In this paper we construct the classical phase space of Jackiw-Teitelboim gravity with positive cosmological constant on spatial slices with circle topology. This turns out to be somewhat more intricate than in the case of negative cosmological constant; this phase space has many singular points and is not even Hausdorff. Nonetheless, it admits a group-theoretic description which is quite amenable to quantization.

United States

Elba Alonso-Monsalve

Daniel Harlow

Patrick Jefferson

Phase space of Jackiw-Teitelboim gravity with positive cosmological constant

Recent arguments based on the quantum extremal surface formula or the gravitational path integral have given fairly compelling evidence that the Hilbert space of quantum gravity in a closed universe is one-dimensional and real. How can this be consistent with the complexity of our own experiences? In this paper we propose that the experiences of any observer $\textit{Ob}$ in a closed universe can be approximately described by a quantum mechanical theory with a Hilbert space whose dimension is roughly $$ {e}^{S_{Ob}} $$ , where $\textit{S}$ is the number of degrees of freedom of $\textit{Ob}$. Moreover we argue that the errors in this description are exponentially small in $\textit{S}$ . We give evidence for this proposal by incorporating it into the gravitational path integral and the coding interpretation of holography in simple models and seeing that it works, and we explain how similar effects arise in black hole physics in appropriate circumstances.

Mykhaylo Usatyuk

Ying Zhao

Quantum mechanics and observers for gravity in a closed universe

Spacetime inversion symmetries such as parity and time reversal play a central role in physics, but they are usually treated as global symmetries. In quantum gravity there are no global symmetries, so any spacetime inversion symmetries must be gauge symmetries. In particular this includes $$\mathcal{C}\mathcal{R}\mathcal{T}$$ symmetry (in even dimensions usually combined with a rotation to become $$\mathcal{C}\mathcal{P}\mathcal{T}$$ ), which in quantum field theory is always a symmetry and seems likely to be a symmetry of quantum gravity as well. In this article we discuss what it means to gauge a spacetime inversion symmetry, and we explain some of the more unusual consequences of doing this. In particular we argue that the gauging of $$\mathcal{C}\mathcal{R}\mathcal{T}$$ is automatically implemented by the sum over topologies in the Euclidean gravity path integral, that in a closed universe the Hilbert space of quantum gravity must be a real vector space, and that in Lorentzian signature manifolds which are not time-orientable must be included as valid configurations of the theory. In particular we give an example of an asymptotically-AdS time-unorientable geometry which must be included to reproduce computable results in the dual CFT.

Japan

Tokiro Numasawa

Gauging spacetime inversions in quantum gravity

Quantum error correction has given us a natural language for the emergence of spacetime, but the black hole interior poses a challenge for this framework: at late times the apparent number of interior degrees of freedom in effective field theory can vastly exceed the true number of fundamental degrees of freedom, so there can be no isometric (i.e. inner-product preserving) encoding of the former into the latter. In this paper we explain how quantum error correction nonetheless can be used to explain the emergence of the black hole interior, via the idea of “non-isometric codes protected by computational complexity”. We show that many previous ideas, such as the existence of a large number of “null states”, a breakdown of effective field theory for operations of exponential complexity, the quantum extremal surface calculation of the Page curve, post-selection, “state-dependent/state-specific” operator reconstruction, and the “simple entropy” approach to complexity coarse-graining, all fit naturally into this framework, and we illustrate all of these phenomena simultaneously in a soluble model.

Chris Akers

Netta Engelhardt

Geoff Penington

Shreya Vardhan

The black hole interior from non-isometric codes and complexity

In this paper we use the covariant Peierls bracket to compute the algebra of a sizable number of diffeomorphism-invariant observables in classical Jackiw-Teitelboim gravity coupled to fairly arbitrary matter. We then show that many recent results, including the construction of traversable wormholes, the existence of a family of SL(2, ℝ) algebras acting on the matter fields, and the calculation of the scrambling time, can be recast as simple consequences of this algebra. We also use it to clarify the question of when the creation of an excitation deep in the bulk increases or decreases the boundary energy, which is of crucial importance for the “typical state” versions of the firewall paradox. Unlike the “Schwarzian” or “boundary particle” formalism, our techniques involve no unphysical degrees of freedom and naturally generalize to higher dimensions. We do a few higher-dimensional calculations to illustrate this, which indicate that the results we obtain in JT gravity are fairly robust.

Jie-qiang Wu

Algebra of diffeomorphism-invariant observables in Jackiw-Teitelboim gravity

In this paper we argue for a close connection between the non-existence of global symmetries in quantum gravity and a unitary resolution of the black hole information problem. In particular we show how the essential ingredients of recent calculations of the Page curve of an evaporating black hole can be used to generalize a recent argument against global symmetries beyond the AdS/CFT correspondence to more realistic theories of quantum gravity. We also give several low-dimensional examples of quantum gravity theories which do not have a unitary resolution of the black hole information problem in the usual sense, and which therefore can and do have global symmetries. Motivated by this discussion, we conjecture that in a certain sense Euclidean quantum gravity is equivalent to holography.

Edgar Shaghoulian

Global symmetry, Euclidean gravity, and the black hole information problem

The covariant phase space method of Iyer, Lee, Wald, and Zoupas gives an elegant way to understand the Hamiltonian dynamics of Lagrangian field theories without breaking covariance. The original literature however does not systematically treat total derivatives and boundary terms, which has led to some confusion about how exactly to apply the formalism in the presence of boundaries. In particular the original construction of the canonical Hamiltonian relies on the assumed existence of a certain boundary quantity “B”, whose physical interpretation has not been clear. We here give an algorithmic procedure for applying the covariant phase space formalism to field theories with spatial boundaries, from which the term in the Hamiltonian involving B emerges naturally. Our procedure also produces an additional boundary term, which was not present in the original literature and which so far has only appeared implicitly in specific examples, and which is already nonvanishing even in general relativity with sufficiently permissive boundary conditions. The only requirement we impose is that at solutions of the equations of motion the action is stationary modulo future/past boundary terms under arbitrary variations obeying the spatial boundary conditions; from this the symplectic structure and the Hamiltonian for any diffeomorphism that preserves the theory are unambiguously constructed. We show in examples that the Hamiltonian so constructed agrees with previous results. We also show that the Poisson bracket on covariant phase space directly coincides with the Peierls bracket, without any need for non-covariant intermediate steps, and we discuss possible implications for the entropy of dynamical black hole horizons.

Covariant phase space with boundaries

Several recent papers have shown a close relationship between entanglement wedge reconstruction and the unitarity of black hole evaporation in AdS/CFT. The analysis of these papers however has a rather puzzling feature: all calculations are done using bulk dynamics which are essentially those Hawking used to predict information loss, but applying ideas from entanglement wedge reconstruction seems to suggest a Page curve which is consistent with information conservation. Why should two different calculations in the same model give different answers for the Page curve? In this note we present a new pair of models which clarify this situation. Our first model gives a holographic illustration of unitary black hole evaporation, in which the analogue of the Hawking radiation purifies itself as expected, and this purification is reproduced by the entanglement wedge analysis. Moreover a smooth black hole interior persists until the last stages the evaporation process. Our second model gives an alternative holographic interpretation of the situation where the bulk evolution leads to information loss: unlike in the models proposed so far, this bulk information loss is correctly reproduced by the entanglement wedge analysis. This serves as an illustration that quantum extremal surfaces are in some sense kinematic: the time-dependence of the entropy they compute depends on the choice of bulk dynamics. In both models no bulk quantum corrections need to be considered: classical extremal surfaces are enough to do the job. We argue that our first model is the one which gives the right analogy for what actually happens to evaporating black holes, but we also emphasize that any complete resolution of the information problem will require an understanding of non-perturbative bulk dynamics.

Simple holographic models of black hole evaporation

In this note we study the 1 + 1 dimensional Jackiw-Teitelboim gravity in Lorentzian signature, explicitly constructing the gauge-invariant classical phase space and the quantum Hilbert space and Hamiltonian. We also semiclassically compute the Hartle-Hawking wave function in two different bases of this Hilbert space. We then use these results to illustrate the gravitational version of the factorization problem of AdS/CFT: the Hilbert space of the two-boundary system tensor-factorizes on the CFT side, which appears to be in tension with the existence of gauge constraints in the bulk. In this model the tension is acute: we argue that JT gravity is a sensible quantum theory, based on a well-defined Lorentzian bulk path integral, which has no CFT dual. In bulk language, it has wormholes but it does not have black hole microstates. It does however give some hint as to what could be added to rectify these issues, and we give an example of how this works using the SYK model. Finally we suggest that similar comments should apply to pure Einstein gravity in 2 + 1 dimensions, which we’d then conclude also cannot have a CFT dual, consistent with the results of Maloney and Witten.

Daniel Jafferis

The factorization problem in Jackiw-Teitelboim gravity

We use the Einstein-Hilbert gravitational path integral to investigate gravita- tional entanglement at leading order O(1/G). We argue that semiclassical states prepared by a Euclidean path integral have the property that projecting them onto a subspace in which the Ryu-Takayanagi or Hubeny-Rangamani-Takayanagi surface has definite area gives a state with a flat entanglement spectrum at this order in gravitational perturbation theory. This means that the reduced density matrix can be approximated as proportional to the identity to the extent that its Renyi entropies Sn are independent of n at this order. The n-dependence of Sn in more general states then arises from sums over the RT/HRT- area, which are generally dominated by different values of this area for each n. This provides a simple picture of gravitational entanglement, bolsters the connection between holographic systems and tensor network models, clarifies the bulk interpretation of alge- braic centers which arise in the quantum error-correcting description of holography, and strengthens the connection between bulk and boundary modular Hamiltonians described by Jafferis, Lewkowycz, Maldacena, and Suh.