Although we do not yet know how long the social distancing related to the Covid-19 pandemic will last, and regardless of the changes that had to be made in the evaluation of the June 2020 session in relation to what is provided for in this learning unit description, new learnig unit evaluation methods may still be adopted by the teachers; details of these methods have been - or will be - communicated to the students by the teachers, as soon as possible.
At the end of this learning unit, the student is able to :
a. Contribution of the teaching unit to the learning outcomes of the programme (PHYS2M and PHS2M1)
1.1, 1.5, 2.1, 2.3, 3.1, 3.2, 3.3, 3.4.
b. Specific learning outcomes of the teaching unit
At the end of the teaching unit, the student will be able to :
1. formulate a symmetry in terms of a group;
2. analyze the consequences of a symmetry by the use of representations of the associated group ;
3. understand the importance of group representations in Physics ;
4. calculate characters of representations ;
5. identify different types of representations ;
6. calculate the reduction of a representation of a finite group in irreducible representations, and identify the associated invariant subspaces ;
7. calculate the algebra of a matrix group and determine its dimension ;
8. characterize a representation of su(2) ;
9. calculate su(2) Clebsch-Gordan coefficients.
The contribution of this Teaching Unit to the development and command of the skills and learning outcomes of the programme(s) can be accessed at the end of this sheet, in the section entitled “Programmes/courses offering this Teaching Unit”.
1. Finite groups:
' basic notions, properties and examples (invariant subgroup, direct and semi-direct product, conjugacy classes, left and right cosets, quotient group, illustrations in symmetric groups) ;
' concept of representation (motivations, definitions, examples, equivalence classes of representations, direct sums, distinction between reducible and irreducible, classification problem) ;
' general results for finite groups (characters, orthogonality relations, irreducible character tables, reduction methods, applications) ;
' tensor product of representations (definition, reduction of tensor products, practical efficiency of tensorial notation, examples) ;
' mathematical characterization and consequences of symmetries in a concrete physical system (calculation of the normal modes of a mechanical system from the identification of irreducible representations in the symmetry group action) ;
' (*) symmetric groups (irreducible representations associated to Young diagrams, dimensions, characters).
2. Lie groups and Lie algebras :
' the group SO(2) (defining representation, infinitesimal generators) ;
' generalization to matrix groups (Lie algebra, exponential map, structure constants, representation of the algebra, group composition law) ;
' the groups SU(2) and SO(3) (group varieties, parametrizations, relation between the two) ;
' the su(2) algebra (irreducible representations, reduction of tensor products, Clebsch-Gordan coefficients) ;
' (*) representations of the su(3) algebra (examples, general structure and classification, reduction of tensor products, applications to the quantum harmonic oscillator) ;
' (*) representations of classical matrix groups (tensorial methods, role of permutation groups, Young diagrams and dimension formulas, peculiarities of the orthogonal groups, application to the Riemann tensor).
The lectures aim at introducing the fundamental concepts of group theory which are central to understand its role in physics. In particular certain concepts developed in other courses of the undergraduate physics program are revisited from a purely group theoretic point of view, thereby showing the full relevance of group methods. The most useful results of group theory are presented and the associated methods are made completely explicit.
The tutorials are meant to get familiar with the theoretical material and the methods presented during the lectures.
Attendance to both the lectures and the tutorials is required.