A Vlasov-Bohm approach to Quantum Mechanics for statistical systems

TL;DR

This paper introduces a novel Vlasov-Bohm approach that integrates Bohmian mechanics with the Vlasov framework to model quantum statistical systems, providing a mean-field theory consistent with quantum mechanics.

Contribution

It presents a new method combining Bohmian mechanics and the Vlasov equation to describe quantum systems at a statistical level.

Findings

The approach reproduces quantum behavior within the RPA.

It captures the corpuscular nature of matter.

Provides a mean-field theory consistent with quantum mechanics.

Abstract

Quantum mechanics is the most successful theory to describe microscopic phenomena. It was derived in different ways over the past 100 years by Heisenberg, Schr\"{o}dinger, and Feynman. At the same time, other interpretations have been suggested, including the Bohm-De Broglie interpretation and the so-called Bohmian mechanics. Here, we show that Bohmian mechanics, which utilizes the concept of the Bohm quantum potential, can also serve as a starting point for quantizing classical non-relativistic systems. By incorporating the Bohm quantum potential into the Vlasov framework, we obtain a mean-field theory that captures the corpuscular nature of matter, in agreement with quantum mechanics within the Random Phase Approximation (RPA).