On $\mathscr{T}$-based orthomodular dynamic algebras

TL;DR

This paper proves a categorical equivalence between complete orthomodular lattices and a new class of $ ext{ extscr}$-based orthomodular dynamic algebras, linking static quantum logic with dynamic quantum actions.

Contribution

It establishes a categorical equivalence between static orthomodular lattices and dynamic algebras, extending the understanding of quantum logic structures.

Findings

Categorical equivalence between $ ext{ extscr}$-based orthomodular dynamic algebras and complete orthomodular lattices.

Extension of the equivalence to Hilbert lattices and broader quantum structures.

Refinement of the connection between static and dynamic quantum logic.

Abstract

This paper establishes a categorical equivalence between the category $\mathbb{COL}$ of complete orthomodular lattices and the category $\mathscr{T}\mathbb{ODA}$ of $\mathscr{T}$-based orthomodular dynamic algebras. Complete orthomodular lattices serve as the static algebraic foundation for quantum logic, modeling the testable properties of quantum systems. In contrast, $\mathcal{T}$-based orthomodular dynamic algebras, which are specialized unital involutive quantales, formalize the composition and quantum-logical properties of quantum actions. This result refines prior connections between orthomodular lattices and dynamic algebras, provides a constructive bridge between static and dynamic quantum logic perspectives, and extends naturally to Hilbert lattices and broader quantum-theoretic structures.