Enrollment year
2018/2019
Academic discipline
FIS/02 (THEORETICAL PHYSICS, MATHEMATICAL MODELS AND METHODS)
Department
DEPARTMENT OF PHYSICS
Curriculum
Fisica teorica
Period
2nd semester (02/03/2020 - 12/06/2020)
Lesson hours
48 lesson hours
Prerequisites
The knowledge of the basic mathematical tools, typically learned during the BSc courses, is required. It is desirable that the student has studied or is studying differential geometry.
Learning outcomes
Learning the basic notions and concepts both of the theory of Lie groups, including homogeneous manifolds and of that of Lie algebras, including their classification. Particular emphasis will be given to the theory of representations on Hilbert spaces.
Course contents
In the first part of the course, the basic structural properties of Lie groups are introduced, in particular the related notion of Lie algebra and its geometrical interpretation. Then, we introduce and study in detail the exponential map. In the second part of the course we prove the Frobenius theorem and we use it both to construct homogeneous manifolds starting from a Lie group and to study their geometric properties. Subsequently the theory of Lie algebras is presented and the Cartan classification of semisimple Lie algebras is outlined. During the third and last part of the course, we develop the theory of representations of Lie groups on Hilbert spaces, giving emphasis to the applications in quantum mechanics and in quantum field theory.
Teaching methods
On account of the theoretical and mathematical nature of the course, all classes will be given at the blackboard and all topics of the programme will be discussed.
Reccomended or required readings
F. Warner "Foundations of differentiable manifolds and Lie groups" (1990) 3ed. Springer-Verlag.
A. W. Knapp "Lie groups: Beyond an introduction" (2005) Birkhäuser
A. O. Barut, R. Raczka "Theory of Group representations and applications" (1986) World Scientific.
Assessment methods
The student has to give an oral exam aimed at verifying that he/she has learned the topics presented during the lectures. Special emphasis will be given to the verification of the ability of the student to present rigorously and with an appopriate technical language all concepts learned.
Further information
The student has to give an oral exam aimed at verifying that he/she has learned the topics presented during the lectures. Special emphasis will be given to the verification of the ability of the student to present rigorously and with an appopriate technical language all concepts learned.
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