Ellerman, David (2013) Quantum Mechanics over sets. [Preprint]

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Abstract

n the tradition of toy models of quantum mechanics in vector spaces over finite fields, e.g., Schumacher and Westmoreland's "modal quantum theory" (MQT), one finite field stands out, ℤ₂, since vectors over ℤ₂ have an interpretation as natural mathematical objects, i.e., sets. This engages a sets to (basis sets of)vector spaces bridge that is part of the mathematical folklore to translate both ways between set concepts and vector space concepts. Using that bridge, the mathematical framework of (finite-dimensional) quantum mechanics can be transported down to sets resulting in quantum mechanics over sets or QM/sets. This approach leads to a different treatment of Dirac's brackets than in MQT, and that gives a full probability calculus (unlike MQT that only has zero-one modalities of impossible and possible). That, in turn, leads to a rather fulsome theory of QM over sets that includes "logical" models of the double-slit experiment, Bell's Theorem, quantum information theory, quantum computing, and much else. Indeed, QM/sets is proposed as the "logic" of QM in the old-fashioned sense of "logic" as giving the simplified essentials of a theory. QM/sets is also a key part of a broader research program to provide an interpretation of QM based on the notion of "objective indefiniteness," a program that grew out the recent development of the logic of partitions mathematically dual to the usual Boolean logic of subsets.

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Item Type:	Preprint
Creators:	Creators Email ORCID Ellerman, David david@ellerman.org
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:	11 Jan 2014 15:55
Last Modified:	11 Jan 2014 15:55
Item ID:	10215
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:	2013
URI:	https://philsci-archive.pitt.edu/id/eprint/10215

Available Versions of this Item

• Quantum Mechanics over sets. (deposited 11 Jan 2014 15:55) [Currently Displayed]
  • Quantum Mechanics over Sets: A pedagogical model with non-commutative finite probability theory as its quantum probability calculus. (deposited 23 Nov 2019 06:56)

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