Relational Quantum Mechanics and Probability

TL;DR

This paper derives the core probability rule of Relational Quantum Mechanics from basic properties of conditional probabilities, showing that the Born rule naturally follows without needing the third postulate.

Contribution

It demonstrates that the third postulate of RQM is unnecessary by deriving the Born rule from the first two postulates using Gleason's theorem and conditional probability properties.

Findings

The Born rule emerges naturally from the first two RQM postulates.

Probability functions are uniquely defined for classical and quantum phenomena.

Interference effects depend on the formulation of conditional probability.

Abstract

We present a derivation of the third postulate of Relational Quantum Mechanics (RQM) from the properties of conditional probabilities.The first two RQM postulates are based on the information that can be extracted from interaction of different systems, and the third postulate defines the properties of the probability function. Here we demonstrate that from a rigorous definition of the conditional probability for the possible outcomes of different measurements, the third postulate is unnecessary and the Born's rule naturally emerges from the first two postulates by applying the Gleason's theorem. We demonstrate in addition that the probability function is uniquely defined for classical and quantum phenomena. The presence or not of interference terms is demonstrated to be related to the precise formulation of the conditional probability where distributive property on its arguments cannot be taken for granted. In the particular case of Young's slits experiment, the two possible argument formulations correspond to the possibility or not to determine the particle passage through a particular path.