Fock representations from U(1) holonomy algebras

TL;DR

This paper analyzes the quantization of U(1) holonomy algebras using algebraic techniques, clarifying the role of smeared loops and Poincare invariance in constructing Fock representations relevant to loop quantum gravity.

Contribution

It provides a detailed clarification of the mathematical structure behind Fock representations from U(1) holonomy algebras, connecting loop quantum gravity methods with traditional Fock space constructions.

Findings

Clarified the role of smeared loops in Fock representations

Re-examined early efforts in linearized gravity and Maxwell theory

Connected loop quantum gravity techniques with Fock space constructions

Abstract

We revisit the quantization of U(1) holonomy algebras using the abelian C* algebra based techniques which form the mathematical underpinnings of current efforts to construct loop quantum gravity. In particular, we clarify the role of ``smeared loops'' and of Poincare invariance in the construction of Fock representations of these algebras. This enables us to critically re-examine early pioneering efforts to construct Fock space representations of linearised gravity and free Maxwell theory from holonomy algebras through an application of the (then current) techniques of loop quantum gravity.