# Group Theory for High Energy Physicists

#### By **Mohammad Saleem**, **Muhammad Rafique**

CRC Press – 2013 – 230 pages

CRC Press – 2013 – 230 pages

Although group theory has played a significant role in the development of various disciplines of physics, there are few recent books that start from the beginning and then build on to consider applications of group theory from the point of view of high energy physicists. **Group Theory for High Energy Physicists** fills that role. It presents groups, especially Lie groups, and their characteristics in a way that is easily comprehensible to physicists.

The book first introduces the concept of a group and the characteristics that are imperative for developing group theory as applied to high energy physics. It then describes group representations since matrix representations of a group are often more convenient to deal with than the abstract group itself. With a focus on continuous groups, the text analyzes the root structure of important groups and obtains the weights of various representations of these groups. It also explains how symmetry principles associated with group theoretical techniques can be used to interpret experimental results and make predictions.

This concise, gentle introduction is accessible to undergraduate and graduate students in physics and mathematics as well as researchers in high energy physics. It shows how to apply group theory to solve high energy physics problems.

**Elements of Group Theory **

Definition of a Group

Some Characteristics of Group Elements

Permutation Groups

Multiplication Table

Subgroups

Power of an Element of a Group

Cyclic Groups

Cosets

Conjugate Elements and Conjugate Classes

Conjugate Subgroups

Normal Subgroups

Centre of a Group

Factor Group

Mapping

Homomorphism

Kernel

Isomorphism

Direct Product of Groups

Direct Product of Subgroups

**Group Representations **

Linear Vector Spaces

Linearly Independent Vectors

Basic Vectors

Operators

Unitary and Hilbert Vector Spaces

Matrix Representative of a Linear Operator

Change of Basis and Matrix Representative of a Linear Operator

Group Representations

Equivalent and Unitary Representations

Reducible and Irreducible Representations

Complex Conjugate and Adjoint Representations

Construction of Representations by Addition

Analysis of Representations

Irreducible Invariant Subspaces

Matrix Representations and Invariant Subspaces

Product Representations

**Continuous Groups **

Definition of a Continuous Group

Groups of Linear Transformations

Order of a Group of Transformations

Lie Groups

Generators of Lie Groups

Real Orthogonal Group in 2 Dimensions: O(2)

Generators of SU (2)

Generators of SU (3)

Generators and Parameterisation of a Group

Matrix Representatives of Generators

Structure Constants

Rank of a Lie Group

Lie Algebras

Commutation Relations between the Generators of a Semi-Simple Lie Group

Properties of the Roots

Structure Constants Nαβ

Classification of Simple Groups

Roots of SU (2)

Roots of SU (3)

Numerical Values of Structure Constants of SU (3)

Weights of a Representation

Computation of the Highest Weight of any Irreducible Representation of SU (3)

Dimension of any Irreducible Representation of SU (n)

Computation of Weights of an Irreducible Representation of SU (3)

Weights of the Irreducible Representation D8 (1,1) of SU(3)

Weight Diagrams

Decomposition of a Product of Two Irreducible Representations

**Symmetry, Lie Groups, and Physics **

Symmetry

Casimir Operators

Symmetry Group and Unitary Symmetry

Symmetry and Physics

Group Theory and Elementary Particles

**Appendices **

**Index**

**Dr. Mohammad Saleem** is a professor emeritus at the University of the Punjab and a professor at the Institute for Basic Research in Florida. He has written more than 150 research papers on high energy physics and is an editor of the *Hadronic Journal*.

**Dr. Muhammad Rafique** was a professor of applied mathematics at University of the Punjab.

Name: Group Theory for High Energy Physicists (Hardback) – CRC Press

Description: By Mohammad Saleem, Muhammad Rafique. Although group theory has played a significant role in the development of various disciplines of physics, there are few recent books that start from the beginning and then build on to consider applications of group theory from the point of view of high...

Categories: Mathematical Physics, Computational Physics, Particle & High Energy Physics