Some Aspects of the Generalized Uncertainty PrincipleTheoretical Development and Applications

Abstract

This thesis investigates some consequences arising from the modification of Heisen- berg’s Uncertainty Principle (HUP), postulated in quantum theory to introduce limit values in the position and momentum uncertainties. In this context, the HUP is re- placed by the so-called Generalized Uncertainty Principle (GUP). Firstly, by focusing on the fundamental physical aspect, we present the arguments of different hypotheses suggesting the existence of upper and lower bounds of some measurable quantities, as well as the related GUPs that have been proposed based on these hypotheses. We focus in particular on three GUPs: the one leading to the existence of a minimal length (suggested in various frameworks, such as quantized space-time theory, string theory and black hole physics), the GUP incorporating a minimal length and a maxi- mal momentum (emerging in doubly special relativity) and the GUP with a maximal length (predicted in cosmology). The formalism of deformed quantum mechanics, which occurs from these GUPs, is studied exhaustively in the second chapter. Es- pecially, we exhibit the modified commutation relations of position and momentum operators, the corresponding Hilbert space representations and the scalar product definition. Moreover, some theoretical developments are discussed by summarizing most important works in the literature. We consider furthermore some applications in statistical physics by focusing on the recent maximal-length GUP. In fact, three systems are investigated, namely, an ideal gas, an ensemble of harmonic oscillators and a relativistic gas. In this framework, the thermodynamic properties of these systems are studied within the canonical ensemble via the quantum and semiclassical approaches. The comparison with the results obtained in the context of the minimal-length GUPs indicates that the maximum length may induce new effects, which become important at high tem- peratures and for large volumes. In particular, a modified equation of state for ideal gases emerges in the scope of this new formalism. By analyzing some experimental data, we argue that the maximal length might be viewed as a macroscopic scale associated with the system under study.