The Born rule -- 100 years ago and today

TL;DR

This paper reviews the historical development and modern formulations of the Born rule in quantum mechanics, highlighting its generalizations, derivations, and domain limitations over the past century.

Contribution

It provides a comprehensive historical overview and introduces a modern derivation of the Born rule from an intuitive detector-based perspective, incorporating recent results.

Findings

The Born rule has evolved significantly since 1926.

Generalizations to POVMs are essential for quantum information.

The validity of the Born rule is domain-specific and can lead to issues outside its scope.

Abstract

Details of the contents and the formulations of the Born rule changed considerably from its inception by Born in 1926 to the present day. This paper traces the early history of the Born rule 100 years ago, its generalization (essential for today's quantum optics and quantum information theory) to POVMs around 50 years ago, and a modern derivation from an intuitive definition of the notion of a quantum detector. It is based to a large extent on little known results from the recent books 'Coherent Quantum Physics' (2019) by A. Neumaier and 'Algebraic Quantum Physics, Vol. 1' (2024) by A. Neumaier and D. Westra, Also discussed is the extent to which the various forms of the Born rule have, like any other statement in physics, a restricted domain of validity, which leads to problems when applied outside this domain.