Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Log inSign up
Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Blog
Job Board
Q3AS

© 2026 Aqora Quantum S.A.S.

TermsPrivacyLegal Notice

arXiv:1208.4724

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

Andreas Doering, Barry Dewitt

Published Aug 23, 2012• 4 citations
arXivHTMLPDF

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.

DOI
10.48550/arxiv.1208.4724

Related papers

Self-adjoint Operators as Functions II: Quantum Probability

Andreas Doering, Barry Dewitt

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.

arXiv:1210.5747•Oct 21, 2012•7 citations

A Quantum Logic Including Unbounded Observables

Qiang Lei, Junde Wu

In quantum mechanics, many observables for a quantum system are represented as self-adjoint linear operators (possibly unbounded) on a complex Hilbert space $\mathcal{H}$. In this paper, we introduce a binary relation $\perp$ on $S(\mathcal{H})$ of all self-adjoint linear operators (including unbounded) on $\mathcal{H}$. We obtain that $A\perp B$ implies that $A,B$ are affiliated with some abelian von Neumann algebra at the same time. Depending on $\perp$$, S(\mathcal{H})$ can be constructed to be a generalized orthoalgebra. Then we present the orthoalgebra order $\preceq$ and the traditional order $\leq$ on $S(\mathcal{H})$ and give some properties about these two orders. We prove that for $A,B\in S(\mathcal{H})$, $A\preceq B$ if and only if $A$ has a value in $\Delta$ implies that $B$ has a value in $\Delta$ for every Borel set $\Delta$ not containing $0$. Moreover, we study the existence of the infimum $A\wedge B$ and supremum $A\vee B$ for $A,B\in S(\mathcal{H})$ with respect to $\preceq$. Finally, we prove that $Q\wedge P=0$ where $Q$ and $P$ are the significant quantum observables corresponding to the position and momentum.}

arXiv:1508.07386?context=math-ph•Aug 29, 2015•0 citations

Towards a Paraconsistent Quantum Set Theory

Benjamin Eva

In this paper, we will attempt to establish a connection between quantum set theory, as developed by Ozawa, Takeuti and Titani, and topos quantum theory, as developed by Isham, Butterfield and Doring, amongst others. Towards this end, we will study algebraic valued set-theoretic structures whose truth values correspond to the clopen subobjects of the spectral presheaf of an orthomodular lattice of projections onto a given Hilbert space. In particular, we will attempt to recreate, in these new structures, Takeuti's original isomorphism between the set of all Dedekind real numbers in a suitably constructed model of set theory and the set of all self adjoint operators on a chosen Hilbert space.

arXiv:1511.01571•Electronic Proceedings in Theoretical Computer Science•Nov 4, 2015•5 citations

Observables as functions: Antonymous functions

Andreas Doering

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.

arXiv:quant-ph/0510102•Oct 13, 2005•6 citations

Bohrification of operator algebras and quantum logic

Chris Heunen, Nicolaas P. Landsman, Bas Spitters

Following Birkhoff and von Neumann, quantum logic has traditionally been based on the lattice of closed linear subspaces of some Hilbert space, or, more generally, on the lattice of projections in a von Neumann algebra A. Unfortunately, the logical interpretation of these lattices is impaired by their nondistributivity and by various other problems. We show that a possible resolution of these difficulties, suggested by the ideas of Bohr, emerges if instead of single projections one considers elementary propositions to be families of projections indexed by a partially ordered set $${\mathcal{C}(A)}$$ of appropriate commutative subalgebras of A. In fact, to achieve both maximal generality and ease of use within topos theory, we assume that A is a so-called Rickart C*-algebra and that $${\mathcal{C}(A)}$$ consists of all unital commutative Rickart C*-subalgebras of A. Such families of projections form a Heyting algebra in a natural way, so that the associated propositional logic is intuitionistic: distributivity is recovered at the expense of the law of the excluded middle. Subsequently, generalizing an earlier computation for n × n matrices, we prove that the Heyting algebra thus associated to A arises as a basis for the internal Gelfand spectrum (in the sense of Banaschewski–Mulvey) of the “Bohrification” $${\underline A}$$ of A, which is a commutative Rickart C*-algebra in the topos of functors from $${\mathcal{C}A}$$ to the category of sets. We explain the relationship of this construction to partial Boolean algebras and Bruns–Lakser completions. Finally, we establish a connection between probability measures on the lattice of projections on a Hilbert space H and probability valuations on the internal Gelfand spectrum of $${\underline A}$$ for A = B(H).

arXiv:0905.2275•Synthese•Mar 28, 2011•30 citations

Complete sets of non-self-adjoint observables: An unbounded approach

Giuseppina Epifanio, Todor S. Todorov, Camillo Trapanı

The notion of completeness of a set 𝒮 of compatible observables represented by maximal symmetric operators is discussed directly in terms of unbounded operators. In contrast with what happens for self-adjoint observables, the present framework forces us to involve some partial algebraic structures such as the partial GW*-algebra generated by 𝒮. In this way the previous approaches based on von Neumann algebras and on O*-algebras are generalized.

Journal of Mathematical Physics•Mar 1, 1996•2 citations
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 "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\mathcal{H}H. In this paper, we introduce a binary relation ⊥\perp⊥ on S(H)S(\mathcal{H})S(H) of all self-adjoint linear operators (including unbounded) on H\mathcal{H}H. We obtain that A⊥BA\perp BA⊥B implies that A,BA,BA,B are affiliated with some abelian von Neumann algebra at the same time. Depending on ⊥\perp⊥,S(H), S(\mathcal{H}),S(H) can be constructed to be a generalized orthoalgebra. Then we present the orthoalgebra order ⪯\preceq⪯ and the traditional order ≤\leq≤ on S(H)S(\mathcal{H})S(H) and give some properties about these two orders. We prove that for A,B∈S(H)A,B\in S(\mathcal{H})A,B∈S(H), A⪯BA\preceq BA⪯B if and only if AAA has a value in Δ\DeltaΔ implies that BBB has a value in Δ\DeltaΔ for every Borel set Δ\DeltaΔ not containing 000. Moreover, we study the existence of the infimum A∧BA\wedge BA∧B and supremum A∨BA\vee BA∨B for A,B∈S(H)A,B\in S(\mathcal{H})A,B∈S(H) with respect to ⪯\preceq⪯. Finally, we prove that Q∧P=0Q\wedge P=0Q∧P=0 where QQQ and PPP are the significant quantum observables corresponding to the position and momentum.}
In this paper, we will attempt to establish a connection between quantum set theory, as developed by Ozawa, Takeuti and Titani, and topos quantum theory, as developed by Isham, Butterfield and Doring, amongst others. Towards this end, we will study algebraic valued set-theoretic structures whose truth values correspond to the clopen subobjects of the spectral presheaf of an orthomodular lattice of projections onto a given Hilbert space. In particular, we will attempt to recreate, in these new structures, Takeuti's original isomorphism between the set of all Dedekind real numbers in a suitably constructed model of set theory and the set of all self adjoint operators on a chosen Hilbert space.
Following Birkhoff and von Neumann, quantum logic has traditionally been based on the lattice of closed linear subspaces of some Hilbert space, or, more generally, on the lattice of projections in a von Neumann algebra A. Unfortunately, the logical interpretation of these lattices is impaired by their nondistributivity and by various other problems. We show that a possible resolution of these difficulties, suggested by the ideas of Bohr, emerges if instead of single projections one considers elementary propositions to be families of projections indexed by a partially ordered set C(A){\mathcal{C}(A)}C(A) of appropriate commutative subalgebras of A. In fact, to achieve both maximal generality and ease of use within topos theory, we assume that A is a so-called Rickart C*-algebra and that C(A){\mathcal{C}(A)}C(A) consists of all unital commutative Rickart C*-subalgebras of A. Such families of projections form a Heyting algebra in a natural way, so that the associated propositional logic is intuitionistic: distributivity is recovered at the expense of the law of the excluded middle. Subsequently, generalizing an earlier computation for n × n matrices, we prove that the Heyting algebra thus associated to A arises as a basis for the internal Gelfand spectrum (in the sense of Banaschewski–Mulvey) of the “Bohrification” A‾{\underline A}A​ of A, which is a commutative Rickart C*-algebra in the topos of functors from CA{\mathcal{C}A}CA to the category of sets. We explain the relationship of this construction to partial Boolean algebras and Bruns–Lakser completions. Finally, we establish a connection between probability measures on the lattice of projections on a Hilbert space H and probability valuations on the internal Gelfand spectrum of A‾{\underline A}A​ for A = B(H).