The Structure of Classical Extensions of Quantum Probability Theory

TL;DR

This paper characterizes classical extensions of quantum mechanics using the Misra-Bugajski map, showing how quantum states can be viewed as reduced classical statistical models on the projective Hilbert space.

Contribution

It proves that all classical extensions of quantum mechanics are essentially described by the Misra-Bugajski reduction map, linking quantum states to classical phase space representations.

Findings

Quantum mechanics can be represented as a reduced classical statistical theory.

The Misra-Bugajski map uniquely characterizes classical extensions.

Technical insights into the topology and Borel structure of the projective Hilbert space.

Abstract

On the basis of a suggestive definition of a classical extension of quantum mechanics in terms of statistical models, we prove that every such classical extension is essentially given by the so-called Misra-Bugajski reduction map. We consider how this map enables one to understand quantum mechanics as a reduced classical statistical theory on the projective Hilbert space as phase space and discuss features of the induced hidden-variables model. Moreover, some relevant technical results on the topology and Borel structure of the projective Hilbert space are reviewed.