Spinning Fields in Lorentzian AdS

TL;DR

The paper constructs explicit higher-spin wave functions in Lorentzian AdS using embedding-space primaries built from a scalar seed and AdS special-conformal generators, and verifies their consistency by computing the quadratic Casimir and the corresponding AdS wave equation with the appropriate mass definition. In the flat-space limit, boosted primaries reduce to massive higher-spin plane waves with transverse polarizations, enabling construction of in/out scattering states, while massless cases reveal nontrivial longitudinal polarization subtleties and potential Goldstone sectors. The work clarifies the role of Inonu–Wigner contraction, provides a concrete framework for the spin-$\ell$ representations via symmetric-traceless tensors, and highlights subtleties in the massless limit and in the decay of traces. These results advance the holographic understanding of higher-spin dynamics in AdS and their flat-space extrapolations, with implications for S-matrix-like interpretations and future extensions to spinors and full extrapolate dictionaries.

Abstract

We construct the higher spin wave functions in the embedding space of anti-de Sitter Lorentzian spacetime. These wave functions are built from a primary wave functions that has a simple structure expressed in terms of the special conformal generator vector fields in AdS. We compute the eigenvalue of the quadratic Casimir for the symmetric traceless states, and show explicitly that these satisfy the higher spin wave equation. We also demonstrate that these wave functions have the right structure in the flat space limit for massive higher spin fields, and can be used to construct $in$ and $out$ states for scattering processes. Spinning states that become massless in the flat limit are extremely subtle. The problem can be isolated to longitudinal polarizations.