Thesis title: Evolutionary Minisuperspace Quantum Dynamics in the WKB approach
This Thesis deals with the application of the Vilenkin idea, firstly proposed in the 1969, to the Bianchi models exploiting in particular two different quantization procedures: the Standard one and the Polymer Quantum Mechanics. The whole work can be divided in three research areas.
The first one is the study of the Taub Cosmology, a particular case of the Bianchi IX model in which there are two equal cosmic scale factors. We developed a suitable technical algorithm to implement a separation of the Minisuperspace configurational variables into quasi-classical and purely quantum degrees of freedom, in the framework of both a Standard quantization procedure and a Polymer quantum Mechanics reformulation of the canonical dynamics. We then implemented the obtained general scheme to the specific case of a Taub Universe with a massless scalar field. In particular, with an original set of the Misner-Chitrè-like variables, we identified the quasi-classical variables in the Universe volume and a suitable function of the scalar field, while the purely quantum degree of freedom corresponds to the Universe anisotropy.
The resulting evolutionary (Schr\"{o}dinger) equation for this anisotropy variable has, in the spirit of this analysis, two main physical implications: first the Taub model is reduced to a cyclical Universe, evolving between a minimum and a maximum value of the Universe volume, offering an intriguing paradigm for the physical implementation of a cosmological history: clearly the maximum volume turning point is expected to live in a classical domain of the Universe dynamics, while the Bounce turning point has a pure quantum character, in the sense of a Polymer regularization; then the Universe anisotropy is always finite in value as a result of the singularity regularization and its specific value in the Bounce turning point depends on the initial conditions of the system, but in principle, it can be restricted to small enough values to make the Bounce dynamics unaffected by their behavior, i.e.\ the applicability of the Born-Oppenheimer approximation is here ensured. This study permitted us to submit an article, whose title is “WKB approximation for the Polymer quantization of the Taub Model”, that is undergoing the refereeing procedure.
In the second part we analysed the Bianchi IX Universe dynamics within the corner region associated to the potential term which the spatial curvature induces in the Minisuperspace. The study was done in two different cases: in the vacuum and in the presence of a massless scalar field plus a cosmological constant term. We investigated the dynamics in terms of WKB scenario for which the isotropic Misner variables (the volume) and one of the two anisotropic ones (and the scalar field when present) are treated on a semi-classical level, while the remaining anisotropy degree of freedom, the one trapped in the corner, is described on a pure quantum level. The quantum dynamics always reduced to the one of a time dependent Schroedinger equation for a harmonic potential with a time dependent frequency. The vacuum case is treated in the limits of both a collapsing and an expanding Universe, while the dynamics in presence of massless scalar field and cosmological constant is studied only in the case of crescent time. In both analysis the quantum dynamics of the anisotropy variable is associated to a decaying standard deviation of its probability density, corresponding to a suppression of the quantum anisotropy associated. In the vacuum case, the corner configuration becomes an attractor for the dynamics and the evolution resembles that one of a Taub cosmology in the limit of a non-singular initial Universe. This suggests that if the Bianchi dynamics enters enough the potential corner then the initial singularity is removed and a Taub picture emerges. The case when the scalar field and the cosmological constant are present well mimics the De-Sitter phase of an inflationary Universe. We showed that both the classical and quantum anisotropies are exponentially suppressed, so that the resulting dynamics corresponded to an isotropic closed Robertson-Walker geometry. This study permitted us to submit an article, whose title is “Quantum dynamics of the corner of the Bianchi IX model in the WKB approximation”, that has been published in Physical Review D 102.