Orthocomplemented subspaces and partial projections on a Hilbert space

TL;DR

This paper introduces orthocomplemented subspaces in Hilbert spaces, linking them to partial projections and offering a new constructive approach to quantum logic that extends classical concepts.

Contribution

It defines orthocomplemented subspaces as a two-dimensional analogue to closed subspaces, connecting them to partial projections and developing a novel constructive quantum logic framework.

Findings

Orthocomplemented subspaces correspond to partial projections.

The approach bypasses the locatedness hypothesis in Hilbert space theory.

Introduces a two-dimensional lattice structure for quantum logic.

Abstract

We introduce the notion of an orthocomplemented subspace of a Hilbert space H, that is, a pair of orthogonal closed subspaces of H, as a two-dimensional counterpart to the one-dimensional notion of a closed subspace of H. Orthocomplemented subspaces are the Hilbert space-analogue to Bishop's complemented subsets. To complemented subsets correspond their characteristic functions, which are partial, Boolean-valued functions. Similarly, to orthocomplemented subspaces of H correspond partial projections on H. Previous work of Bridges and Svozil on constructive quantum logic is an one-dimensional approach to the subject. The lattice-properties of the orthocomplemented subspaces of a Hilbert space is a two-dimensional approach to constructive quantum logic, that we call complemented quantum logic. Since the negation of an orthocomplemented subspace is formed by swapping its components, complemented quantum logic, although constructive, is closer to classical quantum logic than the constructive quantum logic of Bridges and Svozil. The introduction of orthocomplemented subspaces and their corresponding partial projections allows a new approach to the constructive theory of Hilbert spaces. For example, the partial projection operator of an orthocomplemented subspace and the construction of the quotient Hilbert space bypass the standard restrictive hypothesis of locatedness on a subspace. Located subspaces correspond to total orthocomplemented subspaces.