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 four previous attempts to develop a quantum-like model with the base field of $\mathbb{C}$ replaced by $\mathbb{Z}_{2}$ , but they are all forced into a merely "modal" interpretation by requiring the brackets to take values in $\mathbb{Z}_{2}$ (1 = possible, 0 = impossible). But the usual QM brackets give the "overlap" between states ψ and ϕ, so for subsets S,T of U, the natural definition is $=|S{\cap}T|$ . This allows QM/sets to be developed with a full probability calculus that turns out to be the perfectly classical Laplace-Boole finite probability theory. The point is not to clarify finite probability theory but to elucidate quantum mechanics itself by seeing some of its quantum features (e.g., two-slit experiment) in a classical setting.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 four previous attempts to develop a quantum-like model with the base field of $\mathbb{C}$ replaced by $\mathbb{Z}_{2}$, but they are all forced into a merely "modal" interpretation by requiring the brackets to take values in $\mathbb{Z}_{2}$ (1 = possible, 0 = impossible). But the usual QM brackets give the "overlap" between states ψ and ϕ, so for subsets S,T of U, the natural definition is $=|S{\cap}T|$. This allows QM/sets to be developed with a full probability calculus that turns out to be the perfectly classical Laplace-Boole finite probability theory. The point is not to clarify finite probability theory but to elucidate quantum mechanics itself by seeing some of its quantum features (e.g., two-slit experiment) in a classical setting.

arXiv:1502.01048Feb 3, 20152 citations

Quantum Theory, Quantum Logic and the Calculus of Probability

Akten des XIV. Internationalen Kongresses für PhilosophieJan 1, 19691 citation

Local probability model for the Bell correlation based on the statistics of chaotic light and non-commutative processes

Louis Sica

As discussed below, Bell's inequalities and experimental results rule out commutative hidden variable models as a basis for Bell correlations, but not necessarily non-commutative probability models. A local probability model is constructed for Bell correlations based on non-commutative operations involving polarizers. As in the entanglement model, the Bell correlation is obtained from a probability calculus without explicit use of deterministic hidden variables. The probability calculus used is associated with chaotic light. Joint wave intensity correlations at spatially separated polarization analyzers are computed using common information originating at the source. When interpreted as photon count rates, these yield quantum mechanical joint probabilities after the contribution of indeterminate numbers of photon pairs greater than one is subtracted out. The formalism appears to give a local account of Bell correlations.As discussed below, Bell's inequalities and experimental results rule out commutative hidden variable models as a basis for Bell correlations, but not necessarily non-commutative probability models. A local probability model is constructed for Bell correlations based on non-commutative operations involving polarizers. As in the entanglement model, the Bell correlation is obtained from a probability calculus without explicit use of deterministic hidden variables. The probability calculus used is associated with chaotic light. Joint wave intensity correlations at spatially separated polarization analyzers are computed using common information originating at the source. When interpreted as photon count rates, these yield quantum mechanical joint probabilities after the contribution of indeterminate numbers of photon pairs greater than one is subtracted out. The formalism appears to give a local account of Bell correlations.

arXiv:1106.1453Jun 7, 20111 citation

Probability on Real Lie Algebras

Uwe Franz, Nicolas Privault

This monograph is a progressive introduction to non-commutativity in probability theory, summarizing and synthesizing recent results about classical and quantum stochastic processes on Lie algebras. In the early chapters, focus is placed on concrete examples of the links between algebraic relations and the moments of probability distributions. The subsequent chapters are more advanced and deal with Wigner densities for non-commutative couples of random variables, non-commutative stochastic processes with independent increments (quantum Lévy processes), and the quantum Malliavin calculus. This book will appeal to advanced undergraduate and graduate students interested in the relations between algebra, probability, and quantum theory. It also addresses a more advanced audience by covering other topics related to non-commutativity in stochastic calculus, Lévy processes, and the Malliavin calculus.This monograph is a progressive introduction to non-commutativity in probability theory, summarizing and synthesizing recent results about classical and quantum stochastic processes on Lie algebras. In the early chapters, focus is placed on concrete examples of the links between algebraic relations and the moments of probability distributions. The subsequent chapters are more advanced and deal with Wigner densities for non-commutative couples of random variables, non-commutative stochastic processes with independent increments (quantum Lévy processes), and the quantum Malliavin calculus. This book will appeal to advanced undergraduate and graduate students interested in the relations between algebra, probability, and quantum theory. It also addresses a more advanced audience by covering other topics related to non-commutativity in stochastic calculus, Lévy processes, and the Malliavin calculus.

Cambridge University Press eBooksJan 25, 20162 citations

\mathbb{C}

replaced by

\mathbb{Z}_{2}

, but they are all forced into a merely "modal" interpretation by requiring the brackets to take values in

\mathbb{Z}_{2}

(1 = possible, 0 = impossible). But the usual QM brackets give the "overlap" between states ψ and ϕ, so for subsets S,T of U, the natural definition is

=|S{\cap}T|

. This allows QM/sets to be developed with a full probability calculus that turns out to be the perfectly classical Laplace-Boole finite probability theory. The point is not to clarify finite probability theory but to elucidate quantum mechanics itself by seeing some of its quantum features (e.g., two-slit experiment) in a classical setting.