Quantum Set Theory Extending the Standard Probabilistic Interpretation of Quantum Theory (Extended Abstract)

TL;DR

This paper extends the probabilistic interpretation of quantum theory using quantum set theory to define the probability of equality between any two observables, addressing limitations of the standard Born rule.

Contribution

It introduces a novel approach using quantum set theory to assign probabilities to equality of non-commuting observables, expanding foundational quantum theory.

Findings

Defines probability of equality for arbitrary observables

Extends Born rule beyond commuting observables

Discusses implications for measurement theory

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 relation for a pair of 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 probabilistic interpretation of quantum theory to define the probability of equality relation for a pair of arbitrary observables. Applications of this new interpretation to measurement theory are discussed briefly.