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Andrei Khrennikov

Publications and source records attributed to Andrei Khrennikov.

At least 181 records · Page 10Linked to original sources
arXiv2005

Asymptotic expansion of Gaussian integrals of analytic functionals on infinite-dimensional spaces and quantum averages

We study asymptotic expansions of Gaussian integrals of analytic functionals on infinite-dimensional spaces (Hilbert and nuclear Frechet). We obtain an asymptotic equality coupling the Gaussian integral and the trace of the composition of scaling of the covariation operator of a Gaussian measure and the second (Frechet) derivative of a functional. In this way we couple classical average (given by an infinite-dimensional Gaussian integral) and quantum average (given by the von Neumann trace formula). We can interpret this mathematical construction as a procedure of ``dequantization'' of quantum mechanics. We represent quantum mechanics as an asymptotic projection of classical statistical mechanics with infinite-dimensional phase-space. This space can be represented as the space of classical fields, so quantum mechanics is represented as a projection of ``Prequantum Classical Statistical Field Theory''.

quant-ph↗
arXiv2005

Generalized probabilities taking values in non-Archimedean fields and topological groups

We develop an analogue of probability theory for probabilities taking values in topological groups. We generalize Kolmogorov's method of axiomatization of probability theory: main distinguishing features of frequency probabilities are taken as axioms in the measure-theoretic approach. We also present a review of non-Kolmogorovian probabilistic models including models with negative, complex, and $p$-adic valued probabilities. The latter model is discussed in details. The introduction of $p$-adic (as well as more general non-Archimedean) probabilities is one of the main motivations for consideration of generalized probabilities taking values in topological groups which are distinct from the field of real numbers. We discuss applications of non-Kolmogorovian models in physics and cognitive sciences. An important part of this paper is devoted to statistical interpretation of probabilities taking values in topological groups (and in particular in non-Archimedean fields).

math.PR↗
arXiv2005

Quantum mechanics as an asymptotic projection of statistical mechanics of classical fields: derivation of Schrödinger's, Heisenberg's and von Neumann's equations

We show that QM can be represented as a natural projection of a classical statistical model on the phase space $Ω= H\times H,$ where $H$ is the real Hilbert space. Statistical states are given by Gaussian measures on $Ω$ having zero mean value and dispersion of very small magnitude $α$ (which is considered as a small parameter of the model). Such statistical states can be interpreted as fluctuations of the background field, cf. with SED and Nelson's mechanics. Physical variables (e.g., energy) are given by maps $f: Ω\to {\bf R}$ (functions of classical fields). The conventional quantum representation of our prequantum classical statistical model is constructed on the basis of the Taylor expansion (up to the terms of the second order at the vacuum field point $ψ\_{\rm{vacuum}}\equiv 0)$ of variables $f: Ω\to {\bf R}$ with respect to the small parameter $\sqrtα.$ The complex structure of QM is induced by the symplectic structure on the infinite-dimensional phase space $Ω.$ A Gaussian measure (statistical state) is represented in QM by its covariation operator. Equations of Schrödinger, Heisenberg and von Neumann are images of Hamiltonian dynamics on $Ω.$ The main experimental prediction of our prequantum model is that experimental statistical averages can deviate from ones given by QM.

quant-ph↗
arXiv2005

On the notion of a macroscopic quantum system

It is proposed to define "quantumness" of a system (micro or macroscopic, physical, biological, social, political) by starting with understanding that quantum mechanics is a statistical theory. It says us only about probability distributions. The only possible criteria of quantum behaviour are statistical ones. Therefore I propose to consider any system which produces quantum statistics as quantum ("quantumlike"). A possible test is based on the interference of probabilities. I was mainly interested in using such an approach to "quantumness" to extend the domain of applications of quantum mathematical formalism and especially to apply it to cognitive sciences. There were done experiments on interference of probabilities for ensembles of students and a nontrivial interference was really found. One could say that the quantum statistical behaviour might be expected. But the problem was not so trivial. Yes, we might expect nonclassical statistics, but there was no reason to get the quantum one, i.e., cos-interference. But we got it!

quant-ph↗
arXiv2005

Prequantum classical statistical model with infinite dimensional phase-space

We show that quantum mechanics can be represented as an asymptotic projection of statistical mechanics of classical fields. Thus our approach does not contradict to a rather common opinion that quantum mechanics could not be reduced to statistical mechanics of classical particles. Notions of a system and causality can be reestablished on the prequantum level, but the price is sufficiently high -- the infinite dimension of the phase space. In our approach quantum observables, symmetric operators in the Hilbert space, are obtained as derivatives of the second order of functionals of classical fields. Statistical states are given by Gaussian ensembles of classical fields with zero mean value (so these are vacuum fluctuations) and dispersion $α$ which plays the role of a small parameter of the model (so these are small vacuum fluctuations). Our approach might be called {\it Prequantum Classical Statistical Field Theory} - PCSFT. Our model is well established on the mathematical level. However, to obtain concrete experimental predictions -- deviations of real experimental averages from averages given by the von Neumann trace formula - we should find the energy scale $α$ of prequantum classical fields.

quant-ph↗
arXiv2005

Prequantum classical statistical model with infinite dimensional phase-space-2: complex representation of symplectic phase-space model

We show that QM can be represented as a natural projection of a classical statistical model on the phase space $Ω= H\times H,$ where $H$ is the real Hilbert space. Statistical states are given by Gaussian measures on $Ω$ having zero mean value and dispersion of the Planck magnitude -- fluctuations of the ``vacuum field.'' Physical variables (e.g., energy) are given by maps $f: Ω\to {\bf R}$ (functions of classical fields). The crucial point is that statistical states and variables are symplectically invariant. The conventional quantum representation of our prequantum classical statistical model is constructed on the basis of the Teylor expansion (up to the terms of the second order at the vacuum field point $ω=0)$ of variables $f: Ω\to {\bf R}$ with respect to the small parameter $κ= \sqrt{h}.$ A Gaussian symplectically invariant measure (statistical state) is represented by its covariation operator (von Neumann statistical operator). A symplectically invariant smooth function (variable) is represented by its second derivative at the vacuum field point $ω=0.$ From the statistical viewpoint QM is a statistical approximation of the prequantum classical statistical field theory (PCSFT). Such an approximation is obtained through neglecting by statistical fluctuations of the magnitude $o(h), h\to 0,$ in averages of physical variables. Equations of Schrödinger, Heisenberg and von Neumann are images of dynamics on $Ω$ with a symplectically invariant Hamilton function.

quant-ph↗
arXiv2005

Foundations of Probability and Physics-3: Preface of Proceedings and Round Table on Quantum Foundations and Computing

This preprint contains a detailed Preface to Proceedinngs of the International Conference ``Foundations of Probability and Physics-3'' held in Växjö, Sweden, 7-12 June 2004; table of contents and round table. The main theme of the round table was {\it ``Fundamental problems in quantum mechanics, probabilistic description of reality, and quantum information.''} The topics that were specifically discussed were that of Quantum Cryptography, Quantum computing, and Quantum Macroscopic Structures. For each of these topics, the participants were asked to discuss which are the crucial Quantum features required among the following : violation of Bell's inequality, Entanglement, Complementarity, and Interference of Probabilities. Finally, the connection between Mental states and Quantum states was discussed.

quant-ph↗
arXiv2005

Brain as quantum-like computer

We present a contextualist statistical realistic model for quantum-like representations in physics, cognitive science and psychology. We apply this model to describe cognitive experiments to check quantum-like structures of mental processes. The crucial role is played by interference of probabilities for mental observables. Recently one of such experiments based on recognition of images was performed. This experiment confirmed our prediction on quantum-like behaviour of mind. In our approach ``quantumness of mind'' has no direct relation to the fact that the brain (as any physical body) is composed of quantum particles. We invented a new terminology ``quantum-like (QL) mind.'' Cognitive QL-behaviour is characterized by nonzero coefficient of interference $λ.$ This coefficient can be found on the basis of statistical data. There is predicted not only $\cos θ$-interference of probabilities, but also hyperbolic $\cosh θ$-interference. This interference was never observed for physical systems, but we could not exclude this possibility for cognitive systems. We propose a model of brain functioning as QL-computer (there is discussed difference between quantum and QL computers).

quant-ph↗
arXiv2005

Reconstruction of quantum theory on the basis of the formula of total probability

The notion of context (complex of physical conditions) is basic in this paper. We show that the main structures of quantum theory (interference of probabilities, Born's rule, complex probabilistic amplitudes, Hilbert state space, representation of observables by operators) are present in a latent form in the classical Kolmogorov probability model. However, this model should be considered as a calculus of contextual probabilities. In our approach it is forbidden to consider abstract context independent probabilities: ``first context and then probability.'' We start with the conventional formula of total probability for contextual (conditional) probabilities and then we rewrite it by eliminating combinations of incompatible contexts from consideration. In this way we obtain interference of probabilities without to appeal to the Hilbert space formalism or wave mechanics. However, we did not just reconstruct the probabilistic formalism of conventional quantum mechanics. Our contextual probabilistic model is essentially more general and, besides the projection to the complex Hilbert space, it has other projections. The most important new prediction is the possibility (at least theoretical) of appearance of hyperbolic interference.

quant-ph↗
arXiv2005

Principle of supplementarity: contextual probabilistic viewpoint to interference, complementarity and incompatibility

There is presented a contextual statistical model of the probabilistic description of physical reality. Here contexts (complexes of physical conditions) are considered as basic elements of reality. There is discussed the relation with QM. We propose a realistic analogue of Bohr's principle of complementarity. In the opposite to the Bohr's principle, our principle has no direct relation with mutual exclusivity for observables. To distinguish our principle from the Bohr's principle and to give better characterization, we change the terminology and speak about supplementarity, instead of complementarity. Supplementarity is based on the interference of probabilities. It has quantitative expression trough a coefficient which can be easily calculated from experimental statistical data. We need not appeal to the Hilbert space formalism and noncommutativity of operators representing observables. Moreover, in our model there exists pairs of supplementary observables which can not be represented in the complex Hilbert space. There are discussed applications of the principle of supplementarity outside quantum physics.

quant-ph↗
arXiv2005

Wigner's inequality for conditional probabilities and nonexistence of a realistic model for the two dimensional Hilbert space

We perform the analysis of probabilistic assumptions of Bell's approach. We emphasize that J. Bell wrote about probability without to specify the concrete axiomatics of probability theory. The careful analysis demonstrated that (surprisingly) J. Bell did not apply the classical probability model (Kolmogorov) to describe ``classical physical framework.'' In fact, he created his own probabilistic model and compared it with the quantum one. The crucial point is that J. Bell did not pay attention to {\it conditional probabilities.} We show that conditional probability in his model cannot be defined by classical Bayes' formula. We also use the approach based on Bell-type inequalities in the conventional probabilistic approach, Kolmogorov model. We prove an analog of Wigner's inequality for conditional probabilities and by using this inequality show that predictions of the conventional and quantum probability models disagree already in the case of noncomposite systems (even in the two dimensional case).

quant-ph↗
arXiv2004

Probabilistic pathway representation of cognitive information

We present for mental processes the program of mathematical mapping which has been successfully realized for physical processes. We emphasize that our project is not about mathematical simulation of brain's functioning as a complex physical system, i.e., mapping of physical and chemical processes in the brain on mathematical spaces. The project is about mapping of purely mental processes on mathematical spaces. We present various arguments -- philosophic, mathematical, information, and neurophysiological -- in favor of the $p$-adic model of mental space. $p$-adic spaces have structures of hierarchic trees and in our model such a tree hierarchy is considered as an image of neuronal hierarchy. Hierarchic neural pathways are considered as fundamental units of information processing. As neural pathways can go through whole body, the mental space is produced by the whole neural system. Finally, we develop Probabilistic Neural Pathway Model in that Mental States are represented by probability distributions on mental space.

q-bio.NC↗
arXiv2004

Local realist (but contextual) derivation of the EPR-Bohm correlations

The probabilistic structure of quantum mechanics is investigated in the frequency framework. Such an approach can be interpreted as a contextual approach to quantum probabilities. By using rather complicated frequency calculations we reproduce the EPR-Bohm correlation function which is typically derived by using the calculus of probabilities in a Hilbert space. Our frequency probabilistic model of the EPR-Bohm experiment is a realist model -- physical observables are considered as objective properties of physical systems. It is also local -- a measurement over one part of a composite system does not disturb another part of this system. Nevertheless, our result does not contradict to the well known Bell's ``NO-GO'' theorem. J. Bell used the conventional (Kolmogorov) measure-theoretical approach. We use the frequency approach. In the latter approach there are no reasons to assume that the simultaneous probability distribution exists: corresponding frequencies may fluctuate and not approach any definite limit (which Bell would like to use as the probability). The frequency probabilistic derivation demonstrated that incompatibility of observables under consideration plays the crucial role in producing of the EPR-Bohm correlations.

quant-ph↗
arXiv2004

EPR-Bohm experiment, interference of probabilities, imprecision of time

We demonstrate that the EPR-Bohm probabilities can be easily obtained in the classical (but contextual) probabilistic framework by using the formula of interference of probabilities. From this point of view the EPR-Bell experiment is just an experiment on interference of probabilities. We analyse the time structure of contextuality in the EPR-Bohm experiment. The conclusion is that quantum mechanics does not contradict to a local realistic model in which probabilities are calculated as averages over conditionings/measurements for pairs of instances of time $t_1< t_2.$ If we restrict our consideration only to simultaneous measurements at the fixed instance of time $t$ we would get contradiction with Bell's theorem. One of implications of this fact might be the impossibility to define instances of {\it time with absolute precision} on the level of the contextual microscopic realistic model.

quant-ph↗
arXiv2004

Estimation of a classical parameter with gaussian probes: magnetometry with collective atomic spins

We present a theory for the estimation of a classical magnetic field by an atomic sample with a gaussian distribution of collective spin components. By incorporating the magnetic field and the probing laser field as quantum variables with gaussian distributions on equal footing with the atoms, we obtain a very versatile description which is readily adapted to include probing with squeezed light, dissipation and loss and additional measurement capabilities on the atomic system.

quant-ph↗
arXiv2004

$p$-adic discrete dynamical systems and their applications in physics and cognitive sciences

This review is devoted to dynamical systems in fields of $p$-adic numbers: origin of $p$-adic dynamics in $p$-adic theoretical physics (string theory, quantum mechanics and field theory, spin glasses), continuous dynamical systems and discrete dynamical systems. The main attention is paid to discrete dynamical systems - iterations of maps in the field of $p$-adic numbers (or their algebraic extensions): conjugate maps, ergodicity, random dynamical systems, behaviour of cycles, holomorphic dynamics. dynamical systems in finite fields. We also discuss applications of $p$-adic discrete dynamical systems to cognitive sciences and psychology.

nlin.AO↗