Coreflections in Algebraic Quantum Logic

Various generalizations of Boolean algebras are being studied in algebraic quantum logic, including orthomodular lattices, orthomodular po-sets, orthoalgebras and effect algebras. This paper contains a systematic study of the structure in and between categories of such algebras. It does so via a combination of totalization (of partially defined operations) and transfer of structure via coreflections.     

Type-Decomposition of an Effect Algebra

Effect algebras (EAs), play a significant role in quantum logic, are featured in the theory of partially ordered Abelian groups, and generalize orthoalgebras, MV-algebras, orthomodular posets, orthomodular lattices, modular ortholattices, and boolean algebras. We study centrally orthocomplete effect algebras (COEAs), i.e., EAs satisfying the condition that every family of elements that is dominated by an orthogonal family of central elements has a supremum. For COEAs, we introduce a general notion of decomposition into types; prove that a COEA factors uniquely as a direct sum of types I, II, and III; and obtain a generalization for COEAs of Ramsay’s fourfold decomposition of a complete orthomodular lattice.     

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.     

Small Quantum Structures with Small State Spaces

We summarize and extend results about “small” quantum structures with small dimensions of state spaces. These constructions have contributed to the theory of orthomodular lattices. More general quantum structures (orthomodular posets, orthoalgebras, and effect algebras) admit sometimes simplifications, but there are problems where no progress has been achieved.     

Orthosummable orthoalgebras

We introduce notions of orthosummability and-orthosummability for orthoalgebras, which generalize the notions of orthocompleteness and-orthocompleteness for orthomodular posets, and we characterize such orthoalgebras in terms of their chains. We also show how to sum an infinite subset of an orthoalgebra, and we prove a generalized associative law for such sums.     

Quantum Physics and Classical Physics—In the Light of Quantum Logic

In contrast to the Copenhagen interpretation we consider quantum mechanics as universally valid and query whether classical physics is really intuitive and plausible. We discuss these problems within the quantum logic approach to quantum mechanics where the classical ontology is relaxed by reducing metaphysical hypotheses. On the basis of this weak ontology a formal logic of quantum physics can be established which is given by an orthomodular lattice. By means of the Solèr condition and Piron's result one obtains the classical Hilbert spaces. However, this approach is not fully convincing. There is no plausible justification of Solèr's law and the quantum ontology is partly too weak and partly too strong. We propose to replace this ontology by an ontology of unsharp properties and conclude that quantum mechanics is more intuitive than classical mechanics and that classical mechanics is not the macroscopic limit of quantum mechanics.     

Atomic varieties of sets with relative inverses

The generalization of the construction of the lattice of varieties for partial algebras is used for sets with relative inverses. There are many quantum structures representable by sets with relative inverses (orthomodular lattices, orthoalgebras, D-posets, test spaces,...). Varieties covering the trivial variety are investigated for the case of closed (strongest type) subalgebras and closed homomorphisms. Some similar results for weaker types are given. The context with set representation problems is considered for the set-theoretic difference operations.     

Effect algebras and unsharp quantum logics

The effects in a quantum-mechanical system form a partial algebra and a partially ordered set which is the prototypical example of the effect algebras discussed in this paper. The relationships among effect algebras and such structures as orthoalgebras and orthomodular posets are investigated, as are morphisms and group- valued measures (or charges) on effect algebras. It is proved that there is a universal group for every effect algebra, as well as a universal vector space over an arbitrary field.     

Quantum Set Theory

The complete orthomodular lattice of closed subspaces of a Hilbert space is considered as the logic describing a quantum physical system, and called a quantum logic. G. Takeuti developed a quantum set theory based on the quantum logic. He showed that the real numbers defined in the quantum set theory represent observables in quantum physics. We formulate the quantum set theory by introducing a strong implication corresponding to the lattice order, and represent the basic concepts of quantum physics such as propositions, symmetries, and states in the quantum set theory.     

The logic of quantum mechanics derived from classical general relativity

For the first time it is shown that the logic of quantum mechanics can be derived from classical physics. An orthomodular lattice of propositions characteristic of quantum logic, is constructed for manifolds in Einstein’s theory of general relativity. A particle is modelled by a topologically non-trivial 4-manifold with closed timelike curves—a 4-geon, rather than as an evolving 3-manifold. It is then possible for both the state preparationand measurement apparatus to constrain the results of experiments. It is shown that propositions about the results of measurements can satisfy a non-distributive logic rather than the Boolean logic of classical systems. Reasonable assumptions about the role of the measurement apparatus leads to an orthomodular lattice of propositions characteristic of quantum logic.     

Quantum Logics and Relative Lindenbaum Property

The relative Lindenbaum property is considered in orthomodular quantum logic and in partial class ical logic. The properties of these models are connected with the possibility of hidden-variable reconstruction of particular (1/2-spin particle e.g.) quantum physical systems.     

On the reality of spin and helicity

The possibilities of a realistic interpretation of quantum mechanics are investigated by means of a statistical analysis of experiments performed on the simplest type of quantum systems carrying spin or helicity. To this end, fundamental experiments, some new, for measuring polarization are reviewed and (re)analyzed. Theunsharp reality of spin is essential in the interpretation of some of these experiments and represents a natural motivation for recent generalizations of quantum mechanics to a theory incorporating effect-valued measures as unsharp observables and generalized systems of imprimitivity.     

Deduction, Ordering, and Operations in Quantum Logic

We show that in quantum logic of closed subspaces of Hilbert space one cannot substitute quantum operations for classical (standard Hilbert space) ones and treat them as primitive operations. We consider two possible ways of such a substitution and arrive at operation algebras that are not lattices what proves the claim. We devise algorithms and programs which write down any two-variable expression in an orthomodular lattice by means of classical and quantum operations in an identical form. Our results show that lattice structure and classical operations uniquely determine quantum logic underlying Hilbert space. As a consequence of our result, recent proposals for a deduction theorem with quantum operations in an orthomodular lattice as well as a, substitution of quantum operations for the usual standard Hilbert space ones in quantum logic prove to be misleading. Quantum computer quantum logic is also discussed.     

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.     

Nonordered quantum logic and its YES-NO representation

It is shown that an orthomodular lattice is an ortholattice in which aunique operation of bi-implication corresponds to the equality relation and that the ordering relation in the binary formulation of quantum logic as well as the operation of implication (conditional) in quantum logic are completely irrelevant for their axiomatization. The soundness and completeness theorems for the corresponding algebraic unified quantum logic are proved. A proper semantics, i.e., a representation of quantum logic, is given by means of a new YES-NO relation which might enable a proof of the finite model property and the decidability of quantum logic. A statistical YES-NO physical interpretation of the quantum logical propositions is provided.     

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.     

Probability Weights and Measures on Finite Effect Algebras

We study probability weights and measures onfinite effect algebras, thus generalizing the existingtheory for orthomodular posets and orthoalgebras. Ourdevelopment proceeds somewhat more generally in that we study weights and measures associatedwith an antichain in the positive cone of a euclideanvector space with the standard partialordering.     

Toward a formal language for unsharp properties

Some algebraic structures of the set of all effects are investigated and summarized in the notion of a(weak) orthoalgebra. It is shown that these structures can be embedded in a natural way in lattices, via the so-calledMacNeille completion. These structures serve as a model ofparaconsistent quantum logic, orthologic, andorthomodular quantum logic.     

Map for Simultaneous Measurements for a Quantum Logic

In this paper we will study a function of simultaneous measurements for quantum events (s-map) which will be compared with the conditional states on an orthomodular lattice as a basic structure for quantum logic. We will show the connection between s-map and a conditional state. On the basis of the Rényi approach to the conditioning, conditional states, and the independence of events with respect to a state are discussed. Observe that their relation of independence of events is not more symmetric contrary to the standard probabilistic case. Some illustrative examples are included.     

Automata Theory Based on Quantum Logic. (I)

We present a basic framework of automata theory based on quantum logic. Inparticular, we introduce the orthomodular lattice-valued (quantum) predicate ofrecognizability and establish some of its fundamental properties.