Lie Groups and Lie Algebras for Physicists

The following sections are included:

• Dedication
• Preface
• Contents

In this chapter, we introduce the concept of a group and present in some detail various examples of commonly used groups in physics. This is helpful in establishing the terminology as well as the notations commonly used in the study of groups which will also be useful in further development of various ideas associated with groups.

While the structure of the abstract group is quite interesting, in physics the representation of the group itself is more important. The most common representations of (Lie) groups are in terms of finite dimensional n × n matrices, although sometimes the infinite dimensional (unitary) representations may be more meaningful for non-compact groups. In this chapter we will discuss the matrix representations as well as representations of groups by linear operators in a vector space.

The following sections are included:

• Definition of a Lie algebra
• Examples of commonly used Lie algebras in physics
  • Lie algebra of gl(N) and sl(N)
  • Lie algebra of so(N)
  • Lie algebras of u(N) and su(N)
• Structure constants and the Killing form
• Simple and semi-simple Lie algebras
• Universal enveloping Lie algebra
• References

In the last two chapters we discussed the concepts of Lie groups and Lie algebras. In this chapter we will explore systematically the connection between the two. We start in the next section with the concept of an infinitesimal group and its generators and see how it helps derive the Lie algebra from a Lie group.

In chapters 2 and 3, we have constructed simple representations of various Lie groups and Lie algebras. In this chapter, we will discuss how more complicated representations can be constructed and how the Young tableau or the Young diagrams (named after the British mathematician Alfred Young) prove to be of great help in this direction. For simplicity we will restrict ourselves to the U(N) and SU(N) groups.

The following sections are included:

• Clifford algebra
  • Dimension of the representation
  • Reducible representation
  • Irreducible representation and its uniqueness
• Charge conjugation
• Clifford algebra and the O(N) group
• References

The following sections are included:

• Lorentz group
  • Proper orthochronous Lorentz group
  • Orthochronous Lorentz group
  • Improper Lorentz group
• Generalized Clifford algebra
• Dirac equation
  • Charge conjugation
  • Weyl and Majorana particles
• References

In physics gauge theories play an important role since the basic fundamental forces are described by such theories. We are familiar with the Maxwell gauge theory from our study of electrodynamics. This is the simplest of gauge theories based on the symmetry group U(1). However, there are other gauge theories based on larger symmetry groups that, for example, are necessary to describe such fundamental forces as the weak force or the strong force. In this chapter we will discuss how such dynamical theories are constructed based on larger symmetry groups.

The following sections are included:

• SU_F flavor symmetry
• SU_F(3) flavor symmetry breaking
• Some applications in nuclear physics
• References

To start with let us consider, as an example, the u(N) Lie algebra given by

We have derived the irreducible space of the Lie algebra su(N) from the unitary group U(N) by the method of Weyl. However, it may be desirable to obtain the same directly from the Lie algebra structure itself without utilizing the global group structure. This is the goal of the present chapter. The method to be used, for this purpose, is essentially special cases of the more general Cartan-Dynkin theory which is discussed in great detail in the literature and which we review briefly in the next section. The present discussion will allow us to see its connection with the Young tableaux method discussed earlier.

The following section is included:

• Index