  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
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.
