Group Averaging for de Sitter free fields

TL;DR

This work analyzes the viability of group averaging to construct a de Sitter-invariant physical Hilbert space for perturbative quantum fields on de Sitter space. By computing explicit boost matrix elements and deriving convergence criteria, the authors show that the de Sitter group-averaged inner product converges for free fields (scalars of general mass, linear vector, and graviton fields) when the states contain sufficiently many particles, with thresholds depending on the field type and representation (principal, complementary, discrete). The results provide concrete bounds and asymptotic behavior guiding when the physical Hilbert space is well-defined, and clarify how higher-spin sectors can be treated similarly to scalars after accounting for gauge structure and residual symmetries. Collectively, these findings illuminate how gravitational constraints in a de Sitter background shape the admissible quantum states and offer a path to a consistent de Sitter-invariant quantum theory of matter and linearized gravity in arbitrary dimensions.

Abstract

Perturbative gravity about global de Sitter space is subject to linearization-stability constraints. Such constraints imply that quantum states of matter fields couple consistently to gravity {\it only} if the matter state has vanishing de Sitter charges; i.e., only if the state is invariant under the symmetries of de Sitter space. As noted by Higuchi, the usual Fock spaces for matter fields contain no de Sitter-invariant states except the vacuum, though a new Hilbert space of de Sitter invariant states can be constructed via so-called group-averaging techniques. We study this construction for free scalar fields of arbitrary positive mass in any dimension, and for linear vector and tensor gauge fields in any dimension. Our main result is to show in each case that group averaging converges for states containing a sufficient number of particles. We consider general $N$-particle states with smooth wavefunctions, though we obtain somewhat stronger results when the wavefunctions are finite linear combinations of de Sitter harmonics. Along the way we obtain explicit expressions for general boost matrix elements in a familiar basis.