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Linearity and Compound Physical Systems: The Case of Two Separated Spin 1/2 Entities

Diederik Aerts, Frank Valckenborgh

Published Jan 1, 2002• 4 citations
OpenAlex

Abstract

We illustrate some problems that are related to the existence of an underlying linear structure at the level of the property lattice associated with a physical system, for the particular case of two explicitly separated spin 1/2 objects that are considered, and mathematically described, as one compound system. It is shown that the separated product of the property lattices corresponding with the two spin 1/2 objects does not have an underlying linear structure, although the property lattices associated with the subobjects in isolation manifestly do. This is related at a fundamental level to the fact that separated products do not behave well with respect to the covering law (and orthomodularity) of elementary lattice theory. In addition, we discuss the orthogonality relation associated with the separated product in general and consider the related problem of the behavior of the corresponding Sasaki projections as partial state space mappings

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We illustrate some problems that are related to the existence of an underlying linear structure at the level of the property lattice associated with a physical system, for the particular case of two explicitly separated spin 1/2 objects that are considered, and mathematically described, as one compound system. It is shown that the separated product of the property lattices corresponding with the two spin 1/2 objects does not have an underlying linear structure, although the property lattices associated with the subobjects in isolation manifestly do. This is related at a fundamental level to the fact that separated products do not behave well with respect to the covering law (and orthomodularity) of elementary lattice theory. In addition, we discuss the orthogonality relation associated with the separated product in general and consider the related problem of the behavior of the corresponding Sasaki projections as partial state space mappings
We illustrate some problems that are related to the existence of an underlying linear structure at the level of the property lattice associated with a physical system, for the particular case of two explicitly separated spin 1/2 objects that are considered, and mathematically described, as one compound system. It is shown that the separated product of the property lattices corresponding with the two spin 1/2 objects does not have an underlying linear structure, although the property lattices associated with the subobjects in isolation manifestly do. This is related at a fundamental level to the fact that separated products do not behave well with respect to the covering law (and orthomodularity) of elementary lattice theory. In addition, we discuss the orthogonality relation associated with the separated product in general and consider the related problem of the behavior of the corresponding Sasaki projections as partial state space mappings
We prove that a theory that is different from a classical theory and has the property that orthogonal states of a physical system are compatible states (a property called orthomodularity in axiomatics) cannot describe two separated entities. Quantum mechanics has this property and as a consequence it is impossible to describe two separated entities in quantum mechanics. This is essentially due to the fact that quantum mechanics is too specific a theory. We indicate the direction for a generalization where the problem could be solved.
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. In their 1936 founding paper on quantum logic, Birkhoff and von Neumann postulated that the lattice describing the experimental propositions concerning a quantum system is orthocomplemented. We prove that this postulate fails for the lattice Lsep describing a compound system consisting of so called separated quantum systems. By separated we mean two systems prepared in different “rooms ” of the lab, and before any interaction takes place. In that case the state of the compound system is necessarily a product state. As a consequence, Dirac’s superposition principle fails, and therefore Lsep cannot satisfy all Piron’s axioms. In previous works, assuming that Lsep is orthocomplemented, it was argued that Lsep is not orthomodular and fails to have the covering property. Here we prove that Lsep cannot admit and orthocomplementation. Moreover, we propose a natural model for Lsep which has the covering property. PACS numbers: 03.65.Ta, 03.65.Ca 1.
We construct the tensor product for the property lattices of two entities. We give a physical interpretation for this tensor product. We show that the tensor product is never an orthocomplemented lattice if both entities are nontrivial. It also never satisfies the covering law. This is the reason why this tensor product does not exist in quantum mechanics and does not exist in quantum logic. We analyze the relation of the tensor product with the usual description of two entities in quantum mechanics. At the end, we give a mathematical way of constructing this tensor product.
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\mathrel \DeltaΔ and a binary operator d(x,y)d(x,y)d(x,y) that are generalizations of the binary relation C\textrm{C}C and the commutator c(x,y)c(x,y)c(x,y), respectively, known for orthomodular lattices. We characterize orthomodular posets among orthogonal posets. Moreover, we describe connections between the relations Δ\mathrel \DeltaΔ and ↔\leftrightarrow↔ (the latter was introduced by P. Pták and S. Pulmannová) and the operator d(x,y)d(x,y)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\DeltaΔ-blocks with respect to Boolean subalgebras and distributive subposets they include.