Hilbert space in the flat limit of AdS/CFT. I. Massive particles

Abstract

The flat limit of AdS$_4$ spacetime corresponds, at the level of the symmetry algebra, to the In\"on\"u--Wigner contraction $\mathfrak{so}(2,3) \to \mathfrak{iso}(1,3)$. We describe the emergence of the Hilbert space of massive scattering states of arbitrary integer spin $s$ from unitary lowest-weight representations $\mathcal{D}(\Delta,s)$ of the conformal group, using a basis of `pseudo-momentum'states first introduced by Fronsdal. These states nicely reduce to Wigner's momentum eigenstates in the limit of infinite AdS curvature radius $\ell \to \infty$, at fixed $m=\Delta/\ell$. A simple conformal map relates this description to conformal fields in $\mathbb{R}^3$ familiar from radial quantization, with the interior and exterior of the unit ball respectively corresponding to outgoing and ingoing momenta. Under the contraction, and upon appropriate rescaling, the inner product on $\mathcal{D}(\Delta,s)$ converges to the standard Lorentz-invariant inner product of Wigner's massive particles, so that unitarity is preserved throughout. The contraction to massless particles is briefly discussed here, and will be treated in detail in a forthcoming paper.