OVERVIEW

Group Theory (code 63662) has credit value 6 and, staring from the academic year 2021/22, will be taught in the first semester. 

Lectures are usually given in Italian. 

The course aims at introducing the concepts and techniques of group theory and their application to physics.

AIMS AND CONTENT

LEARNING OUTCOMES

The aim of the course is to give the basic properties and constructions of finite dimensional representations of finite and compact groups and (some of) their applications In Quantum Mechanics, Another topic is the structure of Linear Lie Groups and their Lie Algebras

AIMS AND LEARNING OUTCOMES

• At the conclusion of the course the student should know the irreducible representations of the Symmetric Group and of simple Lie groups.
• He should be able to understand the role played by these representations in Quantum Mechanics and in some elemantary particles models
• He should be able to use the theory of group representations in the solution of explicit 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)

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)

   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

TEACHERS AND EXAM BOARD

Exam Board

STEFANO GIUSTO (President)

NICOLA MAGGIORE

SIMONE MARZANI

CAMILLO IMBIMBO (President Substitute)

LESSONS

LESSONS START

Second semester

EXAMS

EXAM DESCRIPTION

Oral exam. During the exam the student will also be asked to discuss the solution of one of the home assignements.  

ASSESSMENT METHODS

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.