Codimension-2 surfaces and their Hilbert spaces: low-energy clues for holography from general covariance

TL;DR

This paper suggests that low-energy, diffeomorphism-invariant quantum field theories imply a Hilbert space structure dependent on codimension-2 surfaces, hinting at holographic principles through area-dependent Hilbert space dimensions.

Contribution

It introduces a novel perspective on Hilbert space dependence on boundary surfaces in diffeomorphism-invariant QFTs, proposing a potential holographic area-proportionality at low energies.

Findings

Hilbert space depends on codimension-2 boundary surfaces

Canonical basis incompatible with assumptions, allowing smaller Hilbert spaces

Hilbert space dimension may decrease with surface area at fixed volume

Abstract

We argue that the holographic principle may be hinted at already from low-energy considerations, assuming diffeomorphism invariance, quantum mechanics and Minkowski-like causality. We consider the states of finite spacelike hypersurfaces in a diffeomorphism-invariant QFT. A low-energy regularization is assumed. We note a natural dependence of the Hilbert space on a codimension-2 boundary surface. The Hilbert product is defined dynamically, in terms of transition amplitudes which are described by a path integral. We show that a canonical basis is incompatible with these assumptions, which opens the possibility for a smaller Hilbert-space dimension than canonically expected. We argue further that this dimension may decrease with surface area at constant volume, hinting at holographic area-proportionality. We draw comparisons with other approaches and setups, and propose an interpretation for the non-holographic space of graviton states at asymptotically-Minkowski null infinity.