A P-lattice is a σ-complete, orthomodular atomic lattice ℒ which is formed by the set of propositions of a physical system. A composition of physical systems is considered, and some concept of locality in a compound physical system is represented in terms of P-lattices. We give a remark toward necessary and sufficient conditions for it to hold, which have been provided implicitly in antecedent studies, and we show that it can be provided under weaker conditions.
A P-lattice is a σ-complete, orthomodular atomic lattice ℒ which is formed by the set of propositions of a physical system. A composition of physical systems is considered, and some concept of locality in a compound physical system is represented in terms of P-lattices. We give a remark toward necessary and sufficient conditions for it to hold, which have been provided implicitly in antecedent studies, and we show that it can be provided under weaker conditions.
A P-lattice is defined as a σ-complete, orthomodular atomic lattice ℒ which is formed by the set of propositions of a physical system. A composition of physical systems in the framework of P-lattices is considered and some notions of locality are given. It is shown that the following statements about compound systems are equivalent. (a) All atoms of a compound system are reducible to those of its subsystems. (b) All pure states of a compound system are separable into those of its subsystems. (c) A compound system has statistical property independence. (d) At least one of the subsystems is classical. (e) Bell-type inequalities hold.
Abstract States of quantum systems correspond to vectors in a Hilbert space and observations to closed subspaces. Hence, this logic corresponds to the algebra of closed subspaces of a Hilbert space. This can be considered as a complete lattice with orthocomplementation, but it is not distributive. It satisfies a weaker condition, the so-called orthomodularity. Later on, it was recognized that joins in this structure need not exist provided the subspaces are not orthogonal. Hence, the resulting structure need not be a lattice but a so-called orthomodular poset, more generally an orthoposet only. For orthoposets, we introduce a binary relation Δ and a binary operator d(x,y) that are generalizations of the binary relation C and the commutator c(x,y), respectively, known for orthomodular lattices. We characterize orthomodular posets among orthogonal posets. Moreover, we describe connections between the relations Δ and ↔ (the latter was introduced by P. Pták and S. Pulmannová) and the operator d(x,y). In addition, we investigate certain orthomodular posets of subsets of a finite set. In particular, we describe maximal orthomodular sublattices and Boolean subalgebras of such orthomodular posets. Finally, we study properties of Δ-blocks with respect to Boolean subalgebras and distributive subposets they include.
Quantum Logic Distributivity And The Algebraic Structure Of Propositions As with many subatomic particles, the question of under what conditions the electron behaves as a particle or as a wave continues to be the subject of research. It was necessary to employ appropriate logic to organise the propositions describing the behaviour of such an object at the subatomic scale into a precise from and to establish coherent relationships between them. Quantum logic, which can be regarded as an interpretation of quantum mechanics, arose from this need. In classical mechanics, the state and properties of a system are described using the propositions of classical logic, which is based on Boolean algebra. In quantum mechanics, however, the fact that the operators representing the measured quantities do not commute has shown that quantum phenomena cannot be described using the propositions of classical logic. It is therefore necessary to have a formal framework suited to the propositions describing quantum phenomena. In our study, we highlight the various differences between classical logic and quantum logic, examine the distributive property and discuss, from the perspective of Birkhoff and von Neumann, the algebraic structure to which quantum logic might correspond. At the conclusion of our research, we find that, in order to assert that all experimental propositions logically equivalent to a given experimental proposition constitute a physical property, the operators corresponding to the measured quantities must commute with one another. Boolean algebra has an algebraic structure that allows classical logic to be formalised. In such algebra, the distributive property holds; thus, distributivity does not pose a problem for classical logic. Consequently, the distributive property does not always hold in quantum logic and quantum systems can be modelled using orthomodular lattices associated with closed subspaces of Hilbert space. We believe that this model must evolve in parallel with the discovery of new quantum phenomena and that it provides a framework conductive to this evolution. Keywords: Quantum logic; distributivity; partially order sets (posets); lattice; orthomodular lattice; Hilbert space; Boolean algebra.
Given a complete orthocomplemented lattice L and a set S of nonnegative real functions on L, sufficient conditions are established that S should fulfill in order that L be atomic. The conditions are investigated under which L may be represented by the lattice of all closed subspaces of a separable Hilbert space. (As is well known, the atomicity of L plays an important role here.) Some unsolved problems are pointed out. In axiomatic quantum mechanics, the lattice L may represent the set of propositions whereas the set of functions S represents the set of physical states. The conditions imposed on the pair (L,S) then have a simple and plausible physical interpretation; an important condition imposed on (L,S) is the existence of the ``maximal'' (i.e., maximally determined) states which appear in the theory as limit constructions.
Abstract: The concept of fractionability or decomposability in parts of a physical system has its mathematical counterpart in the lattice–theoretic concept of orthomodularity. Systems with a finite number of degrees of freedom can be decomposed in different ways, corresponding to different groupings of the degrees of freedom. The orthomodular structure of these simple systems is trivially manifest. The problem then arises as to whether the same property is shared by physical systems with an infinite number of degrees of freedom, in particular by the quantum relativistic ones. The latter case was approached several years ago by Haag and Schroer (1962; Haag, 1992) who started from noting that the causally complete sets of Minkowski spacetime form an orthomodular lattice and posed the question of whether the subalgebras of local observables, with topological supports on such subsets, form themselves a corresponding orthomodular lattice. Were it so, the way would be paved to interpreting spacetime as an intrinsic property of a local quantum field algebra. Surprisingly enough, however, the hoped property does not hold for local algebras of free fields with superselection rules. The possibility seems to be instead open if the local currents that govern the superselection rules are driven by gauge fields. Thus, in the framework of local quantum physics, the request for algebraic orthomodularity seems to imply physical interactions! Despite its charm, however, such a request appears plagued by ambiguities and criticities that make of it an ill–posed problem. The proposers themselves, indeed, concluded that the orthomodular correspondence hypothesis is too strong for having a chance of being practicable. Thus, neither the idea was taken seriously by the proposers nor further investigated
Due to the failure of the classical principles of bivalence and verifunctionality, the logic of experimental propositions relative to quantum systems cannot be interpreted in Boolean algebras. However, we cannot say neither that this logic is captured by orthomodular lattices, as claimed by many authors along the line of Birkhoff's and von Neumann's standard approach. For the alleged violation of distributivity is based on the possibility of combining statements relative to complementary contexts, which does not refer to any experience and, consequently, has no meaning. Indeed, quantum logic should be interpreted in partial, transitive Boolean algebras whose compatibility relation limits the application of the connectives within each of its Boolean sub-algebras, which refer to partial, classical descriptions. Moreover, this approach of quantum logic makes it possible to deal with composite systems, which was not possible to do within the standard approach, and then to deal with the fundamental notion of quantum entanglement. The latter notion can be represented by a series of axioms of the object language that restrict the set of experimental statements bearing on a composite system, while its close link to the notion of complementarity can be expressed in the metalanguage.