2D Quantum Gravity in the Proper-Time Gauge

TL;DR

This paper derives a two-loop amplitude for 2D pure gravity in the proper-time gauge, reducing the problem to quantum mechanics and analyzing boundary length and winding mode dependencies.

Contribution

It provides a continuum formulation of the two-loop amplitude in 2D quantum gravity using the proper-time gauge, solving the constraints explicitly.

Findings

Derived explicit two-loop amplitude in proper-time gauge

Reduced field theory to quantum mechanics via constraints

Analyzed dependence on boundary lengths and winding modes

Abstract

A two-loop (cylinder) amplitude of the 2d pure gravity theory is obtained in the proper-time gauge ($g_{00}=1$, $g_{01}=g_{10}=0$) in the continuum formulation. The constraint $T_{01}=0$ is solved and used to reduce the problem of field theory to that of quantum mechanics. This reduction can also be proved by using a conformal Ward identity. The amplitude depends on the lengths $l_1, l_2$ of the boundaries, the proper time $T$ and a non-negative integer $m$ associated with winding modes around the boundaries.