Lattice Quantum Gravity from Stochastic 3-Geometries

TL;DR

This paper introduces a stochastic approach to 4D Euclidean quantum gravity using Langevin dynamics on 3-geometries, connecting stochastic processes with quantum fluctuations and providing a lattice regularization framework.

Contribution

It formulates a Langevin equation for 3-geometries in Ashtekar's formalism that exactly recovers the 4D quantum gravity Hamiltonian, offering a novel stochastic perspective.

Findings

Langevin equation for 3-geometries reproduces 4D quantum gravity Hamiltonian.

Stochastic time corresponds to Euclidean time in a specific gauge.

Lattice regularization of 4D quantum gravity is developed.

Abstract

I propose the Langevin equation for 3-geometries in the Ashtekar's formalism to describe 4D Euclidean quantum gravity, in the sense that the corresponding Fokker-Planck hamiltonian recovers the hamiltonian in 4D quantum gravity exactly. The stochastic time corresponds to the Euclidean time in the gauge, N=1 and $N^i=0$. In this approach, the time evolution in 4D quantum gravity is understood as a stochastic process where the quantum fluctuation of ` ` triad \rq\rq is characterized by the curvature at the one unit time step before. The lattice regularization of 4D quantum gravity is presented in this context.