87964 - Group Theory for Physics

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

 

Module 1 (Prof. R. Zucchini, 4 credits)

 

1) Quantum mechanics and symmetry

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


2) Formal group theory

Groups
Subgroups
Group homomorphisms and isomorphisms
Function groups
The automorphism group of a group
Group actions
Cosets
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

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


5) 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

 

 

 Module 2 (Prof. I. Brivio, 2 credits)

 

6) Relevant classical groups

The main classical groups
The groups O(2) and SO(2)
The groups O(1,1) and SO(1,1)
The groups U(2) and SU(2)
The groups U(1,1) and SU(1,1)


7) Vector calculus and geometry of R3


8) Pauli matrix formalism


9) The groups SL(2,C), SU(2) and SO(2)

The group SL(2,C) and its Lie algebra
The unitary group SU(2) and its Lie algebra
The unitary group SO(2) and its Lie algebra


10) The rotation group O(3)


11) The Lorentz group O(1,3) and special relativity


12) The basic isomorphisms

Isomorphism SO(3)=SU(2)/Z2
Isomorphism SO_0(3,1)=SL(2,C)/Z2


13) The Poincaré group and its representations

Readings/Bibliography

 

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

Wu-Ki-Tung,
Group Theory in Physics,
World Scientific.
ISBN 9971966565, ISBN 9789971966560

 

M. Hamermesh
Group Theory and Its Application to Physical Problems
Dover Publications
ISBN-13: 978-0486661810, ISBN-10: 0486661814

 

P. Ramond
Group Theory
Cambridge University Press
ISBN 113948964X, ISBN 9781139489645

 

J. Cornwell
Group Theory in Physics: An Introduction
Academic Press
ISBN-10: 0121898008, ISBN-13: 978-0121898007

Teaching methods

 

lectures and tutorial

Assessment methods

 

The exam is oral and is divided into two parts lasting approximately
45 minutes in which the student's learning on the contents of the two course modules is assessed. 

There are no prerequisites for admission to the exam. The exam can be taken starting from the end of the course.

The way the assessment is carried out is the same for the two modules and consists in the presentation of a topic of the program of each module chosen by the student and approved by the teacher of the module and any supplementary questions.

The final grade obtained is equal to the average with identical weights of the grade obtained in the assessment of the learning of the contents of the two modules. The granting of honors is taken into consideration only for those who have demonstrated an uncommon clarity of thought and a degree of knowledge of the subject much higher than the average and must be approved by both teachers of the course.

As a rule, the student can repeat the exam if the grade obtained does not satisfy him/her within the same academic year. In this case, only the last grade obtained 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 grade was achieved. After this deadline, the grade is canceled and the student must repeat the exam.

Teaching tools

 

Lecture notes in English available in Virtuale web site

Office hours

See the website of Roberto Zucchini

See the website of Ilaria Brivio