Path integral games with de Sitter α-vacua

Abstract: The $\alpha$-vacua are a 1-parameter family of quantum field vacua in de Sitter space which are invariant under the $SO(1,d)$ isometry group. In this work we study the $\alpha$-vacua using the path integral. These states can be prepared by acting on the Bunch-Davies vacuum with a certain charge operator. While most conserved charges live on a single codimension-1 manifold, this particular charge lives on a pair of two codimension-1 manifolds which are antipodal mirrors of each other. The rules for the manipulation of this charge as an insertion in the path integral are explained. We also discuss the special $\alpha$-vacua known as the in'' andout'' vacua. It is well known that the in'' andout'' vacua are equal in odd dimensions, but we also show that they are equal in even dimensions when $\sqrt{(d-1)^2/4 - m^2}$ is a half-integer. In addition, we study the wavefunctionals of the $\alpha$-vacua at $\mathcal{I}^+$, and argue that the $\alpha$-vacua cannot be constructed in interacting quantum field theories.