Fibrats principals i equacions de Yang-Mills

Abstract

[en] There has been an increment on the use of mathematics, beyond practical apliactions, in order to understand physical theories that, originally, were described by a set of equations modeling physical phenomena and nowadays are encompassed in more complex and complete mathematical theories. It is the case, for example, of Maxwell’s electrodinamics, that nowadays is part of the gauge theories that Yang-Mills generalized in 1954 and allowed to create the Standard Model theory, a $U (1) × SU (2) × SU (3)$ gauge theory. Mainly, these theories are based on three mathematical concepts, Lie groups, bundle theory and connections. The objective of this work is to define the necessary mathematical concepts in order to, firstly understand the mathematical theories themselves and then understand the results obtained when applied in a physical context. The idea behind these gauge theories is to study, in first place, the simetries of physical phenomena and relate them to elements of Lie groups. After fixing a diferential manifold where the theory will be build on, we define a principal bundle with the simetry group acting on it. That way we can understand connections over this principal bundle as gauge fields of the model. Matter fileds are then related to the associated vector bundles of the principal bundle. The interaction of all these fields is described by the covariant derivative that we will also define in the following sections. We divide this work in three main chapters. In the first one we will define Lie groups and its representations as well as Lie algebras. In the second one we will focus on bundle theory and on the principal bundle case and its connections. Finally we will study Maxwell’s electrodinamics (a $U (1)$-prinicpal bundle) and classic Yang-Mills theory (a $SU (n)$-principal bundle) and we will obtain their corresponding equations.