Preamble

Uspekhi Mat. Nauk 49:5 (1994), 61-70 Russian Math.

Surveys 49:5 (1994), Bogolyubov's f?-operation and the Bogolyubov-Parasyuk theorem Zav'yalov 1. In 1994, when Nikolai Nikolaevich Bogolyubov would have celebrated his birthday, we can reach a more balanced judgement about his scientific achievements. I now believe that the Λ-operation suggested by Bogolyubov in 1956 for removing ultraviolet divergences from Feynman diagrams [l]-[4] and theorem he and Parasyuk then proved (see [4], [5]), namely that the Λ-operation defines a proper generalized function, are Bogolyubov's most important contributions in quantum field theory. I do not to belittle his other results, which were often more ingenious and truly important, but in this instance a cardinal problem was being addressed (and was solved) concerning existence of quantum field theory. the same reason I consider that the Bogolyubov-Parasyuk theorem is quite the most important result in quantum field theory in the period and comparable in significance to the discovery of gauge fields

methods

of their quantization. fact, in that period (essentially up to the present day if no account is taken of superrenormalized models) quantum field theory was really only the aggregate Feynman diagrams.

Various mathematical concepts—the continual integral, the Schwinger-Dyson equations, the Γ-exponent, and so on—were essentially purely ephemeral, playing the role of symbols that in some way made it possible to operate with this aggregate. The mathematical meaning of these symbols was regained only in "hothouse" conditions, when quantum-field Hamiltonians were subject to rigorous infrared ultraviolet prunings and the problem was in fact reduced to the case of finitely degrees of freedom. As far as the so-called constructive approach to simple models of quantum field theory [12] is concerned, although it was of course a highly respectable method it did not reveal a single phenomenon that would not have been seen from second-order perturbation theory.

Of course, Bogolyubov's discovery of the jR-operation was achieved, as always, "on the shoulders of giants". The basic outlines of the renormalization procedure were already clear from the work of Tomonaga [11], Schwinger [7], Dyson [8], Salam [9], and Feynman [10]. Renormalization was interpreted as a This research was supported by the Russian Foundation for Fundamental Research (grant 93-011-147).

68 O.I, Zav'yalov singular overdetermination of coupling constants, masses and functions (hence the name of the procedure). It was also understood that technically the matter reduces to introducing "infinite counterterms" into the Γ-exponent, which assumes a sophisticated passage to the limit.

However, a specific sequence of actions became rapidly more complicated from order to order, and prescriptions applicable to an arbitrary diagram did exist.

There was absolutely no guarantee that renormalization would in every case free diagrams from ultraviolet divergences.

Bogolyubov was the first to understand that an adequate language for this problem was the theory of generalized functions, which at that time was little referred to in the physics literature. In this language the renormalization problem was in each case transformed into determining the product of a certain number of generalized functions defined regularly over the whole coordinate space except for a certain manifold of lower dimension. The arguments relating to "infinite pure mass" and finite "physical mass", which difficult to formulate, thus replaced by explicit calculations in terms of the theory of extensions of functionals. This is what Bogolyubov wrote [1]: "If we reveal the (Feynman) coefficients ... we note that they include the products of singular functions with strong singularities on the light-cone. The rule of integration for each of these functions separately only holds if it is integrated with a sufficiently regular function, and the "integral" of their product needs a special definition. A direct application of the rules of integration ... leads to divergences long known as "ultraviolet catastrophe". ... problem is solved in the following way. We first determine directly the indicated functionals for a special class of functions that, together with all the partial derivatives up to a certain order, vanish when any pair of arguments- coincide.

These linear functionals are extended to the class of arbitrary regular functions. This operation ... is not unique. However, in certain cases this lack of uniqueness corresponds to arbitrariness in the renormalization of particle masses, coupling constants and dimensional units of fields." We shall see below that these words retain their force today.

2. Further commentary requires the introduction of precise formulations. The

Feynman amplitude diagram constructed propagators that are generalized functions in four-dimensional Minkowski space. Propagators most frequently met are those of the scalar field spinor field

Bogolyubov's R-operation and the Bogolyubov-Parasyuk theorem the Yang-Mills field

(3) <?,,(x) = (S-j

m is the mass of the corresponding particle, are Dirac matrices, is a parameter fixing the calibration. If 0 the functional D(x) is given locally for All the other propagators have similar singularities at zero, while outside the origin all these functions are regular. At intermediate stages it is convenient to use smoothed propagators by introducing under the integral in (l)-(3) rapidly decreasing formfactors g(M, q) which tend to unity as oo. We denote such smoothed propagators by the single symbol The Feynman graph containing vertices and lines is given by the incidence matrix ||e/y||, where 1,0,-1, respectively, if line / enters vertex passes round it, emerges from it. The graph is compared with the product of propagators (?/(*, - where corresponds to line /, and and / are the numbers of the vertices that are joined by this line. The Fourier transform Γ(/?ι,... ,ρν) = T{p) of Γ(χι,... is proportional to

(6) Γ(ρ) = J dqi ...

where G/(p) is the Fourier transform of Gi(x). coupling diagram contain spinor lines. number ω = 2L — + 4 is called its divergence index.

Any connected subgraph with a non-negative divergence index is called divergent. If Γ contains at least divergent subgraph, diverges in the limit of the removed regularization oo (underlined letters here and below denote the total aggregate of the corresponding quantities). To eliminate this kind of divergence we can properly use the i?-operation, which we shall now describe.

We consider a subset of vertices of a diagram and the aggregate of its lines joining pairs of vertices of this subset. Suppose that the subgraph obtained is connected.

The coefficient function of is defined naturally as an integral constructed in accordance with (6) applied to y. The "external" impulses of be taken also to be the impulses that are the arguments of certain "internal" lines of the diagram. (From this integral we can usually also single a ^-function of the sum of impulses expressing the general law of conservation of an impulse.) Such a coefficient function can clearly be "built into" the original amplitude (6).

70 O.I. Zav'yalov

one way or another, for any partition

(7) V = Vi * V 2 * · · · * V n

of the set V of vertices of the diagram into non-empty non-intersecting subsets V, it is possible to define on Γ each operator of the form ι)Δ(-γ )...Δ(7η), where the operators A(y,·) act only on the coefficient functions of the subgraphs y, generated by the aggregates V,.

As for the /{-operation, it is given by the recurrent relations: Δ(7ι)Δ( )·--Δ(7η).

Summation here is taken over all partitions of V into non-empty non- intersecting sets V, generating connected diverging subgraphs y,·. In turn each operation A(y,) is defined by

A( 7i ) = 1

if the set V consists of one vertex, and by

^ fc