Antonymous functions are real-valued functions on the Stone spectrum of a von Neumann algebra R. They correspond to the self-adjoint operators in R, which are interpreted as observables in quantum physics. Antonymous functions turn out to be generalized Gelfand transforms, related to de Groote's observable functions.
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.
In the second part of our work on observables we have shown that quantum observables in the sense of von Neumann, i.e.bounded selfadjoint operators in some von Neumann subalgebra R of L(H), can be represented as bounded continuous functions on the Stone spectrum Q(R) of R. Moreover, we have shown that this representation is linear if and only if R is abelian, and that in this case it coincides with the Gelfand transformation of R. In this part we discuss classical observables, i.e. measurable and continuous functions, under the same point of view. We obtain results that are quite similar to the quantum case, thus showing up the common structural features of quantum and classical observables.
In this work we discuss the notion of observable - both quantum and classical - from a new point of view. In classical mechanics, an observable is represented as a function (measurable, continuous or smooth), whereas in (von Neumann's approach to) quantum physics, an observable is represented as a bonded selfadjoint operator on Hilbert space. We will show in the present part II and the forthcoming part III of this work that there is a common structure behind these two different concepts. If R is a von Neumann algebra, a selfadjoint element A∈R induces a continuous function fA:Q(P(R))→R defined on the \emph{Stone spectrum} Q(P(R)) (\cite{deg3}) of the lattice P(R) of projections in R. fA is called the observable function corresponding to A. The aim of this part is to study observable functions and its various characterizations.
In "Self-adjoint Operators as Functions I: Lattices, Galois Connections, and the Spectral Order" [arXiv:1208.4724], it was shown that self-adjoint operators affiliated with a von Neumann algebra N can equivalently be described as certain real-valued functions on the projection lattice P(N) of the algebra, which we call q-observable functions. Here, we show that q-observable functions can be interpreted as generalised quantile functions for quantum observables interpreted as random variables. More generally, when L is a complete meet-semilattice, we show that L-valued cumulative distribution functions and the corresponding L-quantile functions form a Galois connection. An ordinary CDF can be written as an L-CDF composed with a state. For classical probability, one picks L=B(Ω), the complete Boolean algebra of measurable subsets modulo null sets of a measurable space Ω. For quantum probability, one uses L=P(N), the projection lattice of a nonabelian von Neumann algebra N. Moreover, using some constructions from the topos approach to quantum theory, we show that there is a joint sample space for all quantum observables, despite no-go results such as the Kochen-Specker theorem. Specifically, the spectral presheaf Σ of a von Neumann algebra N, which is not a mere set, but a presheaf (i.e., a 'varying set'), plays the role of the sample space. The relevant meet-semilattice L in this case is the complete bi-Heyting algebra of clopen subobjects of Σ. We show that using the spectral presheaf Σ and associated structures, quantum probability can be formulated in a way that is structurally very similar to classical probability.
In quantum mechanics, many observables for a quantum system are represented as self-adjoint linear operators (possibly unbounded) on a complex Hilbert space H. In this paper, we introduce a binary relation ⊥ on S(H) of all self-adjoint linear operators (including unbounded) on H. We obtain that A⊥B implies that A,B are affiliated with some abelian von Neumann algebra at the same time. Depending on ⊥,S(H) can be constructed to be a generalized orthoalgebra. Then we present the orthoalgebra order ⪯ and the traditional order ≤ on S(H) and give some properties about these two orders. We prove that for A,B∈S(H), A⪯B if and only if A has a value in Δ implies that B has a value in Δ for every Borel set Δ not containing 0. Moreover, we study the existence of the infimum A∧B and supremum A∨B for A,B∈S(H) with respect to ⪯. Finally, we prove that Q∧P=0 where Q and P are the significant quantum observables corresponding to the position and momentum.}