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.     

Model-Theoretic Investigations into Consequence Operation (Cn) in Quantum Logics: An Algebraic Approach

In this paper, we present the fundamentals of the so-called algebraic approach to propositional quantum logics. We define the set of formulae describing quantum reality as a free algebra freely generated by the set of quantum proportional variables. We define the general notion of logic as a structural consequence operation. Next, we introduce the concept of logical matrices understood as a model of quantum logics. We give the definitions of two quantum consequence operations defined in these models.     

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.     

Q an algebraic language for quantum-spacetime topology

We propose a third physical logics. The first was classical (C) logics with commutative distributive AND and OR. The second is commutative quantum (CQ) logics, with commutative nondistributive AND and OR. The third logics, Q, has noncommutative nondistributive AND and OR; QCQC. Q predicates are the rays in a Grassmann double algebra of extensors, where CQ predicates are the subspaces of a Hubert space. The AND and OR of Q are projectively represented by Grassmann's progressive and regressive products. Q supports a quantum set theory appropriate to quantum topology. Here Q is applied to a toy theory of the topology of time. It preserves translational invariance and replaces singular delta-function propagators by finite Gaussians.     

Selection maps for quantum logics: applications to the classification of elementary particles

The observed properties of certain groups of elementary particle states are used to construct quantum logics whose pure states correspond to those of the given particles. The pure states can be made into an orthogonality space and the observed properties are representable as frame functions on this structure.This technique generalizes aspects of the existing methods of classification and provides examples of nonstandard logics in physics.     

Quantum logic,state space geometry and operator algebras

The problem of characterising those quantum logics which can be identified with the lattice of projections in a JBW-algebra or a von Neumann algebra is considered. For quantum logics which satisfy the countable chain condition and which have no TypeI ₂ part, a characterisation in terms of geometric properties of the quantum state space is given.     

Solving Problems on Finite Concrete Logics with the Help of a PC

Three routines which may be used in the computertreatment of general finite concrete logics aredescribed. The results for some particular finiteconcrete logics are presented.     

Fuzzy set ideas in quantum logics

Fuzzy set theory language and ideas are used to express basic quantum logic notions. The possibility of replacing probabilistic interpretation of quantum mechanics by interpretation based on infinite-valued logics and fuzzy set theory is outlined. Short review of various structures encountered in the fuzzy set approach to quantum logics is given.     

Quantum Logic for Quantum Computers

The following results obtained within a project of finding the algebra of statesin a general-purpose quantum computer are reported: (1) All operations of anorthomodular lattice, including the identity, are fivefold-defined; (2) there arenonorthomodular models for both quantum and classical logics; (3) there is afour-variable orthoarguesian lattice condition which contains all known orthoarguesianlattice conditions including six- and five-variable ones. Repercussions to quantumcomputers operating as quantum simulators are discussed.     

Universal diffusive decay of correlations in gapped one-dimensional systems

We apply a semiclassical approach to express finite temperature dynamical correlation functions of gapped spin models analytically. We show that the approach of [á. Rapp, G. Zaránd, Phys. Rev. B 74, 014433 (2006)] can also be used for the S = 1 antiferromagnetic Heisenberg chain, whose lineshape can be measured experimentally. We generalize our calculations to O(N) quantum spin models and the sine-Gordon model in one dimension, and show that in all these models, the finite temperature decay of certain correlation functions is characterized by the same universal semiclassical relaxation function.     

Connections among quantum logics. Part 1. Quantum propositional logics

In this paper, we propose a theory of quantum logics which is general enough to enable us to reexamine previous work on quantum logics in the context of this theory. It is then easy to assess the differences between the different systems studied. The quantum logical systems which we incorporate are divided into two groups which we call quantum propositional logics and quantum event logics. We include the work of Kochen and Specker (partial Boolean algebras), Greechie and Gudder (orthomodular partially ordered sets), Domotar (quantum mechanical systems), and Foulis and Randall (operational logics) in quantum propositional logics; and Abbott (semi-Boolean algebras) and Foulis and Randall (manuals) in quantum event logics. In this part of the paper, we develop an axiom system for quantum propositional logics and examine the above structures in the context of this system.     

Connections among quantum logics. Part 2. Quantum event logics

This paper gives a brief introduction to the major areas of work in quantum event logics: manuals (Foulis and Randall) and semi-Boolean algebras (Abbott). The two theories are compared, and the connection between quantum event logics and quantum propositional logics is made explicit. In addition, the work on manuals provides us with many examples of results stated in Part I.     

Measures Defined on Quantum Logics of Sets

We study families formed with subsets of any set X which are quantum logics but which are not Boolean algebras. We consider sequences of measures defined on a sets quantum logics and valued on an effect algebra and obtain a sufficient condition for a sequences of such measures to be uniformly strongly additive with respect to order topology of effect algebras.     

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.     

Fermionic Representation of Two-Dimensional Dimer Models

Dimer models in two dimensions give rise to well-known statistical lattice problems, which can be exactly solved by the same combinatorial techniques associated with the Ising model and which have been used to account for the phase transitions in a number of physically interesting systems. More recently, dimer models have been regarded as classical limits of the quantum ground state of some antiferromagnetic systems. We then revisit an early transfer-matrix calculation for the dimer model on the simple square lattice. We write a spin representation for the transfer matrix associated with the canonical partition function of two paradigmatic dimers models, on the 4–8 lattice, with an Ising-type transition, and on the brick lattice, with a peculiar commensurate–incommensurate transition. Using standard techniques, the problem is reduced to the calculation of the eigenvalues of a system of free fermions.     

Finite element approximation in quantum field theory

Lattices are conventionally used to regulate quantum field theory. However, lattices have a number of drawbacks, one of the most noteworthy being the awkwardness which arises in treating the first difference operator associated with the fermion kinetic energy. We propose here an alternative to the conventional lattice formulation of quantum field theories. This alternative is based on the method of finite elements, which has been a powerful tool in continuum mechanics and fluid dynamics for many years. We believe that this method may be advantageous in quantum field theory because it treats bosons and fermions on a nearly equal footing. This is possible because in the finite element method one does not introduce a finite difference approximation for derivatives. In this paper, we adapt the finite element technique to quantum field theory, and apply it to several elementary field theory models.     

Three-valued derived logics for classical phase spaces

Reichenbach proposed a three-valued logic to describe quantum mechanics. In his development, Reichenbach presented three different negation operators without providing any criteria for choosing among them. In this paper we develop two three-valuedderived logics for classical systems. These logics are derived in that they are based on a theory of physical measurement. In this regard they have some of the characteristics of the quantum logic developed by Birkhoff and von Neumann. The theory of measurement used in the present development is the one used previously in developingbivalent derived logics for classical systems. As these systems are derived logics, many of the ambiguities possessed by systems such as Reichenbach's are avoided.     

Order Topology Orthosummability in Quantum Logics

By using the Antosik–Mikusinski infinite matrix convergence theorem in quantum logics, we prove a theorem on orthosummability with respect to order topology in quantum logics.