Path integral games with de Sitter α-vacua

TL;DR

This work provides a path-integral construction of de Sitter $α$-vacua by acting on the Bunch-Davies vacuum with a non-local antipodal charge $\hat{Q}^A$, yielding a one-parameter family of $SO(1,d)$-invariant states $|α\rangle = e^{i α \hat{Q}^A}|0\rangle$. It develops the Ward identity framework for the antipodal current, explains how the antipodal charge cannot be contracted on the Euclidean half-sphere, and derives Bogoliubov relations that generate $α$-vacua from the BD vacuum. The paper then analyzes the wavefunctionals at future infinity ${\mathcal{I}}^+$ for odd $d$ and large mass, showing how small-$α$ corrections can be obtained via a controlled path-integral deformation and S-matrix evolution. Finally, it argues that in interacting theories the antipodal current is no longer conserved, placing strong constraints on the existence of $SO(1,d)$-invariant α-vacua beyond the free theory, and discusses operator-ordering subtleties that prevent a straightforward finite-$α$ construction. These results illuminate the role of non-local symmetries in curved-spacetime vacua and clarify the limitations of α-vacua in realistic interacting QFTs.

Abstract

The $α$-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 $α$-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 $α$-vacua at $\mathcal{I}^+$ (in the regime that $α$ is small) by deforming the equator of de Sitter space to $\mathcal{I}^+$ / $\mathcal{I}^-$. We also discuss the special $α$-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 $α$-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.

Figures (16)

• Figure 1: Lorentzian de Sitter space $dS_d$ and euclidean de Sitter space $S_d$. The equator $E$, defined at $X^0 = \tilde{X}^0 = 0$, exists on both. The halves of the euclidean sphere above and below $E$ are denoted $S_{d,+}$ and $S_{d,-}$, respectively.
• Figure 2: Definition of the Bunch-Davies wavefunctional $\Psi^E_0[\varphi]$ as a euclidean path integral on the half-sphere with boundary conditions $\eval{\phi}_{E} = \varphi$.
• Figure 3: The Killing vector field $L^{01}$ generates a boost in lorentzian de Sitter and a rotation in euclidean de Sitter.
• Figure 4: Proof of the de Sitter invariance of the Bunch-Davies state. $\epsilon \hat{L}^{01}$ rotates the euclidean half-sphere by angle $\epsilon$, which does not actually change the path integral.
• Figure 5: If we were to add some insertions into the half-sphere path integral, the corresponding state would not be annihilated by $\hat{L}^{01}$ because the rotation of the equator by a tiny angle $\epsilon$ would not leave the path integral invariant.