Canonical Theory of 2d Gravity Coupled to Conformal Matter

A canonical quantization for two dimensional gravity models, including a dilaton gravity model, is performed in a way suitable for the light-cone gauge. We extend the theory developed by Abdalla {\it et.al.}\cite{AM} and obtain the correlation functions by using the screening charges of the symmetry algebra. It turns out that the role of the cosmological constant term in the Liouville theory is played by the screening charge of the symmetry algebra and the resulting theory looks like a chiral part of the Liouville theory or a non-critical open string theory. Moreover, we show that the dilaton gravity theory has a symmetry realized by the centrally extended Poincar\'{e} algebra instead of the $\slr$ Kac-Moody algebra which is a symmetry of an ordinary two dimensional gravity theory.