Massless Scalar Field Vacuum in de Sitter Spacetime

TL;DR

This paper investigates the properties of a shift-invariant vacuum state for a massless scalar field in de Sitter spacetime, demonstrating its cluster-decomposition behavior and potential implications for quantum gravity.

Contribution

It provides explicit calculations of shift-invariant correlation functions in de Sitter space, highlighting their decay properties and discussing implications for de Sitter-invariant quantum states.

Findings

Shift-invariant four-point function obeys cluster decomposition.

Gradient field correlation also exhibits cluster decomposition.

Results may inform quantum gravity models in de Sitter space.

Abstract

As a spacetime with compact spatial sections, de Sitter spacetime does not have a de Sitter-invariant ground state for a minimally-coupled massless scalar field that gives definite expectation values for any observables not invariant under constant shifts of the field. However, if one restricts to observables that are shift invariant, as the action is, then there is a unique vacuum state. Here we calculate the shift-invariant four-point function that is the vacuum expectation value of the product of the difference of the field values at one pair of points and of the difference of the field values at a second pair of points. We show that this vacuum expectation value obeys a cluster-decomposition property of vanishing in the limit that the one pair of points is moved arbitrarily far from the other pair. We also calculate the shift-invariant correlation of the gradient of the scalar field at two different points and show that it also obeys a cluster-decomposition property. Possible relevance to a putative de Sitter-invariant quantum state for gravity is discussed.