  After reviewing the hole argument and its relations with initial value
problem and general covariance, we shall discuss how much freedom one has to
define the physical state of a system in a generally covariant (or gauge
covariant) field theory. We shall show that in gauge covariant theories (and
generally covariant theories with a a compact space) one has no freedom and one
is forced to declare as physically equivalent two configurations which differ
by a gauge transformation (or by a global spacetime diffeomorphism), as it is
usually prescribed. On the contrary, when space is not compact, the result
proven for the compact case does not hold true and one may have different
options to define physically equivalent configurations, still preserving
determinism.
