Hilbert Bundles and Holographic Space-time Models

TL;DR

This paper reformulates Holographic Space-time models using Hilbert bundles over geodesics, connecting quantum models with classical hydrodynamics and proposing a new perspective on the structure of spacetime and holography.

Contribution

It introduces a Hilbert bundle framework for HST models, linking boundary CFTs to Dirac eigenspectra and proposing a novel conjecture for the density matrix and time evolution in quantum spacetime.

Findings

Hilbert bundle formulation of HST models.

Connection between boundary CFTs and Dirac operator eigenspectra.

Conjecture relating 't Hooft relations to Schwinger terms.

Abstract

We reformulate Holographic Space-time (HST) Models as Hilbert bundles over the space of time-like geodesics on a background manifold. The background, following Jacobson, is viewed as a hydrodynamic flow, which the quantum model must reproduce. Work of many authors, but particularly the Verlindes, Carlip and Solodukhin, suggest that the relevant quantum model is a sequence of 1+1 dimensional conformal field theories (CFT) on the boundaries of nested causal diamonds. Finiteness of the entropy suggests the CFTs be built from cutoff fermion fields, and the spin/statistics connection, combined with Connes' demonstration that Riemannian geometry is encoded in the Dirac equation, suggests that these fields are labelled by the cutoff eigenspectrum of the Dirac operator on the holoscreen of each diamond. This leads to a natural conjecture for the density matrix of arbitrary diamonds and,in a subclass of space-times, for the time evolution operator between them. We conjecture that the 't Hooft commutation relations on diamond boundaries are related to Schwinger terms in U(1) currents constructed from the fermion fields. We review the notion of "locality as constraints on holographic variables" discovered by Fiol, Fischler and the present author and, in an appendix, explain how it differs from the notion of locality arising from tensor network constructions in AdS/CFT.