Path integral games with de Sitter {\alpha}-vacua

Abstract

The $\alpha$-vacua are a 1-parameter family of quantum field vacua in de Sitter space which are invariant under the isometry group $SO(1,d)$. In this work give a path integral construction of the de Sitter $\alpha$-vacua. We explain that these states can be prepared by acting on the Bunch-Davies vacuum with a certain non-local charge operator. While most conserved charges live on a single codimension-1 manifold, we show that 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 further explain how this charge can be used to solve for the wavefunctionals of the $\alpha$-vacua at $\mathcal{I}^+$ (in the regime that $\alpha$ is small) by deforming the equator of de Sitter space to $\mathcal{I}^+$ / $\mathcal{I}^-$. We also discuss the special $\alpha$-vacua known as the ``in'' and ``out'' vacua, for both heavy and light scalars. It is well known that the ``in'' and ``out'' 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. Finally, we investigate the question of whether the $\alpha$-vacua can be constructed in interacting quantum field theories. Using the fact that the antipodal charge operator is only conserved in a free theory, we give a symmetry based argument that in general they cannot be, explaining why interactions and $SO(1,d)$ invariance of the vacua are in tension.