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Quantum Mechanics over Sets: A pedagogical model with non-commutative finite probability theory as its quantum probability calculus

Ellerman, David (2019) Quantum Mechanics over Sets: A pedagogical model with non-commutative finite probability theory as its quantum probability calculus. [Preprint]

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Abstract

This paper shows how the classical finite probability theory (with equiprobable outcomes) can be reinterpreted and recast as the quantum probability calculus of a pedagogical or toy model of quantum mechanics over sets (QM/sets). There have been several previous attempts to develop a quantum-like model with the base field of C replaced by Z2. Since there are no inner products on vector spaces over finite fields, the problem is to define the Dirac brackets and the probability calculus. The previous attempts all required the brackets to take values in Z2. But the usual QM brackets ⟨ψ|ϕ⟩ give the ”overlap” between states ψ and ϕ, so for subsets S, T ⊆ U, the natural definition is ⟨S|T ⟩ = |S ∩ T | (taking values in the natural numbers). This allows QM/sets to be developed with a full probability calculus that turns out to be a non-commutative extension of classical Laplace-Boole finite probability theory. The pedagogical model is illustrated by giving simple treatments of the indeterminacy principle, the double-slit experiment, Bell’s Theorem, and identical particles in QM/Sets. A more technical appendix explains the mathematics behind carrying some vector space structures between QM over C and QM/Sets over Z2.


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Item Type: Preprint
Creators:
CreatorsEmailORCID
Ellerman, Daviddavid@ellerman.org0000-0002-5718-618X
Keywords: Quantum mechanics, finite field 2, objective indefiniteness, quantum information theory, quantum computing over 2
Subjects: Specific Sciences > Computation/Information
Specific Sciences > Physics > Quantum Mechanics
General Issues > Realism/Anti-realism
General Issues > Structure of Theories
General Issues > Thought Experiments
Depositing User: David Ellerman
Date Deposited: 23 Nov 2019 06:56
Last Modified: 23 Nov 2019 06:56
Item ID: 16654
Official URL: http://arxiv.org/abs/1310.8221
Subjects: Specific Sciences > Computation/Information
Specific Sciences > Physics > Quantum Mechanics
General Issues > Realism/Anti-realism
General Issues > Structure of Theories
General Issues > Thought Experiments
Date: 22 November 2019
URI: http://philsci-archive.pitt.edu/id/eprint/16654

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