By Andrei Y. Khrennikov

Quantum-like constitution is current essentially in every single place. Quantum-like (QL) versions, i.e. types in keeping with the mathematical formalism of quantum mechanics and its generalizations should be effectively utilized to cognitive technology, psychology, genetics, economics, funds, and online game theory.

This ebook isn't really approximately quantum mechanics as a actual concept. the fast assessment of quantum postulates is as a result usually of ancient price: quantum mechanics is simply the 1st instance of the profitable software of non-Kolmogorov possibilities, step one in the direction of a contextual probabilistic description of ordinary, organic, mental, social, comparatively cheap or monetary phenomena. A common contextual probabilistic version (Växjö version) is gifted. it may be used for describing possibilities in either quantum and classical (statistical) mechanics in addition to within the above pointed out phenomena. This version may be represented in a quantum-like method, particularly, in complicated and extra normal Hilbert areas. during this means quantum chance is completely demystified: Born's illustration of quantum chances via advanced likelihood amplitudes, wave features, is just a unique illustration of this type.

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Itamar Pitowsky found these results and compared them with Bell’s inequality, see [259, 260] and also the preface in [167]. , [165, 167, 5, 6]. The main problem of the classical probabilistic community was concentration on mathematical problems related to a single Kolmogorov space, especially various limit theorems. In such a situation even the idea that something could not be embedded in such a space was not especially welcome. Vorob’ev’s works were not highly estimated by the Soviet probabilistic community (which was one of the strongest in the world) and, as a result, not by the international community either.

It is natural to suppose that cognitive and social systems should take into account (to survive in the process of evolution) the mentioned possibility of violation of FTP. Thus they should develop the ability to use a more general probabilistic model than the classical model. We speculate that they use special representations of the contextual statistical model. They may be able to apply QLRA and to represent contextual probabilities in complex or more general linear spaces. 2) are contextual. They depend on a complex of conditions, context C, for measurements of observables a and b.

50) 36 2 Classical (Kolmogorovian) and Quantum (Born) Probability (here aeka = αk eka ). Thus the conditional probability in this case does not depend on the original state ψ. We can say that the memory of the original state has been destroyed. If also the operator b has a nondegenerate spectrum then we have Pψ (b = βm |a = αk ) = | emb , eka |2 and Pψ (a = αk |b = βm ) = | eka , emb |2 . 2 Let both operators a and b have purely discrete nondegenerate spectra and let Pka ψ = 0 and Pmb ψ = 0. Then conditional probability is symmetric and it does not depend on the original state ψ : Pψ (b = βm |a = αk ) = Pψ (a = αk |b = βm ) = | emb , eka |2 .