  Jackiw-Teitelboim (JT) gravity is a 1+1-dimensional toy model for quantum
gravity in four spacetime dimensions. In the absence of matter, JT gravity is a
topological field theory and there are no local observables. The introduction
of a boundary changes the situation. What was a un-physical gauge direction
before turns into a physical boundary degree of freedom (a gravitational edge
mode). Starting from the BF formulation of JT gravity, we develop a twistor
representation for the boundary charges of JT gravity. We introduce point
sources in the bulk and study the coupled gravity plus matter system at the
quantum level. Eigenstates of quasi-local energy are built from the tensor
product of unitary irreducible representations of SL(2,R). Empty patches of
AdS2 are characterised by the continuous series representations, point sources
are related to the discrete series. Physical states are constructed by fusing
the SL(2,R) representations into an SL(2,R) invariant singlet. The singlet lies
in the kernel of the constraints, which are the analogue of the Wheeler-DeWitt
equations in the presence of distributional matter sources.
