Stueckelberg massive electromagnetism in de Sitter and anti–de Sitter spacetimes: Two-point functions and renormalized stress-energy tensors

By considering Hadamard vacuum states, we first construct the two-point functions associated with Stueckelberg massive electromagnetism in de Sitter and anti–de Sitter spacetimes. Then, from the general formalism developed by Belokogne and Folacci [Phys. Rev. D 93, 044063 (2016)], we obtain an exact analytical expression for the vacuum expectation value of the renormalized stress-energy tensor of the massive vector field propagating in these maximally symmetric spacetimes.

Figure 1

Carter-Penrose diagram of ${\mathrm{dS}}^4$. Without loss of generality, the point $x$ can be taken to be any point in spacetime. $Px$ denotes its associated antipodal point. The left and right edges of this diagram must be identified along the dashed lines. ${\mathcal{I}}^{+}$ and ${\mathcal{I}}^{-}$ denote, respectively, the future and past spacelike infinities for timelike and null geodesics. The hatched area is the set of points $x^{\prime }$ which cannot be reached by geodesics from $x$ and for which $\sigma (x,x^{\prime })$ is not defined.

Figure 2

Carter-Penrose diagram of ${\mathrm{AdS}}^4$. Without loss of generality, the point $x$ can be taken to be any point in spacetime. $Px$ denotes its associated antipodal point. Here, we only display an elementary cell of the universal covering of anti–de Sitter spacetime. The whole manifold is obtained by gluing along the dashed lines the top edge of one cell with the bottom edge of another one and by replicating, ad nauseam, this process. $\mathcal{I}$ denotes timelike infinity. The hatched area is the set of points $x^{\prime }$ which cannot be reached by geodesics from $x$ and for which $\sigma (x,x^{\prime })$ is not defined.