Categorical Equivalence Between Finitary Orthomodular Dynamic Algebras and Orthomodular Lattices

TL;DR

This paper establishes a categorical equivalence between finitary orthomodular dynamic algebras and orthomodular lattices, linking quantum logic structures and extending to Hilbert lattices.

Contribution

It introduces a new categorical equivalence connecting two quantum logic structures, including finitary variants and their applications to Hilbert lattices.

Findings

Categorical equivalence between finitary orthomodular dynamic algebras and orthomodular lattices.

Extension of the equivalence to Hilbert lattices of closed subspaces.

Bridging unital involutive m-semilattices with various quantum formalisms.

Abstract

This paper reveals a categorical equivalence connecting two distinct quantum logic structures. The first is the orthomodular lattice, an algebraic system designed to formalize the properties of quantum systems. The second is a finitary orthomodular dynamic algebra, a specialized development of the orthomodular dynamic algebra where the underlying quantum actions are restricted to be finitary. The applicability of the result extends to more specialized lattices, such as Hilbert lattices of closed subspaces of a Hilbert space, beyond general orthomodular lattices. As these lattice structures exhibit connections to a diverse array of quantum structures, the established equivalence categorically bridges unital involutive m-semilattices with a broad spectrum of quantum formalisms.