Higher-loop norm of the no-boundary state

TL;DR

The paper proves that the leading contribution to the de Sitter no-boundary state has vanishing norm to all orders in perturbation theory for $d\ge 3$, even when coupled to matter. By fixing diffeomorphisms with a transverse-traceless gauge and constructing field-dependent residual diffeomorphisms that realize $\mathfrak{so}(d,1)$ on the gauge slice, the authors show the loop-corrected wavefunction remains invariant under these transformations and that the norm is always proportional to $1/\text{vol}(\mathrm{SO}(d,1))$, which vanishes for the noncompact group. This extends prior one-loop and $d=3$ all-loop results to general dimensions and demonstrates a persistent subtlety: the no-boundary norm does not coincide with the sphere partition function, due to contour choices and complex metrics in the path integral. The work also explains that coupling to matter does not remove the residual gauge structure, preserving the all-loop vanishing result and prompting discussions about the physical and cosmological implications of a zero-norm leading contribution in quantum cosmology.

Abstract

The leading contribution to the de Sitter no-boundary state comes from geometries with spherical spatial slices, including the Hartle-Hawking geometry and fluctuations around it. Recent work showed that this leading contribution has vanishing norm at one loop. Here we show that the norm in fact vanishes to all orders in perturbation theory.