Monographic course on Group Theory and their representations. Main definitions and theorems. Finite groups, congruence classes, representations. Applications to crystallographic groups. Elements of the theory of characters. Lie Groups, Lie Algebras, and differential-geometric meaning. Classification of Lie Algebras. Representations of Lie Algebras, products and branching rules. Several applications to Physics of Elementary Particles.
The course aims to provide the students with the foundations of Group Theory (both finite and infinite groups) and of their representations. These notions fall in the category of mathematical methods for Physics and are essential for every curriculum in Theoretical Physics.
At the end of the course, the student will master the concept of abstract Algebra involved in Group Theory and will understand their relevance in the applications to Physics. He will be able to single out connections among the various topics taught, and to autonomously apply the methods to physics contexts different from those seen in classes.
KNOWLEDGE AND UNDERSTANDING:
The student is supposed to understand in depth the content of the course, so to be able to create connections among different themes. For this it is necessary that students make the exercises periodically assigned in classes, and that they autonomously ask themselves questions about the compatibility of what they are learning with what they already know.
APPLYING KNOWLEDGE AND UNDERSTANDING:
The student is supposed to be able to apply the methods learned to any physics context, and thus be able to export the logic seen in classes to different and more complex situations. At any rate, he is supposed to single out the physics questions which can be addressed and solved by means of Group Theory.
The student will have to learn how to abstract the notions learned and hence adapt them to the physics problem is needs to solve, each time evaluating the appropriateness and effectiveness of the techniques to be used. In order to acquire such a skill, he needs to deepen at least some of the topics discussed, in particular making spontaneous searches in literature and not limiting himself to the material suggested by the teacher.
The student will have to learn how to communicate what he learned in classes and autonomously to a public of specialists and non-specialists. In particular, he should be able to identify all relevant information, to convey it by means of an appropriate and precise language, and to comfortably explain complex concepts by way of examples and connections.
The student is supposed to read with ease the majority of textbooks on Group Theory available in the literature, and to possess all necessary tools to comprehend the large use of Group Theory in the recent and non-recent scientific publications in Theoretical Physics (in particular High Energy Physics).