Quantization of Jackiw-Teitelboim gravity with a massless scalar

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