On the lapse contour in the gravitational path integral

TL;DR

This paper investigates the correct contour for the lapse integration in the gravitational path integral, addressing essential singularities and proposing a contour that aligns with physical and thermodynamic consistency.

Contribution

It introduces a new perspective on the lapse contour in gravitational path integrals, linking complex analysis with physical constraints and entropy calculations.

Findings

Lapse contour should pass below zero in the complex plane.

The proposed contour aligns with vacuum fluctuation amplitudes.

Supports deriving Bekenstein-Hawking entropy from Lorentzian path integrals.

Abstract

The gravitational path integral is usually implemented with a covariant action by analogy with other gauge field theories, but the gravitational case is different in important ways. A key difference is that the integrand has an essential singularity, which occurs at zero lapse where the spacetime metric degenerates. The lapse integration contour required to impose the local time reparametrization constraints must run from $-\infty$ to $+\infty$, yet must not pass through zero. This raises the question: what is the correct integration contour, and why? We study that question by starting with the reduced phase space path integral, which involves no essential singularity. We observe that if the momenta are to be integrated before the lapse, to obtain a configuration space path integral, the lapse contour should pass below the origin in the complex lapse plane. This contour is also consistent with the requirement that quantum field fluctuation amplitudes have the usual short distance vacuum form, and with obtaining the Bekenstein-Hawking horizon entropy from a Lorentzian path integral. We close with a discussion of related issues, including potential obstacles to deriving a nonperturbative covariant gravitational path integral.