Quantum mechanics over sets: a pedagogical model with non-commutative finite probability theory as its quantum probability calculus

2021

J Philos Logic

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ion turns equivalence into identity, but there are two ways to do it. Given the equivalence relation of parallelness on lines, the #1 way to turn equivalence into identity by abstraction is to consider equivalence classes of parallel lines. The #2 way is to consider the abstract notion of the direction of parallel lines. This paper developments simple mathematical models of both types of abstraction and shows, for instance, how …nite probability theory can be interpreted using #2 abstracts as "superposition events" in addition to the ordinary events. The goal is to use the second notion of abstraction to shed some light on the notion of an inde…nite superposition in quantum mechanics.

Section: The Connection To Interpreting Symmetry Operationsmentioning

confidence: 99%

Section: From Incidence To Density Matricesmentioning

confidence: 99%

On Abstraction in Mathematics and Indefiniteness in Quantum Mechanics

2021

J Philos Logic

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“…The above machinery from the classical cases formulated over Z n 2 gives a pedagogical (or "toy") model of QM-"quantum mechanics over sets." [15]…”

Section: The Quantum Casementioning

confidence: 99%

New Light on the Objective Indefiniteness or Literal Interpretation of Quantum Mechanics

2018

SSRN Journal

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The development of the new logic of partitions (= equivalence relations) dual to the usual Boolean logic of subsets, and its quantitative version as the new logical theory of information provide the basic mathematical concepts to describe distinctions/indistinctions, definiteness/indefiniteness, and distinguishability/indistinguishability. They throw some new light on the objective indefiniteness or literal interpretation of quantum mechanics (QM) advocated by Abner Shimony. This paper shows how the mathematics of QM is the math of indefiniteness and thus, literally and realistically interpreted, it describes an objectively indefinite reality at the quantum level. In particular, the mathematics of wave propagation is shown to also be the math of the evolution of indefinite states that do not change the degree of indistinctness between states. This corrects the historical wrong turn of seeing QM as "wave mechanics" rather than the mechanics of particles with indefinite/definite properties. For example, the so-called "wave-particle duality' for particles is the juxtaposition of the evolution of a particle having an indefinite position ("wave-like" behavior) with a particle having a definite position (particle-like behavior).

“…Given two DSDs = fV i g i2I and = fW j g j2J , their proto-join is the set of non-zero subspaces fV i \ W j jV i \ W j 6 = f0gg (i;j)2I J (which do not necessarily form a DSD). If the two DSDs and were de…ned as the eigenspace DSDs of two diagonalizable operators, then the space spanned by the proto-join would be the space spanned by the simultaneous eigenvectors of the two operators, and that space is the kernel of the commutator of the two operators [6]. If the two operators commuted, then their commutator is the zero operator whose kernel is the whole space so the proto-join would span the whole space.…”

Section: Compatibility Of Dsdsmentioning

confidence: 99%

“…For instance, in the pedagogical model of "quantum mechanics over sets" or QM/Sets [6], the vector space is Z n 2 so the only diagonalizable operators are projection operatorsP : Z n 2 ! Z n 2 .…”

Section: Introductionmentioning

confidence: 99%

The quantum logic of direct-sumdecompositions: the dual to the quantum logic of subspaces

Logic Journal of the IGPL

2017

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Since the pioneering work of Birkho¤ and von Neumann, quantum logic has been interpreted as the logic of (closed) subspaces of a Hilbert space. There is a progression from the usual Boolean logic of subsets to the "quantum logic" of subspaces of a general vector space-which is then specialized to the closed subspaces of a Hilbert space. But there is a "dual" progression. The set notion of a partition (or quotient set or equivalence relation) is dual (in a categorytheoretic sense) to the notion of a subset. Hence the Boolean logic of subsets has a dual logic of partitions. Then the dual progression is from that logic of set partitions to the quantum logic of direct-sum decompositions (i.e., the vector space version of a set partition) of a general vector space-which can then be specialized to the direct-sum decompositions of a Hilbert space. This allows the quantum logic of direct-sum decompositions to express measurement by any self-adjoint operators. The quantum logic of direct-sum decompositions is dual to the usual quantum logic of subspaces in the same sense that the logic of partitions is dual to the usual Boolean logic of subsets.