AQV VI. A holonomy groupoid formulation

TL;DR

This paper explores the use of holonomy groupoids and path connections to generalize the duality between connections and holonomies, proposing new algebraic structures for Loop Quantum Gravity.

Contribution

It introduces a holonomy groupoid formulation and new quantum variable algebras, extending the duality framework in Loop Quantum Gravity.

Findings

Proposes a holonomy groupoid approach to quantum gravity

Develops new algebraic structures for quantum variables

Links path connections with quantum geometric formulations

Abstract

The philosophy of the Loop Quantum Gravity approach is to construct the canonical variables by using the duality of infinitesimal connections and holonomies along loops. Based on this fundamental property for example the holonomy-flux *-algebra has been formulated. A generalisation of the one-to-one correspondence between infinitesimal objects: connections and curvature and path based objects: holonomy maps and parallel transports is used to replace the configuration space of the theory. This generalised duality is related to the concept of path connections and holonomy groupoids, which originally has been invented by Mackenzie and which is presented shortly in this article. Finally these objects are used to propose some new algebras of quantum variables for Loop Quantum Gravity.