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Journal of Mathematical Physics

On the coupling of logics

Antonio Zecca A

Published Jun 1, 1978• 19 citations
Publisher

Abstract

A scheme for the coupling of physical systems is proposed within an axiomatic approach to quantum mechanics.
DOI
10.1063/1.523816

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In this work we present an intuitive construction of the quantum logical axiomatic system provided by George Mackey. The goal of this work is a detailed discussion of the results from the paper 'Physical justification for using the tensor product to describe two quantum systems as one joint system' [1] published by Diederik Aerts and Ingrid Daubechies. This means that we want to show how certain composed physical systems from classical and quantum mechanics should be described logically. To reach this goal, we will, like in [1], discuss a special class of axiomatically defined composed physical systems. With the help of certain results from lattice and c-morphism theory (see [2] and [23]), we will present a detailed proof of the statement, that in the quantum mechanical case, a composed physical system must be described via a tensor product space.
The logic approach to axiomatic quantum mechanics via orthocomplemented partial ordered sets of yes–no measurements, which constitute the observing part of a concretely realizable experiment on microworld, has been criticized from the empirical point of view by Mielnik, which on the contrary privileges the convex scheme linked to the preparing part. In this work we do assume that a description of quantum phenomenology must take into account both these two parts in which every elementary experiment can be decomposed. According to this predecision, we develop an axiomatic approach based on indistinguishability principles of a quantum information system. The very general concept of yes–no measurement or ‘‘question’’ is accepted, and then the set of all questions is classified according to the behavior with respect to a phenomenological orthogonality relation. In particular, we single out the set F of fuzzy events or effects and the set E⊆F of exact events. The Mielnik critique is then refused since it regards the order structure of E using counterexamples which pertain to F/E. The notions of physical property and noperty are then introduced and an axiomatic foundation of quantum mechanics based on a pre-Hilbert space is discussed.