The notion of Isolated Horizons has played an important role in gravitational
physics, being useful from the characterization of the endpoint of black hole
mergers to (quantum) black hole entropy. In particular, the definition of {\it
weakly} isolated horizons (WIHs) as quasilocal generalizations of event
horizons is purely geometrical, and is independent of the variables used in
describing the gravitational field. Here we consider a canonical decomposition
of general relativity in terms of connection and vierbein variables starting
from a first order action. Within this approach, the information about the
existence of a (weakly) isolated horizon is obtained through a set of boundary
conditions on an internal boundary of the spacetime region under consideration.
We employ, for the self-dual action, a generalization of the Dirac algorithm
for regions with boundary. While the formalism for treating gauge theories with
boundaries is unambiguous, the choice of dynamical variables on the boundary is
not. We explore this freedom and consider different canonical formulations for
non-rotating black holes as defined by WIHs. We show that both the notion of
horizon degrees of freedom and energy associated to the horizon is not unique,
even when the descriptions might be self-consistent. This represents a
generalization of previous work on isolated horizons both in the exploration of
this freedom and in the type of horizons considered. We comment on previous
results found in the literature.
