Quantum Set Theory Extending the Standard Probabilistic Interpretation of Quantum Theory

TL;DR

This paper introduces a quantum set theory framework that extends the probabilistic interpretation of quantum mechanics to define the probability of equality between any two observables, regardless of their commutativity.

Contribution

It develops a systematic extension of the standard interpretation using quantum set theory to address equality between arbitrary observables in any state.

Findings

Defines probability of equality for non-commuting observables

Establishes a logical basis for measurement and determinateness

Extends probabilistic interpretation beyond commuting observables

Abstract

The notion of equality between two observables will play many important roles in foundations of quantum theory. However, the standard probabilistic interpretation based on the conventional Born formula does not give the probability of equality between two arbitrary observables, since the Born formula gives the probability distribution only for a commuting family of observables. In this paper, quantum set theory developed by Takeuti and the present author is used to systematically extend the standard probabilistic interpretation of quantum theory to define the probability of equality between two arbitrary observables in an arbitrary state. We apply this new interpretation to quantum measurement theory, and establish a logical basis for the difference between simultaneous measurability and simultaneous determinateness.