Large field renormalization. I: The basic step of the \({\mathbb{R}}\) operation
The article is the continuation of the author's investigations in the field of lattice gauge theories [ibid. 109, 249-301 (1987; Zbl 0611.53080); ibid. 116, No.1, 1-22 (1988; Zbl 0688.53045); ibid. 119, No.2, 243-285 (1988; Zbl 0664.58050)]. It is known that for the four-dimensional gauge theories the bare couplings constant behaves asymptotically as \((a+b \log \epsilon^{- 1})^{-1/2},\) for \(\epsilon\to 0\), with some positive constants a, b. Then the Wilson action has the bound which does not give any positive power of \(\epsilon\). This leads to the impossibility to control expressions arising in the large field regions for all steps of the procedure, i.e. until we reach the unit lattice. In such situations we have to change the procedure, i.e. we have to be able to renormalize the expression corresponding to the large field region. The renormalization operation \({\mathbb{R}}\) is constructed by the author. The expression of this operation is written in the following form \[ ({\mathbb{R}}\rho)(V)=\sum_{Z''}\rho (Z'',V)\quad \exp \sum {\mathbb{R}}(X,V) \] where V is a gauge field variable, \(\rho\) is a density, Z is a large field region which is decomposed into disjoint subregions \(Z''\) and \(Z'\), \(Z''\) is a union of components of the region Z, for which the small factors connected with large field control some number of next steps, \(Z'\) is a union of remaining components, i.e. components for which the corresponding expressions require a renormalization, \(X\cap Z^{''c}=\emptyset.\) The densities \(\rho (Z'',V)\) have enough small factors to control the given number of steps and the expression in the exponential can be renormalized. This renormalization makes the whole renormalization group procedure convergent. The mathematical methods used are the theory of operators in spaces of measures, the renormalization theory and the lattice theory.
scientific article; zbMATH DE number 4155534