Concerning a natural compatibility condition between the action and the renormalized operator product

Abstract

In this article we note that in a number of situations the operator product and the classical action satisfy a natural compatibility condition. We consider the interest of this condition to be twofold: First, the naturality (functoriality) of the compatibility condition suggests that it be used for geometrical applications of renormalized functional integration. Second, the compatibility can be used as the definition of a category; consideration of this category as the central object of study in quantum field theory seems to have quite some advantages over previously introduced theories of the type ``S-matrix theory'', ``Vertex operator algebras'', since this seems to be the only category in which both the action and the expectation values enter, the two being linked roughly speaking by a combination of the Frobenius property and the renormalized Schwinger-Dyson equation.