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, 3 credits)


1) Quantum mechanics and symmetry

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


2) Formal group theory

Groups
Subgroups
Group homomorphisms and isomorphisms


3) Representation theory

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


4) Vector calculus and geometry of R3


5) The rotation group O(3)

Axis angle parametrization
Euler angle parametrization


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



Module 2 (Prof. L. Lin, 3 credits)


7) Lie groups and Lie algebras

Groups with manifold structures
Invariant vector fields and Lie algebras
The exponential map
Matrix groups as Lie groups


8) Representations of Lie algebras

Derivatives of Lie group homomorphisms
The adjoint representation
Semi-simple and reductive Lie algebras
Representations of Lie groups vs Lie algebras


9) Root decomposition

Lie algebras and their complexification
Diagonalization of the adjoint representation and Cartan subalgebras
Root decomposition of semi-simple Lie algebras
Simple roots and their geometric interpretation
Root decomposition for su(3)


10) Weights and representations

Group/algebra representations from highest weights
Cartan matrices and geometry of weight lattices
Examples of weight diagrams


11) Applications

Tensor products and Clebsch—Gordan decomposition
Meson and Baryon multiplets
Group theoretic aspects of gauge theories
Classification of Lie algebras

Readings/Bibliography

H. Weyl

The Theory of Groups and Quantum Mechanics,

Dover

ISBN-10: 1614275807, ISBN-13: 978-1614275800

 

W.-K.Tung

Group Theory in Physics,

World Scientific

ISBN 9971966565, ISBN 9789971966560

 

M. Hamermesh

Group Theory and Its Application to Physical Problems

Dover Publications

ISBN-10: 0486661814, ISBN-13: 978-0486661810

 

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

 

B. C. Hall

Lie Groups, Lie Algebras, and Representations

Springer

SBN-10: 3319134663, ISBN-13: 978-3319134666

 

W. Fulton and J. Harris

Representation Theory: a First Course

Springer

ISBN-10: 0387974954, ISBN-13: 978-0387974958

 

H. Georgi

Lie Algebras in Particle Physics

CRC Press

ISBN-10: 0738202339, ISBN-13: 978-0738202334

 

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 lasting 45 minutes 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 Ling Lin