On Representations of the Lorentz Group

Abstract

The Lorentz group is a non-compact group. Consequently, it's representations cannot be expected to be equivalent to representations of a unitary group. Actually, they act on a large-component space and a separated small-component space, in some sense analogous to 4-vectors. In contrast to representations of compact groups state vectors carry the actual value of the non-compact variables, the boost-vector. In the non-boosted state the small components vanish and the large components transform according to a representation of the rotation subgroup. Application of a boost then generates small components, a process that preserves norms. However, the norm now has a growing positive contribution from the large-components and a negative contribution from the small-components, growing absolutely to keep the total unchanged. General transformations are described in detail. The freedom to assign boost directions to the phases of small components leads to a topological symmetry with avor-generating representations for two sheeted representations.