Stochastic Quantization on Lorentzian Manifolds

TL;DR

This paper develops a non-perturbative stochastic quantization framework on Lorentzian manifolds, deriving equations for scalar particles and their coupling to gravity, advancing quantum mechanics in curved spacetime.

Contribution

It embeds Nelson's stochastic quantization into second order geometry, deriving new stochastic differential equations and Schrödinger equations on pseudo-Riemannian manifolds.

Findings

Massive scalar particles must be conformally coupled to gravity.

Derived stochastic differential equations for particles in curved spacetime.

Established a non-perturbative quantum mechanics framework on Lorentzian manifolds.

Abstract

We embed Nelson's stochastic quantization in the Schwartz-Meyer second order geometry framework. The result is a non-perturbative theory of quantum mechanics on (pseudo)-Riemannian manifolds. Within this approach, we derive stochastic differential equations for massive spin-0 test particles charged under scalar potentials, vector potentials and gravity. Furthermore, we derive the associated Schr\"odinger equation. The resulting equations show that massive scalar particles must be conformally coupled to gravity in a theory of quantum gravity. We conclude with a discussion of some prospects of the stochastic framework.