Deformed Heisenberg algebras and minisuperspace formulations

This thesis focuses on the development of semiclassical and quantum formalisms that could provide new suitable frameworks to address the dynamics of homogeneous cosmological models in the proximity of their initial singularity. Such theories are constructed in such a way as to subsume into their foundational structure effects and corrections that supposedly arise when a quantum nature of spacetime is considered. The main physical idea underlying this effective approach is based on the belief that these corrections should come into play roughly at the same scale at which the cosmological systems of interest display quantum features. For the present study, the desired framework is provided by Generalized Uncertainty Principle (GUP) theories. GUP theories are quantum non-relativistic theories stemming from a deformation of the ordinary Heisenberg algebra. Within these frameworks, two different remarkable properties can arise: on the one hand, the appearance of a nonzero absolute minimal uncertainty in the coordinate operators, which equips the theory with a "natural" minimal scale; on the other, the emergence of a non-commutative "geometry", manifesting itself in the non-commutativity of the coordinate operators. The resulting new quantum theory is then configured as an extension that is able to represent, at the effective level, a different fundamental structure of spacetime and provide - in first approximation - a "correct" quantization procedure for the degrees of freedom of the gravitational field, with the possibility of inducing a faithful transposition at the semiclassical level as well. In view of this, the first part of the work is devoted to the analysis of the formal structure of GUP theories, in both their one-dimensional formulation and their more complicated n-dimensional non-commutative extensions. At the quantum level, a pivotal role is played by the investigation of the localization properties of the theories emerging in configuration space - properties that are clearly altered by the presence of the minimal scale and non-commutativity - and by the development of new proposals to recover information on position. Alongside this formulation, a classical definition of GUP theories is given and their symplectic structure is carefully defined in order to correctly provide a semiclassical Hamiltonian language for the dynamics of the systems of interest. Building on the theoretical framework established in the first part, the second part of this work investigates the deformed dynamics of two distinct cosmological systems using all the tools developed. More specifically, we first examine the non-commutative GUP quantization of the Bianchi I model, which represents the simplest cosmological anisotropic model. When considering the probabilistic interpretation of the wave function, the evolution of the anisotropy degrees of freedom presents a markedly different picture compared to the ordinary scenario, from which it clearly emerges how, in particular, the GUP corrections affect the probabilistic persistence of the initial condition of the Universe. We then undertake a semiclassical non-commutative GUP study of the celebrated Mixmaster model. In the standard case, this model describes the chaotic behavior of the Bianchi IX model as it approaches the initial singularity, serving as a prototype for the general inhomogeneous General Relativity solution in this regime, in accordance with the Belinski–Khalatnikov–Lifshitz (BKL) conjecture. In this setting, we are able to appreciate how the GUP effects drastically change the behavior of the cosmological model, leading to a replacement of the ergodic and chaotic dynamics mentioned above with a quasi-periodic oscillatory behavior in proximity to the initial singularity. As a final consideration, this thesis aims to be a clear demonstration of how suitable effective models can provide valuable insights into the dynamics of quantum gravitational systems, in particular quantum cosmological models, and consequently into what we should and could expect from a purely quantum phase of our Universe and, more generally, from a fundamental quantum theory of gravity.