ON THE EXACT OPERATOR-FORMALISM OF 2-DIMENSIONAL LIOUVILLE QUANTUM-GRAVITY IN MINKOWSKI SPACE-TIME

A detailed re-examination is made of the exact operator formalism of t wo-dimensional Liouville quantum gravity in Minkowski space-time with the cosmological term fully taken into account. Making use of the cano nical mapping from the interacting Liouville field into a free field, we focus on the problem of how the Liouville exponential operator shou ld be properly defined. In particular, the condition of mutual localit y among the exponential operators is carefully analyzed, and a new sol ution, which is neither smoothly connected nor relatively local to the existing solution, is found. Our analysis indicates that, in Minkowsk i space-time, coupling gravity to matter with central charge d < 1 is problematical. For d = 1, our new solution appears to be the appropria te one; for this value of d, we demonstrate that the operator equation of motion is satisfied to all orders in the cosmological constant wit h a certain regularization. As an application of the formalism, an att empt is made to study how the basic generators of the ground ring get modified due to the inclusion of the cosmological term. Our investigat ion, although incomplete, suggests that in terms of the canonically ma pped free field the ground ring is not modified.