Ellerman, David (2019) Quantum Mechanics over Sets: A pedagogical model with noncommutative 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 quantumlike 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 ⟨ψϕ⟩ give the ”overlap” between states ψ and ϕ, so for subsets S, T ⊆ U, the natural deﬁnition is ⟨ST ⟩ = 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 noncommutative extension of classical LaplaceBoole ﬁnite probability theory. The pedagogical model is illustrated by giving simple treatments of the indeterminacy principle, the doubleslit 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  

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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/Antirealism 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/Antirealism General Issues > Structure of Theories General Issues > Thought Experiments 

Date:  22 November 2019  
URI:  https://philsciarchive.pitt.edu/id/eprint/16654 
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Quantum Mechanics over sets. (deposited 11 Jan 2014 15:55)
 Quantum Mechanics over Sets: A pedagogical model with noncommutative ﬁnite probability theory as its quantum probability calculus. (deposited 23 Nov 2019 06:56) [Currently Displayed]
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