ABSTRACT
We have already seen that Lorentz transformations are the allowed spacetime transformations which satisfy the physical requirements of special relativity. Lorentz transformations contain the usual space rotations and Lorentz boosts and intertwine them into a unified relativistic formalism. In this chapter, we derive the Lorentz transformations group-theoretically and study the corresponding Lie group. Our focus is on the restricted Lorentz group which excludes space and time inversions. For both the Lorentz group and the Lorentz algebra, we derive the existence of spinors. We then derive the finite-dimensional tensorial representations for scalars, vectors, and tensors, as well as the representations for Weyl and Dirac spinors. Finally, we derive the infinite-dimensional orbital representations where the Lorentz group acts on functions of spacetime coordinates.
