The Generalized Hartle-Hawking Initial State: Quantum Field Theory on Einstein Conifolds

TL;DR

This paper extends the Hartle-Hawking initial state to include conifold histories, showing that quantum field theory on Einstein conifolds emerges in the semiclassical limit, with implications for scalar fields on de Sitter space.

Contribution

It introduces a generalized Hartle-Hawking state incorporating conifold topologies, enabling quantum field theory on Einstein conifolds to be derived from gravitational path integrals.

Findings

Quantum field theory on Einstein conifolds is recovered in the semiclassical limit.

The state on $RP^3$ de Sitter spacetime differs from but relates to the Bunch-Davies vacuum.

The generalized state cannot be obtained from the original Hartle-Hawking formulation as no Einstein manifold with round $RP^3$ boundary exists.

Abstract

Recent arguments have indicated that the sum over histories formulation of quantum amplitudes for gravity should include sums over conifolds, a set of histories with more general topology than that of manifolds. This paper addresses the consequences of conifold histories in gravitational functional integrals that also include scalar fields. This study will be carried out explicitly for the generalized Hartle-Hawking initial state, that is the Hartle-Hawking initial state generalized to a sum over conifolds. In the perturbative limit of the semiclassical approximation to the generalized Hartle-Hawking state, one finds that quantum field theory on Einstein conifolds is recovered. In particular, the quantum field theory of a scalar field on de Sitter spacetime with $RP^3$ spatial topology is derived from the generalized Hartle-Hawking initial state in this approximation. This derivation is carried out for a scalar field of arbitrary mass and scalar curvature coupling. Additionally, the generalized Hartle-Hawking boundary condition produces a state that is not identical to but corresponds to the Bunch-Davies vacuum on $RP^3$ de Sitter spacetime. This result cannot be obtained from the original Hartle-Hawking state formulated as a sum over manifolds as there is no Einstein manifold with round $RP^3$ boundary.