The Heisenberg Uncertainty Principle as an Endogenous Equilibrium Property of Stochastic Optimal Control Systems in Quantum Mechanics

Jussi Lindgren,Jukka Liukkonen

doi:10.3390/sym12091533

Abstract

We provide a natural derivation and interpretation for the uncertainty principle in quantum mechanics from the stochastic optimal control approach. We show that, in particular, the stochastic approach to quantum mechanics allows one to understand the uncertainty principle through the “thermodynamic equilibrium”. A stochastic process with a gradient structure is key in terms of understanding the uncertainty principle and such a framework comes naturally from the stochastic optimal control approach to quantum mechanics. The symmetry of the system is manifested in certain non-vanishing and invariant covariances between the four-position and the four-momentum. In terms of interpretation, the results allow one to understand the uncertainty principle through the lens of scientific realism, in accordance with empirical evidence, contesting the original interpretation given by Heisenberg.

Highlights

This research article aims to provide an intuitive explanation for the quantum phenomenon of measurement uncertainty and it explains why the Born rule should hold, i.e., why the probability of finding a test particle should be proportional to the square of the wave function
Even though we have shown how relativistic and non-relativistic wave equations of quantum mechanics are the special cases of the corresponding Hamilton–Jacobi–Bellman transport equation for the value function, little consideration was given to the Born rule and the Heisenberg uncertainty principle
We believe that the Heisenberg uncertainty principle can be understood in a fruitful manner by considering it as a stationarity property of stochastic or statistical mechanics

Summary

Introduction

This research article aims to provide an intuitive explanation for the quantum phenomenon of measurement uncertainty and it explains why the Born rule should hold, i.e., why the probability of finding a test particle should be proportional to the square of the wave function. The article builds on the basic framework presented in the authors’ recent article [1] and it improves the framework there. The basic framework is built on the assumption that quantum mechanics should be seen through the framework of stochastic optimal control theory; stochastic dynamic optimization in a coordinate-invariant manner on the Minkowski spacetime. Even though we have shown how relativistic and non-relativistic wave equations of quantum mechanics are the special cases of the corresponding Hamilton–Jacobi–Bellman transport equation for the value function, little consideration was given to the Born rule and the Heisenberg uncertainty principle. We consider these issues in more detail as an extension and follow-up

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