On the Exact Operator Formalism of Two-Dimensional Liouville Quantum Gravity in Minkowski Spacetime

TL;DR

This paper reexamines the exact operator formalism of two-dimensional Liouville quantum gravity in Minkowski spacetime, focusing on defining exponential operators, analyzing mutual locality, and exploring implications for matter coupling and ground ring modifications.

Contribution

It introduces a new solution for the exponential operator definition that is not smoothly connected to previous solutions, and analyzes the problem of matter coupling and ground ring modifications in this formalism.

Findings

A new solution for exponential operators is found, not connected to existing solutions.

Coupling gravity to matter with central charge d<1 is problematic in Minkowski spacetime.

For d=1, the new solution satisfies the operator equation of motion to all orders.

Abstract

A detailed reexamination is made of the exact operator formalism of two-dimensional Liouville quantum gravity in Minkowski spacetime 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 spacetime, 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.