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arXiv:quant-ph/0503083

Decomposition Theorem for State Property Systems

Diederik Aerts, Didier Deses, Bart D’Hooghe

Published Mar 8, 2005• 1 citation
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Abstract

We prove a decomposition theorem for orthocomplemented state property systems. More specifically we prove that an orthocomplemented state property system is isomorphic to the direct union of the non classical components of this state property system over the state space of the classical state property system of this state property system.

DOI
10.48550/arxiv.quant-ph/0503083

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We prove a decomposition theorem for orthocomplemented state property systems. More specifically we prove that an orthocomplemented state property system is isomorphic to the direct union of the non classical components of this state property system over the state space of the classical state property system of this state property system.
In earlier work a description of a physical entity is given by means of a state property system and it is proven that any state property system is equivalent to a closure space. In the present paper we investigate the relations between classical properties and connectedness for closure spaces. The main result is a decomposition theorem, which allows us to split a state property system into a number of 'pure nonclassical state property systems' and a 'totally classical state property system'
In quantum physics the tests of most properties do not have predetermined outcomes. The latter have nevertheless well-defined probabilities of being realized during a test. Following Popper we interpret these probabilities as physical propensities. A first purpose of the present article is to formalize the propensity interpretation in the framework of state-property structures. Next, Gleason’s theorem asserts that in the Hilbert space there exists a unique propensity function (i.e., one probability measure for each state vector); the propensities are thus uniquely determined by the state vector. Conversely, we prove that if the state-property structure admits one and only one propensity function, then the set ℒ of all properties is a complete atomic orthomodular lattice. We point out that according to our assumption the probabilistic aspect of the system is entirely determined by its deterministic aspect. Assuming furthermore that each property can be ideally tested, it follows that ℒ is isomorphic to the direct union of Hilbertian space lattices. We recover thus the purely classical and purely quantum frameworks as the two extreme cases. The intermediate cases correspond to quantum mechanics with—possibly continuous—superselection variables. Finally, we prove that a system is classical, i.e., all properties are mutually compatible, if and only if the propensity function is dispersion free. In our approach the quantum probabilities appear thus as a generalization of classical determinism rather than a generalization of classical probabilities.
Abstract It is well known that the closed subspaces of a Hilbert space form an orthomodular lattice. Elements of this orthomodular lattice are the propositions of a quantum mechanical system represented by the Hilbert space, and by Gleason’s theorem atoms of this orthomodular lattice correspond to pure states of the system. Wigner’s theorem says that the automorphism group of this orthomodular lattice corresponds to the group of unitary and anti-unitary operators of the Hilbert space. This result is of basic importance in the use of group representations in quantum mechanics. The closed subspaces A of a Hilbert space H H{\mathcal H}H correspond to direct product decompositions H ≃ A × A ⊥ H≃A×A⊥\mathcal{H}\simeq A\times A^\perpH≃A×A⊥ of the Hilbert space, a result that lies at the heart of the superposition principle. In [10] it was shown that the direct product decompositions of any set, group, vector space, and topological space form an orthomodular poset. This is the basis for a line of study in foundational quantum mechanics replacing Hilbert spaces with other types of structures. It is the purpose of this note to prove a version of Wigner’s theorem: for an infinite set X, the automorphism group of the orthomodular poset Fact(X) of direct product decompositions of X is isomorphic to the permutation group of X. The structure Fact(X) plays the role for direct product decompositions of a set that the lattice of equivalence relations plays for surjective images of a set. So determining its automorphism group is of interest independent of its application to quantum mechanics. Other properties of Fact(X) are determined in proving our version of Wigner’s theorem, namely that Fact(X) is atomistic in a very strong way.
Purpose To elaborate a theory for modeling concepts that incorporates how a context influences the typicality of a single exemplar and the applicability of a single property of a concept. To investigate the structure of the sets of contexts and properties. Design/methodology/approach The effect of context on the typicality of an exemplar and the applicability of a property is accounted for by introducing the notion of “state of a concept”, and making use of the state‐context‐property formalism (SCOP), a generalization of the quantum formalism, whose basic notions are states, contexts and properties. Findings The paper proves that the set of context and the set of properties of a concept is a complete orthocomplemented lattice, i.e. a set with a partial order relation, such that for each subset there exists a greatest lower bound and a least upper bound, and such that for each element there exists an orthocomplement. This structure describes the “and”, “or”, and “not”, respectively for contexts and properties. It shows that the context lattice as well as the property lattice are non‐classical, i.e. quantum‐like, lattices. Originality/value Although the effect of context on concepts is widely acknowledged, formal mathematical structures of theories that incorporate this effect have not been successful. The study of this formal structure is a preparation for the elaboration of a theory of concepts that allows the description of the combination of concepts.