Observables, Disassembled

TL;DR

This paper explores the conditions under which non-self-adjoint operators can serve as quantum observables, expanding the traditional framework by analyzing four classes of such operators and their implications.

Contribution

It introduces a classification of non-self-adjoint operators as observables and shows how some extend the standard quantum theory framework.

Findings

Normal operators provide an equivalent formulation of quantum theory.

Symmetric and real spectrum operators extend quantum theory.

Operators with none of these properties offer new observable frameworks.

Abstract

This paper argues that non-self-adjoint operators can be observables. There are only four ways for this to occur: non-self-adjoint observables can either be normal operators, or be symmetric, or have a real spectrum, or have none of these three properties. I explore each of these four classes of observables, arguing that the class of normal operators provides an equivalent formulation of quantum theory, whereas the other classes considerably extend it.