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# Group Theory in Physics

## Module PH2116

This module handbook serves to describe contents, learning outcome, methods and examination type as well as linking to current dates for courses and module examination in the respective sections.

### Module version of WS 2017/8

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 2022/3SS 2021WS 2019/20WS 2018/9WS 2017/8WS 2010/1

### Basic Information

PH2116 is a semester module in English language at Master’s level which is offered irregular.

This module description is valid to SS 2019.

If not stated otherwise for export to a non-physics program the student workload is given in the following table.

150 h 75 h 5 CP

Responsible coordinator of the module PH2116 in the version of WS 2017/8 was Antonio Vairo.

### Content, Learning Outcome and Preconditions

#### Content

The course 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, Physics application: 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.

#### Learning Outcome

The student learns about the most important notions and theorems in group theory  for  discrete  groups, Lie groups and their representations (in particular SU(2) and SU(3)) .

Emphasis is put on physical applications in particle physics.

#### Preconditions

Linear algebra,  basic quantum mechanics

### Courses, Learning and Teaching Methods and Literature

#### Learning and Teaching Methods

Oral presentation with writing on black board.

#### 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 and 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,   e-Print: arXiv:0710.0468

### Module Exam

#### Description of exams and course work

There will be a written exam of 90 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: The exam will consist of general theory questions covering all course subjects and possibly of simple exercises discussed during the lectures.

#### Exam Repetition

The exam may be repeated at the end of the semester.

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