[Paper Review] Quantum gravity tomography

William Donnelly|arXiv (Cornell University)|Jun 14, 2018

Black Holes and Theoretical Physics27 references5 citations

TL;DR

This paper proposes a method to reconstruct the full quantum state of a single particle in perturbative quantum gravity in flat spacetime from measurements of the gravitational field at spatial infinity. Using a relativistic generalization of the Wigner function and the Poincaré charges (momentum and center-of-energy operators), it demonstrates that the asymptotic metric encodes complete quantum information holographically, providing a concrete mechanism for bulk-to-boundary state reconstruction without relying on anti-de Sitter geometry.

ABSTRACT

The holographic principle posits that all quantum information in a region of spacetime is encoded on its boundary. While there is strong evidence for this principle in certain models of quantum gravity in asymptotically anti-de Sitter spacetime, it is yet to be established whether holography is a generic feature of quantum gravity, or a peculiar property of these models. The goal of the present work is to present a model of holographic reconstruction in the framework of perturbative quantum gravity in flat spacetime. Specifically, we consider a state in the single-particle sector of a quantum field theory and give a method to completely reconstruct the quantum state from measurement of the metric at spatial infinity. Our argument uses a relativistic generalization of the quantum-mechanical Wigner function, and gives an explicit mechanism by which the gravitational constraints encode quantum information holographically on the boundary. Moreover, it suggests how information about more general states might be recovered from soft charges at null infinity, with applications to the black hole information loss paradox.

Motivation & Objective

• To establish whether the holographic principle—encoding bulk quantum information on a boundary—holds in perturbative quantum gravity in asymptotically flat spacetime, beyond the anti-de Sitter context.
• To investigate whether the set of observables at spatial infinity is tomographically complete for reconstructing quantum states in quantum gravity.
• To provide an explicit, constructive mechanism for recovering a single-particle density matrix from asymptotic gravitational field measurements.
• To explore how gravitational constraints and global charges (Poincaré and potentially supertranslation/superrotation charges) encode quantum information on the boundary.

Proposed method

• The paper introduces a relativistic Wigner function defined via displacement operators built from the momentum operator $\hat{P}^\mu$ and the center-of-energy operator $\hat{K}^\mu$, forming a dual pair of complementary observables.
• It constructs the characteristic function $\tilde{W}(u,v)$ as the expectation value of the operator $e^{i u \cdot \hat{K} + i v \cdot \hat{P}}$, which is measurable via asymptotic field measurements.
• The inverse Radon transform is applied to reconstruct the characteristic function from measured data, enabling full state reconstruction via $\hat{\rho} = \int d^d u \, d^d v \, \tilde{W}(u,v) \, \hat{\mathcal{O}}(u,v)$.
• The method relies on the linearized Einstein equations and the constraint $\bra{\phi}(\hat{G}_{0\mu} - 8\pi G \hat{T}_{0\mu})\ket{\psi} = 0$, which ensures that physical states are gravitationally dressed and entangled with the metric.
• The framework uses perturbative quantum gravity in Minkowski spacetime, assuming no higher-order corrections beyond $h^2$, and treats the metric perturbation $h_{\mu\nu}$ as a quantum field.
• The reconstruction is shown to be faithful and complete for single-particle states, with the Wigner function encoding the full density matrix.

Experimental results

Research questions

• RQ1 Can the full quantum state of a single particle in quantum gravity be reconstructed from measurements of the asymptotic gravitational field in flat spacetime?
• RQ2 Is the set of Poincaré charges—specifically momentum and center-of-energy—tomographically complete for single-particle states?
• RQ3 How do gravitational constraints in linearized gravity encode quantum information on the boundary, and what is the role of diffeomorphism invariance in this process?
• RQ4 Can the holographic encoding of quantum information in asymptotically flat spacetime be extended to multiparticle states using extended global symmetries like supertranslations and superrotations?

Key findings

• The relativistic Wigner function defined via $\hat{P}^\mu$ and $\hat{K}^\mu$ provides a complete and faithful encoding of the single-particle density matrix in perturbative quantum gravity.
• The characteristic function $\tilde{W}(u,v)$ is measurable via asymptotic field operators and can be inverted using the inverse Radon transform to reconstruct the density matrix $\hat{\rho}$.
• The method demonstrates that the asymptotic gravitational field at spatial infinity contains complete quantum information about the bulk single-particle state, establishing a concrete mechanism for holographic reconstruction in flat spacetime.
• The reconstruction relies solely on perturbative quantum gravity and does not require anti-de Sitter boundary conditions or gauge/gravity duality.
• The Poincaré charges (momentum and boost/center-of-energy) form a tomographically complete set for single-particle states, with the Wigner function serving as the key tool.
• The work suggests that supertranslation and superrotation charges may provide a natural extension to multiparticle states, potentially resolving the black hole information loss paradox via holography.