An Alternate Path Integral for Quantum Gravity

TL;DR

This paper introduces a semi-classical path integral for quantum gravity with Neumann boundary conditions, relating it to traditional approaches and deriving black hole and horizon thermodynamics.

Contribution

It presents a novel Neumann boundary condition formulation for quantum gravity path integrals and connects it with established thermodynamic laws and ensembles.

Findings

Reproduces black hole and horizon entropy using the new formulation

Derives the generalized Smarr formula and first law in this framework

Establishes equivalence with covariant and Hamiltonian approaches in flat space

Abstract

We define a (semi-classical) path integral for gravity with Neumann boundary conditions in $D$ dimensions, and show how to relate this new partition function to the usual picture of Euclidean quantum gravity. We also write down the action in ADM Hamiltonian formulation and use it to reproduce the entropy of black holes and cosmological horizons. A comparison between the (background-subtracted) covariant and Hamiltonian ways of semi-classically evaluating this path integral in flat space reproduces the generalized Smarr formula and the first law. This "Neumann ensemble" perspective on gravitational thermodynamics is parallel to the canonical (Dirichlet) ensemble of Gibbons-Hawking and the microcanonical approach of Brown-York.