M\"obius randomness in the Hartle-Hawking state

TL;DR

This paper explores the quantum cosmology of toroidal universes, linking the Hartle-Hawking state to automorphic forms, the Riemann zeta zeros, and the Langlands program, suggesting deep connections between cosmology and number theory.

Contribution

It provides a novel representation of the Hartle-Hawking state using Langlands spectral decomposition, connecting quantum cosmology to automorphic forms and the Riemann zeta function.

Findings

Expresses the Hartle-Hawking state as a sum over Riemann zeta zeros.

Shows near singularity dynamics governed by the Hilbert-Pólya Hamiltonian.

Reveals a M"obius average of CFT partition functions related to de Sitter entropy.

Abstract

We consider quantum cosmology for toroidal universes in d+1 dimensions. The Hilbert space is the space of square-integrable automorphic forms for GL(d). The Hartle-Hawking state is defined as a Poincar\'e sum over the no-boundary geometries. We obtain its representation in the Langlands spectral decomposition. This leads to an expression as a sum over the Riemann zeta zeros and implies that its near singularity dynamics is governed by the Hilbert-P\'olya Hamiltonian. It also takes the form of a M\"obius average of CFT partition functions which suggests a similar interpretation for the de Sitter entropy. We briefly discuss the relationship between quantum cosmology and the Langlands program.