Wheeler-DeWitt Equation in Two-dimensional Quantum Gravity with $c=1$ Conformal Matter Field

TL;DR

This paper derives the Wheeler-DeWitt equation in two-dimensional quantum gravity with c=1 conformal matter, showing how the transition amplitude relates to string path-integrals with specific boundary conditions and the modular parameter acting as time.

Contribution

It demonstrates that the transition amplitude in 2D quantum gravity with c=1 matter satisfies the Wheeler-DeWitt equation, highlighting the role of the modular parameter as a time variable.

Findings

Non-zero modes cancel with c=1 matter and Neumann boundary conditions.

Transition amplitude obeys the minisuperspace Wheeler-DeWitt equation.

Modular parameter on annulus acts as a time variable.

Abstract

In the context of two-dimensional quantum cosmology, we consider the path-integral of a string on annulus which contains the Liouville field and conformal matter fields. We show that, in the transition amplitude of the string universe, the non-zero modes of the fields are all cancelled out only when we take the $c=1$ conformal matter field and impose the Neumann boundary condition on the system. The transition amplitude obtained obeys the minisuperspace Wheeler-DeWitt equation. In our treatment, the modular parameter on annulus plays the role of time variable to integrate out.