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BIRLA INSTITUTE OF TECHNOLOGY AND SCIENCE PILANI

K.K.BIRLA GOA CAMPUS

Instruction Division

First Semester 2014-2015

Course Handout

Course No. : PHY F422/425 August 04, 2014Course Title: Group theory & Applications / Advanced Mathematical Methods of PhysicsInstructor-in-charge: Madhu Kallingalthodi

Course Description :

The course intends to introduce the mathematics of groups with emphasis on problems in Physics.

Prerequisites : Quantum Mechanics II or Mathematical Methods of PhysicsThe first part of the course will focus on the mathematical aspects of groups. Basic features of groupswill be described by using mostly discrete groups as examples. The notions of cosets, conjugacy classes andinvariant subgroup will be discussed. Cayleys theorem, Lagranges theorem and their implications will bedealt with. A significant number of permutation groups (symmetry groups) used in solid state physics willbe discussed. We will also discuss direct product and semidirect product groups.

Aspects of group representation will be discussed. Faithful representations, reducible/irreducible repre-sentations will be defined with examples. Young diagrams will be introduced as a method to represent andconstruct group representations.

Lie groups will make up the major part of the course. Lie algebra will be defined and discussed. Importantclasses of simple, semi-simple Lie algebra will be explained. Various representations of Lie algebra will beintroduced including the adjoint representation. SU(2) and SU(3) algebras will be discussed in detail Afterintroducing subalgebra, Cartan subalgebra will be discussed. Weights and roots of a Lie algebra will bediscussed. Dynkin diagrams as a method to represent the Lie algebras will be introduced. Classification ofLie groups/algebras based on this will be introduced.

Final part of the course will discuss spinors as a representations of Lorentz groups and SU(2). Weyl, Ma-jorana represntations will be introduced. Clifford algebra and supersymmetry algebra will also be discussedif time permits.

Textbooks :

T1. Group Theory and Its Applications to Physical Problems, Morton Hamermesh, Dover Publications.

T2. Semi-simple Lie Algebra and Their Representations, R. N. Cahn, Benjamin/Cummins.

References :

R1. Brian G. Wybourne, Classical Groups for Physicists, John Wiley & Sons.

R2. H. Georgi, Lie Algebras in Particle Physics, Addison-Wesley.

R3. M. Artin, Algebra, Prentice Hall.

R4. W. Ledermann, Introduction to Group Theory, Longman.

R5. I. N. Herstein, Topics in Algebra, Wiley International.

R6. I. N. Herstein, Abstract Algebra, Prentice Hall.R7. D. J. S. Robinson, A Course in the Theory of Groups, Springer-Verlag.

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Evaluation Scheme :

Evaluation Component Duration Weightage (%) Date & Time Nature

Test 1 60 minutes 20 16-09-2014

14:00 to 15:00

Open book

Test 2 60 minutes 20 21-10-2014

14:00 to 15:00 Open book

Problem sets + Tutorials 15 + 5 Common Hour

Compre. Exam 180 minutes 40 02-12-2014

14:00 to 17:00 Open book

Course plan :

Lecture No. Learning Ob jectives Topics to be covered Reference

1-12 Algebra of groups

Symmetry groups

Groups; isomorphisms, subgroups;Cayleys and Lagranges theorems;

coset, conjugacy class, normalsubgroup; Simple/semi-simple groups;

direct,semidirect product;Symmetry group, point group andlattices

Hamermesh

13-18 RepresentationsFaithful, Irreducible representations,

Schurs lemma, Young tableaux,Characters

Hamermesh

19-35 Lie groups

Lie algebras

Lie groups, Lie algebra,Fundamental & Adjoint representations

Casimirs, Cartan subalgebra,Weights, Roots, SU(2), SU(3)

Dynkin diagrams, classifications

Cahn, Georgi

36-44 Spinors

Tensor and spinor representationsWeyl, Majorana representations

Clifford algebraSupersymmetry algebra

TBA

Consultation hours : Schedule and venue to be discussed in class.

Make-up Policy : Make-up will be given only in genuine cases. No make-up for the tutorials.

Instructor-in-charge