Follow the Math!: The Mathematics of Quantum Mechanics as the Mathematics of Set Partitions Linearized to (Hilbert) Vector Spaces

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.

A New Approach to Understanding Quantum Mechanics: Illustrated Using a Pedagogical Model over ℤ2

“…Our simplified model of QM is based on set notions and, where possible, the set notions connected by the Yoga to the vector spaces ℘(U) over Z 2 . When the vector space V is a finite-dimensional Hilbert vector space over C, then the Yoga shows how the machinery in the simplified model corresponds to the full-blown mathematical machinery of QM [1]. But when V = Z n 2 , then only a characteristic function χ S : U → {0, 1} defines a linear operator P [S] : Z n 2 → Z n 2 , but a general numerical attribute f : U → R still defines a partition f −1 on U and the DSD ℘ f −1 (r) r∈ f (U) of Z n 2 .…”

“…That movement from an indefinite state to a more definite state, like the arrow in Figure 3 is the skeletal representation of the infamous quantum jump in full QM. It is easily shown in the general case, [1], that:…”

“…Case 1. With detection at the slits, the superposition state {a, c} is reduced to {a} (i.e., going through Slit 1) with probability 1 2 or to {c} (i.e., going through Slit 2) with probability 1 2 so we have:…”

“…A treatment of Bell's Theorem [33] intended for a popular audience can be reworked into QM/Sets. Still more applications include the treatment of indistinguishable particles and group representation theory [1]. But even this introduction to QM/Sets raises some interesting questions that go far beyond its pedagogical use.…”

“…A new approach to understanding quantum mechanics (QM) has been developed elsewhere [1,2] that corroborates an interpretation of QM supported by Werner Heisenberg [3], Abner Shimony [4], R. I. G. Hughes [5], Gregg Jaeger [6], and many others who consider quantum reality as involving indefinite, blurred, unsharp, smeared, or indeterminate states. The new approach is based on showing that the distinctive mathematical formalism in QM is the linearization of the mathematics of partitions (or equivalence relations) on a set-which is the set-level mathematics to represent indistinctions (equivalences) and distinctions (inequivalences).…”