Gruppentheorie in der Physik
Group Theory in Physics

The module is structured in the following way:

Basic notions of group theory: Symmetries in physics, Magmas, Semigroups, Groups, Homomorphism and Isomorphism, Realizations, Representations

Discrete groups: Order, Period, The cyclic group Cn (and Zn), Rearrangement theorem, The symmetric group Sn, The Dihedral group Dn, The alternating group An, Cayley's theorem, Cosets, Lagrange's theorem, Normal subgroups, Quotient groups, Homomorphism theorem, Conjugacy classes, Representations of finite groups, Irreducible and reducible representations, Unitary representations , Schur's lemmas, Great orthogonality theorem, Characters and character table, Decomposition theorem
Physics application: spontaneous symmetry breaking and the example of super conductivity

Lie groups in physics: Rotations in quantum mechanics and projective representation, Basic properties of matrices: exponential, determinant, BCH formula, SO(3), Generators and structure constants of SO(3), Lie groups: definitions and basic theorems, Lie algebras: definitions and basic theorems, Adjoint representation, Direct sums, Direct products and decompositions, Representations of SO(3), SU(2), The SO(3) ~ SU(2) homomorphism, Generators and structure constants of SU(2), Spinors in SU(2), Representations of SU(2)
Physics application: the hydrogen atom, Isospin, Cartan subalgebras, roots and weights, SU(3), Generators and structure constants of SU(3), Basic properties of the Gell-Mann matrices, Weights of SU(3), Fundamental representation, Complex conjugate representation,  Adjoint representation, Casimir operator, Simple roots and basic theorems, Dynkin diagrams, Heighest weight and classification of all SU(3) representations, SU(n), Tensor method, Young tableaux and classification of representations, Decomposition of products of representations in SU(n), Physics application: SU(3) and the strong interaction, Classification scheme of all Lie algebras in terms of Dynkin diagrams: An, Bn, Cn, Dn, G2, F4, E6, E7, E8

Introduction to the Lorentz and Poincaré groups and representations.

After successful completion of the module the students are able to:

• remember the most important notions and theorems in group theory for discrete groups
• classify finite Abelian groups
• understand the origin of spin groups
• remember the definition of a Lie group and distinguish Lie groups and Lie algebras
• to construct character tables for the representations of finite groups
• to classify the representations of SU(2) and SU(3) by weight diagrams
• to explain irreducible representations of the Lorentz group.

Emphasis is put on physical applications in particle physics.

Linear algebra (e.g. MA9201), basic quantum mechanics (e.g. PH0007)

In der Vorlesung werden die Inhalte (in der Regel) durch Tafelvortrag vermittelt. Der Fokus liegt auf den mathematischen Grundlagen dieses Anwendungsgebiets der theoretischen Physik. Durch Beispiele werden die Anwendungen der Gruppentheorie in der theoretischen Physik klar gemacht. Graphische Methoden wie Gewichtsdiagramme dienen der Veranschaulichung der mathematischen Konzepte.

Vorlesung, Präsentation, Tafelanschrieb

Books:

M. Hamermesh: Group Theory and its Application to Physical Problems, Addison Wesley, (1962)

J.F. Cornwell: Group Theory in Physics, Academic Press, (1984)

T. Cheng & L. Li: Gauge Theory of Elementary Particle Physics, Oxford University Press, (1984)

H. Georgi: Lie Algebras in Particle Physics, 2nd edition, Westview Press, (1999)

P. Ramond: Group Theory: A Physicist's Survey, Cambridge University Press, (2010)

S. Scherer: Symmetrien und Gruppen in der Teilchenphysik, Springer Spektrum, (2016)

Lecture notes:

F.J. Yndurain: Elements of Group Theory, Cornell University, (2007)  e-Print: arXiv:0710.0468

Es findet eine mündliche Prüfung von 25 Minuten Dauer statt. Darin wird das Erreichen der im Abschnitt Lernergebnisse dargestellten Kompetenzen mindestens in der dort angegebenen Erkenntnisstufe exemplarisch durch Rechenaufgaben und Verständnisfragen überprüft.

Prüfungsaufgabe könnte beispielsweise sein:

• Nennen Sie die grundlegenden Definitionen und Begriffsbildungen in der Gruppentheorie
• Wenden Sie die allgemeinen Sätze auf konkrete Beispiele
• Nennen Sie Beispiele für orthogonale und unitäre Gruppen