Quantum Mechanics (QM) is primarily a probabilistic theory. The standard Kolmogorov probability theory is not suitable for this goal since it does not offer the possibility of the reversible time evolution. This implies the need to construct a new probability theory which is able to model the reversible time evolution. I have proposed such an Extended probability theory. The main goal of this project is to understand what QM is. In this note I shall describe the program of the research which is already partially realized.Quantum Mechanics (QM) is primarily a probabilistic theory. The standard Kolmogorov probability theory is not suitable for this goal since it does not offer the possibility of the reversible time evolution. This implies the need to construct a new probability theory which is able to model the reversible time evolution. I have proposed such an Extended probability theory. The main goal of this project is to understand what QM is. In this note I shall describe the program of the research which is already partially realized.

Jul 1, 20150 citations

Quantum Mechanics as a Theory of Probability

Itamar Pitowsky

arXiv:quant-ph/0510095Springer eBooksNov 2, 2006129 citations

Kolmogorovian versus Non-Kolmogorovian Probabilities in Contextual Theories

Claudio Garola

Most scholars maintain that quantum mechanics (QM) is a contextual theory and that quantum probability does not allow for an epistemic (ignorance) interpretation. By inquiring possible connections between contextuality and non-classical probabilities we show that a class TμMP of theories can be selected in which probabilities are introduced as classical averages of Kolmogorovian probabilities over sets of (microscopic) contexts, which endows them with an epistemic interpretation. The conditions characterizing TμMP are compatible with classical mechanics (CM), statistical mechanics (SM), and QM, hence we assume that these theories belong to TμMP. In the case of CM and SM, this assumption is irrelevant, as all of the notions introduced in them as members of TμMP reduce to standard notions. In the case of QM, it leads to interpret quantum probability as a derived notion in a Kolmogorovian framework, explains why it is non-Kolmogorovian, and provides it with an epistemic interpretation. These results were anticipated in a previous paper, but they are obtained here in a general framework without referring to individual objects, which shows that they hold, even if only a minimal (statistical) interpretation of QM is adopted in order to avoid the problems following from the standard quantum theory of measurement.Most scholars maintain that quantum mechanics (QM) is a contextual theory and that quantum probability does not allow for an epistemic (ignorance) interpretation. By inquiring possible connections between contextuality and non-classical probabilities we show that a class TμMP of theories can be selected in which probabilities are introduced as classical averages of Kolmogorovian probabilities over sets of (microscopic) contexts, which endows them with an epistemic interpretation. The conditions characterizing TμMP are compatible with classical mechanics (CM), statistical mechanics (SM), and QM, hence we assume that these theories belong to TμMP. In the case of CM and SM, this assumption is irrelevant, as all of the notions introduced in them as members of TμMP reduce to standard notions. In the case of QM, it leads to interpret quantum probability as a derived notion in a Kolmogorovian framework, explains why it is non-Kolmogorovian, and provides it with an epistemic interpretation. These results were anticipated in a previous paper, but they are obtained here in a general framework without referring to individual objects, which shows that they hold, even if only a minimal (statistical) interpretation of QM is adopted in order to avoid the problems following from the standard quantum theory of measurement.

arXiv:1911.02481EntropyJan 18, 20212 citations

Quantum Mechanics as Quantum Probability

Bassano Vacchini

Graduate texts in physicsJan 1, 20240 citations

On classical finite probability theory as a quantum probability calculus

David Ellerman

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

\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.