  We study the Hamiltonian formulation of the Ashtekar-Olmedo-Singh model for
the description of the interior geometry of non-rotating, uncharged black
holes. This model incorporates loop quantum effects through the introduction of
two regularization parameters. We consider an extended phase space formalism
proposed by the creators of the model that includes such parameters as
configuration variables, constrained to be functions of the black hole mass. We
generalize this restriction, allowing for an off-shell phase space dependence.
We then introduce a gauge fixing procedure and reduce the system, proving that
the reduced symplectic structure cannot reproduce the standard relativistic one
in terms of the densitized triad and the Ashtekar-Barbero connection. Actually,
the reduced structure precisely compensates the modifications that arise in the
Hamilton equations when the regularization parameters are treated as phase
space functions, rather than as numbers, attaining a consistent Hamiltonian
derivation of the dynamics. We then choose the extended phase space formalism
as starting point to address the loop quantization of the model. Taking the
definition of certain geometric operators as the only basic ingredient and
adopting prescriptions that have proven successful in loop quantum cosmology,
we construct a polymer representation of all the constraints and deduce the
formal expression of the physical states, assuming reasonable spectral
properties for the constraint operators. The physical states turn out to be
characterized by a wave function of the black hole mass with support on a very
specific set. We finally discuss conditions that guarantee the existence of
physical states in the region of large black hole masses. This is a first step
in the development of a new loop quantum theory of black holes.
