Andrei Y. Khrennikov - Böcker

This text is devoted to the study of non-Archimedean - and especially p-adic - mathematical physics. Basic questions about the nature and possible applications of such a theory are investigated. Interesting physical models are developed, such as the p-adic universe, where distances can be infinitely large p-adic numbers, energies and momentums. Two types of measurement algorithms are shown to exist, one generating real values and one generating p-adic values. The mathematical basis for the theory is a well developed non-Archimedean analysis, and subjects that are treated include non-Archimedean valued distributions using analytic test functions, Gaussian and Feynman non-Archimedean distributions with applications to quantum field theory, differential and pseudo-differential equations, infinite-dimensional non-Archimedean analysis, and p-adic valued theory of probability and statistics. This volume will appeal to a wide range of researchers and students whose work involves mathematical physics, functional analysis, number theory, probability theory, stochastics, statistical physics or thermodynamics.

This volume can be recommended as an extensive course on p-adic mathematics, treating subjects such as a p-adic theory of probability and stochastic processes; spectral theory of operators in non-Archimedean Hilbert spaces; dynamic systems; p-adic fractal dimension, infinite-dimensional analysis and Feynman integration based on the Albeverio-Hoegh-Rohn approach; both linear and nonlinear differential and pseudo-differential equations; and complexity of random sequences and a p-adic description of chaos. The book also explores the concept of using fields of p-adic numbers and their corresponding non-Archimedean analysis, a p-adic solution of paradoxes in the foundations of quantum mechanics, and especially the famous Einstein-Podolsky-Rosen paradox to create an epistemological framework for scientific use. This work should be of interest to postgraduate students and researchers with an interest in such diverse disciplines as mathematics, physics, biology and philosophy.

This work could be recommended as a course in superanalysis, the theory of functions of commuting and anticommuting variables. It follows the so-called functional superanalysis which was developed by J. Schwinger, B. De Witt, A. Rogers, V.S. Vladimirov and I.V. Volovich, Yu. Kobayashi and S. Nagamashi, M. Batchelor, U. Buzzo and R. Cianci and the present author. In this approach, superspace is defined as a set of points on which commuting and anticommuting coordinates are given. Thus functional superanalysis is a natural generalization of Newton's analysis (on real space) and strongly differs from the so-called algebraic analysis which has no functions of superpoints, and where "functions" are just elements of Grassmann algebras. It offers the possibility of extending the notion of space, and of operating on spaces which are described by noncommuting coordinates. These supercoordinates, which are described by an infinite number of ordinary real, complex or p-adic coordinates, are interpreted as creation or annihilation operators of quantum field theory.Subjects treated include differential calculus, including Cauchy-Riemann conditions, on superspaces over supercommutative Banach and topological superalgebras; integral calculus, including integration of differential forms; theory of distributions and linear partial differential equations with constant coefficients; calculus of pseudo-differential operators; analysis on infinite-dimensional superspaces over supercommutative Banach and topological supermodules; infinite-dimensional superdistributions and Feynman integrals with applications to superfield theory; noncommutative probabilities (central limit theorem); and non-Archimedean superanalysis.

This book develops a new physical/mathematical model for the functioning of the human brain, based, not on the modern Newton-Einstein view of physical reality, but on 'information reality'. The work is devoted to the physical-mathematical modeling of (conscious) cognitive phenomena. The most important distinguishing feature of the theory presented here is a new model of mental space, the so-called p-adic hierarchic tree space, and the development of mental analogs of classical and quantum mechanics. Mental processes and more general information processes are handled as a kind of new physical processes. In particular, the procedure of information quantization and an information analog of Bohmian mechanics are developed. Here, mind is a singularity in the mental pilot wave. Applications to neurophysiology, localization of mental function and brain ablations, and psychology (in particular, Freud's psychoanalysis) are considered. Audience: This book will be of interest to researchers working on physical, mathematical, cognitive, neurophysical, psychological and philosophical aspects of human consciousness.

This book provides an overview of the theory of p-adic (and more general non-Archimedean) dynamical systems. The main part of the book is devoted to discrete dynamical systems. It presents a model of probabilistic thinking on p-adic mental space based on ultrametric diffusion. Coverage also details p-adic neural networks and their applications to cognitive sciences: learning algorithms, memory recalling.

The aim of this book is to show that the probabilistic formalisms of classical statistical mechanics and quantum mechanics can be unified on the basis of a general contextual probabilistic model. By taking into account the dependence of (classical) probabilities on contexts (i.e. complexes of physical conditions), one can reproduce all distinct features of quantum probabilities such as the interference of probabilities and the violation of Bell’s inequality. Moreover, by starting with a formula for the interference of probabilities (which generalizes the well known classical formula of total probability), one can construct the representation of contextual probabilities by complex probability amplitudes or, in the abstract formalism, by normalized vectors of the complex Hilbert space or its hyperbolic generalization. Thus the Hilbert space representation of probabilities can be naturally derived from classical probabilistic assumptions. An important chapter of the book critically reviews known no-go theorems: the impossibility to establish a finer description of micro-phenomena than provided by quantum mechanics; and, in particular, the commonly accepted consequences of Bell’s theorem (including quantum non-locality). Also, possible applications of the contextual probabilistic model and its quantum-like representation in complex Hilbert spaces in other fields (e.g. in cognitive science and psychology) are discussed.

This book mathematically analyzes the basic problems of biology, decision making and psychology within the framework of the theory of open quantum systems. In recent years there has been an explosion of interest in applications of quantum theory in fields beyond physics.

This book mathematically analyzes the basic problems of biology, decision making and psychology within the framework of the theory of open quantum systems. In recent years there has been an explosion of interest in applications of quantum theory in fields beyond physics.

Quantum-like structure is present practically everywhere. Quantum-like (QL) models, i.e. models based on the mathematical formalism of quantum mechanics and its generalizations can be successfully applied to cognitive science, psychology, genetics, economics, finances, and game theory.This book is not about quantum mechanics as a physical theory. The short review of quantum postulates is therefore mainly of historical value: quantum mechanics is just the first example of the successful application of non-Kolmogorov probabilities, the first step towards a contextual probabilistic description of natural, biological, psychological, social, economical or financial phenomena. A general contextual probabilistic model (Växjö model) is presented. It can be used for describing probabilities in both quantum and classical (statistical) mechanics as well as in the above mentioned phenomena. This model can be represented in a quantum-like way, namely, in complex and more general Hilbert spaces. In this way quantum probability is totally demystified: Born's representation of quantum probabilities by complex probability amplitudes, wave functions, is simply a special representation of this type.

Quantum-like structure is present practically everywhere. Quantum-like (QL) models, i.e. models based on the mathematical formalism of quantum mechanics and its generalizations can be successfully applied to cognitive science, psychology, genetics, economics, finances, and game theory.This book is not about quantum mechanics as a physical theory. The short review of quantum postulates is therefore mainly of historical value: quantum mechanics is just the first example of the successful application of non-Kolmogorov probabilities, the first step towards a contextual probabilistic description of natural, biological, psychological, social, economical or financial phenomena. A general contextual probabilistic model (Växjö model) is presented. It can be used for describing probabilities in both quantum and classical (statistical) mechanics as well as in the above mentioned phenomena. This model can be represented in a quantum-like way, namely, in complex and more general Hilbert spaces. In this way quantum probability is totally demystified: Born's representation of quantum probabilities by complex probability amplitudes, wave functions, is simply a special representation of this type.

This book is devoted to the study of non-Archimedean, and especially p-adic mathematical physics. Basic questions about the nature and possible applications of such a theory are investigated. Interesting physical models are developed like the p-adic universe, where distances can be infinitely large p-adic numbers, energies and momentums. Two types of measurement algorithms are shown to exist, one generating real values and one generating p-adic values. The mathematical basis for the theory is a well developed non-Archimedean analysis, and subjects that are treated include non-Archimedean valued distributions using analytic test functions, Gaussian and Feynman non-Archimedean distributions with applications to quantum field theory, differential and pseudo-differential equations, infinite-dimensional non-Archimedean analysis, and p-adic valued theory of probability and statistics. This volume will appeal to a wide range of researchers and students whose work involves mathematical physics, functional analysis, number theory, probability theory, stochastics, statistical physics or thermodynamics.

This book develops a new physical/mathematical model for the functioning of the human brain, based, not on the modern Newton-Einstein view of physical reality, but on 'information reality'. The work is devoted to the physical-mathematical modeling of (conscious) cognitive phenomena. The most important distinguishing feature of the theory presented here is a new model of mental space, the so-called p-adic hierarchic tree space, and the development of mental analogs of classical and quantum mechanics. Mental processes and more general information processes are handled as a kind of new physical processes. In particular, the procedure of information quantization and an information analog of Bohmian mechanics are developed. Here, mind is a singularity in the mental pilot wave. Applications to neurophysiology, localization of mental function and brain ablations, and psychology (in particular, Freud's psychoanalysis) are considered.

This book provides an overview of the theory of p-adic (and more general non-Archimedean) dynamical systems. The main part of the book is devoted to discrete dynamical systems. It presents a model of probabilistic thinking on p-adic mental space based on ultrametric diffusion. Coverage also details p-adic neural networks and their applications to cognitive sciences: learning algorithms, memory recalling.

The aim of this book is to show that the probabilistic formalisms of classical statistical mechanics and quantum mechanics can be unified on the basis of a general contextual probabilistic model. By taking into account the dependence of (classical) probabilities on contexts (i.e. complexes of physical conditions), one can reproduce all distinct features of quantum probabilities such as the interference of probabilities and the violation of Bell’s inequality. Moreover, by starting with a formula for the interference of probabilities (which generalizes the well known classical formula of total probability), one can construct the representation of contextual probabilities by complex probability amplitudes or, in the abstract formalism, by normalized vectors of the complex Hilbert space or its hyperbolic generalization. Thus the Hilbert space representation of probabilities can be naturally derived from classical probabilistic assumptions. An important chapter of the book critically reviews known no-go theorems: the impossibility to establish a finer description of micro-phenomena than provided by quantum mechanics; and, in particular, the commonly accepted consequences of Bell’s theorem (including quantum non-locality). Also, possible applications of the contextual probabilistic model and its quantum-like representation in complex Hilbert spaces in other fields (e.g. in cognitive science and psychology) are discussed.

defined as elements of Grassmann algebra (an algebra with anticom­ muting generators). The derivatives of these elements with respect to anticommuting generators were defined according to algebraic laws, and nothing like Newton's analysis arose when Martin's approach was used. Later, during the next twenty years, the algebraic apparatus de­ veloped by Martin was used in all mathematical works. We must point out here the considerable contribution made by F. A. Berezin, G 1. Kac, D. A. Leites, B. Kostant. In their works, they constructed a new division of mathematics which can naturally be called an algebraic superanalysis. Following the example of physicists, researchers called the investigations carried out with the use of commuting and anticom­ muting coordinates supermathematics; all mathematical objects that appeared in supermathematics were called superobjects, although, of course, there is nothing "super" in supermathematics. However, despite the great achievements in algebraic superanaly­ sis, this formalism could not be regarded as a generalization to the case of commuting and anticommuting variables from the ordinary Newton analysis. What is more, Schwinger's formalism was still used in practically all physical works, on an intuitive level, and physicists regarded functions of anticommuting variables as "real functions" == maps of sets and not as elements of Grassmann algebras. In 1974, Salam and Strathdee proposed a very apt name for a set of super­ points. They called this set a superspace.