Representations and Realizations of so(4)

Abstract

In this chapter we first develop two approaches to the general representation theory of the Lie algebra so(4). In the first approach the results of Chapter 3 for the matrix representation of a vector operator are extended by specifying the commutators [V _j , V _k ] which close out the commutation relations among the six components of J and V to give the defining commutation relations of three important Lie algebras: so(4), the Lie algebra of the 4-dimensional rotation group, so(3,1), the Lie algebra of the Lorentz group with three space and one time dimension, and sl(3), the Lie algebra of the 3-dimensional Euclidean group. The advantage of the first approach (Section 5.2) is that the representation theory of all three Lie algebras can be obtained in a unified manner. We are interested only in so(4) so our final results will be presented only for this case. A general discussion of so(n) and so(p,q) is given in Appendix B.