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. 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”.
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Notes
The slogan “Follow the money” means that finding the source of an organization’s or person’s money may reveal their true nature. In a similar sense, we use the slogan “Follow the math!” to mean that finding “where the mathematics of QM comes from” reveals a good deal about the key concepts and machinery of the theory.
Since our purpose is conceptual clarity, not mathematical generality, we will stick to the finite sets and dimensions throughout.
Many of the older texts [5] presented the “lattice of partitions” upside down, i.e., with the opposite partial order, so the join and meet as well as the top and bottom were interchanged.
In his sympathetic interpretation of Aristotle’s treatment of substance and form, Heisenberg refers to the substance as: “a kind of indefinite corporeal substratum, embodying the possibility of passing over into actuality by means of the form” [26, p. 148]. Heisenberg’s “potentiality” “passing over into actuality by means of the form” should be seen as the actual indefinite “passing over into” the actual definite by being objectively in-formed through the making of distinctions.
The indefiniteness interpretation of qubit \(a\left| 0\right\rangle +b\left| 1\right\rangle\) is more routine in quantum information and computation theory [33] as opposed to the being simultaneously \(\left| 0\right\rangle\) and \(\left| 1\right\rangle\) version of superposition.
There is at least an analogy between superposition in QM and abstraction in mathematics. In Frege’s example of a set of parallel directed line segments oriented in the same way, the abstraction “direction” is definite on what is common between the lines and indefinite on how they differ [18]. In QM, the emphasis is on the indefiniteness between the definite eigenstates (glass half-empty) while in abstraction, the emphasis is on the common definiteness between the instances (glass half-full).
There is the linearization map (functor) which takes a set U to the vector space \(\mathbb {C} ^{U}\) where u lifts to the basis vector \(\delta _{u}=\chi _{\left\{ u\right\} }:U\rightarrow \mathbb {C}\), but we apply the Yoga to many different concepts.
Heisenberg’s German word was “Unbestimmtheit” which could well be translated as “indefiniteness” or “indeterminacy” rather than ”uncertainty”.
This is particularly clear in the pedagogical model of the double-slit experiment in QM over \(\mathbb {Z} _{2}\) [16] where the interference in the evolving 0, 1-vectors has no resemblence to waves.
In the general vector space case, for two commuting (diagonalizable) operators, \(FG=GF\), if both are represented in a basis of F-eigenvectors and \(\left( u_{i},u_{k}\right)\) is a qudit of F, then \(G_{ik}=0\) [44, p. 4].
The Yoga is used to generate vector space concepts corresponding to set concepts. There is no implication that every instance of a vector space concept, e.g., a commuting F, must come from an instance of the set concept, e.g., a commuting f..
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Ellerman, D. Follow the Math!: The Mathematics of Quantum Mechanics as the Mathematics of Set Partitions Linearized to (Hilbert) Vector Spaces. Found Phys 52, 100 (2022). https://doi.org/10.1007/s10701-022-00608-3
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DOI: https://doi.org/10.1007/s10701-022-00608-3