The meet of incommutable projection operators contradicts Burnside's theorem

TL;DR

This paper investigates the mathematical structure of incommutable projection operators in finite Hilbert spaces, revealing contradictions with Burnside's theorem and challenging assumptions about their lattice properties.

Contribution

It demonstrates that in finite Hilbert spaces, the column spaces of incommutable projection operators cannot form a single partially ordered set, contradicting expectations based on Burnside's theorem.

Findings

Incommutable projection operators' column spaces do not form a lattice in finite Hilbert spaces.

Contradicts the assumption that all projection operators' column spaces can be ordered.

Highlights limitations of representing conjunctions of incommutable projections mathematically.

Abstract

In contrast to conjunctions of commutable projection operators unambiguously represented by their meets, the mathematical representation of conjunctions of incommutable projection operators is a question that has yet to be solved. This question relates to another asking whether the set of the column spaces of the projection operators, commutable and incommutable alike, forms a lattice. As it is demonstrated in the paper, if the Hilbert space is finite, the column spaces of the incommutable projection operators cannot be elements of one partially ordered set in accordance with Burnside's theorem on matrix algebras.