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On the covering property in physical theories

Ivanov, Al.

Published May 16, 1994• 0 citations
Publisher

Abstract

A class of quantum logics (orthomodular atomic lattices) is defined and studied. The obtained results suggest that the covering property acts as a selection criterion for theories in the sense that, given a set of orthomodular atomic lattices, those which may be theories for a fixed set of systems may be selected by verifyng if they have the covering property or not. A relation between physical meanings of the covering property, lattice - algebraic operations and the commutativity relation is also suggested

DOI
10.5169/seals-116642

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A class of quantum logics (orthomodular atomic lattices) is defined and studied. The obtained results suggest that the covering property acts as a selection criterion for theories in the sense that, given a set of orthomodular atomic lattices, those which may be theories for a fixed set of systems may be selected by verifyng if they have the covering property or not. A relation between physical meanings of the covering property, lattice - algebraic operations and the commutativity relation is also suggested
The classical theories, represented by a special type of Boolean orthomodular lattices, are defined. By using the notion of classical theory, several reasons are given, permitting to assume that any orthomodular lattice which may be considered as a physical theory is «constructed» from a set of «classical components». This assumption leads to an interpretation of the atomicity of physical theories/orthomodular lattices. Arguments for the existence of nonclassical theories are examined by employing nonclassical observables, which are also defined in the paper
In the logico-algebraic foundation of quantum mechanics one often deals with the orthomodular lattices (OML) which enjoy state-separating properties of noncompatible pairs (see e.g. [18], [9] and [15]). These properties usually guarantee reasonable “richness” of the state space—an assumption needed in developing the theory of quantum logics. In this note we consider these classes of OMLs from the universal algebra standpoint, showing, as the main result, that these classes form quasivarieties. We also illustrate by examples that these classes may (and need not) be varieties. The results supplement the research carried on in [1], [3], [4], [5], [6], [11], [12], [13] and [16].
The paper is the contribution to quantum toposophy focusing on the abstract orthomodular structures (following Dunn-Moss-Wang terminology). Early quantum toposophical approach to "abstract quantum logic" was proposed based on the topos of functors [E,Sets]\mathsf{[E,Sets]}[E,Sets] where E\mathsf{E}E is a so-called orthomodular preorder category – a modification of categorically rewritten orthomodular lattice (taking into account that like any lattice it will be a finite co-complete preorder category). In the paper another kind of categorical semantics of quantum logic is discussed which is based on the modification of the topos construction itself – so called quantosquantosquantos – which would be evaluated as a non-classical modification of topos with some extra structure allowing to take into consideration the peculiarity of negation in orthomodular quantum logic. The algebra of subobjects of quantos is not the Heyting algebra but an orthomodular lattice. Quantoses might be apprehended as an abstract reflection of Landsman's proposal of "Bohrification", i.e., the mathematical interpretation of Bohr's classical concepts by commutative C∗C^*C∗-algebras, which in turn are studied in their quantum habitat of noncommutative C∗C^*C∗-algebras – more fundamental structures than commutative C∗C^*C∗-algebras. The Bohrification suggests that topos-theoretic approach also should be modified. Since topos by its nature is an intuitionistic construction then Bohrification in abstract case should be transformed in an application of categorical structure based on an orthomodular lattice which is more general construction than Heyting algebra – orthomodular lattices are non-distributive while Heyting algebras are distributive ones. Toposes thus should be studied in their quantum habitat of "orthomodular" categories i.e. of quntoses. Also an interpretation of some well-known systems of orthomodular quantum logic in quantos of functors [E,QSets]\mathsf{[E,QSets]}[E,QSets] is constructed where QSets\mathsf{QSets}QSets is a quantos (not a topos) of quantum sets. The completeness of those systems in respect to the semantics proposed is proved.