Résumé:
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.
