Holography from the Wheeler-DeWitt equation

TL;DR

This work provides a perturbative analysis of the Wheeler-DeWitt constraints around AdS, revealing that the asymptotic $T$-component of the metric is correlated with the energy of bulk matter and gravitons. By solving the integrated Hamiltonian constraint and then uplifting to full pointwise constraints, the authors construct a dressed Fock-space description for gravitons and matter and define a compatible inner product. They prove that boundary data in an infinitesimal time band, through correlators of boundary observables including the boundary Hamiltonian $H_ ext{∂}$, uniquely determine the bulk state, yielding a perturbative version of holography of information in AdS. This establishes a concrete mechanism by which gravitational constraints enforce nonlocal information encoding and provides a framework to explore subregion duality and extensions to flat space in future work.

Abstract

In a theory of quantum gravity, states can be represented as wavefunctionals that assign an amplitude to a given configuration of matter fields and the metric on a spatial slice. These wavefunctionals must obey a set of constraints as a consequence of the diffeomorphism invariance of the theory, the most important of which is known as the Wheeler-DeWitt equation. We study these constraints perturbatively by expanding them to leading nontrivial order in Newton's constant about a background AdS spacetime. We show that, even within perturbation theory, any wavefunctional that solves these constraints must have specific correlations between a component of the metric at infinity and energetic excitations of matter fields or transverse-traceless gravitons. These correlations disallow strictly localized excitations. We prove perturbatively that two states or two density matrices that coincide at the boundary for an infinitesimal interval of time must coincide everywhere in the bulk. This analysis establishes a perturbative version of holography for theories of gravity coupled to matter in AdS.

Figures (1)

• Figure 1: A common misunderstanding of "holography" is that it only tells us that data on the timelike boundary of AdS can be used to reconstruct physics at the bulk point P as shown in the left subfigure. But this statement follows from the causal structure of AdS and does not require holography. Gravitational theories are genuinely holographic. In this paper we show how, in gravity, data on an infinitesimal time band (right subfigure) can be used to reconstruct physics in the bulk.