Course Unit Page

Academic Year 2018/2019

Learning outcomes

At the end of the course, the student will have a basic knowledge of the main applications of group theory to physics, acquire the elements of the theory of Lie groups, algebras and their representations, with an emphasis on the unitary and orthogonal groups and in particular the rotation and Lorentz groups.

Course contents

1) Quantum mechanics and symmetry

States and observables
Symmetry groups
Quantum formalism
Symmetry groups action
Projective representations
Representations and energy eigenvectors classification

2) Formal group theory

Group homomorphisms and isomorphisms
Function groups
The automorphism group of a group
Group actions
Conjugacy classes
Normal subgroups and quotient groups

3) Classical groups

The general linear groups GL(V)
Volume forms and the special linear groups SL(V)
Metrics and the orthogonal and unitary groups O(V), U(V), O(V), SU(V)
Symplectic form and the symplectic groups Sp(V)

4) Representation theory

Operations with representations
Equivalent representations
Reducible representations
The Schur lemma
Unitary representations and the Weyl theorem
Characters of a representation

5) Finite groups

Finite groups
Representation theory of a finite group
The group algebra of a finite group

6) Lie groups and Lie algebras

Lie algebras
Lie algebra homomorphisms
Lie algebra representations
Lie groups
The Lie algebra of a Lie group
Lie group homomorphisms
Lie group representations
The exponential of an endomorphism
The exponential map of a Lie group
The Lia algebras of the classical groups

7) Groups relevant to physics

The group SL(2,C)
The unitary group SU(2)
The rotation group O(3)
The Lorentz group O(3,1)
Isomorphism SO(3)=SU(2)/Z2
Isomorphism SO_0(3,1)=SL(2,C)/Z2
Other physically relevant groups


H. Weyl,
The Theory of Groups and Quantum Mechanics,
ISBN-10: 1614275807
ISBN-13: 978-1614275800

Group Theory in Physics,
World Scientific.
ISBN 9971966565, ISBN 9789971966560

Teaching methods

lectures and tutorial

Assessment methods

The exam is oral and includes the exposition of a topic of the program of the course chosen by the student and approved by the teacher and supplementary questions as required.

The exam is about one hour long.

There are no prerequisites for admission to the exam.

The exam can be taken from the end of the course on.

The award of a cum laude grade is taken into account only for the student who has demonstrated an uncommon clarity of thought and a degree of knowledge of the matter far above the average.

As a rule, the student may repeat the exam if the grade he/she obtained does not satisfy him/her within the same academic year. In this case, only the last vote can be registered, even if it is lower than that received in previous attempts. The student can accept a previously rejected grade within the academic year during which the vote was achieved. Beyond that term, the vote is canceled and the student must repeat the exam.

Teaching tools

Lecture notes in English available in AMS Campus web site

Office hours

See the website of Roberto Zucchini