Abstract
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.
Highlights
A related point is that the entire formation and evaporation of a small black hole in AdS is spacelike to some boundary time slice, and must be describable purely via the Hamiltonian constraints
We semiclassically compute the HartleHawking wave function in two different bases of this Hilbert space. We 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
Whether or not the Euclidean path integral defines a thermal partition function, we can always use it to define a natural family of states in the Hilbert space of the two-boundary system which we constructed in section 3.1 above: these states are labelled by a real parameter β, and are collectively called the Hartle-Hawking state [52, 53]
Summary
The Jackiw-Teitelboim action on a 1 + 1 dimensional asymptotically-AdS spacetime M is given by. The obvious choice is to fix the induced metric γμν and dilaton Φ at the AdS boundary, which we can do by imposing γtt|∂M = rc Φ|∂M = φbrc (2.6). Φb is analogous to the AdS radius in Planck units in higher dimensions, it will be large in the semiclassical limit These boundary conditions are only preserved by the subset of infinitesimal diffeomorphisms ξμ which approach an isometry of the boundary metric, γμαγν β∇(αξβ)|∂M = 0,. On we specialize to boundary conditions where there are precisely two asymptotically-AdS boundaries: we are restricting to spacetimes with topology R × [0, 1], on which the metric and dilaton obey (2.6) at each asymptotic boundary, and the full Hamiltonian H is the sum of left and right ADM Hamiltonians HL and HR
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