Generalization of Uncertainty Relation for Quantum and Stochastic Systems

TL;DR

This paper derives a generalized uncertainty relation applicable to both quantum and stochastic systems, revealing that finite minimum uncertainty is a common feature, not exclusive to quantum mechanics, with implications for microscopic systems.

Contribution

The paper introduces a unified uncertainty relation within the stochastic variational method that applies to various physical systems, including quantum, fluid, and microscopic models.

Findings

Reproduces the quantum uncertainty inequality

Applies to Gross-Pitaevskii and Navier-Stokes-Fourier equations

Suggests finite minimum uncertainty is a universal stochastic feature

Abstract

The generalized uncertainty relation applicable to quantum and stochastic systems is derived within the stochastic variational method. This relation not only reproduces the well-known inequality in quantum mechanics but also is applicable to the Gross-Pitaevskii equation and the Navier-Stokes-Fourier equation, showing that the finite minimum uncertainty between the position and the momentum is not an inherent property of quantum mechanics but a common feature of stochastic systems. We further discuss the possible implication of the present study in discussing the application of the hydrodynamic picture to microscopic systems, like relativistic heavy-ion collisions.