Based on the book "Grundlagen Kontinuierlicher Symmetrien -- von der Raumzeit zur Quantenmechanik" by Franck Laloë (Wiley VCH 2024), the lecture gives a "bottom-up" presentation of the typical symmetry groups in physics.
Outline
Basics of group theory, discrete and topological groups. Action of a group on objects (e.g. point mass, atoms in a molecule, vectorial observables).
Classical symmetries of space time: translations, rotations, boosts (Galilei and Lorentz), Noether theorem.
Symmetries in classical mechanics: Poisson brackets, canonical transformations (symplectic groups).
Representations of groups (matrix transformations) and their generating algebras (matrices, differential operators). Wigner theorem on (anti)unitary representations.
Point groups and classification of electronic states in molecules (with Foudhil Bouakline). Irreducible tensors and applications of the Wigner-Eckart-Theorem in light-matter interactions.
Relativistic transformations: the spinors of Klein-Gordon, Dirac, and Weyl, and their wave equations. The spin in relativity.
Internal symmetries: local gauge invariance, the U(2)xSU(3) groups of the standard model. Isospin and multiplets of mesons and baryons.
Broken symmetries.
- Kursleiter*in: Carsten Henkel