Quantum Holonomy Fields

TL;DR

This paper develops a framework for quantum gauge theories using non-commutative algebraic structures called H-algebras, introducing quantum holonomy fields and generalizing the master field concept.

Contribution

It introduces a general definition of quantum holonomy fields over H-algebras and constructs these fields from quantum Lévy processes, extending gauge theory concepts.

Findings

Defined quantum holonomy fields over H-algebras

Constructed fields from quantum Lévy processes

Proposed higher-dimensional generalizations of the master field

Abstract

We investigate lattice and continuous quantum gauge theories on the Euclidean plane with a structure group that is replaced by a $H$-algebra; non-commutative analogues of groups and contain the class of Voiculescu's dual groups. We are interested in non-commutative analogues of random gauge fields, which we describe through the random Holonomy that they induce. We propose a general definition of a Quantum Holonomy Hield over a $H$-algebra and construct such fields starting from a quantum L\'evy process on a $H$-algebra. As an application, we define higher-dimensional generalizations of the so-called master field.