Hilbert space in the flat limit of AdS/CFT. II. Massless particles

Abstract

Building on Part I of this series, where we described the emergence of massive scattering states, we demonstrate how unitary lowest-weight representations $\mathcal D(Δ,s)$ of the conformal group with fixed dimension $Δ$ and arbitrary integer spin $s$ contract to massless Poincaré particle representations. This requires combining the Inönü--Wigner contraction $\mathfrak{so}(2,3) \to \mathfrak{iso}(1,3)$ with an infinite boost, in which case massless scattering states naturally emerge in a carrollian basis. For $Δ>s+1$, we find that $\mathcal D(Δ,s)$ contracts to the direct sum of massless particles of all spins $s'\leq s$. At the unitarity bound, lower-spin states become null so that $\mathcal D(s+1,s)$ contracts to a single massless particle representation carrying the two maximal helicities $J=\pm s$. In both cases, the inner product converges to the Poincaré-invariant one after a single helicity-independent rescaling, so that unitarity is preserved throughout the contraction.