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arXiv:math/0010088

An Introduction to the Neutrosophic Probability Applied in Quantum Physics

Florentín Smarandache

Published Oct 10, 2000• 18 citations
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Abstract

In this paper one generalizes the classical probability and imprecise probability to the notion of "neutrosophic probability" in order to be able to model Heisenberg's Uncertainty Principle of a particle's behavior, Schr"dinger's Cat Theory, and the state of bosons which do not obey Pauli's Exclusion Principle (in quantum physics). Neutrosophic probability is close related to neutrosophic logic and neutrosophic set, and etymologically derived from "neutrosophy".

DOI
10.48550/arxiv.math/0010088

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Bulletin of the American Physical Society•Oct 17, 2008•1 citation
In this paper one generalizes the classical probability and imprecise probability to the notion of "neutrosophic probability" in order to be able to model Heisenberg's Uncertainty Principle of a particle's behavior, Schr"dinger's Cat Theory, and the state of bosons which do not obey Pauli's Exclusion Principle (in quantum physics). Neutrosophic probability is close related to neutrosophic logic and neutrosophic set, and etymologically derived from "neutrosophy".
This article discusses Neutrosophic Logic interpretation of the Schrodinger’s cat paradox. We argue that this paradox involves some degree of indeterminacy (unknown) which Neutrosophic Logic could take into consideration, whereas other methods including Fuzzy Logic could not. For a balanced discussion, other interpretations have also been discussed. 1 Schr¨ odinger equation As already known, Schr¨ odinger equation is the most used equation to describe non-relativistic quantum systems. Its relativistic version was developed by Klein-Gordon and Dirac, but Schr¨
We introduce refined concepts for neutrosophic quantum computing such as neutrosophic quantum states and transformation gates, neutrosophic Hadamard matrix, coherent and decoherent superposition states, entanglement and measurement notions based on neutrosophic quantum states. We also give some observations using these principles. We present a number of quantum computational matrix transformations based on neutrosophic logic and clarify quantum mechanical notions relying on neutrosophic states. The paper is intended to extend the work of Smarandache by introducing a mathematical framework for neutrosophic quantum computing and presenting some results.
This thesis is a review of the quantum logic approach to quantum probability theory which was first studied by Birkhoff and von Neumann in 1936. Our study is based on the more general quantum structure which is known as the sigma\\sigmasigma-orthocomplete orthomodular poset or quantum logic. We introduce the definitions of states and observables and show that the quantum probability theory is a generalization of the classical probability theory due to Kolmogorov. Discussions about joint distributions of observables are given, and we also give the general expression of the uncertainty relation in the logico-algebraic sense which generalizes the well-known Heisenberg uncertainty relation. Two examples, the classical probability structure of Kolmogorov and the Hilbert space quantum logic, are given as illustrations throughout the whole discussion.
One of the aspects of quantum theory which has attracted the most general attention, is the novelty of the logical notions which it presupposes. It asserts that even a complete mathematical description of a physical system S does not in general enable one to predict with certainty the result of an experiment on S, and that in particular one can never predict with certainty both the position and the momentum of S, (Heisenberg’s Uncertainty Principle). It further asserts that most pairs of observations are incompatible, and cannot be made on S, simultaneously (Principle of Non-commutativity of Observations).
Quantum mechanics reveals a reality where uncertainty is not merely epistemic but ontological, challenging classical logics that rely on determinacy. We introduce neutrosophic logic as a novel framework to formalise this intrinsic indeterminacy. Unlike probability or fuzzy systems, neutrosophy treats truth (T), falsity (F), and indeterminacy (I) as independent dimensions, enabling a richer representation of quantum superposition, entanglement, and measurement collapse. We show that neutrosophic triplets capture the undefined yet real state of qubits before observation and provide a coherent description of entangled correlations beyond probabilistic models. This approach offers a unifying formalism that accommodates both structural indeterminacy and outcome probabilities, bridging foundational debates in quantum theory with practical advances in quantum computation and information science. By explicitly integrating indeterminacy into mathematical modelling, neutrosophy advances our understanding of quantum reality and suggests new avenues for experimental validation and quantum technology design.