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On the Perturbative Stability of Quantum Field Theories in de Sitter Space

Daniel Boyanovsky, Richard Holman

TL;DR

The paper develops a field-theoretic Weisskopf-Wigner (WW) approach to track real-time evolution of the Bunch-Davies vacuum and excitations in de Sitter space, avoiding reliance on asymptotic states. It shows that a massless, conformally coupled λφ^4 theory leaves the BD vacuum stable, while non-conformal interactions such as λφ^3 induce a time-dependent, volume-scaling vacuum wavefunction renormalization and vacuum decay, accompanied by particle production and entangled superhorizon pairs. A non-perturbative self-consistent screening mechanism is proposed, potentially yielding a Hilbert-space fixed point where the vacuum evolves only by a phase. The framework connects to dynamical renormalization group ideas and provides a real-time tool to probe non-gaussianity and infrared dynamics in expanding spacetimes, with several open questions for correlation functions and beyond-Markovian effects.

Abstract

We use a field theoretic generalization of the Wigner-Weisskopf method to study the stability of the Bunch-Davies vacuum state for a massless, conformally coupled interacting test field in de Sitter space. We find that in $λφ^4$ theory the vacuum does {\em not} decay, while in non-conformally invariant models, the vacuum decays as a consequence of a vacuum wave function renormalization that depends \emph{singularly} on (conformal) time and is proportional to the spatial volume. In a particular regularization scheme the vacuum wave function renormalization is the same as in Minkowski spacetime, but in terms of the \emph{physical volume}, which leads to an interpretation of the decay. A simple example of the impact of vacuum decay upon a non-gaussian correlation is discussed. Single particle excitations also decay into two particle states, leading to particle production that hastens the exiting of modes from the de Sitter horizon resulting in the production of \emph{entangled superhorizon pairs} with a population consistent with unitary evolution. We find a non-perturbative, self-consistent "screening" mechanism that shuts off vacuum decay asymptotically, leading to a stationary vacuum state in a manner not unlike the approach to a fixed point in the space of states.

On the Perturbative Stability of Quantum Field Theories in de Sitter Space

TL;DR

The paper develops a field-theoretic Weisskopf-Wigner (WW) approach to track real-time evolution of the Bunch-Davies vacuum and excitations in de Sitter space, avoiding reliance on asymptotic states. It shows that a massless, conformally coupled λφ^4 theory leaves the BD vacuum stable, while non-conformal interactions such as λφ^3 induce a time-dependent, volume-scaling vacuum wavefunction renormalization and vacuum decay, accompanied by particle production and entangled superhorizon pairs. A non-perturbative self-consistent screening mechanism is proposed, potentially yielding a Hilbert-space fixed point where the vacuum evolves only by a phase. The framework connects to dynamical renormalization group ideas and provides a real-time tool to probe non-gaussianity and infrared dynamics in expanding spacetimes, with several open questions for correlation functions and beyond-Markovian effects.

Abstract

We use a field theoretic generalization of the Wigner-Weisskopf method to study the stability of the Bunch-Davies vacuum state for a massless, conformally coupled interacting test field in de Sitter space. We find that in theory the vacuum does {\em not} decay, while in non-conformally invariant models, the vacuum decays as a consequence of a vacuum wave function renormalization that depends \emph{singularly} on (conformal) time and is proportional to the spatial volume. In a particular regularization scheme the vacuum wave function renormalization is the same as in Minkowski spacetime, but in terms of the \emph{physical volume}, which leads to an interpretation of the decay. A simple example of the impact of vacuum decay upon a non-gaussian correlation is discussed. Single particle excitations also decay into two particle states, leading to particle production that hastens the exiting of modes from the de Sitter horizon resulting in the production of \emph{entangled superhorizon pairs} with a population consistent with unitary evolution. We find a non-perturbative, self-consistent "screening" mechanism that shuts off vacuum decay asymptotically, leading to a stationary vacuum state in a manner not unlike the approach to a fixed point in the space of states.

Paper Structure

This paper contains 17 sections, 185 equations, 4 figures.

Table of Contents

  1. Introduction
  2. The Wigner-Weisskopf Method
  3. Markovian approximation
  4. Exact Solutions
  5. The Quantum Field Theoretic Wigner-Weisskopf Method in Minkowski Space
  6. Vacuum Amplitude in $\phi^4$
  7. Vacuum amplitude and $1$-particle decay in $\varphi \chi^2$
  8. Particle decay and particle production
  9. On the wave function renormalization of the vacuum state:
  10. The Quantum Field Theoretic Wigner-Weisskopf Method in de Sitter Space
  11. De Sitter Vacuum Amplitude in $\phi^4$
  12. Vacuum amplitude and $1$-particle decay in $\varphi^3$
  13. Pair Production
  14. Non-perturbative self-consistent screening?
  15. Conclusions and further questions
  16. ...and 2 more sections

Figures (4)

  • Figure 1: Transitions $|A\rangle \leftrightarrow |\kappa\rangle$ in first order in $H_I$.
  • Figure 2: $\varphi^4$ theory: left: transition matrix element $\langle \kappa|H_I|0\rangle$ with the four particle state $|\kappa\rangle= |1_{\vec{k}_1};1_{\vec{k}_2};1_{\vec{k}_3};1_{\vec{k}_4}\rangle$, right: $\langle 0|H_I|\kappa\rangle \langle \kappa|H_I|0\rangle$.
  • Figure 3: $\varphi \,\chi^2$ vacuum to vacuum amplitude. Thin lines correspond to $\varphi$, thick lines to $\chi$.
  • Figure 4: $\varphi\,\chi^2$ one $\varphi$ particle amplitude. Thin lines correspond to $\varphi$, thick lines to $\chi$.