We embed Nelson’s theory of 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.

A preprint version of the article is available at ArXiv.

中文翻译：

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洛伦兹流形上的随机量化

我们将纳尔逊的随机量化理论嵌入到Schwartz-Meyer二阶几何框架中。结果是（伪）黎曼流形上的量子力学非微扰理论。在这种方法中，我们导出了在标量电势，矢量电势和重力作用下带电的大量spin-0测试粒子的随机微分方程。此外，我们推导了相关的薛定ding方程。所得方程式表明，在量子引力理论中，大量标量粒子必须与引力共形耦合。最后，我们讨论了随机框架的一些前景。

该文章的预印本可在ArXiv上获得。