THE C-MATRIX AND THE REALITY CLASSIFICATION OF THE REPRESENTATIONS OF THE HOMOGENEOUS LORENTZ GROUP .1. IRREDUCIBLE REPRESENTATIONS OF SO(3, 1)

摘要

A basis-independent criterion for the classification of irreducible group representations, into potentially-real, pseudo-real and essentially-complex representations, is given for an arbitrary group which may also possess infinite-dimensional representations. These considerations are applied, in particular, to the finite- and infinite-dimensional representations D(j(0), c) of the orthochronous proper Lorentz group SO(3, 1) and it is shown that the irreps which are neither unitary nor pseudo-unitary are essentially-complex. Further, among the unitary and pseudo-unitary irreps of SO(3, 1), those irreps with a half-odd-integer j(0) are shown to be pseudo-real, while the others with an integer j(0) (including zero) are potentially-real. [References: 10]