On the applicability of Kolmogorov's theory of probability to the description of quantum phenomena. Part I: foundations

TL;DR

This paper explores whether classical probability theory, based on Kolmogorov's axioms, can be rigorously applied to quantum systems, challenging the notion that quantum mechanics fundamentally requires a generalized probability framework.

Contribution

It constructs a mathematically rigorous classical probability model for quantum systems that reproduces key quantum predictions, questioning the necessity of non-classical probability in quantum theory.

Findings

Classical probability models can replicate central quantum predictions.

A rigorous construction of a Kolmogorov-based quantum probability theory is provided.

The approach opens pathways for empirical testing and extension to other quantum models.

Abstract

By formulating the axioms of quantum mechanics, von Neumann also laid the foundations of a "quantum probability theory". As such, it is regarded a generalization of the "classical probability theory" due to Kolmogorov. Outside of quantum physics, however, Kolmogorov's axioms enjoy universal applicability. This raises the question of whether quantum physics indeed requires such a generalization of our conception of probability or if von Neumann's axiomatization of quantum mechanics was contingent on the absence of a general theory of probability in the 1920s. This work argues in favor of the latter position. In particular, it shows how to construct a mathematically rigorous theory for non-relativistic $N$-body quantum systems subject to a time-independent scalar potential, which is based on Kolmogorov's axioms and physically natural random variables. Though this theory is provably distinct from its quantum mechanical analog, it nonetheless reproduces central predictions of the latter. Further work may make an empirical comparison possible. Moreover, the approach can in principle be adapted to other classes of quantum-mechanical models. Part II of this series discusses the empirical violation of Bell inequalities in the context of this approach. Part III addresses the projection postulate and the question of measurement.