  We regularize compact and non-compact Abelian Chern-Simons-Maxwell theories
on a spatial lattice using the Hamiltonian formulation. We consider a doubled
theory with gauge fields living on a lattice and its dual lattice. The Hilbert
space of the theory is a product of local Hilbert spaces, each associated with
a link and the corresponding dual link. The two electric field operators
associated with the link-pair do not commute. In the non-compact case with
gauge group $\mathbb{R}$, each local Hilbert space is analogous to the one of a
charged "particle" moving in the link-pair group space $\mathbb{R}^2$ in a
constant "magnetic" background field. In the compact case, the link-pair group
space is a torus $U(1)^2$ threaded by $k$ units of quantized "magnetic" flux,
with $k$ being the level of the Chern-Simons theory. The holonomies of the
torus $U(1)^2$ give rise to two self-adjoint extension parameters, which form
two non-dynamical background lattice gauge fields that explicitly break the
manifest gauge symmetry from $U(1)$ to $\mathbb{Z}(k)$. The local Hilbert space
of a link-pair then decomposes into representations of a magnetic translation
group. In the pure Chern-Simons limit of a large "photon" mass, this results in
a $\mathbb{Z}(k)$-symmetric variant of Kitaev's toric code, self-adjointly
extended by the two non-dynamical background lattice gauge fields. Electric
charges on the original lattice and on the dual lattice obey mutually anyonic
statistics with the statistics angle $\frac{2 \pi}{k}$. Non-Abelian $U(k)$
Berry gauge fields that arise from the self-adjoint extension parameters may be
interesting in the context of quantum information processing.
