Bohmian measures and their classical limit

TL;DR

This paper investigates the classical limit of Bohmian phase space measures, comparing them with Wigner measures, and explores their relation to Young measures to better understand quantum-to-classical transition.

Contribution

It introduces a detailed analysis of Bohmian measures' classical limit, connecting them with Wigner and Young measures, and provides new insights into oscillation and concentration effects in semi-classical wave functions.

Findings

Bohmian measures' classical limit depends on oscillation and concentration scales.

Established connections between Bohmian, Wigner, and Young measures.

Provided new results on the behavior of Wigner measures in semi-classical analysis.

Abstract

We consider a class of phase space measures, which naturally arise in the Bohmian interpretation of quantum mechanics (when written in a Lagrangian form). We study the so-called classical limit of these Bohmian measures, in dependence on the scale of oscillations and concentrations of the sequence of wave functions under consideration. The obtained results are consequently compared to those derived via semi-classical Wigner measures. To this end, we shall also give a connection to the, by now classical, theory of Young measures and prove several new results on Wigner measures themselves. We believe that our analysis sheds new light on the classical limit of Bohmian quantum mechanics and gives further insight on oscillation and concentration effects of semi-classical wave functions.