Concrete quantum logics with covering properties

LetL be a concrete (=set-representable) quantum logic. Letn be a natural number (or, more generally, a cardinal). We say thatL admits intrinsic coverings of the ordern, and writeL C _n , if for any pairA, BL we can find a collection {C _i iI}, where cardI<n andC _i L for anyiI, such thatA B= _il C _i . Thus, in a certain sense, ifLC _n , then the rate of noncompatibility of an arbitrary pairA,BL is less than a given numbern. In this paper we first consider general and combinatorial properties of logics ofC _n and exhibit typical examples. In particular, for a givenn we construct examples ofL C _n+1\C _n . Further, we discuss the relation of the classesC _n to other classes of logics important within the quantum theories (e.g., we discover the interesting relation to the class of logics which have an abundance of Jauch-Piron states). We then consider conditions on which a class of concrete logics reduce to Boolean algebras. We conclude with some open questions.     

Higher-order quantum logics

Until now quantum logics has been first-order, but physics requires higher-order logics. We construct a natural higher-order languageQ for quantum physics.Q is a finitistic logic based on Peano set theory and Grassmann algebra. Higher-order predicates are identified with their extensions, higher-rank sets. QAND and QOR (the AND and OR ofQ) are naturally noncommutative but reduce to the commutative lattice operations for the first-order part of the language. We form higher-order predicates and sets by a setting operator similar to Peano'st that forms a simple extensort = }} from any extensor. In a note added in proof, we correctQ so that a bond like {{, }} between two fermions and is a quasiboson, as the application to lattice chromodynamics strongly suggests.     

Paraconsistent quantum logics

Paraconsistent quantum logics are weak forms of quantum logic, where the noncontradiction and the excluded-middle laws are violated. These logics find interesting applications in the operational approach to quantum mechanics. In this paper, we present an axiomatization, a Kripke-style, and an algebraic semantical characterization for two forms of paraconsistent quantum logic. Further developments are contained in Giuntini and Greuling's paper in this issue.     

Unsharp quantum logics

The event-structure of a state-event system, containing unsharp elements, can be described either as aregular involutive bounded poset, or alternatively as anunsharp orthoalgebra (called alsodifference poset oreffect algebra). Such structures give rise to different forms ofunsharp quantum logics.     

h-Fuzzy quantum logics

Families of fuzzy subsets equipped by continuous fuzzy connectives which are quantum logics in a traditional sense are studied. As a special case, we obtain a generalized fuzzy quantum logic introduced recently by Pykacz.     

Axiomatization of quantum logics

Starting with a quantum logic (a -orthomodular poset)L, a set of probabilistically motivated axioms is suggested to identifyL with a standard quantum logicL(H) of all closed linear subspaces of a complex, separable, infinite-dimensional Hilbert space. Attention is paid to recent results in this field.     

Automorphisms of quantum logics

Automorphisms of quantum logics are studied. If a quantum logic, i.e. an orthomodular complete lattice of propositions concerning a physical system, is represented as the lattice of all projections in a von Neumann algebra, then each automorphism of the logic can be represented as a Jordan automorphism in the algebra. Groups of transformations of a physical system are represented as groups of ¹-automorphisms in a von Neumann algebra, provided certain continuity conditions are fulfilled.     

Coupling of quantum logics

A quantum logic is a couple (L, M), whereL is a logic andM is a quite full set of states onL. A tensor product in the category of quantum logics is defined and a comparison with the definition of free orthodistributive product of orthomodular σ lattices is given. Several physically important cases are treated.     

Automata simulating quantum logics

The idea of computational complementarity is developed further. A special class of macroscopic automata to imitate quantum and classical systems is described. The simplest automaton imitating a spin-1/2 particle is completely considered.     

Symmetries on quantum logics

We study conditions under which the group of symmetries of a quantum logic is isomorphic to the group of symmetries on certain subsets of the state space of the logic. The notions of Jordan–Hahn decomposition and ultrafulness of the set of states under consideration play a fundamental role in these investigations. They are used to establish a connection between the elements of the logic and the weak¹-exposed points or extreme points of the unit interval of the Banach dual of the signed state space. The results are then interpreted in the standard logic of quantum mechanics.     

Symmetries in quantum logics

A symmetry in the quantum logic (L, M) is defined as a pair of bijections :L L andv :M M such that the probabilities are preserved. Some properties of the symmetries are investigated.     

Boolean transfer from coherent quantum logics to quantum logics with continuous superselection rules

Quantum logics with continuous superselection rules are shown to be Booleanvalued coherent quantum logics. Since modern set theory provides a transfer principle from standard mathematics to Boolean-valued mathematics, this makes it possible to transfer automatically well-known results on coherent quantum logics to quantum logics with continuous superselection rules. Many illustrations are given.     

Macroscopic realizations of quantum logics

Summary Two-particle quantum systems with spin can be simulated by classical automata described by graphs. These graphs are associated with nondistributive property lattices of these quantum systems. We emphasize that to non-local properties of a quantum system being in a certain eigenstate of the permutation operator there correspond merely some additional vertices in the graph which have nothing nonlocal in their nature. This leads to the possibility of violating Bell's inequalities in classical systems described by graphs (see Section 6) without violating relativity theory.The subjective interpretation of quantum mechanics of von Neumann, London, and Bauer can be connected with the Boolean nature of mind grasping the non-Boolean nature of the world, which results in the projection postulate: wave packet reduction. A simple example is given for a two-particle system with spin.     

Convexity and finite quantum logics

The notion of a superposition of a set of states and that of a Jauch-Piron state are geometrically interpreted in the context of the facial structure of the state space of a finite quantum logic.     

Probabilistic forcing in quantum logics

It is shown that an orthomodular lattice can be axiomatized as an ortholattice with aunique operation of identity (bi-implication) instead of the operation of implication, and a corresponding algebraic unified quantum logic is formulated. A statisticalyes-no physical interpretation of the quantum logical propositions is then provided to establish a support for a novelyes-no representation of quantum logic which prompts a conjecture about a possible completion of quantum logic by means of probabilistic forcing.     

Irreversible evolution in quantum logics

The notion of dynamical semigroup is introduced in the quantum logic scheme on the set of the states. Under suitable nonempty mathematical assumptions it is shown that a Heisenberg picture exists equivalent to the Schrödinger one and having many aspects similar to those of the Hilbert case.     

Modular almost orthogonal quantum logics

Almost orthogonal quantum logics, i.e., atomic orthomodular lattices in which to every atom there exist only finitely many nonorthogonal atoms, are studied. It is shown that an almost orthogonal quantum logic is modular if and only if it has the exchange property if and only if it can be embedded into a direct product of finite modular quantum logics. The class of almost orthogonal modular OMLs is the largest subclass of the class of atomic modular OMLs in which the conditions commutator-finite and block-finite are equivalent. A finite faithful valuation on an almost orthogonal quantum logicL exists if and only ifL is modular and the set of all atoms ofL is at most countable.     

Stochastic processes on quantum logics

Stochastic processes on quantum logics are defined and the properties of a Brownian motion process are studied. A stochastic integral with respect to this Brownian motion process is constructed.