  In this paper, we explore the question of how different gauge choices in a
gauge theory affect the tensor product structure of the Hilbert space in
configuration space. In particular, we study the Coulomb gauge and observe that
the naive gauge potential degrees of freedom cease to be local operators as
soon as we impose the Dirac brackets. We construct new local set of operators
and compute the entanglement entropy according to this algebra in $2+1$
dimensions. We find that our proposal would lead to an entanglement entropy
that behave very similar to a single scalar degree of freedom if we do not
include further centers, but approaches that of a gauge field if we include
non-trivial centers. We explore also the situation where the gauge field is
Higgsed, and construct a local operator algebra that again requires some
deformation. This should give us some insight into interpreting the
entanglement entropy in generic gauge theories and perhaps also in
gravitational theories.
