• Loop
• Quantum
• Gravity

The main aim of this thesis is to give a geometrical interpretation of ``spacetime grains'' at Planck scales in the framework of Loop Quantum Gravity.

My work consisted in analyzing the details of the interpretation of the quanta of space in terms of polytopes. The main results I obtained are the following:

We clarified details on the relation between polytopes and interwiners, and concluded that an intertwiner can be seen unambiguously as the state of a \emph{quantum polytope}.

Next we analyzed the properties of these polytopes: studying how to reconstruct the solid figure from LQG variables, the possible shapes and the volume. We adapted existing algorithms to express the geometry of the polytopes in terms of the holonomy-fluxes variables of LQG, thus providing an explicit bridge between the original variables and the interpretation in terms of polytopes of the phase space.

Finally we present some direct application of this geometrical picture. We defined a volume operator such as in the large spin limit it reproduce the geometrical volume of a polytope, we computed numerically his spectrum for some elementary cases and we pointed out some asymptotic property of his spectrum. We discuss applications of the picture in terms of polytopes to the study of the semiclassical limit of LQG, in particular commenting a connection between the quantum dynamics and a generalization of Regge calculus on polytopes.