Taming the alpha-vacuum

TL;DR

This work shows that in de Sitter space, the α-vacua can be tamed by a double-source propagator derived from a new α-dependent time-ordering, enabling causal and renormalizable perturbation theory at one loop. By formulating a Schwinger-Keldysh generating functional with sources at points x and x_A, the authors demonstrate renormalizability for local and special antipodal interactions, and argue that a broad class of antipodal couplings Φ^p(x) Φ^q(x_A) with p+q ≤ 4 remain renormalizable. The analysis relies on internal lines being Euclidean, with external α-dependence, and extends to the inflationary patch where mode functions and contour propagators are explicitly constructed. The results imply a larger landscape of renormalizable quantum field theories in curved spacetime and open avenues for exploring whether inflationary dynamics might dynamically favor the Bunch-Davies vacuum under interactions.

Abstract

An interacting scalar field theory in de Sitter space is non-renormalizable for a generic alpha-vacuum state. This pathology arises since the usual propagator used allows for a constructive interference among propagators in loop corrections, which produces divergences that are not proportional to standard counterterms. This interference can be avoided by defining a new propagator for the alpha-vacuum based on a generalized time-ordering prescription. The generating functional associated with this propagator contains a source that couples to the field both at a point and at its antipode. To one loop order, we show that a set of theories with very general antipodal interactions is causal and renormalizable.

Figures (5)

• Figure 1: The contour used to evaluate the evolution of operators over a finite time interval. Here we have shown the contour doubling for global coordinates, Eq. (\ref{['globalcoords']}), but an analogous prescription is used for any coordinate system. The initial state is specified at $\tau_0$ and is evolved until $\tau$. We double the field content so that separate copies of the fields are used for the upper and lower parts of the contour.
• Figure 2: The Penrose diagram for de Sitter space showing the region covered by inflationary coordinates which has been left unshaded. The past and future null infinity surfaces are denoted by ${\cal I}^-$ and ${\cal I}^+$. The Schwinger-Keldysh approach evolves a matrix element for an initially specified state at $\eta_0$ forward to an arbitrary later time $\eta$.
• Figure 3: The one loop self-energy correction in a $\Phi^3$ theory.
• Figure 4: The counterterm graphs corresponding to the possible quadratic terms. The labels $m,\alpha$ indicate whether a propagator is mixed or $\alpha$.
• Figure 5: A one loop self-energy correction in a general cubic theory with an arbitrary propagator structure. Note that the external legs are either mixed or $\alpha$ propagators. The counterterm diagram removes the logarithmic divergence of the loop.