  Several proposals to deal with the dynamics of general relativity involve
gauge fixings or the introduction matter fields in terms of which the theory is
deparameterized. The resulting theories have true Hamiltonians for their
evolution that usually involve square roots, and this poses certain challenges
for their implementation as self-adjoint quantum operators. We show in the
context of a simple model of totally constrained theory that one can introduce
related, well defined operators that reproduce semiclassically the same physics
as the original ones, at least for states peaked in the regions of phase space
where their associated classical quantities are well defined.
