Docente
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FRANCIA DARIO
(programma)
Part I
Introduction: Physics and symmetries
Finite Groups: Definition of group; groups of order 2; the Cayley table; groups of order 3; cyclic groups; subgroups; direct products; Lagrange's theorem; groups of order 4; higher-order groups: the dihedral series, the quaternionic group; pre- sentations; the symmetric group Sn; the alternating group An; cycles; Cayley's theorem; representations: basic definitions; the regular representation; conjugation and cosets; normal subgroups; factor groups; simple and semisimple groups; clas- sification of simple groups (without proof); characters; review of finite-dim Hilbert spaces; unitary representations; group-averaged inner product; equivalence of finite- dim reps to unitary reps; reducible and fully reducible reps; reps of finite groups and full reducibility; Schur's Lemmas; the orthogonality relations; the character table; real, pseudoreal and complex reps; Kronecker products and Clebsch-Gordan series. 2 Lie Groups and Lie Algebras: group manifold; Lie groups; the classical ma- trix groups; Lie algebras: definition and first properties; structure constants; ad- joint representation; Cartan-Killing form; simple and semiimple Lie algebras; fully antisymmetric structure constants; Casimir operators; quadratic Casimir; highest (lowest)-weight representations; su(2); su(3); Gell-Mann matrices; roots; α-basis, ω basis, Chevalley basis; general Lie algebras: Cartan subalgebra and structure contants in the Cartan basis; roots; the map adH(X); α-chains; Weyl reflections; root systems for semisimple Lie algebras: relative angles and lengths between roots; positive roots; simple roots; Dynkin diagrams; rank-two algebras; the Cartan cata- logue of simple Lie algebras; representation theory: highest weights and construction of representation space; level vector; the adjoint representation and maximal roots; Weyl dimension formula.
Part II
Partial Differential Equations: Generalities; linear, quasilinear, semilinear and non linear PDE; well-posedness; initial data vs boundary data; types of boundary conditions; the Cauchy-Kovalevskaya theorem; first-order equations; characteris- tics: parametric solutions and Lagrange's method; second-order PDE: classifica- tion of quasi-linear eqs: hyperbolic, parabolic and elliptic PDE; characteristics and canonical forms; the wave equation: d'Alembert solution and Fourier solution by separation of variables; normal modes; the heat equation: solution by separation of variables for insulated ends and for ends at finite temperatures T1 and T2.
(testi)
Referenze:
Parte I
- Ramond P, Group theory - a physicist’s survey (Cambridge University Press, 2010). - Gilmore R, Lie Groups, Physics, and Geometry (Cambridge University Press, 2008). - Fuchs J and Schweigert C, Symmetries, Lie algebras and representations (Cambridge University Press, 2003). - Slansky R, Group theory for unified model building (Physics Reports 791, Elsevier, 1981). - Hamermesh M, Group theory and its application to phyical problems (Dover, 1962).
Parte II
- Pradisi G, Lezioni di metodi matematici della fisica (Edizioni della Normale, Pisa, 2012). - Evans L C, Partial differential equations (American Mathematical Society, 1998-2010). - Courant R and Hilbert D, Methods of mathematical physics II (Wiley, 1962).
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