Double groups and projective representations

Abstract

Some problems are discussed in relation to the usual treatment of improper groups through their double groups, in particular the identification (rather than the mere isomorphism) of such groups as C 3v and D 3. The enhancement of SU(2) by the addition of the inversion is analysed for this purpose. This requires a careful discussion of the behaviour of spinors under inversion and two types of spinors are defined, Cartan and Pauli spinors, that behave differently with respect to inversion, although it is shown that this difference merely entails a choice of gauge in the language of projective representations. A distinction is proposed between the inversion operation and the parity operator: when the former is realized as a binary rotation in 4-space, the latter can be identified with its infinitesimal generator. The passage from SO(3) to O(3) (group of all proper and improper rotations) is studied and a hitherto unknown faithful projective representations of O(3) is given. It is shown how spinor representations can be constructed for improper point groups in either the Cartan or Pauli gauges. A choice of gauge is proposed to ensure agreement with current practice in angular momentum theory and with that in single point groups. As an example, Clebsch-Gordan coefficients are constructed for C 3v .