Follow the Math!: The mathematics of quantum mechanics as the mathematics of set partitions linearized to (Hilbert) vector spaces

DOI10.1007/S10701-022-00608-3zbMATH Open1510.81010arXiv2208.00384WikidataQ114226451 ScholiaQ114226451MaRDI QIDQ6406567

Publication date: 31 July 2022

Abstract: The purpose of this paper is to show that the mathematics of quantum mechanics (QM) is the mathematics of set partitions (which specify indefiniteness and definiteness) linearized to vector spaces, particularly in Hilbert spaces. That is, the math of QM is the Hilbert space version of the math to describe objective indefiniteness that at the set level is the math of partitions. The key analytical concepts are definiteness versus indefiniteness, distinctions versus indistinctions, and distinguishability versus indistinguishability. The key machinery to go from indefinite to more definite states is the partition join operation at the set level that prefigures at the quantum level projective measurement as well as the formation of maximally-definite state descriptions by Dirac's Complete Sets of Commuting Operators (CSCOs). This development is measured quantitatively by logical entropy at the set level and by quantum logical entropy at the quantum level. This follow-the-math approach supports the Literal Interpretation of QM--as advocated by Abner Shimony among others which sees a reality of objective indefiniteness that is quite different from the common sense and classical view of reality as being "definite all the way down."

Mathematics Subject Classification ID

Hilbert and pre-Hilbert spaces: geometry and topology (including spaces with semidefinite inner product) (46C05) General and philosophical questions in quantum theory (81P05) Quantum measurement theory, state operations, state preparations (81P15) Logical foundations of quantum mechanics; quantum logic (quantum-theoretic aspects) (81P10) Measures of information, entropy (94A17) Quantum state spaces, operational and probabilistic concepts (81P16)

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