La quantizzazione del monopolo di Dirac: un approccio geometrico

This thesis presents an introduction to the geometric foundations of classical gauge theories. Starting from the theory of differentiable manifolds and their tangent and cotangent structures, the necessary tools are progressively constructed: vector bundles for describing fields on curved spaces, the calculus of differential forms, which find a natural application in the geometric reformulation of Maxwell's equations, and the theory of Lie groups and Lie algebras for the description of continuous symmetries. These tools converge in the study of connections on principal bundles, where the Hopf fibration provides the natural setting to construct the Dirac magnetic monopole and derive its charge quantization condition as a consequence of the non trivial topology of the bundle. The treatment shows how differential geometry provides not merely a convenient language for physics, but a framework in which deep physical constraints emerge from purely mathematical structures.