Second Order Langevin Equation and Definition of Quantum Gravity By Stochastic Quantisation

TL;DR

This paper explores a novel approach to quantum gravity using second order Langevin equations in stochastic quantisation, proposing a transition from quantum to classical cosmology and implications for black hole formation and dark matter.

Contribution

It introduces a second order Langevin equation framework for stochastic quantisation of quantum gravity and explicitly defines second order stochastic quantisation in a solvable model.

Findings

Transition from oscillating quantum phase to semi-classical phase with Lorentz time emergence

Black hole formation and decay consistent with inflation and dark matter observations

Explicit calculation of second order Langevin equation in a zero-dimensional model

Abstract

Euclidean quantum gravity might be defined by stochastic quantisation that is governed by a higher order Langevin equation rather than a first order stochastic equation. In a transitory phase where the Lorentz time cannot be defined, the parameter that orders the evolution of quantum gravity phenomena is the stochastic time. This changes the definition of causality in the period of primordial cosmology. The prediction of stochastically quantised gravity is that there will a transition from an oscillating quantum phase to a semi-classical one, when the Lorentz time emerges. The end of the transition, as it can be observed from now and described by inflation models, is a diluted Universe, following the inflation phenomenological evolution. It is filled at the beginning with scattered classical primordial black holes. The smallest ones will quickly decay in matter, with a standard quantum field theory evolution till our period. The stable heavier black holes will remain, forming a good fraction of the dark matter and the large black holes observed in the galaxies. In a theoretically related way, this framework suggests the possibility of a gravitational parton content for "point-like" particles, in the same five dimensional quantum field theory context as in the primordial cosmology, with a (+----) signature for the 5d metrics. The very precise and explicit result expressed in this paper is actually far more modest than its motivation. We compute explicitly the meaning of a second order Langevin equation in zero dimensions and define precisely what is second order stochastic quantisation in a soluble case.