|SCIENTIFIC DISCIPLINARY SECTOR||FIS/02|
Advanced mathematical methods of physics (code 61843) has credit value 6 and it is taught in the first semester of the first or second year of the LM.
Lectures are usually given in Italian.
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.
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.
Traditional: chalk and blackboard. Home assignements will be handed out weekly.
General properties of groups
Definition of group
Examples of finite and infinite (continous) groups: the cyclic group, the permutation group, the dihedral group, SO(3)
Subgroups, Cayley and Lagrange theorems
Conjugation classes, invariant subgroups, cosets, simple and semisimple groups
Direct and semidirect products
Representations of finite groups
Definition of representation
Examples: the trivial representation, regular representation, the sign and natural representation of Sn
Equivalent representations, characters
Decomposable, reducible and irredicible representations
Decomposition of reducible representations: the regular representation; the number of conjugation classes and of irreps
The character table
Real, pseudoreal and complex representations
Representations of Sn and Young Tableaux (a sketch)
the normal modes of molecules from group theory
Lie groups and Lie algebras
Definition of Lie group
Groups of matrices
The invariant measure, compact and non-compact groups
Lie algebras, exponential map, commutators and structure constants; the BCH formula (a sketch)
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
Simple and semisimple algebras, the Cartan-Killing metric
Representations of Lie groups: generalities
Examples: fundamental and adjoint representations, SU(2) irreps
Sum and product of representations
Compact groups, unitary representations, reducible and irreducible representations
Group and algebra representations
Classification of simple Lie algebras
Roots, weights and Wey group
Examples: su(N), so(2N+1), sp(2N), so(2N) algebras
General properties of root systems
Dynkin diagrams and classification
Reconstructing the algebra from the Dynkin diagram
Representations of simple Lie algebras
Highest weight representations
Examples: some representation of su(3) and applications to the theory of hadrons
su(N) irreps and Young tableaux (a sketch)
Office hours: Students can request an appointment by email: firstname.lastname@example.org
STEFANO GIUSTO (President)
Oral exam. During the exam the student will also be asked to discuss the solution of one of the home assignements.
A list of problems will be handed out weekly. To verify that students are able to apply the techniques of group theory to problem solving, students will be asked to present the solution of one of the home assignements during the oral exam. The exam also aims at assessing the knwoledge and comprehension of the results derived in class.