A Maximally Symmetric Vector Propagator

TL;DR

This work derives a de Sitter invariant Feynman propagator for a massive vector (Proca) field in $D$ dimensions, correcting a missing inhomogeneous term in the classic Allen–Jacobson solution to ensure proper massless and flat-space limits. By recasting the propagator in a $y$-basis and solving for a master function $\\gamma(y)$, the authors obtain a closed form in terms of hypergeometric functions with parameter $\\nu = \sqrt{((D-3)/2)^{2} - m^{2}/H^{2}}$, and demonstrate that the retarded Green's function reproduces the correct classical response to a point source in the appropriate limit. The analysis covers several key limits (flat space, coincidence, massless, and $D=4$), provides explicit expressions for the massless propagator, and shows how to extract physically meaningful, gauge-consistent results for derivative-interaction theories in de Sitter space. The results facilitate perturbative calculations in curved backgrounds and have implications for quantum field theory in expanding universes.

Abstract

We derive the propagator for a massive vector field on a de Sitter background of arbitrary dimension. This propagator is de Sitter invariant and possesses the proper flat spacetime and massless limits. Moreover, the retarded Green's function inferred from it produces the correct classical response to a test source. Our result is expressed in a tensor basis which is convenient for performing quantum field theory computations using dimensional regularization.