arXiv subjects
Publications and source records attributed to A. Yu. Khrennikov.
Free evolution for quantum particle in general ultrametric space is considered. We find that if mean zero wave packet is localized in some ball in the ultrametric space then its evolution remains localized in the same ball.
Examples of games between two partners with mixed strategies, calculated by the use of the probability amplitude as some vector in Hilbert space are given. The games are macroscopic, no microscopic quantum agent is supposed. The reason for the use of the quantum formalism is in breaking of the distributivity property for the lattice of yes-no questions arising due to the special rules of games. The rules of the games suppose two parts: the preparation and measurement. In the first part due to use of the quantum logical orthocomplemented non-distributive lattice the partners freely choose the wave functions as descriptions of their strategies. The second part consists of classical games described by Boolean sublattices of the initial non-Boolean lattice with same strategies which were chosen in the first part. Examples of games for spin one half are given. New Nash equilibria are found for some cases. Heisenberg uncertainty relations without the Planck constant are written for the "spin one half game".
We continue the development of a so called contextual statistical model (here context has the meaning of a complex of physical conditions). It is shown that, besides contexts producing the conventional trigonometric $\cos$-interference, there exist contexts producing the hyperbolic $\cos$-interference. Starting with the corresponding interference formula of total probability we represent such contexts by hyperbolic probabilistic amplitudes or in the abstract formalism by normalized vectors of a hyperbolic analogue of the Hilbert space. There is obtained a hyperbolic Born's rule. Incompatible observables are represented by noncommutative operators.
A family of orthonormal bases, the ultrametric wavelet bases, is introduced in quadratically integrable complex valued functions spaces for a wide family of ultrametric spaces. A general family of pseudodifferential operators, acting on complex valued functions on these ultrametric spaces is introduced. We show that these operators are diagonal in the introduced ultrametric wavelet bases, and compute the corresponding eigenvalues. We introduce the ultrametric change of variable, which maps the ultrametric spaces under consideration onto positive half-line, and use this map to construct non-homogeneous generalizations of wavelet bases.
Examples of games between two partners with mixed strategies, calculated by the use of the probability amplitude are given. The first game is described by the quantum formalism of spin one half system for which two noncommuting observables are measured. The second game corresponds to the spin one case. Quantum logical orthocomplemented nondistributive lattices for these two games are presented. Interference terms for the probability amplitudes are analyzed by using so called contextual approach to probability (in the von Mises frequency approach). We underline that our games are not based on using of some microscopic systems. The whole scenario is macroscopic.
We consider a classical analogue of the well known quantum two-slit experiment. Charged particles are scattered on flat screen with two slits and hit the second screen. We show that the probability distribution on the second screen when both slits are open is not given by the sum of distributions for each slit separately, but has an extra interference term that is given with the quantum rule of the addition of probabilistic alternatives. We show that the proposed classical model has a context dependence and could be adequately described with contextual formalism.