Path integrals, spontaneous localisation, and the classical limit

TL;DR

This paper explores how spontaneous localisation models, specifically GRW, modify the path integral formulation to explain the quantum to classical transition and the absence of macroscopic superpositions.

Contribution

It provides two new derivations of the path integral formulation for the GRW model and demonstrates the quantum to classical transition rigorously.

Findings

Modified path integral formulation for GRW model

Derivation of von Neumann and Liouville equations from the path integral

Rigorous demonstration of the quantum-classical transition

Abstract

We recall that in order to obtain the classical limit of quantum mechanics one needs to take the $\hbar\rightarrow 0$ limit. In addition, one also needs an explanation for the absence of macroscopic quantum superposition of position states. One possible explanation for the latter is the Ghirardi-Rimini-Weber (GRW) model of spontaneous localisation. Here we describe how spontaneous localisation modifies the path integral formulation of density matrix evolution in quantum mechanics. (Such a formulation has been derived earlier by Pearle and Soucek; we provide two new derivations of their result). We then show how the von Neumann equation and the Liouville equation for the density matrix arise in the quantum and classical limit, respectively, from the GRW path integral. Thus we provide a rigorous demonstration of the quantum to classical transition.