Chen Integrals, Generalized Loops and Loop Calculus

TL;DR

This paper develops a rigorous mathematical framework using Chen integrals to study loops, leading to a topological group of generalized loops and a loop calculus applicable to gauge theories and quantum gravity.

Contribution

It introduces a topological algebra of separating functions on the group of loops, constructs a topological group of generalized loops, and develops a loop calculus with derivatives for gauge theories and quantum gravity.

Findings

Constructed a Hopf algebra structure on separating functions.

Defined a topological group of generalized loops.

Developed a loop calculus with endpoint and area derivatives.

Abstract

We use Chen iterated line integrals to construct a topological algebra ${\cal A}_p$ of separating functions on the {\it Group of Loops} ${\bf L}{\cal M}_p$. ${\cal A}_p$ has an Hopf algebra structure which allows the construction of a group structure on its spectrum. We call this topological group, the group of generalized loops $\widetilde {{\bf L}{\cal M}_p}$. Then we develope a {\it Loop Calculus}, based on the {\it Endpoint} and {\it Area Derivative Operators}, providing a rigorous mathematical treatment of early heuristic ideas of Gambini, Trias and also Mandelstam, Makeenko and Migdal. Finally we define a natural action of the "pointed" diffeomorphism group $Diff_p({\cal M})$ on $ \widetilde {{\bf L}{\cal M}_p}$, and consider a {\it Variational Derivative} which allows the construction of homotopy invariants. This formalism is useful to construct a mathematical theory of {\it Loop Representation} of Gauge Theories and Quantum Gravity. Figures available by request.