Stochastic Quantization 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 Schrödinger 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.

Stochastic Quantization of Relativistic Theories

“…e.g. Ref [9]. The derivation of the Schrödinger equation in the massless case is similar to the derivation in the massive case, which can be found in Ref.…”

“…The derivation of the Schrödinger equation in the massless case is similar to the derivation in the massive case, which can be found in Ref. [9] and references therein. If a probability density ρ(x, η) associated to the stochastic process X exists, one can construct the wave function…”

“…In this letter, we remedy this and show that stochastic quantization can be made into a relativistic quantization scheme. Here, we build on our previous work [9], where we extended stochastic quantization to (pseudo-)Riemannian geometry. In this letter, we restrict this general framework to a special class of theories, namely the relativistic theories defined on Lorentzian manifolds.…”