[Paper Review] Algebras and States in JT Gravity

Geoff Penington, Edward Witten|ArXiv.org|Jan 18, 2023

Noncommutative and Quantum Gravity Theories25 citations

TL;DR

The paper analyzes boundary observables in Jackiw–Teitelboim gravity with and without matter, showing that coupling to matter yields a Type II∞ boundary algebra with a unique trace up to a constant and that entropy from this algebra matches Euclidean replica results; it also studies wormholes and baby universes and their impact on boundary vs bulk Hilbert spaces.

ABSTRACT

We analyze the algebra of boundary observables in canonically quantised JT gravity with or without matter. In the absence of matter, this algebra is commutative, generated by the ADM Hamiltonian. After coupling to a bulk quantum field theory, it becomes a highly noncommutative algebra of Type II$_\infty$ with a trivial center. As a result, density matrices and entropies on the boundary algebra are uniquely defined up to, respectively, a rescaling or shift. We show that this algebraic definition of entropy agrees with the usual replica trick definition computed using Euclidean path integrals. Unlike in previous arguments that focused on $\mathcal{O}(1)$ fluctuations to a black hole of specified mass, this Type II$_\infty$ algebra describes states at all temperatures or energies. We also consider the role of spacetime wormholes. One can try to define operators associated with wormholes that commute with the boundary algebra, but this fails in an instructive way. In a regulated version of the theory, wormholes and topology change can be incorporated perturbatively. The bulk Hilbert space $\mathcal{H}_\mathrm{bulk}$ that includes baby universe states is then much bigger than the space of states $\mathcal{H}_\mathrm{bdry}$ accessible to a boundary observer. However, to a boundary observer, every pure or mixed state on $\mathcal{H}_\mathrm{bulk}$ is equivalent to some pure state in $\mathcal{H}_\mathrm{bdry}$.

Motivation & Objective

• Understand the algebra of boundary observables in JT gravity with and without matter.
• Characterize the transition from a commutative boundary algebra to a Type II∞ algebra upon coupling to bulk QFT.
• Relate boundary algebra entropy to Euclidean path integral replica results.
• Investigate the role of wormholes and baby universes in the boundary/bulk Hilbert spaces.
• Assess background independence of the JT gravity algebra and its implications for entropy and states.

Proposed method

• Define the boundary algebras A_L and A_R for JT gravity with matter.
• Show that A is a Type II∞ factor with trivial center and a well-defined trace.
• Compute entropy via the algebraic trace and compare to the replica trick in Euclidean path integrals.
• Analyze baby universe operators and demonstrate their incompatibility with commuting boundary algebras.
• Incorporate wormhole corrections perturbatively and study their effect on H_bulk and H_bdry.
• Discuss multi-boundary generalizations and observer-accessibility of bulk versus boundary states.

Experimental results

Research questions

• RQ1 What is the nature of the boundary observable algebra in JT gravity with and without bulk matter?
• RQ2 Does coupling matter render the boundary algebra a Type II∞ factor, and how does this affect entropy definitions?
• RQ3 How do wormholes and baby universes modify the relation between bulk and boundary Hilbert spaces?
• RQ4 Can operators commuting with the boundary algebra (baby universe operators) exist, and how does this fail?
• RQ5 Are boundary-observer-defined entropies consistent with Euclidean replica calculations across temperatures?

Key findings

• Without matter, the boundary algebra is commutative and generated by the ADM Hamiltonian.
• With bulk matter, the boundary algebra becomes a Type II∞ factor with trivial center, and its entropy is defined up to an additive constant.
• The boundary-algebra entropy agrees with the replica trick entropy computed via Euclidean path integrals, up to a state-independent additive constant.
• Wormhole corrections preserve the Type II∞ structure and require interpreting H_bulk and H_bdry as related but distinct under a boundary-facing lens.
• Baby universe operators commuting with the boundary algebra fail in a precise, instructive way, and wormholes expand the bulk Hilbert space while leaving the boundary description effectively small.
• For multiple asymptotic boundaries, the same boundary algebra acts on bulk Hilbert spaces with any number of open universes, and boundary observers cannot detect the number of other boundaries to the order considered.