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Journal of the Australian Mathematical Society

TYPE DECOMPOSITION OF A PSEUDOEFFECT ALGEBRA

David J. Foulis, Sylvia Pulmannová, Elena Vinceková

Published Dec 1, 2010• 10 citations
Publisher

Abstract

Abstract Effect algebras, which generalize the lattice of projections in a von Neumann algebra, serve as a basis for the study of unsharp observables in quantum mechanics. The direct decomposition of a von Neumann algebra into types I, II, and III is reflected by a corresponding decomposition of its lattice of projections, and vice versa. More generally, in a centrally orthocomplete effect algebra, the so-called type-determining sets induce direct decompositions into various types. In this paper, we extend the theory of type decomposition to a (possibly) noncommutative version of an effect algebra called a pseudoeffect algebra. It has been argued that pseudoeffect algebras constitute a natural structure for the study of noncommuting unsharp or fuzzy observables. We develop the basic theory of centrally orthocomplete pseudoeffect algebras, generalize the notion of a type-determining set to pseudoeffect algebras, and show how type-determining sets induce direct decompositions of centrally orthocomplete pseudoeffect algebras.

DOI
10.1017/s1446788711001042

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Abstract Effect algebras, which generalize the lattice of projections in a von Neumann algebra, serve as a basis for the study of unsharp observables in quantum mechanics. The direct decomposition of a von Neumann algebra into types I, II, and III is reflected by a corresponding decomposition of its lattice of projections, and vice versa. More generally, in a centrally orthocomplete effect algebra, the so-called type-determining sets induce direct decompositions into various types. In this paper, we extend the theory of type decomposition to a (possibly) noncommutative version of an effect algebra called a pseudoeffect algebra. It has been argued that pseudoeffect algebras constitute a natural structure for the study of noncommuting unsharp or fuzzy observables. We develop the basic theory of centrally orthocomplete pseudoeffect algebras, generalize the notion of a type-determining set to pseudoeffect algebras, and show how type-determining sets induce direct decompositions of centrally orthocomplete pseudoeffect algebras.
Abstract An effect algebra (EA) is a partial algebraic structure, originally formulated as an algebraic base for unsharp quantum measurements. The class of EAs includes, as special cases, several partially ordered algebraic structures, including orthomodular lattices (OMLs) and orthomodular posets (OMPs), hitherto used as mathematical models for experimentally verifiable propositions pertaining to physical systems. Moreover, MV-algebras, which are mathematical models for many-valued logics, are special cases of EAs. The present paper studies generalizations to EAs of the hull mapping featured in L. Loomis’s dimension theory for complete OMLs and develops a theory of direct decomposition for EAs with a hull mapping. A. Sherstnev and V. Kalinin have extended Loomis’s dimension theory to orthocomplete OMPs, and here it is further extended to orthocomplete EAs; moreover, a corresponding direct decomposition into types I, II, and III is obtained using the hull mapping induced by the dimension equivalence relation.
One of von Neumann’s motivations for developing the theory of operator algebras and his and Murray’s 1936 classification of factors was the question of possible decompositions of quantum systems into independent parts. For quantum systems with a finite number of degrees of freedom the simplest possibility, i.e., factors of type I in the terminology of Murray and von Neumann, are perfectly adequate. In relativistic quantum field theory (RQFT), on the other hand, factors of type III occur naturally. The same holds true in quantum statistical mechanics of infinite systems. In this brief review some physical consequences of the type III property of the von Neumann algebras corresponding to localized observables in RQFT and their difference from the type I case will be discussed. The cumulative effort of many people over more than 30 years has established a remarkable uniqueness result: The local algebras in RQFT are generically isomorphic to the unique, hyperfinite type III1 factor in Connes ’ classification of 1973. Specific theories are characterized by the net structure of the collection of these isomorphic algebras for different space-time regions, i.e., the way they are embedded into each other. John von Neumann was the father of the Hilbert space formulation of quantum mechanics [1] that has been the basis of almost all mathematically rigorous investigations of the theory to this day. We start by recalling the main concepts and explaining some notations.
The axioms of measurements introduced by Ludwig are formulated and studied in the framework of operator algebras. It is shown that a concrete C*-algebra with identity satisfies the axiom of sensitivity increase of effects if and only if it is a von Neumann algebra; although a von Neumann algebra satisfies the axiom of decompossability of ensembles, however, the axiom of components of the mixtures of two ensembles is true only if a von Neumann algebra is a factor of type In (n < + ∞). It is also verified that the set of decision effects, which is proved to be a subset of projections of a von Neumann algebra, has similar lattice structure of quantum mechanics, and its connection with quantum logic in the sense of Varadarjan is also figured out.
List of Figures. List of Tables. Preface. Acknowledgements. I: Mathematical and Physical Background. Introduction. 1. The mathematical scenario of quantum theory and von Neumann's axiomatization. 1.1. Algebraic structures. 1.2. The geometry of quantum theory. 1.3. The axiomatization of orthodox QT. 1.4. The 'logic' of the quantum events. 1.5. The logico-algebraic approach to QT. 2. Abstract axiomatic foundations of sharp QT. 2.1. Mackey's minimal axiomatization of QT. 2.2. Events. 2.3. Event-state systems. 2.4. Event-state systems and preclusivity spaces. 3. Back to Hilbert space. 3.1. Events as closed subspaces. 3.2. Events as projections. 3.3. Hilbert event-state systems. 3.4. From abstract orthoposets of events to Hilbert lattices. 4. The emergence of fuzzy events in Hilbert space quantum theory. 4.1. The notion of effect. 4.2. Effect-Brouwer Zadeh posets. 4.3. MacNeille completions. 4.4. Unsharp preclusivity spaces. 5. Effect algebras and quantum MV algebras. 5.1. Effect algebras and Brouwer Zadeh effect algebras. 5.2. The Lukasiewwicz operations. 5.3. MV algebras and QMV algebras. 5.4. Quasi-linear QMV algebras and effect algebras. 6. Abstract axiomatic foundations of unsharp quantum theory. 6.1. A minimal axiomatization of unsharp QT. 6.2. The algebraic structure of abstract effects. 6.3. The sharply dominating principle. 6.4. Abstract unsharp preclusivity spaces. 6.5. Sharp and unsharp abstract quantum theory. 7. To what extent is quantum ambiguity ambiguous? 7.1. Algebraic notions of 'sharp'. 7.2. Probabilistic definitions of 'sharpness'. II: Quantum Logics as Logic. Introduction. 8. Sharp quantum logics. 8.1. Algebraic and Kripkean semantics for sharp quantum logics. 8.2. Algebraic and Kripkean realizations of Hilbert event-state systems. 8.3. The implication problem in quantum logic. 8.4. Five polynomial conditions. 8.5. The quantum logical conditional as a counterfactual conditional. 8.6. Implication-connectives. 9. Metalogical properties and anomalies of quantum logic. 9.1. The failure of the Lindenbaum properety. 9.2. A modal interpretation of sharp quantum logics. 10. An axiomatization of OL and OQL. 10.1. The calculi of OL and OQL. 10.2. The soundness and completeness theorems. 11. The metalogical intractability of orthomodularity. 11.1. Orthomodularity is not elementary. 11.2. The embeddability problem. 11.3. Hilbert quantum logic and the orthomodular law. 12. First-order quantum logics and quantum set theories. 12.1. First-order semantics. 12.2. Quantum set theories. 13. Partial classical logic, the Lindenbaum property and the hidden variable problem. 13.1. Partial classical logic. 13.2. Partial classical logic and the Lindenbaum property. 13.3. States on partial Boolean algebras. 13.4. The Lindenbaum property and the hidden variable problem. 14. Unsharp quantum logics. 14.1.
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\mathcal{R}R is a von Neumann algebra, a selfadjoint element A∈RA \in \mathcal{R}A∈R induces a continuous function fA:Q(P(R))→Rf_{A} : \mathcal{Q}(\mathcal{P(R)}) \to \mathbb{R}fA​:Q(P(R))→R defined on the \emph{Stone spectrum} Q(P(R))\mathcal{Q}(\mathcal{P(R)})Q(P(R)) (\cite{deg3}) of the lattice P(R)\mathcal{P(R)}P(R) of projections in R\mathcal{R}R. fAf_{A}fA​ is called the observable function corresponding to AAA. The aim of this part is to study observable functions and its various characterizations.
We relate notions of complementarity in three layers of quantum mechanics: (i) von Neumann algebras, (ii) Hilbert spaces, and (iii) orthomodular lattices. Taking a more general categorical perspective of which the above are instances, we consider dagger monoidal kernel categories for (ii), so that (i) become (sub)endohomsets and (iii) become subobject lattices. By developing a ‘point-free’ definition of copyability we link (i) commutative von Neumann subalgebras, (ii) classical structures, and (iii) Boolean subalgebras.