Witten’s de Sitter pairing positivity conjecture

Establish positivity of the sesquilinear pairing obtained from Witten’s de Sitter transition amplitude by applying CPT conjugation to one argument, namely $(j,i)_W=\mathcal M(\Theta j,i)$, in quantum gravity.

Background

The paper discusses Witten’s proposal to interpret the de Sitter transition amplitude as a bilinear map between initial and final boundary-data spaces. CPT conjugation supplies an antilinear map from initial to final data, converting the amplitude into a sesquilinear pairing on the initial-data space.

The positivity of this pairing would provide a norm-like structure for de Sitter quantum gravity. The present paper does not attempt to resolve the conjecture; instead, it retains the structural idea that a de Sitter inner product may involve a reflection and studies an analogous question for a free scalar one-particle representation on a fixed de Sitter background.