On the exact operator formalism of two-dimensional Liouville Quantum Gravity in Minkowski space-time

A detailed re-examination is made of the exact operator formalism of two-dimensional Liouville quantum gravity in Minkowski space-time with the cosmological term fully taken into account. Making use of the canonical mapping from the interacting Liouville field into a free field, we focus on the problem of how the Liouville exponential operator should be properly defined. In particular, the condition of mutual locality among the exponential operators is carefully analyzed, and a new solution, which is neither smoothly connected nor relatively local to the existing solution, is found. Our analysis indicates that, in Minkowski space-time, coupling gravity to matter with central charge d<1 is problematical. For d=1, our new solution appears to be the appropriate one; for this value of d, we demonstrate that the operator equation of motion is satisfied to all orders in the cosmological constant with a certain regularization. As an application of the formalism, an attempt is made to study how the basic generators of the ground ring get modified due to the inclusion of the cosmological term. Our investigation, although incomplete, suggests that in terms of the canonically mapped free field the ground ring is not modified.