# Group Theory

Wintersemester 2021/2022
Dozent: Maurits W. Haverkort and Martin Braß
104 Teilnehmer/innen
Symmetry is an essential concept in physics. The momentum of a particle is conserved whilst space is translational invariant. In a crystal translational symmetry is broken. What is left is a discrete translational symmetry by the lattice vectors. As such momentum is not conserved, but crystal momentum, i.e. the momentum of particles modulus the reciprocal lattice vectors, is conserved. Furthermore, symmetry allows one to explain why some particles do not interact with each other, why some optical transitions between states have zero intensity or if a phase transition from a simple cubic material to a material that shows biaxial optical birefringence is physically possible or not.

In this lecture we will discuss group theory, the mathematics behind the physical description of symmetry. We will start from discrete point-groups and space-groups and use these to introduce the basic concepts of symmetry, group- and representation-theory. We will discuss several examples from condensed matter physics where symmetry is used. Then we will move on to Lie-groups and finite dimensional representations of compact Lie-groups and their Lie-algebras.

Having knowledge of basics from quantum mechanics (PTP4) and linear algebra is recommended.

The lecture takes place on Friday from 11:15 to 13:00 in the lecture hall 2 of INF 227. Please note that you need to be vaccinated, cured or daily tested to participate in the lectures.

## Übungsblätter

• Übungsblatt 1
• Übungsblatt 2
• Übungsblatt 3
• Übungsblatt 4
• Übungsblatt 5

## Literature

(list will be expanded)

• W.Fulton and J. Harris, Representation Theory
• Philip H. Butler, Point Group Symmetry Applications
• C.J. Bradley and A.P. Cracknell, The Mathematical Theory of Symmetry in Solids, representation theory for point groups and space groups.
• Brian C. Hall, Lie Groups, Lie Algebras and Representations
• J. Walcher https://www.mathi.uni-heidelberg.de/~walcher/teaching/wise1516/lie_groups/LieGruppenWS1516.pdf (German)
• O. Baues https://www.math.kit.edu/iag2/~baues/media/lie.pdf (German)
• a script of the lecture will be provided
Group Theory
Wintersemester 2021/2022
Haverkort / Braß