Bell as the Copernicus of Probability

TL;DR

This paper highlights the pivotal role of Bell's theorem in shifting from classical to non-Kolmogorovian probability models in quantum physics, paralleling the historical shift from Euclidean to non-Euclidean geometry.

Contribution

It draws an analogy between Bell's role in quantum probability and Lobachevsky's in geometry, emphasizing the non-Kolmogorovian nature of quantum models and critiquing standard interpretations.

Findings

Bell's theorem indicates classical probability cannot model quantum phenomena.

Quantum probability requires a non-Boolean, linear subspace event structure.

Violation of Bell's inequality challenges classical logical frameworks.

Abstract

Our aim is to emphasize the role of mathematical models in physics, especially models of geometry and probability. We briefly compare developments of geometry and probability by pointing to similarities and differences: from Euclid to Lobachevsky and from Kolmogorov to Bell. In probability Bell played the same role as Lobachevsky in geometry. In fact, violation of Bell's inequality implies the impossibility to apply the classical probability model of Kolmogorov (1933) to quantum phenomena. Thus quantum probabilistic model (based on Born's rule) is an example of non-Kolmogorovian model of probability, similarly to the Lobachevskian model -- the first example of non-Euclidean model of geometry. We also discuss coupling of the classical probabilistic model with classical (Boolean) logic. The Kolmogorov model of probability is based on the set-theoretic presentation of the Boolean logic. In this framework violation of Bell's inequality implies the impossibility to use the Boolean structure of events for quantum phenomena; instead of it, events have to be represented by linear subspaces. This is the "probability model" interpretation of violation of Bell's inequality. We also criticize the standard interpretation -- an attempt to add to rigorous mathematical probability models additional elements such as (non)locality and (un)realism. Finally, we compare embeddings of non-Euclidean geometries into the Euclidean space with embeddings of the non-Kolmogorovian probabilities (in particular, quantum probability) into the Kolmogorov probability space. As an example, we consider the CHSH-test.