  We implement Polymer Quantum Mechanics on the Hamiltonian formulation of the
isotropic Universe in both the representations of the standard
Ashtekar-Barbero-Immirzi connection and of a new generalized coordinate
conjugate to the Universe volume. The resulting dynamics is a bouncing
cosmology; when quantizing the volume-like variable the Big Bounce is an
intrinsic cut-off on the cosmological dynamics, while when using the standard
connection the Bounce density results to be dependent on the initial conditions
of the prepared wave packet. Then we compare the nature of the resulting Bounce
with what emerges in Loop Quantum Cosmology, where the dependence of the
critical density on the initial conditions is present when the minimum area
eigenvalue is implemented in a comoving representation instead of the physical
one. We conclude that, if one hand the preferable scenario should be a Big
Bounce whose density depends on initial conditions in view of the privileged
SU(2) character that the Ashtekar-Barbero-Immirzi connection possesses in the
full Loop Quantum Gravity, on the other hand the equivalence demonstrated in
the context of polymer cosmology can be a hint in favour of the viability of
the improved scheme of Loop Quantum Cosmology even though it is not expressed
through the privileged set of variables.
