Algebras and States in JT Gravity

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 ∞ 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 O(1) fluctuations to a black hole of specified mass, this Type II ∞ 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 H bulk that includes baby universe states is then much bigger than the space of states H bdry accessible to a boundary observer. However, to a boundary observer, every pure or mixed state on H bulk is equivalent to some pure state in H bdry .

Notes on the type classification of von Neumann algebras

“…In particular, I have in mind Witten's paper "Gravity and the crossed product" [1], which includes an illuminating comment on page two that the type of a von Neumann factor can be thought of in terms of whether "[the algebra has] pure states, as well as... density matrices," "[the] algebra does not have pure states, but it does have density matrices," or "[the] algebra does not have pure states [or] density matrices." Throughout much of Witten's recent work both alone and with other authors [1][2][3][4][5][6][7], and in the work by Leutheusser and Liu that inspired it [8,9], other important properties of the type classification of von Neumann algebras have appeared; in particular, the properties of traces on a von Neumann factor of a given type. These notions are certainly not new to physics, as the original papers on von Neumann algebra classification [10][11][12][13] were motivated in part by a desire to classify the algebraic structures that can appear in quantum mechanical systems.…”

Quantization of Jackiw-Teitelboim gravity with a massless scalar

“…In addition to the Euclidean path integral approach [9], this boundary picture allows the canonical analysis in Lorentzian setup [10]. This analysis may be extended to the case of JT gravity including matter as far as the matter field does not couple directly to the dilaton field [11].…”

“…It is noticeable that, even at the level of the algebra, the algebra A is not a tensor product of A l and A r . In the JT gravity with matter, it is shown that the type of von Neumann algebra is type II ∞ and the corresponding algebra A l/r is fully specified by the boundary Hamiltonian H l/r and the boundary matter operator φl/r derived from the bulk matter operator [11].…”

“…However there is no inconsistency since these higher derivative terms are constrained by a gauge symmetry of SL(2, R) which leaves the full geometry including the cutoff boundaries invariant. By imposing the corresponding gauge constraints with an appropriate gauge-fixing, the quantization of JT theory with and without matter has been carried out in [11], which will be reviewed in the following. Upon quantization, the reduced Hilbert space exhibits a genuinely two-sided nature while the Hamiltonian H l/r generates the left and right time evolution respectively.…”

“…In section 2, we give our basic setup of JT gravity. In section 3, we review the canonical quantization of JT gravity with matter following [11]. In section 4 , we specialize in JT gravity with a massless scalar field and provide full details of quantization.…”