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

Synthese (12) (2017)
  Copy   BIBTEX


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.

Similar books and articles

An Introduction to Partition Logic.David Ellerman - 2014 - Logic Journal of the IGPL 22 (1):94-125.
Sensible quantum mechanics: Are probabilities only in the mind?Don N. Page - 1996 - International Journal of Modern Physics D 5:583-96.
What Is Fuzzy Probability Theory?S. Gudder - 2000 - Foundations of Physics 30 (10):1663-1678.
From physics to information theory and back.Wayne C. Myrvold - 2010 - In Alisa Bokulich & Gregg Jaeger (eds.), Philosophy of Quantum Information and Entanglement. Cambridge University Press. pp. 181--207.
Modal Quantum Theory.Benjamin Schumacher & Michael D. Westmoreland - 2012 - Foundations of Physics 42 (7):918-925.


Added to PP

245 (#72,691)

6 months
81 (#46,579)

Historical graph of downloads
How can I increase my downloads?