We study a lattice gauge theory in Wilsons Hamiltonian formalism. In view of the realization of a quantum simulator for QED in one dimension, we introduce an Abelian model with a discrete gauge symmetry $mathbb{Z}_n$, approximating the $U(1)$ theory for large $n$. We analyze the role of the finiteness of the gauge fields and the properties of physical states, that satisfy a generalized Gausss law. We finally discuss a possible implementation strategy, that involves an effective dynamics in physical space.
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