Quantum Isotropic Universe in RQM Analogy: the Cosmological Horizon

We investigate the quantum dynamics of the isotropic Universe in the presence of a free massless scalar field, playing the role of a physical clock. The Hilbert space is constructed via a direct analogy between the Wheeler-DeWitt equation in the minisuperspace and a relativistic scalar one in physical space. In particular, we show how the introduction of a "turning point" in the Universe evolution allows to overcome an intrinsic ambiguity in representing the expanding and collapsing Universe. In this way, the positive and negative frequencies are simply identified with time reversed states. The main subject of the present analysis is the construction of a horizon operator, whose quantum behavior is investigated when Polymer Quantum Mechanics is implemented to describe the asymptotic evolution near the initial singularity. The reason of this choice is motivated by the intrinsic spreading of localized wavepackets when the polymer dispersion relation governs the quantum dynamics. The evidence that the mean value of the quantum horizon operator follows its semiclassical behavior (corrected for polymerization) is a clear indication that a concept of causality can be restored also in the quantum cosmological picture.