Stochastic Metric Space and Quantum Mechanics

TL;DR

This paper introduces a stochastic metric space framework for quantizing dynamic systems and spacetime, providing a mathematically consistent model that unifies classical and quantum theories using stochastic calculus and path integrals.

Contribution

It proposes a novel stochastic metric space approach for quantization, integrating stochastic calculus and path integrals to model quantum mechanics and gravity.

Findings

Path integrals are well-defined on stochastic metric spaces.

Quantum effects of gravity are analyzed within this framework.

Stochastic quantization is consistent with canonical commutation relations.

Abstract

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.