Remarks on Lightlike Continuous Spin and Spacelike Representations of the Poincaré Group

Certain classes of unitary representations of the lightlike continuous spin and the spacelike cases are constructed. The generators in these representations involve operators forming the ``intrinsic'' algebras (i.e., commuting with the orbital parts) E3 and O(3, 1) for P2 = 0 and P2 < 0, respectively. The parallel construction for P2 > 0 involving an intrinsic O4 algebra is indicated. Equivalence relations with certain other forms are given through a unitary transformation. The physical significance of the ``translation'' generators of E3 is brought out in terms of the projections of W orthogonal to P. Corresponding results for the O(3, 1) and O(4) algebras are given. For the continuous-spin case, these operators are shown to provide a basis with extremely simple transformation properties, related to a certain symmetric toplike behavior even under Lorentz transformations. With a view to future use, the matrix elements of W on the energy-rotation basis are calculated in a unified manner for all the three cases, P2⋛0. A deformation formula leading from zero-mass continuous-spin representations to spacelike ones is studied. Certain types of nonunitary representations are briefly introduced at the end.