Lectures on Selected Topics in Mathematical Physics

Further applications of Lie theory

By (author) William A Schwalm

Publication date:

07 March 2019

Publisher

Morgan & Claypool Publishers

ISBN-13: 9781643273488

BOOK DESCRIPTION
TABLE OF CONTENTS

This book is the sequel to Lectures on Selected Topics in Mathematical Physics: Introduction to Lie Theory with Applications. This volume is devoted mostly to Lie groups, Lie algebras and generating functions, both for standard special functions and for the solution of certain types of physical problems. It is an informal treatment of these topics, which is intended for physics graduate students or others with a physics background who want a brief and informal introduction to the subjects addressed, in a style and vocabulary that is not completely unfamiliar.

Contents

1 Generating Functions

1.1 The basic idea

1.2 Elementary examples of generating functions

2 Groups

2.1 Introduction

2.2 Groups in general and nite groups in particular

2.3 Continuous groups

2.4 Group action and innitesimal generators

2.5 Three examples of generating functions from one-parameter groups

2.6 Quantum oscillator example

2.7 Bessel functions by factoring

2.8 Bessel function generator

2.9 Multiparameter Lie groups

3 Lie algebras

3.1 Algebras

3.2 Associative algebras are essentially matrix algebras

3.3 Lie algebras are commutator sub algebras

3.4 Ideals and classication of complex Lie algebras

3.5 Levi's decomposition

3.6 The Killing form

3.7 Cartan subalgebra

3.8 Root geometry, Weyl group and brief comment on classication

3.9 Representations and Casimir operators

4 Examples and Applications

4.1 The algebra so(5)

4.2 2D Oscillator in a magnetic eld

4.3 Generating functions for spherical harmonics

5 Concluding Remarks