Introduction to Groups, Invariants and Particles.pdf

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Introduction
Groups, Invariants
and
Particles
Frank W. K. Firk, Professor Emeritus of Physics, Yale University
2000
to
ii
iii
CONTENTS
Preface
v
1. Introduction
1
2. Galois Groups
4
3. Some Algebraic Invariants
15
4. Some Invariants of Physics
23
5. Groups
-
Concrete and Abstract
37
6. Lie’s Differential Equation, Infinitesimal Rotations,
and Angular Momentum Operators
50
7. Lie’s Continuous Transformation Groups
61
8. Properties of n-Variable, r-Parameter Lie Groups
71
9. Matrix Representations of Groups
76
10. Some Lie Groups of Transformations
87
11. The Group Structure of Lorentz Transformations
100
12. Isospin
107
13. Groups and the Structure of Matter
120
14. Lie Groups and the Conservation Laws of the Physical Universe 150
15. Bibliography
155
iv
v
PREFACE
This introduction to Gro up Theory, with its emp hasis on Lie Gro ups
and their application to the study of symmetries of the fundamental
constituents of matter, has its origin in a one-semester course that I taught
at Yale University for more tha n ten years. The course was developed for
Seniors, and advanced Juniors, majoring in the Phy sical Sciences. The
students had generally completed the core courses for their majors, and
had tak en intermediate level courses in Linear Algebra, Real and Complex
Analysis, Ord inary Linear Differential Equ ations, and some of the Special
Fun ctions of Phy sics. Gro up Theory was not a mathematical requirement
for a degree in the Phy sical Sciences. The majority of existing
undergra duate textbooks on Gro up Theory and its applications in Phy sics
tend to be either highly qualitative or highly mathematical. The pur pose of
this introduction is to steer a middle course that pro vides the student with
a sound mathematical basis for studying the symmetry pro perties of the
fundamental particles. It is not generally appreciated by Phy sicists that
continuous tra nsformation groups (Lie Groups) originated in the Theory of
Differential Equ ations. The infinitesimal generators of Lie Gro ups
therefore have forms that involve differential operators and their
commutators , and these operators and their algebraic properties have found,
and continue to find, a natural place in the development of Quantum Physics.
Guilford, CT.
June, 2000.

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