Quantization of Jackiw-Teitelboim gravity with a massless scalar

We study canonical quantization of Jackiw-Teibelboim (JT) gravity coupled to a massless scalar field. provide concrete expressions matter SL(2,{\,\bf R}) charges and the boundary operators in terms creation annihilation The are represented form an oscillator (Jordon-Schwinger) realization algebra. also show how gauge constraints implemented classically, by matching explicitly classical solutions Schwarzian dynamics with bulk solutions. introduce $n$-point transition functions defined insertions along two-sided Lorentzian evolution, which may fully spell out quantum presence matter. For Euclidean case, we proceed picture disk geometry consider $2$-point correlation function where initial final states arranged inserting specific way. some simple states, evaluate perturbatively. discuss basic features additional evolution.