Aplicacions dels grups i àlgebres de Lie a a física de partícules

Abstract

[en] Over the 1850-60’s, twenty years before Sophus Lie and Friedrich Engel started their research on Lie group theory, mathematicians such as A. Cayley, W. R. Hamilton and J. J. Sylvester introduced matrices and matrix groups thinking they had invented something of no possible use for natural scientists. On the contrary, Lie groups have taken an essential role in the theories of modern physics, and many questions have arisen. For example, why do Lie groups have such a fundamental role in physics? Why do they show up so often? How can they be used? The main objective of this research is to study of Lie groups and Lie algebras, based on the works of mathematicians such as Lie, Cartan, Killing and Weyl, in order to identify those features that make them essentially different and that will help us classify them. The second objective is to observe how the systems studied by physics are related to Lie group theory and to see that we can obtain new results thanks to the mathematical approach. This work is based on a bibliographic research to present a description of Lie group theory, with the posterior practical analysis of some specific cases. The first chapter is a historical retrospective to introduce the reader to the issue. We start looking at what the Standard Model of Particle Physics is and how it arose, then we follow with the history of the birth and consolidation of group theory as a mathematical theory, and we finish looking at how it was introduced, during the 20th century, the formulation of group theory in different aspects of science. In the second chapter we introduce the definition of Lie groups and Lie algebras and how they relate through the exponential function. Moreover, we study its basic concepts, such as the reducibility, and others which will be of great use afterwards. In the third chapter we apply these concepts and properties to classify of Lie groups using a simplification of the structure constants, which will allow us to see the underlying structure of the group. This structure will be represented with Root diagrams and Dynkin diagrams, which will reveal some constraints on the configuration of the possible diagrams, limiting the existence of Lie groups essentially different to a reduced number. In the fourth chapter we define the representation of a group as a map from an abstract group to a matrix group. We specifically introduce the adjoint representation of a group as the one that carries the structure constants in its form. In the fifth chapter we apply the concepts studied in Chapter 3 in order to analyze some particular Lie groups. Finally, the sixth chapter shows some examples of the form in which representations of different Lie groups appear in particle physics, evidencing some kind of symmetry that manifests the existence of conserved quantities.