Course Syllabus

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Teoria dei gruppi e applicazioni alla fisica teorica.

Teoria dei gruppi e delle algebre di Lie; loro rappresentazioni.

Definizioni di base

  • Definizione di gruppo, sottogruppo, omomorfismo, rappresentazione.
  • Gruppo coniugato, sottogruppi invarianti; gruppo quoziente

Gruppi finiti

  • Gruppo ciclico, gruppi diedrali, gruppo delle permutazioni

Rappresentazioni

  • Rappresentazioni unitarie, classificazione, ortonormalità e completezza. Rappresentazione regolare. Rappresentazioni irriducibili di SO(3).

Gruppi di Lie

  • Varietà, gruppi di Lie, algebre di Lie, generatori, mappa esponenziale.
  • Esempi di gruppi di Lie: ortogonale, unitario, Lorentz, Poincaré. SU(2) e SO(3)

Algebre di Lie

  • Definizioni, semplice, semi-semplice. Forma di Killing.

Classificazione delle Algebre di Lie

  • Sottoalgebra di Cartan, sistemi di radici, diagrammi di Dynkin.

I corsi del triennio.

Lezione frontale (6 CFU). Questo insegnamento sara’ tenuto in inglese.

Note del docente caricate sul sito del corso.

Teoria dei gruppi:

  • Wu-Ki Tung, Group Theory in Physics
  • Georgi, Lie Algebras in Particle Physics
  • Keski-Vakkuri-Montonen-Panero, Mathematical Methods for Physics - An Introduction to Group Theory, Topology and Geometry
  • Fulton-Harris, Representation theory, Springer.

Testi addizionali:

  • Gilmore, Lie Groups Lie Algebras and some of their applications, Dover.
  • Gilmore, Lie Groups, Physics and Geometry, Cambridge.
  • Cornwell, Group Theory in Physics, Academic Press.

Primo semestre

Esame orale. Domande aperte su tutti i contenuti del corso svolti a lezione.

Su appuntamento, scrivendo un e-mail a mattia.bruno@unimib.it

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Group theory and its applications to theoretical physics.

Lie groups, Lie algebras; their representations.

Basic definitions

  • Definition of a group; subgroups, homomorphisms, representations.
  • conjugate, invariant subgroups; quotient group;

Finite Groups

  • Cyclic group, dihedral group, group of permutations

Representations

  • Unitary representations, their classification, orthonormality and completeness, Regular representation. Irreps of SO(3)

Lie groups

  • Manifolds, Lie groups, Lie algebras, generators, exponential map.
  • Examples of Lie groups: orthogonal, unitary, Lorentz, Poincaré. SU(2) and SO(3)

Lie algebras

  • Definition, simple and semi-simple algebras. Killing form.

Classification of Lie Algebras

  • Cartan subalgebra, Root systems, Dynkin diagrams.

Undergraduate degree in math or physics

Lessons (6 CFU), This course will be taught in English.

Lecture notes uploaded on the course webpage.

Group Theory:

  • Wu-Ki Tung, Group Theory in Physics
  • Georgi, Lie Algebras in Particle Physics.
  • Keski-Vakkuri-Montonen-Panero, Mathematical Methods for Physics - An Introduction to Group Theory, Topology and Geometry
  • Fulton-Harris, Representation theory, Springer.

Further readings:

  • Gilmore, Lie Groups Lie Algebras and some of their applications, Dover.
  • Gilmore, Lie Groups, Physics and Geometry, Cambridge.
  • Cornwell, Group Theory in Physics, Academic Press.

First semester

Oral exam. Open questions on all course's topics covered during the lectures.

By appointment, by sending an e-mail to mattia.bruno@unimib.it

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Key information

Field of research
FIS/02
ECTS
6
Term
First semester
Activity type
Mandatory to be chosen
Course Length (Hours)
42
Degree Course Type
2-year Master Degreee
Language
English

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