  A connection modulo gauge symmetry on the trivial principal bundle $M\times
G$ is a morphism from the loop group of $M$ into $G$. Thus, considering only
loops around the 2-cells of a distinguished family of progressively refined
cellular structures on $M$, the observable algebra $A$ of an abelian gauge
field can be presented as an inductive limit of quotients of polynomial
algebras. In that context, it turns out that the state
$\mu_\lambda:A\rightarrow\mathbb{C}$ of the Yang-Mills field on the sphere can
be written $\mu_\lambda = \mu_0\mathrm{e}^{\lambda L}$ with $\lambda$ an
interaction strength parameter, $L:A\rightarrow A$ an explicit second-order
partial differential operator and $\mu_0$ the state of an almost surely flat
connection. Extrapolating, we provide analogous states for the case of abelian
gauge fields on $\mathbb{R}^d$.
