Quantization of Jackiw-Teitelboim gravity with a massless scalar (2303.05057v2)

Abstract: We study canonical quantization of Jackiw-Teibelboim (JT) gravity coupled to a massless scalar field. We provide concrete expressions of matter SL(2,{\,\bf 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,{\,\bf 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.