Differential Geometry for the Space of Connections Modulo Gauge Transformations

TL;DR

This paper extends projective limit techniques to develop a differential geometric framework for the space of connections modulo gauge transformations, enabling new operator constructions and quantum representations.

Contribution

It introduces a differential geometric approach using projective limits for the space of connections, facilitating the construction of operators and their quantum representations.

Findings

Defined a commutator algebra of vector fields

Introduced divergence and quantum representations of vector fields

Constructed regularized loop operators and Laplace operators

Abstract

(This short article is a continuation of a longer, review work, in the same volume of Proceedings, by Ashtekar, Marolf and Mour\~ao [gr-qc/9403042]. All the details and other results are to be found in joint papers of the author with Abhay Ashtekar.) The projective limit technics derived for spaces of connections are extended to a new framework which for the associated projective limit plays a role of the differential geometry. It provides us with powerfull technics for construction and studding various operators. In particular, we introduce the commutator algebra of `vector fields', define a divergence of a vector field and find for them a quantum representation. Among the vector fields, there are operators which we identify as regularised Rovelli-Smolin loop operators linear in momenta. Another class of operators which comes out naturally are Laplace operators.