Some recent stochastic quantum mechanics

[Philip Thrift]

Stochastic Metric Space and Quantum Mechanics

Yoshimasa Kurihara

(Submitted on 13 Dec 2016 (v1), last revised 22 Dec 2017 (this version, v5))

https://arxiv.org/abs/1612.04228

A new idea for the quantization of dynamic systems, as well as space time itself, using a stochastic metric is proposed. The quantum mechanics of a mass point is constructed on a space time manifold using a stochastic metric. A stochastic metric space is, in brief, a metric space whose metric tensor is given stochastically according to some appropriate distribution function. A mathematically consistent model of a space time manifold equipping a stochastic metric is proposed in this report. The quantum theory in the local Minkowski space can be recognized as a classical theory on the stochastic Lorentz-metric-space. A stochastic calculus on the space-time manifold is performed using white noise functional analysis. A path-integral quantization is introduced as a stochastic integration of a function of the action integral, and it is shown that path-integrals on the stochastic metric space are mathematically well-defined for large variety of potential functions. The Newton--Nelson equation of motion can also be obtained from the Newtonian equation of motion on the stochastic metric space. It is also shown that the commutation relation required under the canonical quantization is consistent with the stochastic quantization introduced in this report.

The quantum effects of general relativity are also analyzed through natural use of the stochastic metrics. Some example of quantum effects on the universe is discussed.

Stochastic path integrals can be derived like quantum mechanical path integrals

John J. Vastola, William R. Holmes

(Submitted on 28 Sep 2019)

https://arxiv.org/abs/1909.12990

Stochastic mechanics---the study of classical stochastic systems governed by things like master equations and Fokker-Planck equations---exhibits striking mathematical parallels to quantum mechanics. In this article, we make those parallels more transparent by presenting a quantum mechanics-like formalism for deriving a path integral description of systems described by stochastic differential equations. Our formalism expediently recovers the usual path integrals (the Martin-Siggia-Rose-Janssen-De Dominicis and Onsager-Machlup forms) and is flexible enough to account for different variable domains (e.g. real line versus compact interval), stochastic interpretations, arbitrary numbers of variables, explicit time-dependence, dimensionful control parameters, and more. We discuss the implications of our formalism for stochastic biology.

ref: https://en.wikipedia.org/wiki/Stochastic_quantum_mechanics

@philipthrift