Singleton field theory and Flato - Fronsdal dipole equation

Abstract

We study solutions of the equations $(\triangle -λ)ϕ= 0$ and $(\triangle -λ)^2ϕ= 0$ in global coordinates on the covering space $CAdS_d$ of the $d$-dimensional Anti de-Sitter space subject to various boundary conditions and their connection to the unitary irreducible representations of $\widetilde{SO}(d-1,2)$. The ``vanishing flux'' boundary conditions at spatial infinity lead to the standard quantization scheme for $CAdS_d$ in which solutions of the second- and the fourth-order equations are equivalent. To include fields realizing the singleton unitary representation in the bulk of $CAdS_d$ one has to relax the boundary conditions thus allowing for the nontrivial space of solutions of the dipole equation known as the Gupta - Bleuler triplet. We obtain explicit expressions for the modes of the Gupta - Bleuler triplet and the corresponding two-point function. To avoid negative-energy states one must also introduce an additional constraint in the space of solutions of the dipole equation.