Mode-sum construction of the two-point functions for the Stueckelberg vector fields in the Poincar\'e patch of de Sitter space

TL;DR

This paper constructs and analyzes the two-point functions for massive vector fields in de Sitter space using mode-sum quantization, clarifying their relation to previous results and limits.

Contribution

It provides a comprehensive mode-sum construction of the two-point functions for Stueckelberg vector fields in de Sitter space, connecting various limits and prior works.

Findings

Reproduces flat-space propagator in the appropriate limit

Shows the two limits of the two-point function correspond to previous works

Clarifies the relation between different vector field two-point functions

Abstract

We perform canonical quantization of the Stueckelberg Lagrangian for massive vector fields in the conformally flat patch of de Sitter space in the Bunch-Davies vacuum and find their Wightman two-point functions by the mode-sum method. We discuss the zero-mass limit of these two-point functions and their limits where the Stueckelberg parameter $\xi$ tends to zero or infinity. It is shown that our results reproduce the standard flat-space propagator in the appropriate limit. We also point out that the classic work of Allen and Jacobson for the two-point function of the Proca field and a recent work by Tsamis and Woodard for that of the transverse vector field are two limits of our two-point function, one for $\xi \to \infty$ and the other for $\xi \to 0$. Thus, these two works are consistent with each other, contrary to the claim by the latter authors.