Group Theory for Physicists

The following sections are included:

• Preface
• Preface to the First Edition
• List of Symbols
• Contents

The main mathematical tool in group theory is linear algebras. In this chapter, we will review some fundamental concepts and calculation methods in linear algebras, which are often used in group theory.

Group theory is a powerful tool for studying the symmetry of a physical system. In this chapter, the mathematical definition of a group will be abstracted from the common property of the sets of the symmetry transformations of physical systems. Some simple examples of groups are explained to give the readers a concrete understanding on groups. The subsets in a group, the isomorphism and homomorphism of two groups, and the direct product of groups will be discussed.

The theory of linear representations of groups is the foundation of group theory. In this chapter we will first introduce the definition of the representation of a group and the concepts of the equivalent representations and the irreducible representations. Then, through a few theorems we will study the methods for finding all inequivalent irreducible representations of a finite group, and demonstrate the fundamental steps for application of group theory to physics by an example. At last, we will calculate the irreducible bases in the group algebra of a finite group.

The irreducible bases in the group algebra $\mathcal{L}$ of a finite group G can be calculated directly from Eq. (3.114) if all inequivalent irreducible representations of G have known. The order of the permutation group S_n is very large such that its irreducible representations are hard to calculate generally. In this chapter we will study an opposite problem that the representation theory of the permutation group is established from its irreducible bases. This method is so-called the Young operator method that is an elegant theory and is widely used in modern physics.

The spherical symmetry is the most important symmetry both in the classical mechanics and in the quantum mechanics, because not only a system with the spherical symmetry is easier to deal with, but also many systems in physics have approximative spherical symmetry. A system with the spherical symmetry has a symmetry center, usually chosen as the origin, so that this system is isotropic with respect to the origin. This system is invariant under all rotations about any axis across the origin. The symmetry group of this system is the three-dimensional rotation group SO(3). The group SO(3) is the simplest Lie group. Therefore, the study on the group SO(3) is very important both in physics and in mathematics.

The study on the symmetry of crystals is a typical example of application of group theory to physics. Due to the translation symmetry of crystals, through a systematic study by group theory, all the crystals are classified completely into 7 crystal systems, 14 Bravais lattices, and 230 distinct types of crystal structures. There are 32 crystallographic point groups (11 of which are proper point groups), and 230 space groups (73 of which are symmorphic space groups). In this chapter we will study the symmetry group of crystals, their representations, and the classification of crystals.

In Chap. 5 the fundamental concepts on Lie groups have been introduced through the SO(3) group and its covering group SU(2). In this chapter we will study the general theory on Lie groups and Lie algebras, such as the property of simple and semisimple Lie algebras, the regular commutative relations of the Cartan–Weyl bases of a semisimple Lie algebra, and the classification of the simple Lie algebras.

In this chapter we will explain the methods of calculating the basis states and the matrix elements of the generators in an irreducible representation of a simple Lie algebra $\mathcal{L}$ with dimension larger than one. Three Chevalley bases of generators, E_μ, F_μ and H_μ with the same subscript μ, span an A₁ subalgebra. The multiplets of those A₁ subalgebras embed in the representation space of $\mathcal{L}$ by a complicated way. The method of the block weight diagram analyzes the complicated overlapping of those multiplets and calculate the representation matrices of the generators. The Gel’fand’s method and its generalization gives a certain rule to define the orthonormal basis states in the representation space of $\mathcal{L}$. The study on the physical background of the basis states is left to the subsequent chapters…

The generalized Gel’fand’s method gives a certain rule to define the orthonormal basis states in an irreducible representation of a simple Lie algebra $\mathcal{L}$, but does not give the tensor structure of the basis states, which are important for calculating wave functions in physics. In the present chapter we study the tensor structure and the wave functions of the orthonormal basis states belonging to the given irreducible representation of the unitary group SU(N) by combining with the method of block weight diagrams and the Gel’fand’s method, and briey introduce some applications to the particle physics. The tensor structure of the orthonormal basis tensors for the irreducible representations of the SO(N) group and the USp(2ℓ) group will be studied in the subsequent chapters.

In this chapter we will study the tensor representations and the spinor representations of the SO(N) groups, and calculate the basis functions in those irreducible representations.

The proper Lorentz group L_p is a noncompact Lie group, and the SO(4) group is a compact Lie group. Two groups have the same Lie algebra but their real Lie algebras are different. In this chapter the irreducible representations of the proper Lorentz group is studied through those of the SO(4) group. The study of the Lorentz group provides a typical example in mathematics for studying the noncompact Lie groups.

In subsection 7.4.3 we have introduced the fundamental property of the USp(2ℓ) group and the Sp(2ℓ, R) group. In this chapter we will study the irreducible representations of USp(2ℓ) and their applications to physics.

The following sections are included:

• Bibliography
• Index