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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 (9 sections, 53 equations, 2 figures)

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.