Cylinder Measures in Renormalization Theory

TL;DR

This paper introduces a new approach to renormalization using cylinder measures, aiming for a more rigorous mathematical foundation and providing explicit solutions for scalar boson theories.

Contribution

It develops a framework for renormalization based on cylinder measures and reduces the problem to strongly coupled theories, with explicit solutions for scalar bosons.

Findings

Reduced renormalization to strongly coupled theories.

Constructed explicit cylinder measures for scalar bosons.

Provided formal expressions for local field measures.

Abstract

The standard approach to renormalization relies, technically, on the asymptotic perturbation of Gaussian measures embodied in Feynman diagram theory. From a mathematical standpoint this is not good enough, because thereby solving the renormalization problem does not immediately traduce into having a rigorous construction of the corresponding theory. Here, we start developing an approach to renormalization based on cylinder measures. After explaining how renormalization can be mathematically better understood in terms of them, we argue that a renormalization problem can, under certain hypothesis, be reduced to that of the corresponding strongly coupled theory, and obtain a family of solutions for the case of scalar bosons whose interaction Lagrangian does not contain derivatives. As a further application, we produce an explicit, formal expression for the cylinder measure of a local field with effectively quartic interaction at any given, fixed scale, in arbitrary dimension.