Singleton Field Theory and Flato–Frønsdal Dipole Equation

Abstract

We study solutions of the equations (△ -λ )φ = 0 and (△ -λ )2φ = 0 in global coordinates on the covering space CAdSd of the d-dimensional Anti de-Sitter space subject to various boundary conditions and their connection to the unitary irreducible representations of \(\overline {{\text{SO}}} _0 \)(d-1,2). The ‘vanishing flux’ boundary conditions at spatial infinity lead to the standard quantization scheme for CAdSd 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.