Brouwer-Zadeh logic and the operational approach to quantum mechanics

This paper is concerned with a logical system, called Brouwer-Zadeh logic, arising from the BZ poset of all effects of a Hilbert space. In particular, we prove a representation theorem for Brouwer-Zadeh lattices, and we show that Brouwer-Zadeh logic is not characterized by the MacNeille completions of all BZ posets of effects.     

A unified framework for the algebra of unsharp quantum mechanics

On the basis of the concrete operations definable on the set of effect operators on a Hilbert space, an abstract algebraic structure of sum Brouwer-Zadeh (SBZ)-algebra is introduced. This structure consists of a partial sum operation and two mappings which turn out to be Kleene and Brouwer unusual orthocomplementations. The Foulis-Bennett effect algebra substructure induced by any SBZ-algebra, allows one to introduce the notions of unsharp “state” and “observable” in such a way that any “state-observable” composition is a standad probability measure (classical state). The Cattaneo-Nisticò BZ substructure induced by any SBZ-algebra permits one to distinguish, in an equational and simple way, the sharp elements from the really unsharp ones. The family of all sharp elements turns out to be a Foulis-Randall orthoalgebra. Any unsharp element can be “roughly” approximated by a pair of sharp elements representing the best sharp approximation from the bottom and from the top respectively, according to an abstract generalization introduced by Cattaneo of Pawlack “rough set” theory (a generalization of set theory, complementary to fuzzy set theory, which describes approximate knowledge with applications in computer sciences). In both the concrete examples of fuzzy sets and effect operators the “algebra” of rough elements shows a weak SBZ structure (weak effect algebra plus BZ standard poset) whose investigation is set as an interesting open problem.     

Brouwer-Zadeh (Fuzzy-intuitionistic) posets for unsharp quantum mechanics

The partial ordered structure which plays for unsharp quantum mechanics the same role of orthomodular lattices for ordinary quantum mechanics is introduced. Differently from the unsharp case, in which one can identify quantum propositions (i.e., Hilbert space subspaces) with yes-no devices (i.e., orthogonal projections) they are tested by, in the unsharp case this identification is broken down: every quantum generalized proposition (i.e., pair of mutually orthogonal subspaces) is tested by many different yes-no devices (i.e., Hilbert space effects). The set of all quantum effects has a structure of Brouwer-Zadeh poset, canonically embeddable in a (minimal) Brouwer-Zadeh lattice, whereas the set of all quantum generalized propositions has a structure of Brouwer-Zadeh complete lattice.A Brouwer-Zadeh poset is defined as a partially ordered structure equipped with two nonusual orthocomplementations: a regular degenerate (Zadeh or fuzzy-like) one and a weak (Brouwer or intuitionistic-like) one linked by an interconnection rule. Using these two orthocomplementations it is possible to introduce the two modal-like operators of necessity and possibility.     

From Effect Algebras to Algebras of Effects (Sum Brouwer—Zadeh Algebras)

The algebraic structures arising in the axiomatic framework of unsharp quantummechanics based on effect operators on a Hilbert space are investigated. It isstressed that usually considered effect algebras neglect the unitary Brouwerianmap of complementation, and the main results based on this complementationare collected, showing the enrichment produced into the theory by its introduction.In particular, in these structures two notions of sharpness can be considered: K-sharpness induced by the usual complementation of effect algebrasand B-sharpness induced by this new complementation. Quantum (resp., classical) SBZalgebras are then characterized by the condition of B-coherence (resp., B-coherence plusB-compatibility), showing that in this case the poset of all B-sharp elements is orthomodular (resp., Boolean algebra). In the unsharp contextof effect operators, the finite dimensionality of the Hilbert space or the finitenessof a von Neumann algebra are both characterized by a de Morgan property ofthe Brouwer complementation. Moreover, since effect operators on a pre-Hilbertspace give rise to a standard model of effect algebras, a characterization ofcompleteness of pre-Hilbert spaces is given making use of the Brouwercomplement.     

Regularity in Quantum Logic

In 1996, Harding showed that the binarydecompositions of any algebraic, relational, ortopological structure X form an orthomodular poset FactX. Here, we begin an investigation of the structuralproperties of such orthomodular posets of decompositions.We show that a finite set S of binary decompositions inFact X is compatible if and only if all the binarydecompositions in S can be built from a common n-arydecomposition of X. This characterization ofcompatibility is used to show that for any algebraic,relational, or topological structure X, the orthomodularposet Fact X is regular. Special cases of this result include the known facts that theorthomodular posets of splitting subspaces of an innerproduct space are regular, and that the orthomodularposets constructed from the idempotents of a ring are regular. This result also establishes theregularity of the orthomodular posets that Mushtariconstructs from bounded modular lattices, theorthomodular posets one constructs from the subgroups ofa group, and the orthomodular posets oneconstructs from a normed group with operators. Moreover,all these orthomodular posets are regular for the samereason. The characterization of compatibility is also used to show that for any structure X, thefinite Boolean subalgebras of Fact X correspond tofinitary direct product decompositions of the structureX. For algebraic and relational structures X, this result is extended to show that the Booleansubalgebras of Fact X correspond to representations ofthe structure X as the global sections of a sheaf ofstructures over a Boolean space. The above results can be given a physical interpretation as well.Assume that the true or false questions of a quantum mechanical system correspond tobinary direct product decompositions of the state spaceof the system, as is the case with the usual von Neumanninterpretation of quantum mechanics. Suppose S is asubset of . Then a necessary andsufficient condition that all questions in S can beanswered simultaneously is that any two questions in S can be answeredsimultaneously. Thus, regularity in quantum mechanicsfollows from the assumption that questions correspond todecompositions.     

Testing for Classicality of a Physical System

Often quantum logics are algebraically modelled by orthomodular posets. The physical system described by such a quantum logic is classical if and only if the corresponding orthomodular poset is a Boolean algebra. We provide an easy testing procedure for this case. Moreover, we characterize orthomodular posets which are lattices and consider orthomodular posets which admit a full set of states and hence represent so-called spaces of numerical events. This way further test procedures are obtained.     

Noncommutative lattices as finite approximations

Lattice discretizations of continuous manifolds are common tools used in a variety of physical contexts. Conventional discrete approximations, however, cannot capture all aspects of the original manifold, notably its topology. In this paper we discuss an approximation scheme due to Sorkin (1991) which correctly reproduces important topological aspects of continuum physics. The approximating topological spaces are partially ordered sets (posets), the partial order encoding the topology. Now, the topology of a manifold M can be reconstructed from the commutativè C^*algebra C(M) of continuous functions defined on it. In turn, this algebra is generated by continuous probability densities in ordinary quantum physics on M. The latter also serves to specify the domains of observables like the Hamiltonian. For a poset, the role of this algebra is assumed by a noncommutative C^*-algebra A. This fact makes any poset a genuine ‘noncommutative’ (‘quantum’) space, in the sense that the algebra of its ‘continuous functions’ is a noncommutative C^*-algebra. We therefore also have a remarkable connection between finite approximations to quantum physics and noncommutative geometries. We use this connection to develop various approximation methods for doing quantum physics using A.     

A New Light on Nets of C*-Algebras and Their Representations

The present paper deals with the question of representability of nets of C*-algebras whose underlying poset, indexing the net, is not upward directed. A particular class of nets, called C*-net bundles, is classified in terms of C*-dynamical systems having as group the fundamental group of the poset. Any net of C*-algebras has a canonical morphism into a C*-net bundle, the enveloping net bundle, which generalizes the notion of universal C*-algebra given by Fredenhagen to nonsimply connected posets. This allows a classification of nets; in particular, we call injective those nets such that the canonical morphism is faithful. Injectivity turns out to be equivalent to the existence of faithful representations. We further relate injectivity to a generalized Čech cocycle of the net, and this allows us to give examples of nets exhausting the above classification.     

Jauch-Piron logics with finiteness conditions

We show that there are no non-Boolean block-finite orthomodular posets possessing a unital set of Jauch-Piron states. Thus, an orthomodular poset representing a quantum physical system must have infinitely many blocks.     

Difference posets and the histories approach to quantum theories

Direct limits and tensor products of difference posets are studied. In the spirit of a recent paper by Isham, a potential model for an unsharp histories approach to quantum theory based on difference posets as abstract models for the set of effects is considered. It is shown that the set of all histories in this approach has an algebraic structure of a difference poset.     

Contraexamples in difference posets and orthoalgebras

We show that every orthoalgebra (difference orthoposet) uniquely determines a difference orthoalgebraic structure. We give examples of posets on which there exist more than one difference operation. In spite of that, every finite chain is a uniquely determined difference poset. On a difference poset there need not exist any orthoalgebraic operation, but the category of difference orthoposets is isomorphic with the category of orthoalgebras. But a difference poset which is also an orthoposet need not be a difference orthoposet. Moreover, there exist complete lattices on which there does not exist any difference operation. Finally, we show that difference operations and orthoalgebraic operations need not be extendable on a MacNeille completion of the base poset.     

Physical and geometrical interpretation of the Jordan-Hahn and the Lebesgue decomposition property

The Jordan-Hahn decomposition and the Lebesgue decomposition, two basic notions of classical measure theory, are generalized for measures on orthomodular posets. The Jordan-Hahn decomposition property (JHDP) and the Lebesgue decomposition property (LDP) are defined for sections of probability measures on an orthomodular poset L. If L is finite, then these properties can be characterized geometrically in terms of two parallelity relations defined on the set of faces of . A section is shown to have the JHDP if and only if every pair of f-parallel faces is p-parallel; it is shown to have the LDP if and only if every pair of disjoint faces is p-parallel. It follows from these results that the LDP is stronger than the JHDP in the setting of finite orthomodular posets. Mielnik's convex scheme of quantum theory provides the frame for a physical interpretation of these results.     

Algebraic Structures Arising in Axiomatic Unsharp Quantum Physics

This article presents and compares various algebraic structures that arise in axiomatic unsharp quantum physics. We begin by stating some basic principles that such an algebraic structure should encompass. Following G. Mackey and G. Ludwig, we first consider a minimal state-effect-probability (minimal SEFP) structure. In order to include partial operations of sum and difference, an additional axiom is postulated and a SEFP structure is obtained. It is then shown that a SEFP structure is equivalent to an effect algebra with an order determining set of states. We also consider -SEFP structures and show that these structures distinguish Hilbert space from incomplete inner product spaces. Various types of sharpness are discussed and under what conditions a Brouwer complementation can be defined to obtain a BZ-poset is investigated. In this case it is shown that every effect has a best lower and upper sharp approximation and that the set of all Brouwer sharp effects form an orthoalgebra.     

Difference posets,effects, and quantum measurements

Difference posets as generalizations of quantum logics, orthoalgebras, and effects are studied. Observables and measures generalizing normalized POV-measures and generalized measures on sets of effects are introduced. Characterization of orthomodularity of subsets of a difference poset in terms of triangle closedness and regularity of these subsets enables us to characterize observables with a Boolean range. Boolean powers of difference posets are investigated; they have similar properties to that of tensor products, and their connection with quantum measurements is studied.     

Topological Properties of Spatial Coherence Function

The topological properties of the spatial coherence function are investigated rigorously. The phase singular structures （coherence vortices） of coherence function can be naturally deduced from the topological current, which is an abstract mathematical object studied previously. We find that coherence vortices are characterized by the Hopf index and Brouwer degree in topology. The coherence flux quantization and the linking of the closed coherence vortices are also studied from the topological properties of the spatial coherence function.     

Topological Structure of Vortex Solution in Jackiw-Pi Model

By using o-mapping method, we discuss the topological structure of the self-duality solution in Jackiw-Pi model in terms of gauge potential decomposition. We set up relationship between Chern-Simons vortex solution and topological number, which is determined by Hopf index and Brouwer degree. We also give the quantization of flux in this case. Then, we study the angular momentum of the vortex, which can be expressed in terms of the flux.     

Topological Structure of Vortex Solution in Jackiw-Pi Model

By using φ-mapping method, we discuss the topological structure of the self-duality solution in Jackiw-Pi model in terms of gauge potential decomposition. We set up relationship between Chern-Simons vortex solution and topological number, which is determined by Hopf index and Brouwer degree. We also give the quantization of flux in this case. Then, we study the angular momentum of the vortex, which can be expressed in terms of the flux.     

Vortex Lines and Monopoles in Electrically Conducting Plasmas

Based on the φ-mapping topological current theory and the decomposition of gauge potential theory, the vortex lines and the monopoles in electrically conducting plasmas are studied. It is pointed out that these two topological structures respectively inhere in two-dimensional and three-dimensional topological currents, which can be derived from the same topological term n^→·（Эin^→×Эjn^→）, and both these topological structures axe characterized by the φ-mapping topological numbers-Hopf indices and Brouwer degrees. Furthermore, the spatial bifurcation of vortex lines and the generation and annihilation of monopoles are also discussed. At last, we point out that the Hopf invaxiant is a proper topological invaxiant to describe the knotted solitons.     

Commutativity in orthomodular posets

An abstract characterization of the commutation relation in orthomodular posets is given. This characterization is a generalization of Guz's result. In particular, if an orthomodular poset P is Boolean, then aCb iff a ∧ b exists in P.A method of constructing nonregular Boolean orthomodular posets is presented.     

Vortex lines in a ferromagnetic spin-triplet superconductor

Based on Duan's topological current theory, we show that in a ferromagnetic spin-triplet superconductor there is a topological defect of string structures which can be interpreted as vortex lines. Such defects are different from the Abrikosov vortices in one-component condensate systems. We investigate the inner topological structure of the vortex lines. The topological charge density, velocity, and topological current of the vortex lines can all be expressed in terms of δ function, which indicates that the vortices can only arise from the zero points of an order parameter field. The topological charges of vortex lines are quantized in terms of the Hopf indices and Brouwer degrees of φ-mapping. The divergence of the self-induced magnetic field can be rigorously determined by the corresponding order parameter fields and its expression also takes the form of a δ-like function. Finally, based on the implicit function theorem and the Taylor expansion, we conduct detailed studies on the bifurcation of vortex topological current and find different directions of the bifurcation.