Towards an Analytic Theory of Stochastic and Quantum Fields

TL;DR

This paper introduces a rigorous method for constructing functional measures in stochastic and quantum field theories, providing new insights into the structure of renormalized interactions and their mathematical foundations.

Contribution

It presents a novel approach to defining functional measures using flows on measure spaces, with a focus on Boson fields and polynomial interactions, advancing the mathematical understanding of quantum fields.

Findings

Renormalized interaction Lagrangian acts as a generator of measure flows.

Generalized Appell polynomials characterize the density of measures.

Provides conceptual insights into the structure of quantum field measures.

Abstract

We propose a method for the rigorous construction of physically relevant functional measures. In shaping it we get several conceptual insights, which can perhaps be summarized by the following statement: the renormalized interaction Lagrangian should be the generator of a flow on a space of asymptotically free cylinder functional measures with density given, in the case of Boson fields with polynomial self-interaction, by a generalized form of the Appell polynomials.