By Andrei Y. Khrennikov

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

This booklet isn't really approximately quantum mechanics as a actual thought. the fast assessment of quantum postulates is accordingly frequently of historic worth: quantum mechanics is simply the 1st instance of the profitable program of non-Kolmogorov possibilities, step one in the direction of a contextual probabilistic description of normal, organic, mental, social, low-priced or monetary phenomena. A normal 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 will be represented in a quantum-like method, particularly, in complicated and extra basic Hilbert areas. during this means quantum likelihood is completely demystified: Born's illustration of quantum possibilities by means of complicated likelihood amplitudes, wave services, is just a distinct illustration of this type.

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N ), φ = (φ1 , . . , φn ) ∈ Hn . 29) j=1 Self-adjoint operators can be represented by Hermitian matrices, a = (ai j ), such that akm = amk , where z = x + iy → z = x − iy is the operation of complex conjugation. The spectrum, Spec(a), is nothing else than the set of eigenvalues: 30 2 Classical (Kolmogorovian) and Quantum (Born) Probability aψ = αψ. We remark that all eigenvalues are real. Eigenvectors corresponding to the same eigenvalue α form a linear subspace. Its dimension gives the degree of degeneration of α.

Thus Spec(a) = {α1 , . . , αm , . }. 8 ˜ Thus states are given by elements of S. 3 Quantum Probabilistic Model 31 Postulate 4 (Born’s rule – in formalization of Dirac and von Neumann) Let a physical observable a be represented by a self-adjoint operator a with purely discrete spectrum. The probability Pψ (a = αm ) of obtaining the eigenvalue αm of a for measurement of a in a state ψ is given by Pψ (a = αm ) = Pma ψ 2 . 32) If the operator a has nondegenerate (purely discrete) spectrum, then each αm is associated with a one-dimensional subspace.

Consider two nondegenerate observables. Set pβ|α = P(b = β|a = α). ), is not only stochastic but doubly stochastic. It is easy to see that pβ|α = α α | eβb , eαa |2 = eβb , eβb = 1. Double stochasticity is also a very specific property of quantum probability, cf. the Kolmogorovian model and my model Chap. 3. In fact, condition DS-DS holds: both matrices of transition probabilities Pa|b and Pb|a are doubly stochastic. 18). In the quantum framework independent observables are considered in Sect. 2.