[Paper Review] Self-adjoint Operators as Functions I: Lattices, Galois Connections, and the Spectral Order

Andreas Doering, Barry Dewitt|arXiv (Cornell University)|Aug 23, 2012

Advanced Operator Algebra Research14 references6 citations

TL;DR

This paper introduces q-observable functions—real-valued functions on the projection lattice of a von Neumann algebra—that provide a dual characterization of self-adjoint operators via Galois connections with their spectral families. The key contribution is establishing an order-isomorphism between the lattice of self-adjoint operators under the spectral order and the lattice of q-observable functions, offering a new order-theoretic framework for quantum observables and connecting to daseinisation in the topos approach to quantum theory.

ABSTRACT

Observables of a quantum system, described by self-adjoint operators in a von Neumann algebra or affiliated with it in the unbounded case, form a conditionally complete lattice when equipped with the spectral order. Using this order-theoretic structure, we develop a new perspective on quantum observables. In this first paper (of two), we show that self-adjoint operators affiliated with a von Neumann algebra can equivalently be described as certain real-valued functions on the projection lattice of the algebra, which we call q-observable functions. Bounded self-adjoint operators correspond to q-observable functions with compact image on non-zero projections. These functions, originally defined in a similar form by de Groote, are most naturally seen as adjoints (in the categorical sense) of spectral families. We show how they relate to the daseinisation mapping from the topos approach to quantum theory. Moreover, the q-observable functions form a conditionally complete lattice which is shown to be order-isomorphic to the lattice of self-adjoint operators with respect to the spectral order. In a subsequent paper, we will give an interpretation of q-observable functions in terms of quantum probability theory, and using results from the topos approach to quantum theory, we will provide a joint sample space for all quantum observables.

Motivation & Objective

• To reformulate self-adjoint operators in von Neumann algebras as real-valued functions on their projection lattices, termed q-observable functions.
• To establish that these functions form a conditionally complete lattice isomorphic to the lattice of self-adjoint operators under the spectral order.
• To connect the q-observable function construction to the daseinisation maps in the topos approach to quantum theory, particularly outer and inner daseinisation.
• To provide a categorical and order-theoretic foundation for quantum observables, emphasizing spectral order over linear structure.
• To lay the groundwork for interpreting q-observable functions in quantum probability, as continued in the sequel paper.

Proposed method

• Define q-observable functions as the left adjoint (in the categorical sense) of the spectral family map $E^{ ilde{A}}:\overline{\mathbb{R}} \to \mathcal{P}(\mathcal{N})$, using Galois connections between meet-semilattices.
• Show that the image of a q-observable function $o^{\tilde{A}}$ on non-zero projections equals the spectrum of $\tilde{A}$, establishing a direct link to spectral theory.
• Characterize q-observable functions abstractly as functions on $\mathcal{P}(\mathcal{N})$ satisfying specific order-theoretic and monotonicity conditions.
• Demonstrate that the lattice of q-observable functions is order-isomorphic to the lattice of self-adjoint operators under the spectral order.
• Use restriction and extension of domains to characterize outer and inner daseinisation maps as restrictions of q-observable functions to subalgebras.
• Establish a limited functional calculus: for monotone $f:\overline{\mathbb{R}} \to \overline{\mathbb{R}}$, $o^{f(\tilde{A})} = f \circ o^{\tilde{A}}$.

Experimental results

Research questions

• RQ1 How can self-adjoint operators affiliated with a von Neumann algebra be equivalently described as functions on the projection lattice?
• RQ2 What is the order-theoretic structure of the set of such functions, and how does it relate to the spectral order on operators?
• RQ3 How do q-observable functions relate to the daseinisation maps used in the topos approach to quantum theory?
• RQ4 Can a functional calculus be defined for q-observable functions, and under what conditions does it preserve operator functions?
• RQ5 What is the role of q-antonymous functions, and how do they relate to the negative of an observable?

Key findings

• The q-observable function $o^{\tilde{A}}$ is uniquely determined by the spectral family $E^{\tilde{A}}$, and vice versa, establishing a one-to-one correspondence.
• The image of $o^{\tilde{A}}$ on non-zero projections is exactly the spectrum of $\tilde{A}$, confirming its physical relevance.
• The lattice of q-observable functions is order-isomorphic to the lattice of self-adjoint operators under the spectral order, providing a new representation of quantum observables.
• Outer and inner daseinisation maps correspond to the restriction of $o^{\tilde{A}}$ to subalgebras $\mathcal{M} \subset \mathcal{N}$, with $o^{\tilde{A}}|_{\mathcal{P}(\mathcal{M})} = o^{\delta^o(\tilde{A})_{\mathcal{M}}}$.
• A limited functional calculus holds: for monotone $f$, $o^{f(\tilde{A})} = f \circ o^{\tilde{A}}$, preserving functional relations in the function representation.
• The construction generalizes to unbounded operators via affiliation with von Neumann algebras, and the spectral order ensures that approximations via daseinisation preserve spectral subsets: $\operatorname{sp}(\delta^o(\tilde{A})_{\mathcal{M}}) \subseteq \operatorname{sp}(\tilde{A})$.