Maximal Beable Subalgebras of Quantum-Mechanical Observables

TL;DR

This paper characterizes maximal subalgebras of quantum observables where the state's restriction is a mixture of dispersion-free states, providing a mathematical foundation for key Copenhagen interpretation principles.

Contribution

It introduces the concept of maximal beable subalgebras in quantum algebras and extends the theory to unbounded observables, linking to foundational quantum interpretation.

Findings

Characterization of maximal beable subalgebras in bounded quantum observables

Extension of results to algebras of unbounded observables

Mathematical support for Copenhagen interpretation principles

Abstract

Given a state on an algebra of bounded quantum-mechanical observables (the self-adjoint part of a C*-algebra), we investigate those subalgebras that are maximal with respect to the property that the given state's restriction to the subalgebra is a mixture of dispersion-free states---what we call maximal "beable" subalgebras (borrowing a terminology due to J. S. Bell). We also extend our investigation to the theory of algebras of unbounded observables (as developed by R. Kadison), and show how our results articulate a solid mathematical foundation for central tenets of the orthodox Copenhagen interpretation of quantum theory (such as the joint indeterminacy of canonically conjugate observables, and Bohr's defense of the completeness of quantum theory against the argument of Einstein, Podolsky, and Rosen).