CODE 61843 ACADEMIC YEAR 2022/2023 CREDITS 6 cfu anno 2 FISICA 9012 (LM-17) - GENOVA 6 cfu anno 1 FISICA 9012 (LM-17) - GENOVA SCIENTIFIC DISCIPLINARY SECTOR FIS/02 TEACHING LOCATION GENOVA 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 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 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 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 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 STEFANO GIUSTO 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 ADVANCED MATHEMATICAL METHODS IN PHYSICS 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.
