Continuous Degenerate Representations of Noncompact Rotation Groups. II

Abstract

Three principal continuous series of most degenerate unitary irreducible representations of an arbitrary noncompact rotation group SO(p, q) have been derived and their properties discussed in detail. The corresponding harmonic functions have been constructed.

Tall tales from de Sitter space II: Field theory dualities

“…The discrete series then corresponds to special values of the mass in this range where a subset of the modes display the convergent h + behavior at both I ± -see Appendix A and refs. [24,25,26,27]. However, we emphasize that even in these special cases, the full space of solutions still includes modes diverging at both asymptotic boundaries.…”

“…These three regimes also appear in discussions in the mathematics literature -see, e.g., [24,25,26,27]. There the scalar field is classified according to M 2 regarded as its SO(n + 1, 1) Casimir.…”

“…Finally we briefly consider the tachyonic or discrete series [25,26,27]. Recall that in this case with M 2 < 0, h − < 0 so that the modes scaling as e ±h − t diverge as one approaches either I − or I + , respectively.…”

Lie algebra bases for the pseudo-orthogonal groups

“…The Lie algebra so (p, q) of SO(p, q) may then be defined as the set of real (p + q) x (p + q) matrices a such that tr a = 0 and ag+ga=0 (1.1) Even in two closely related problems it can happen that the so (p, q) basis is the most useful for one problem while the canonical basis is the most convenient for the other. For example, the recent determination of the maximal solvable subgroups of SO(p, q) by Patera, Winternitz & Zassenhaus (1974) employs the so(p, q) basis, while for embeddings of SO(p, q) with other semi-simple Lie groups (Cornwell, 1971a(Cornwell, , 1971bEkins & Comwell, 1974a, 1974b, 1974c) the canonical basis is the most convenient.…”

Eigenfunction Expansions of Functions Describing Systems with Symmetries

“…The operator −F on L 2 (H 4 + ) coincides with the Laplace operator ∆(H 4 + ) in the corresponding coordinate system (see [37]). …”