The Lorentz Group

We elaborate on an alternative approach to construct the representations of the Poincaré group to those that have been discussed in the previous chapters, which is mainly based on the construction of the little groups in a covariant manner. For this purpose, the quantities that remain invariant under Lorentz transformations are to be found along with the Casimir operators. The Casimir operators of the Lorentz group, which are quite different from those of the Poincaré group, commute with the six generators of SL(2, c). Thus, all possible finite dimensional representations of SL(2, c) are presented. We discuss the metric and the normalization useful for unitary representation of the Dirac spinors. The solutions of the covariant harmonic oscillator can serve both as the representations of Poincaré group and the Lorentz group depending on the coordinate system in which they are written along with their respective Casimir operators. It is pointed out that, as in O(3) and SL(2, c), the step-up and step-down operators may be useful in constructing representations of the Lorentz group. The hyperbolic coordinate system serves well for the Lorentz group. Traditional methods existing in the literature are also discussed.