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A note on alpha-vacua and interacting field theory in de Sitter space

Kevin Goldstein, David A. Lowe

TL;DR

This work addresses whether quantum field theory in de Sitter space can be consistently extended to the one-parameter family of $\alpha$-vacua for interacting scalars. It develops an imaginary-time (Euclidean) perturbation framework in which the $\alpha$-vacuum is realized as a squeezed state on the Euclidean vacuum via a unitary operator $${\cal U}_\alpha$$, enabling a renormalizable theory and enabling a causal continuation to real time. Key results include a generalized Wick's theorem for $\alpha$-vacua, a renormalizable perturbation theory with a non-local but constrained interaction structure, and a spectral theorem ensuring causal real-time two-point functions; the renormalized stress-energy tensor acquires $\alpha$-dependent counterterms but has no imaginary part at leading order, indicating stability. The findings support the viability of $\alpha$-vacua in inflationary contexts and suggest potential observable consequences while providing a concrete framework for analyzing squeezed-state vacua in curved spacetime.

Abstract

We set up a consistent renormalizable perturbation theory of a scalar field in a nontrivial alpha vacuum in de Sitter space. Although one representation of the effective action involves non-local interactions between anti-podal points, we show the theory leads to causal physics, and we prove a spectral theorem for the interacting two-point function. We construct the renormalized stress energy tensor and show this develops no imaginary part at leading order in the interactions, consistent with stability.

A note on alpha-vacua and interacting field theory in de Sitter space

TL;DR

This work addresses whether quantum field theory in de Sitter space can be consistently extended to the one-parameter family of -vacua for interacting scalars. It develops an imaginary-time (Euclidean) perturbation framework in which the -vacuum is realized as a squeezed state on the Euclidean vacuum via a unitary operator , enabling a renormalizable theory and enabling a causal continuation to real time. Key results include a generalized Wick's theorem for -vacua, a renormalizable perturbation theory with a non-local but constrained interaction structure, and a spectral theorem ensuring causal real-time two-point functions; the renormalized stress-energy tensor acquires -dependent counterterms but has no imaginary part at leading order, indicating stability. The findings support the viability of -vacua in inflationary contexts and suggest potential observable consequences while providing a concrete framework for analyzing squeezed-state vacua in curved spacetime.

Abstract

We set up a consistent renormalizable perturbation theory of a scalar field in a nontrivial alpha vacuum in de Sitter space. Although one representation of the effective action involves non-local interactions between anti-podal points, we show the theory leads to causal physics, and we prove a spectral theorem for the interacting two-point function. We construct the renormalized stress energy tensor and show this develops no imaginary part at leading order in the interactions, consistent with stability.

Paper Structure

This paper contains 9 sections, 53 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Free fields
  3. Generalized Wick's theorem
  4. Interacting fields
  5. Real-time correlators and causality
  6. Stress-Energy Tensor
  7. Conclusions
  8. Appendix: Squeezed states
  9. Appendix: Some useful facts

Figures (2)

  • Figure 1: Feynman diagram for (\ref{['eq:free13']}). The thick line represents the $\alpha$-vacuum propagator $G_\alpha(x,y)$. Thin lines represent Euclidean vacuum propagators with a factor of $N_\alpha^2$, and the other factors are shown explicitly. Grey dots denote points that appear in propagators as antipodes. We have defined $\gamma=e^\alpha$.
  • Figure 2: Feynman diagram for propagator in $\lambda \phi^3$. The indices $i_k$ label end-points of the propagators $G_{i j}$. All indices are to be summed over. Vertex factors carry no $i_k$ dependence.