Regularity in Quantum Logic

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Abstract

In 1996, Harding showed that the binary decompositions of any algebraic, relational, or topological structure X form an orthomodular poset Fact X. Here, we begin an investigation of the structural properties of such orthomodular posets of decompositions. We show that a finite set S of binary decompositions in Fact X is compatible if and only if all the binary decompositions in S can be built from a common n-ary decomposition of X. This characterization of compatibility is used to show that for any algebraic, relational, or topological structure X, the orthomodular poset Fact X is regular. Special cases of this result include the known facts that the orthomodular posets of splitting subspaces of an inner product space are regular, and that the orthomodular posets constructed from the idempotents of a ring are regular. This result also establishes the regularity of the orthomodular posets that Mushtari constructs from bounded modular lattices, the orthomodular posets one constructs from the subgroups of a group, and the orthomodular posets one constructs from a normed group with operators. Moreover, all these orthomodular posets are regular for the same reason. The characterization of compatibility is also used to show that for any structure X, the finite Boolean subalgebras of Fact X correspond to finitary direct product decompositions of the structure X. For algebraic and relational structures X, this result is extended to show that the Boolean subalgebras of Fact X correspond to representations of the structure X as the global sections of a sheaf of structures over a Boolean space. The above results can be given a physical interpretation as well. Assume that the true or false questions 𝒬 of a quantum mechanical system correspond to binary direct product decompositions of the state space of the system, as is the case with the usual von Neumann interpretation of quantum mechanics. Suppose S is a subset of 𝒬. Then a necessary and sufficient condition that all questions in S can be answered simultaneously is that any two questions in S can be answered simultaneously. Thus, regularity in quantum mechanics follows from the assumption that questions correspond to decompositions.

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