Vacua and infrared radiation in de Sitter quantum field theory

TL;DR

This paper develops a gauge-invariant framework to understand infrared behavior of quantum fields on de Sitter space via the horizon memory observable. It proves a sharp criterion: the de Sitter-invariant Bunch-Davies vacuum exists in the free theory iff the memory commutes with all local observables; applying this to the massive scalar, electromagnetism, and linearized gravity yields BD vacua, while the massless minimally coupled scalar lacks a normalizable BD state and requires a direct-integral Hilbert space construction across memory sectors. It further shows that horizons render long-lived sources capable of producing large, potentially infinite, infrared radiation, though fully interacting theories likely select a single Hilbert space representation. The work provides a unified, gauge-invariant treatment of IR structure across free fields in de Sitter and clarifies the role of memory in both vacuum structure and radiation production.

Abstract

We analyze the asymptotic behavior of quantum fields and perturbative quantum gravity in de Sitter space. We show that the necessary and sufficient condition for the existence of a de Sitter invariant vacuum state in the free theory is if the local field observables commute with the ``memory observable'' on any cosmological horizon. This criterion yields simple, gauge-invariant proofs of the existence of a de Sitter invariant vacuum for the (source-free) massive scalar field, electromagnetic field and linearized gravitational field. However, the massless, minimally coupled scalar does not satisfy this criterion and, consequently, there is no de Sitter invariant vacuum state. In this case, we construct the Hilbert space of normalizable states that have ``square integrable'' distributions in the memory and the conjugate constant mode. This Hilbert space has a unitary representation of the de Sitter symmetry group but no de Sitter invariant normalizable state. However even for free theories with a normalizable de Sitter invariant vacuum we show, by a simple example, that in the presence of a source a large number of infrared particles can be produced if the source interacts with the field for timescales much longer than the Hubble time. In the limit as the source persists forever it emits an infinite number of infrared particles.

Figures (1)

• Figure 1: The Carter-Penrose diagram for global de Sitter spacetime. Each point on the diagram represents a $2$-sphere in spacetime, except on the vertical lines where it represents a point in the spacetime. In global coordinates (detailed in the text), a constant time slice is depicted by the horizontal dashed line representing a $3$-sphere. The "doubled" horizontal lines at the bottom and top of the diagram at $\eta = 0$ and $\eta = \pi$, represent past ($\mathscr I^-$) and future ($\mathscr I^+)$ conformal infinities, respectively. A cosmological horizon $\mathcal{H}$ corresponding to $\eta = \chi$ is shown as a diagonal line. The gray shaded region is the associated static patch.

Theorems & Definitions (2)

• Remark 1: $\mathscr{H}_\textrm{dS}$ as a tensor product
• Remark 2: Existence and non-existence of states invariant under de Sitter subgroups