Ubiquitous Quantum Structure
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Ubiquitous Quantum Structure Andrei Khrennikov Ubiquitous Quantum Structure From Psychology to Finance 123 Prof. Andrei Khrennikov University of Vaxj¨ o¨ International Center for Mathematical Modeling in Physics and Cognitive Science Vejdes Plats 7 SE-351 95 Vaxj¨ o¨ Sweden andrei.khrennikov@vxu.se ISBN 978-3-642-05100-5 e-ISBN 978-3-642-05101-2 DOI 10.1007/978-3-642-05101-2 Springer Heidelberg Dordrecht London New York Library of Congress Control Number: 2009943303 c Springer-Verlag Berlin Heidelberg 2010 This work is subject to copyright. All rights are reserved, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilm or in any other way, and storage in data banks. Duplication of this publication or parts thereof is permitted only under the provisions of the German Copyright Law of September 9, 1965, in its current version, and permission for use must always be obtained from Springer. Violations are liable to prosecution under the German Copyright Law. The use of general descriptive names, registered names, trademarks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. Cover design: deblik, Berlin Printed on acid-free paper Springer is part of Springer Science+Business Media (www.springer.com) Preface The aim of this book is to demonstrate that quantum-like (QL) models, i.e., models based on the mathematical formalism of quantum mechanics (QM) and its general- izations, can be successfully applied to cognitive science, psychology, genetics, the economy, finances, and game theory. This book is not about quantum mechanics as a physical theory. The short review of quantum postulates has merely historical value: quantum mechanics is just the first example of successful application of non-Kolmogorovian probabilities, the first step towards a contextual probabilistic description of natural, biological, psycholog- ical, social, economic or financial phenomena. I have developed a general contextual probabilistic model (Vaxj¨ o¨ model) that can be used to describe probabilities in both quantum and classical (statistical) mechanics as well as in the above-mentioned phe- nomena. This model can be represented in a QL way, namely, in complex and more general Hilbert spaces. In particular, 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. QL representation of data is very convenient; it can be used in any domain of science. I have presented [180, 198] a fundamental conjecture that some biological systems might develop the ability to create QL representations of external and internal worlds. Start- ing with this conjecture, QL models of cognitive and psychological processes are developed. A simple statistical test of QL probabilistic behavior based on interference of probabilities was elaborated [180, 176] and corresponding experiments have been performed by Conte et al. [66, 67]. The conjecture on the QL processing of infor- mation in the brain was confirmed: ensembles of students performing incompati- ble recognition tasks for recognition of ambiguous figures demonstrated nontrivial interference of probabilities, see Chapter 6. Recently a professor of cognitive psychology, Jerome Busemeyer, conjectured that probabilistic data obtained in famous experiments on the so-called disjunction effect by Tversky and Shafir [295, 275] cannot be described by the conventional Markovian probabilistic model, and he speculated that the disjunction effect can be described by quantum formalism. This viewpoint was elaborated in works by Busemeyer et al. [48–50]. We recall that the disjunction effect is by definition (given v vi Preface by Tversky and Shafir1) an exhibition of violation of Savage’s Sure Thing Principle (STP) [271]. The latter is the basis of the modern theory of rational decision making. In particular, by Savage’s axiomatics, economic processes are based on actions of rational agents of the market (in particular, of the financial market), i.e., on the STP. In this book I apply the contextual approach to the disjunction effect. A QL model is presented in Chapter 7; see [208, 206, 213, 209] for original publications. It seems that, in spite of Busemeyer’s conjecture, the conventional quantum formalism is too restrictive to describe the disjunction effect. It is impossible to describe this effect not only by classical (Kolmogorovian) probability theory, but even by conventional quantum probability (defined by Born’s rule in Dirac–von Neumann’s formalism of QM). I apply a generalization of the QM-formalism that is naturally generated in the contextual probabilistic framework. My model is based on the assumption that, in the process of evolution, cognitive systems developed the ability to repre- sent contexts by probabilistic amplitudes (complex and even more general). Such amplitudes form a linear space. Thus the brain is able to linearize probabilistic images of contexts. The dynamics is described by a linear evolution equation, a mental Schodinger’s¨ equation. Consequently, decision making is described mathe- matically by quantum (and more general QL) theory, see, e.g., Holevo [147, 148], Helstrom [141] or Marley and Hornstein [238]. We also mention that the interest in applications of quantum and QL methods to decision making in cognitive sci- ence, psychology and economics is very large, see publications by Danilov and Lambert-Mogiliansky [73–76] (utility theory), La Mura [223] (utility theory) (see also [222]), Franco [109–114] (cognitive psychology), and Haven and Khrennikov [137] (a fundamental work covering all possible paradoxes in cognitive psychology related to the disjunction effect). The crucial point is that QL probabilistic behavior, e.g., in research on brain func- tioning, need not be a consequence of special physical conditions. For example, the hot brain can still produce interference of probabilities. This is a significant advan- tage of the QL approach compared with quantum physical reductionism. I have named this approach the QL paradigm. A detailed presentation of this paradigm can be found in the first chapter of this book. This book is truly intended to be accessible to psychologists and researchers working in cognitive science, sociology or economics. Therefore, the first chapter provides a detailed review of all the basic ideas and methods used in the following chapters, without containing any mathematics. It might even be useful for philoso- phers interested in quantum foundations and the QL description of physical, biolog- ical, and mental phenomena. Vaxj¨ o,¨ Moscow Andrei Khrennikov 2009 1 See also the excellent experimental work of Croson [71] confirming and generalizing results [295, 275]. Acknowledgements I thank L. Accardi, D. Aerts, J. Busemeyer, E. Conte, A. Elitzur, A. Ezhov, R. Franco, W. Freudenberg, A. Grib, S. Hameroff, E. Haven, A. Lambert-Mogiliansky, M. Ohya, P. Suppes, and K. Svozil for fruitful discussions on applications of quan- tum probability outside of physics. This book was completed during my visits to the University of Lecce and the Centro Vito Volterra of the University of Rome 2, June 2008, and the Steklov Mathematical Institute of the Russian Academy of Science and the Institute of Information Security of the Russian State University for Humanities, September–December 2008. I thank C. Garola, L. Accardi, I. Volovich, S. Kozyrev, and V. Maksimov for hospitality. vii Contents 1 Quantum-like Paradigm ........................................ 1 1.1 Applications of Mathematical Apparatus of QM Outside ofPhysics............................................... 1 1.2 Irreducible Quantum Randomness, Copenhagen Interpretation . 2 1.3 Quantum Reductionism in Biology and Cognitive Science ...... 3 1.4 Statistical(orEnsemble)InterpretationofQM ................ 4 1.5 No-Go Theorems . ................................... 5 1.6 Einstein’sandBohr’sViewsonRealism...................... 6 1.7 Quantum and Quantum-like Models . ........................ 7 1.8 Quantum-like Representation Algorithm – QLRA ............. 7 1.9 Non-Kolmogorov Probability . ........................ 8 1.10 Contextual Probabilistic Model – Vaxj¨ o¨ Model . ............. 9 1.11 Experimental Verification . ............................... 10 1.12 Violation of Savage’s Sure Thing Principle . ............. 11 1.13 Quantum-like Description of the Financial Market ............. 11 1.14 Quantum and Quantum-like Games . ........................ 13 1.15 Terminology: Context, Contextual Probability, Contextuality . 14 1.16 Formula of Total Probability ............................... 15 1.17 Formula of Total Probability with Interference Term . ...... 15 1.18 Quantum-like Representation of Contexts .................... 16 2 Classical (Kolmogorovian) and Quantum (Born) Probability ........ 19 2.1 Kolmogorovian Probabilistic Model . ........................ 19 2.1.1 Probability Space . ............................... 19 2.1.2 Conditional Probability . ........................ 22 2.1.3 Formula of Total Probability ........................ 24 2.2 Probabilistic Incompatibility: Bell–Boole Inequalities . ...... 25 2.2.1 Views of Boole, Kolmogorov, and Vorob’ev . ...... 26 2.2.2 Bell’s and Wigner’s Inequalities . .................... 28 2.2.3 Bell-type Inequalities