LEARNING OUTCOMES

The calculus of variations is a general method to derive differential and partial differential equations 

used in physics.

We will show how to solve these equations using the theory of distributions and the Green's function method.

AIMS AND LEARNING OUTCOMES

The course introduces group theory, the mathematical formalism used to decribe symmetries.

It will cover finite and continuous groups and their representations.

Emphasis will be given to fundamental concepts and computational tools, rather than to generality and mathematical rigour.

At the end of the course students shoudl be able to apply the theory of representations to the solution of physical problems.

PREREQUISITES

• Linear Algebra for finite-dimensional spaces, including the Spectral Theorem for an arbitrary set of commuting linear transformations on a finite-dimensional vector space.
• Basic notions of Quantum Mechanics, and in particular the theory of angular momentum. 

TEACHING METHODS

Traditional: chalk and blackboard. Home assignements will be handed out weekly.

SYLLABUS/CONTENT

1. General properties of groups

   1. Definition of group

   2. Examples of finite and infinite (continous) groups: the cyclic group, the permutation group, the dihedral group, SO(3)

   3. Subgroups, Cayley and Lagrange theorems

   4. Conjugation classes, invariant subgroups, cosets, simple and semisimple groups

   5. Direct and semidirect products 

2. Representations of finite groups

   1. Definition of representation

   2. Examples: the trivial representation, regular representation, the sign and natural representation of Sn

   3. Equivalent representations, characters

   4. Decomposable, reducible and irredicible representations 

   5. Unitary representations 

   6. Schur's lemmas

   7. Orthogonality theorems

   8. Decomposition of reducible representations: the regular representation; the number of conjugation classes and of irreps

   9. The character table

   10.  Real, pseudoreal and complex representations

   11. Representations of Sn and Young Tableaux (a sketch)

   12. the normal modes of molecules from group theory

3. Lie groups and Lie algebras

   1. Definition of Lie group

   2. Groups of matrices

   3. The invariant measure, compact and non-compact groups

   4. Lie algebras, exponential map, commutators and structure constants; the BCH formula (a sketch)

   5. Local and global properties of a Lie group: relation between SO(3) and SU(2), SO(3,1) and SL(2,C), algebra complexification and compactness

   6. Simple and semisimple algebras, the Cartan-Killing metric

4. Representations of Lie groups: generalities

   1. Examples: fundamental and adjoint representations, SU(2) irreps

   2. Sum and product of representations 

   3. Compact groups, unitary representations, reducible and irreducible representations 

   4. Group and algebra representations

5. Classification of simple Lie algebras

   1. Cartan subalgebra

   2. Roots, weights and Wey group

   3. Examples: su(N), so(2N+1), sp(2N), so(2N) algebras

   4. General properties of root systems

   5. Dynkin diagrams and classification

   6. Reconstructing the algebra from the Dynkin diagram

6. Representations of simple Lie algebras

   1. Highest weight representations

   2. Examples: some representation of su(3) and applications to the theory of hadrons

   3. su(N) irreps and Young tableaux (a sketch)

RECOMMENDED READING/BIBLIOGRAPHY

• A. Zee, Group Theory in a Nutshell for Physicists, Princeton University Press 2016
• H. Georgi, Lie Algebras in Particle Phyics, CRC Press 1999
• M. Hamermesh, Group Theory and its applications to physical problems, Dover Publications 1962
• S. Sternberg, Group theory and physics, Cambridge University Press 1994
• B. Hall, Lie groups Lie algebras and representations, Springer 2004
• Class notes will be made available to the students