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Classical theories, atomicity and nonclassical theories

Ivanov, Al.

Published Aug 5, 1992• 0 citations
Publisher

Abstract

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

DOI
10.5169/seals-116507

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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
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
This work employs the empirical logic approach of D. Foulis and C. Randall to compare non-relativistic, conservative, non-compound physical systems (universes) in classical and quantum mechanics. We present definitions of test spaces, propositions, proposition logics, statistical states, observables, and dynamical systems. Then we try to identify their mathematical representatives for both, classical and quantum universes. Except for the case of the dynamical system, which is not defined for classical universes, the definitions prove to be general enough to apply to both cases. Necessary mathematical tools not generally known to physicists are also treated in a concise fashion. These include some basic theory of partially ordered sets (posets) and lattices, in particular boolean and orthomodular lattices.
This paper deals with the foundations of quantum mechanics. We start by outlining the characterisation, due to Birkhoff and Von Neumann, of the logical structures of the theories of classical physics and quantum mechanics, as boolean and modular lattices respectively. We then derive these descriptions from what we claim are basic properties of any physical theory - i.e. the notion that a quantity in such a theory may be analysed into parts and that the results of this analysis may be treated in languages with an underlying boolean structure. We shall see that in the course of constructing a model of a theory with these properties different indistinguishable possibilities will arise for how the elements of the model may be named, that is to say different possibilities arise for how they can be associated with points from Set. Taking a particular collection of possibilities gives the usual boolean lattice of the propositions of classical physics. Taking all possibilities - in a sense, the set of all things that may be described by physical theories - gives the lattice of quantum mechanical propositions. This gives an interpretation of quantum mechanics as the complete set of such possible descriptions, the complete physical description of the world.
We analyze the difference between classical mechanics and quantum mechanics. We come to the conclusion that this difference can be found in the nature of the observables that are considered for the physical system under consideration. Classical mechanics can only describe a certain kind of what we called ‘‘classical observable.’’ Quantum mechanics can only describe another kind of observable; it cannot describe, however, classical observables. To perform this analysis, we use a theory where every kind of observable can be treated and which is in a natural way a generalization of both classical and quantum mechanics. If in a study of a physical system in this theory we restrict ourselves to the classical observables, we rediscover classical mechanics as a kind of first study of the physical system, where all the nonclassical properties are hidden. If we find that this first study is too rough we can also study the nonclassical part of the physical system by a theory which is eventually quantum mechanics.
The purpose of this paper is to prove that the construction of a quantum logic is a task of specific interest for non-classical logic, but that it is relevant neither to the methodology of quantum mechanics nor to the formalisation of the alternative yes-no process. The main argument is that quantum structures (for example, orthocomplemented partially ordered sets, orthomodular lattices and partial algebras) are not systems of logic proper, but rather are perfectly formalizable structures in classical logic. In § 1, the orthodox formulation of Jauch and Piron are analyzed. I affirm that the objections made by these authors against “quantum logic” are not eliminated by the introduction of a conditional in the lattice structures, since the true reason why quantum “logic” is not really logic lies in its formulation, which is not that of a propositional system, but rather that of a specific theory whose model is these lattices. It is argued that the collection of the properties satisfied by certain objects of a structure does not justify their presentation as purely logical properties, nor does the presence of certain operations of the least upper bound and greatest lower bound permit us to call symbols introduced by analogy in propositionaI logic “connectives”. In any case, if that formalization is indeed possible, it does not imply the adoption of a new logic as a formal instrument for quantum mechanics. In § 2, aspects of quantum theories which are especially semantic in nature are analyzed. I agree with van Fraassen’s statement in the sense that that semantic formulation makes specific concepts of the theory clearer, but it does not attempt to construct its own logic. I deduce that it is non-classical logic which can actually take advantage of quantum semantics, as a model which, eventually, could throw light on the metalogical properties of certain special propositional systems. Finally, by using Benioff’s profound set theory analysis, I advance the hypothesis that classical logic is possibly the only one that can account for the richness of the meanings that appear in quantum mechanics.[Summary by Carlos Lungarzo]