth \((5)\) functions \(g(x)\), not including \(\theta\)-functions. As a result, for the boson self-energy diagram, which is the most strongly divergent diagram in renormalizable theories, the passage to the limit \(g\to\theta_\tau\) in the ordinary sense is impossible. However, the limiting expression can be given a quite definite meaning if the transition \(g\to\theta_\tau\) is understood not in the ordinary, but in the improper sense, i.e., as the definition of \(\Lambda_2(x,y)\) on a broader class of functions including \(\theta\)-functions.*

The transformations carried out above from (2) to (4) are precisely an attempt at such a definition. In doing so, we have succeeded in separating, from the entire expression of interest to us, the term in which the transition to the limit \(g\to\theta_\tau\) is realized directly, and a term requiring additional study. For the convenience of carrying it out, we introduce the notation

\[ f(x^0)=-\frac{1}{2}\frac{e^2}{2!}B_2\int d\bar{x}\,\frac{\partial\varphi^2}{\partial x^0}; \tag{5} \]

where \(f(x^0)\), as is known, is a sufficiently smooth function. Then the term in (4) that interests us takes the form

\[ \lim_{g\to\theta_\tau}\int dx^0\, f(x^0)\,[g'(\tau-x^0)]^2. \tag{6} \]

Let us emphasize that carrying out a direct definition of the operator \(\Lambda_2(x,y)\) on a class of functions including \(\theta\)-functions proves to be a very difficult problem, since the expression for \(\Lambda_2(x,y)\) is very rigidly fixed by general physical requirements \((5)\). Therefore we shall use here the following auxiliary device. Instead of directly defining \(\Lambda_2(x,y)\) on a class of functions including \(\theta\)-functions, we first consider a somewhat different problem, namely the problem of defining in formula (6) a new generalized function of one variable, which we shall symbolically denote by \(\delta^2(\tau-x^0)\), on the corresponding class of functions \(f(x^0)\).

It is not difficult to show that if the continuously differentiable function \(f(x^0)\) vanishes at the point \(x^0=\tau\), then the improper limit \(g\to\theta_\tau\) in (6) exists, i.e. \(\delta^2(\tau-x^0)\) is integrable on the subclass of such functions. Indeed, represent \(f(x^0)\) in the form

\[ f(x^0)=(x^0-\tau)f_1(x^0), \tag{7} \]

* In other words, the limiting expression can be given a quite definite meaning if it is possible to extend the generalized function \(\Lambda_2(x,y)\), considered as a linear continuous functional on the space of sufficiently smooth functions \(g(x)\), to a space of functions including \(\theta\)-functions.

where \(f_1(x^0)\) is a continuous function. Then

\[ \lim_{g\to\theta_\tau}\int dx^0\,(x^0-\tau)f_1(x^0)\,[g'(\tau-x_0)]^2 = \int dx^0\,(x^0-\tau)f_1(x^0)\,\delta^2(\tau-x^0) = \]

\[ = f_1(\tau)\int dx^0\,(x^0-\tau)\delta^2(\tau-x^0)\equiv 0. \tag{8} \]

Thus, \(\delta^2(\tau-x^0)\) is defined on the subclass of continuously differentiable functions that vanish at the point \(x^0=\tau\). Now the problem is to define it in such a way that it is integrable over the entire class of continuously differentiable functions. From the Hahn—Banach theorem (see, for example, \((^6)\)) it is known that such a definition of a generalized function, or, in other words, such an extension of the functional, is always possible, but is not unique.

In view of (8), it is natural to choose, as a particular method of such an extension of the functional, the identical zero; that is, to set

\[ \int dx^0 f(x^0)\delta^2(\tau-x^0)\equiv 0, \tag{9} \]

where \(f(x^0)\) is an arbitrary continuously differentiable function. To find the arbitrariness that arises under such an extension of the functional, it is enough to solve the operator equation immediately following from (8),

\[ (x^0-\tau)\delta^2(\tau-x^0)=0. \tag{10} \]

It can be shown \((^6)\) that the general solution of this equation has the form*

\[ \delta^2(\tau-x^0)=\alpha\delta(\tau-x^0), \tag{11} \]

where \(\alpha\) is an arbitrary finite constant; and, taking (9) into account, formula (11) gives us the desired definition of \(\delta^2(\tau-x^0)\) as an integrable function in the class of arbitrary continuously differentiable functions.

Having solved the auxiliary problem of defining \(\delta^2(\tau-x^0)\) on the class of continuously differentiable functions, and having clarified the character of the arbitrariness that thereby arises, we shall use the results obtained to define the quasilocal operator \(\Lambda_2(x,y)\) on the class of functions \(g(x)\) including \(\theta\)-functions, which would have been difficult to do directly. Namely, it is natural to choose, as such a definition, the expression

\[ \widetilde H_2(\tau) = -\int d\bar x\,\widetilde L_2(\bar x,\tau;1) - \]

\[ -\frac12\,\frac{e^2}{2!}B_2 \int d\bar x\, \lim_{g\to\theta_\tau} \int dx^0 \left[ :\frac{\partial\varphi^2(x)}{\partial x^0}: - :\frac{\partial\varphi^2(\bar x,\tau)}{\partial \tau}: \right] [g'(\tau-x^0)]^2 - \]

\[ -\frac12\,\frac{e^2}{2!}\alpha B_2 \int d\bar x:\frac{\partial\varphi^2(\bar x,\tau)}{\partial\tau}:, \tag{12} \]

where the second term is already finite as \(g\to\theta_\tau\) and, by virtue of (8), is identically equal to zero. In other words:

\[ \widetilde H_2(\tau) = -\int d\bar x\,\widetilde L_2(\bar x,\tau;1) - \frac12\,\frac{e^2}{2!}\alpha B_2 \int d\bar x:\frac{\partial\varphi^2(\bar x,\tau)}{\partial\tau}:, \tag{13} \]

which solves the problem of obtaining the ordinary Hamiltonian for the second-order proper-energy diagram of a boson.

1. Thus, in defining \(\Lambda_2(x,y)\) on a broader class of functions, we may proceed in the same way as is customary \((^5)\) when defining

* Indeed, by virtue of (10) \(\delta^2(\tau-x^0)=0\) if \(x^0\ne\tau\), and such generalized functions are representable \((^6)\) in the form \(\sum \alpha_i\delta^{(i)}(\tau-x^0)\). In the present case, only the first term remains from the whole sum.

of \(T\)-products at coincident arguments. The only difference here is that, since it is difficult to give a direct definition of \(\Lambda_2(x,\) we have from the very beginning explicitly singled out from \(\widetilde{H}_2(\tau)\) the term of the form
\[ - \lim_{g\to\theta_\tau}\int dx \times \widetilde{L}_2(x;1)\,[g^2(\tau-x^0)]', \]
which remains finite when the intermediate regularization is removed, i.e., when \(g\to\theta_\tau\), and subsequently considered a certain auxiliary problem that allowed us to regularize the remaining divergent term. It is, however, quite clear that this difference is inessential. Another difference between “surface” regularization and the ordinary one is connected with the problem of generalizing the consideration carried out here to higher orders. In the case of the usual regularization \({}^{(5)}\), this problem is far from trivial, since in different orders different nonintegrable expressions arise, and the solution of the question of the possibility of their successive definition, i.e., the construction of the \(R\)-operation \({}^{(5)}\), represents a serious problem. In the present case, however, by virtue of the symmetry of the quasilocal operators \({}^{(4)}\) and because, in renormalizable theories, there are no diagrams divergent in the ordinary sense more strongly than quadratically, in all orders there appear undefined expressions of the same type as those described above, or close to it. Thus the problem of constructing a new \(R\)-operation is considerably simplified, and it will be considered by us later.

Thus, the problem of obtaining the Hamiltonian \(H(\tau)\) in Bogoliubov’s method \({}^{(5)}\) can in principle be solved if, in the necessary case (i.e., for the boson self-energy diagram), we regard the passage to the limit \(g\to\theta_\tau\) in (1) as improper, i.e., define the limiting expression for \(H(\tau)\) as an integrable generalized function in the corresponding class of functions.

The finite arbitrariness that arises in this case has a “surface” nature, i.e., in solving the Tomonaga–Schwinger equation it contributes only to \(S(\sigma,-\infty)\), but not to the \(S\)-matrix, which is in agreement with \({}^{(7)}\). However, in order finally to clarify the question of the role of finite “surface” counterterms in \(S(\sigma,-\infty)\), it is necessary to obtain an expression for the Hamiltonian \(H(\tau)\) in arbitrary order, which will be done in the following work.

In conclusion I express my deep gratitude to Acad. N. N. Bogoliubov and V. S. Vladimirov, conversations with whom helped clarify for the author a number of fundamental questions.

Mathematical Institute named after V. A. Steklov
Academy of Sciences of the USSR

Received
28 III 1962

References Cited

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\({}^{2}\) E. C. G. Stueckelberg, Phys. Rev., 81, 130 (1951).
\({}^{3}\) A. D. Sukhanov, ZhETF, 41, 1915 (1961).
\({}^{4}\) A. D. Sukhanov, ZhETF, 43, No. 8 (1962).
\({}^{5}\) N. N. Bogoliubov, D. V. Shirkov, Introduction to the Theory of Quantized Fields, 1957.
\({}^{6}\) I. M. Gelfand, G. E. Shilov, Spaces of Basic and Generalized Functions, 1958.
\({}^{7}\) D. A. Slavnov, A. D. Sukhanov, ZhETF, 41, 1940 (1961).