Group Averaging of massless scalar fields in 1+1 de Sitter

TL;DR

This paper investigates the construction of de Sitter-invariant quantum states for a massless scalar field in 1+1 dimensions using group averaging, including zero-modes and counting invariant states by energy.

Contribution

It extends previous work by explicitly including zero-modes and providing a detailed count of invariant states based on energy in the toy model.

Findings

De Sitter-invariant states can be constructed via group averaging.

Zero-modes play a crucial role in the state space.

Number of invariant states depends on the defined energy.

Abstract

Perturbative gravity in global de Sitter space is subject to so-called linearization stability constraints: If they are to couple consistently to the gravitational field, quantum states must be invariant under the de Sitter isometries. While standard Fock spaces contain no de Sitter-invariant states apart from (possibly) the vacuum, a full Hilbert space of de Sitter-invariant quantum states can be constructed via group averaging techiniques. We re-examine the simple toy model of de Sitter group averaging given by the free 1+1 scalar field, expanding on an earlier analysis by Higuchi. Our purpose is twofold: to include the scalar zero-mode, and to explicitly count the number of de Sitter-invariant states as a function of an appropriately defined energy.