On Representations of the Conformal Group Which When Restricted to Its Poincaré or Weyl Subgroups Remain Irreducible

Abstract

Unitary irreducible representations of the conformal group which, when restricted to its Poincaré or Weyl (= Poincaré group extended by dilatations) subgroups, remain irreducible are found. In particular it is proved that the continuous spin representations (p_μ·p^μ = 0) of the Poincaré group cannot be extended to the conformal group and that, on the other hand, a known extension in the discrete spin case is unique (up to a unitary equivalence). Similar results hold for Weyl group for which, in addition, extensions also exist in the case p_μ·p^μ ≠ 0. Namely, each unitary irreducible representation of the Weyl group characterized by the sign of p_μ·p^μ (≠0) and by invariants of the corresponding little group can be extended to a one‐parameter family of irreducible representations of the conformal group. Finally, it is shown that, besides the above mentioned extensions of unitary irreducible representations of the Weyl group, there are no others.