Möbius randomness in the Hartle-Hawking state

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 Poincaré 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-Pólya Hamiltonian. It also takes the form of a Möbius 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.

63 pages, 7 figures, v2: typos and minor corrections