Non-Locality induces Isometry and Factorisation in Holography

TL;DR

The work shows that non-local gravitational corrections—encoded as replica wormholes arising from state averaging over time-shifted TFD microstates—reduce an apparently infinite BH Hilbert space to a finite dimension D = e^{S_BH}, restoring bulk-boundary isometry. By counting linearly independent microstates with a resolvent analysis of the Gram matrix, it demonstrates a transition from a type III_1 to a type I_D von Neumann algebra, providing a concrete mechanism for factorization. The approach unifies non-isometric bulk-boundary mappings and the factorisation puzzle through non-perturbative, non-local gravitational effects inherent to wormholes. This links Euclidean and Lorentzian notions of non-locality and supports the black hole information paradox resolutions within a single framework grounded in gravitational path integrals and algebraic QFT.

Abstract

In holography, two manifestations of the black hole information paradox are given by the non-isometric nature of the bulk-boundary map and by the factorisation puzzle. By considering time-shifted microstates of the eternal black hole, we demonstrate that both these puzzles may be simultaneously resolved by taking into account non-local quantum corrections that correspond to wormholes arising from state averaging. This is achieved by showing, using a resolvent technique, that the resulting Hilbert space for an eternal black hole in Anti-de Sitter space is finite-dimensional with a discrete energy spectrum. The latter gives rise to a transition to a type I von Neumann algebra.

Figures (12)

• Figure 1: Visualization of the black hole microstates \ref{['eq:generalised_TFD']}. The left figure shows the Kruskal diagram of an eternal black hole. The vertical arrows indicate that time runs in different directions for the left and right boundary, while the black hole horizon is depicted by dashed lines. The TFD state \ref{['eq:TFD_state']} corresponds to this geometry, together with an identification of the left and right boundary times along the Cauchy slice drawn in blue. This identification is preserved under the evolution with the difference of boundary Hamiltonians, shown on the right hand side. Evolving with the sum of the boundary Hamiltonians, as indicated by the red line, induces a time shift $\delta$, and the TFD state is changed to a generalized TFD state \ref{['eq:generalised_TFD']}.
• Figure 2: Geometric representation of the right hand side of \ref{['eq:derivation_overlaps']}. The first term corresponds to two disconnected topologies, while the second represents a replica wormhole arising from state averaging. The averaging over the states labeled by $\gamma$ is depicted through the dashed line in this diagram.
• Figure 3: At large $N$, the black-hole Hilbert space $\mathcal{H}_{\text{BH}}$ splits into GNS Hilbert spaces around the TFD state, labelled by $\mathcal{H}_{\text{GNS}}$, and the microstates \ref{['eq:generalised_TFD']}, labelled $\mathcal{H}^{\alpha/\gamma}_{\text{GNS}}$. Due to \ref{['eq:evolved_TFD']}, the left boundary Hamiltonian allows us to transition between different GNS Hilbert spaces.
• Figure 4: The left-hand side shows the boundary conditions imposed in the gravitational path integral when preparing the bra or the ket of a generalized TFD state \ref{['eq:generalised_TFD']}. These correspond to the upper or lower half of a thermal circle that is cut open along the $\tau=0$ point in Euclidean time. The phases enter the boundary conditions via the respective identification between left and right time. The boundary condition imposed on the path integral, when computing the overlap between different microstates \ref{['eq:orthogonal']} corresponds to a single thermal circle, obtained by gluing both halves of the circle together. The leading geometry contributing to the gravitational path integral, which is consistent with this boundary condition is given by the Euclidean black hole, which is shown on the right hand side. Since different phases $\alpha/\gamma$ correspond to different identifications of boundary times this path integral gives zero, unless the phases coincide.
• Figure 5: Leading geometries contributing to the second moment of the Gram matrix \ref{['eq:Gram_matrix']}. The first contribution corresponds to two copies of a Euclidean bkack hole, each being proportional to the Gibbons-Hawking partition function $Z_1$. The second contribution corresponds to a wormhole, with partition function $Z_2$. The averaging over the phase shifted states is depicted through the dashed line in this diagram.