Improper double groups and the bases for the representations of O(3)

Abstract

Improper double point groups can be defined under two alternative multiplication rules in which the square of the inversion is identified with the identity and with a turn by 2 pi respectively, although the first rule is the standard one. It is shown that the second multiplication rule can be transformed away into the first one but that, in this process, transformation rules of spinors for j=1/2 are obtained which differ from the standard ones, thus solving the following problem. In the standard method, spinors for j=1/2 are defined as gerade (even with respect to inversion) and thus their tensor products are incapable of generating a complete set of irreducible bases of O(3), the ungerade harmonics for j=1 having to be postulated outside the tensor hierarchy. It is shown that in the new scheme, once the standard multiplication rule for the inversion is defined, spinors for j=1/2 appear which are ungerade as well as gerade and thus their tensor products span a complete set of bases of O(3). Comparison is made with the treatment of this problem by projective representations.