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 the lecture the content is (usually) presented on the blackboard. Main focus is on the mathematical foundations of this field of application in theoretical Physics. Numerous examples clarify applications of group theory in theoretical Physics. Graphical methods such as weight diagrams serve to illustrate the mathematical concepts.

Oral presentation with writing on black board.

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

There will be an oral exam of 25 minutes duration. Therein the achievement of the competencies given in section learning outcome is tested exemplarily at least to the given cognition level using calculation problems and comprehension questions.

For example an assignment in the exam might be:

• give basic definitions and notions in group theory
• application of general theorems to specific examples
• give examples for orthogonal and unitary groups