Convolution product construction of interactions in probabilistic physical models

TL;DR

This paper introduces a probabilistic method for constructing interactions in physical models, especially quantum field theories, using convolution products of probability measures to model interactions.

Contribution

It presents a novel probabilistic convolution approach to model interactions in quantum field theories, enabling generalizations to infinite-dimensional Gaussian spaces.

Findings

Interaction terms are independent of free theories

Different free theories can share the same interaction

Construction applies to infinite-dimensional Gaussian spaces

Abstract

This paper aims to give a probabilistic construction of interactions which may be relevant for building physical theories such as interacting quantum field theories. We start with the path integral definition of partition function in quantum field theory which recall us the probabilistic nature of this physical theory. From a Gaussian law considered as free theory, an interacting theory is constructed by nontrivial convolution product between the free theory and an interacting term which is also a probability law. The resulting theory, again a probability law, exhibits two proprieties already present in nowadays theories of interactions such as Gauge theory : the interaction term does not depend on the free term, and two different free theories can be implemented with the same interaction. The direct use of Gaussian measures allows to generalize the present construction for infinite dimensional spaces equipped with Gaussian measures.