Unitary Symmetry and Elementary Particles

Unitary Symmetry and Elementary Particles discusses the role of symmetry in elementary particle physics. The book reviews the theory of abstract groups and group representations including Eigenstates, cosets, conjugate classes, unitary vector spaces, unitary representations, multiplets, and conservation laws. The text also explains the concept of Young Diagrams or Young Tableaux to prove the basis functions of the unitary irreducible representations of the unitary group SU(n). The book defines Lie groups, Lie algebras, and gives some examples of these groups. The basis vectors of irreducible unitary representations of Lie groups constitute a multiplet, which according to Racah (1965) and Behrends et al. (1962) can have properties of weights. The text also explains the properties of Clebsch-Gordan coefficients and the Wigner-Eckart theorem. SU(3) multiplets have members classified as hadrons (strongly interacting particles), of which one characteristic show that the mass differences of these members have some regular properties. The Gell-Mann and Ne-eman postulate also explains another characteristic peculiar to known multiplets. The book describes the quark model, as well as, the uses of the variants of the quark model. This collection is suitable for researchers and scientists in the field of applied mathematics, nuclear physics, and quantum mechanics.

Preface

1. Introduction

1.1 Uses of Symmetry

1.2 Symmetries and Conservation Laws

1.3 Symmetries and Groups

1.4 Eigenstates, Quantum Numbers, and Selection Rules

1.5 A Listing of Symmetries

2. Some Properties of Groups

2.1 Elementary Notions

2.2 Homomorphism, Isomorphism, and Subgroups

2.3 Infinite Groups

2.4 Cosets, Conjugate Classes, and Invariant Subgroups

3. Symmetry, Group Representations, and Particle Multiplets

3.1 Linear and Unitary Vector Spaces

3.2 Operators

3.3 Some Properties of Representations

3.4 Unitary Representations, Multiplets, and Conservation Laws

4. The Symmetric Group and Identical Particles

4.1 Two- and Three-Particle States

4.2 Standard Arrangements of Young Tableaux

4.3 Basis Functions of S3

5. Lie Groups and Lie Algebras

5.1 Some Definitions and Examples

5.2 Generators of Lie Groups

5.3 Simple and Semisimple Lie Algebras

5.4 Standard Form of Lie Algebras

6. Multiplets

6.1 Diagonal Generators and Weights

6.2 Generators of SU(2) and SU(3)

6.3 Properties of the Weights

6.4 Weight Diagrams of SU(3)

6.5 Casimir Operators and the Labeling of States

6.6 Tensor Operators

7. Young Tableaux and Unitary Symmetry

7.1 Dimensionality of Multiplets of SU(n)

7.2 Dimensionality Formulas

7.3 Multiplets of the SU(n-1) Subgroups of SU(n)

7.4 Decomposition of Products of Irreducible Representations

7.5 Classes of Representations

7.6 Multiplets of U(n)

8. Clebsch—Gordan Coefficients

8.1 Some Properties of the Coefficients

8.2 Raising and Lowering Operators

8.3 Matrix Representation of the Algebra of SU(n)

8.4 Clebsch—Gordan Coefficients of SU(2)

8.5 Clebsch—Gordan Coefficients of SU(3)

8.6 Wigner—Eckart Theorem

9. The Eightfold Way

9.1 SU(3) and Hadrons

9.2 Baryon Multiplets

9.3 Meson Multiplets

9.4 U-Spin

9.5 Tests of U-Spin Invariance

9.6 Gell-Mann—Okubo Mass Formula

9.7 Meson—Baryon Coupling

9.8 Hadron Decays

9.9 Weak Hadron Decays

10. Approximate SU(6)

10.1 Dynamical Symmetry

10.2 Classification of Hadrons

10.3 Matrix Generators of SU(6)

10.4 Troubles with SU(6)

11. The Quark Model

11.1 Sakata Triplets

11.2 Properties of Quarks

11.3 Baryon and Meson Wave Functions

11.4 Baryon Magnetic Moments

11.5 Hadron Mass Splittings

11.6 Quark Model and SU(6)

11.7 Orbital Excitations

11.8 High Energy Scattering

11.9 Troubles with the Quark Model

12. Variants of the Quark Model

12.1 Examples of Models

12.2 Two-Particle Model of Baryons

12.3 Dyon Model

12.4 Usefulness of the Various Models

References

Index