Quantum mechanics over sets

a pedagogical model with non-commutative finite probability theory as its quantum probability calculus

David Ellerman

pp. 4863-4896

in: Matteo Colombo, Raoul Gervais, Jan Sprenger (eds), Objectivity in science, Synthese 194 (12), 2017.

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 ({mathbb {C}} ) replaced by ({mathbb {Z}}_{2}). 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 ({mathbb {Z}}_{2}). But the usual QM brackets (leftlangle psi |varphi ight angle ) give the “overlap” between states (psi ) and (varphi ), so for subsets (S,Tsubseteq U), the natural definition is (leftlangle S|T ight angle =left| Scap T ight| ) (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 ({mathbb {C}} ) and QM/Sets over ({mathbb {Z}}_{2}).