Fock representations from $U\left(1\right)\mathrm{}$ holonomy algebras

We examine the quantization of $U\left(1\right)\mathrm{}$ 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 Poincaré invariance in the construction of Fock representations of these algebras. This enables us to critically reexamine early pioneering efforts to construct Fock space representations of linearized gravity and free Maxwell theory from holonomy algebras through an application of the (then current) techniques of loop quantum gravity.