A discrete series gauge field at the late-time boundary of $dS_4$
We study the free Maxwell field on the planar patch of four-dimensional de Sitter spacetime ($dS_4$). We review its bulk canonical quantization in the Bunch-Davies vacuum, and we give a representation-theoretic viewpoint by studying the transformation properties of single-particle states under infinitesimal dS transformations. By taking the late-time limit, we identify two late-time operators with scaling dimensions $Δ=1$ (leading) and $Δ=2$ (subleading). We introduce CFT-inspired inner products invariant under the dS group ($SO(4,1)$) for states created by late-time operators acting on the Bunch-Davies vacuum. We explain how the unitary discrete series representations of $SO(4,1)$ associated with the photon are furnished by both operators, in contrast with the Maxwell field on $AdS$ where the $Δ=1$ operator corresponds to a non-normalizable mode. We also explain how the corresponding discrete series representations of $SO(4,1)$ split into a direct sum of representations corresponding to two helicities $+1$ and $-1$. This is achieved by taking advantage of the dS invariance of the self-dual and anti-self-dual sectors of the photon field strength. In our outlook, we draw inspiration from a recent proposal for the microscopic description of $dS_4$ higher-spin gravity where conformal gauge fields in 3 dimensions play a central role, and we investigate a possible connection of the $Δ=1$ late-time operator with a conformal spin-1 gauge field in 3 dimensions.