Group Theory in Physics
Module PH2116
Module version of WS 2019/20 (current)
There are historic module descriptions of this module. A module description is valid until replaced by a newer one.
Whether the module’s courses are offered during a specific semester is listed in the section Courses, Learning and Teaching Methods and Literature below.
available module versions  

WS 2019/20  WS 2018/9  WS 2017/8  WS 2010/1 
Basic Information
PH2116 is a semester module in English language at Master’s level which is offered irregular.
This Module is included in the following catalogues within the study programs in physics.
 Specific catalogue of special courses for nuclear, particle, and astrophysics
 Complementary catalogue of special courses for condensed matter physics
 Complementary catalogue of special courses for Biophysics
 Complementary catalogue of special courses for Applied and Engineering Physics
 Specialization Modules in EliteMaster Program Theoretical and Mathematical Physics (TMP)
If not stated otherwise for export to a nonphysics program the student workload is given in the following table.
Total workload  Contact hours  Credits (ECTS) 

300 h  90 h  10 CP 
Responsible coordinator of the module PH2116 is Björn Garbrecht.
Content, Learning Outcome and Preconditions
Content
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 GellMann 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.
Learning Outcome
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.
Preconditions
Linear algebra (e.g. MA9201), basic quantum mechanics (e.g. PH0007)
Courses, Learning and Teaching Methods and Literature
Courses and Schedule
Type  SWS  Title  Lecturer(s)  Dates  Links 

VO  4  Group Theory in Physics  Garbrecht, B. 
Fri, 12:00–14:00, PH 3343 Tue, 10:00–12:00, PH 3343 

UE  2  Exercise to Group Theory in Physics 
Responsible/Coordination: Garbrecht, B. 
Fri, 14:00–16:00, PH 1121 
Learning and Teaching Methods
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.
Media
Oral presentation with writing on black board.
Literature
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) ePrint: arXiv:0710.0468
Module Exam
Description of exams and course work
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
Exam Repetition
The exam may be repeated at the end of the semester.