  We analyze the canonical quantum dynamics of the isotropic Universe in a
metric approach by adopting a self-interacting scalar field as relational time.
When the potential term is absent we are able to associate the the expanding
and collapsing dynamics of the Universe with the positive and negative
frequency modes that emerge in the Wheeler-DeWitt equation. On the other side,
when the potential term is present a non-zero transition amplitude from
positive to negative frequency states arises, as in the standard relativistic
scattering theory below the particle creation threshold. In particular, we are
able to compute the transition probability for an expanding Universe that
emerges from a collapsing regime both in the standard quantization procedure
and in the polymer formulation. The probability distribution results similar in
the two cases and its maximum takes place when the mean values of the momentum
essentially coincide in the in-going and out-going wave packets, as it would
take place in a semiclassical Big Bounce dynamics.

General Relativity and Quantum Cosmology