Entropy vs. Action in the (2+1)-Dimensional Hartle-Hawking Wave Function

TL;DR

This paper demonstrates that in (2+1)-dimensional quantum gravity, entropy effects from multiple instantons outweigh the action's influence, challenging the common saddle point approximation in calculating the Hartle-Hawking wave function.

Contribution

It reveals that considering only the least action extremum is insufficient in (2+1)D gravity, as entropy from multiple instantons dominates the path integral.

Findings

Topologically inequivalent instantons with larger actions dominate the path integral.

Entropy effects are crucial in (2+1)D quantum gravity calculations.

The saddle point approximation may be inadequate in certain quantum gravity contexts.

Abstract

In most attempts to compute the Hartle-Hawking ``wave function of the universe'' in Euclidean quantum gravity, two important approximations are made: the path integral is evaluated in a saddle point approximation, and only the leading (least action) extremum is taken into account. In (2+1)-dimensional gravity with a negative cosmological constant, the second assumption is shown to lead to incorrect results: although the leading extremum gives the most important single contribution to the path integral, topologically inequivalent instantons with larger actions occur in great enough numbers to predominate. One can thus say that in 2+1 dimensions --- and possibly in 3+1 dimensions as well --- entropy dominates action in the gravitational path integral.