CODE  61843 

ACADEMIC YEAR  2022/2023 
CREDITS 

SCIENTIFIC DISCIPLINARY SECTOR  FIS/02 
TEACHING LOCATION 

SEMESTER  1° Semester 
TEACHING MATERIALS  AULAWEB 
OVERVIEW
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.
AIMS AND CONTENT
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 finitedimensional spaces, including the Spectral Theorem for an arbitrary set of commuting linear transformations on a finitedimensional 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

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

Unitary representations

Schur's lemmas

Orthogonality theorems

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 noncompact 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 CartanKilling 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

Cartan subalgebra

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)

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
Ricevimento: Students can request an appointment by email: stefano.giusto@ge.infn.it
Exam Board
STEFANO GIUSTO (President)
ANDREA AMORETTI
PIERANTONIO ZANGHI'
NICODEMO MAGNOLI (President Substitute)
LESSONS
LESSONS START
26/9/22
Class schedule
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.