Interpolations between Bosonic and Fermionic Relations given by Generalized Brownian Motions (revised version)

TL;DR

This paper introduces a new framework called generalized Brownian motion that interpolates between bosonic and fermionic relations, expanding the mathematical understanding of quantum field relations.

Contribution

It develops the theory of generalized Brownian motions and links them to Voiculescu's free product, providing a novel mathematical tool for quantum relations.

Findings

Established a generalized pairing rule for moments of fields.

Connected generalized Brownian motions with free product concepts.

Laid foundational theory for interpolating quantum relations.

Abstract

We present an interpolation between the bosonic and fermionic relations. This interpolation is given by an object which we call `generalized Brownian motion' and which is characterized by a generalization of the pairing rule for the calculation of the moments of bosonic and fermionic fields. We develop some basic theory for such generalized Brownian motions and consider more closely one example, which turns out to be intimately connected with Voiculescu's concept of `free product'.