Contextual Approach to Quantum Formalism
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) probab
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Fundamental Theories of Physics
An International Book Series on The Fundamental Theories of Physics: Their Clarification, Development and Application
Series Editors: GIANCARLO GHIRARDI, University of Trieste, Italy VESSELIN PETKOV, Concordia University, Canada TONY SUDBERY, University of York, UK ALWYN VAN DER MERWE, University of Denver, CO, USA
Volume 160 For other titles published in this series, go to www.springer.com/series/6001
Andrei Khrennikov
Contextual Approach to Quantum Formalism
Prof. Andrei Khrennikov University of Växjö International Center for Mathematical Modeling in Physics and Cognitive Science Vejdes Plats 7 35195 Växjö Sweden E-mail: [email protected]
ISBN 978-1-4020-9592-4
e-ISBN 978-1-4020-9593-1
Library of Congress Control Number: 2009920940 © Springer Science + Business Media B.V. 2009 No part of this work may be reproduced, stored in a retrieval system, or transmitted in any form or by any means, electronic, mechanical, photocopying, microfilming, recording or otherwise, without written permission from the Publisher, with the exception of any material supplied specifically for the purpose of being entered and executed on a computer system, for exclusive use by the purchaser of the work. Printed on acid-free paper 9 8 7 6 5 4 3 2 1 springer.com
Dedication
To memory of Richard von Mises whose theory of collectives opened the way to contextual probability.
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Preface
Calculus of quantum probabilities is one of the most important sectors of the mathematical formalism of quantum mechanics. This calculus is based on the representation of probabilities by complex probability amplitudes (or normalized complex vectors in the abstract Hilbert space formalism). This algebraic representation was created in the process of the development of quantum mechanics. Since that time it has been successfully applied to the statistical analysis of data obtained in various experiments with quantum systems. Applications to quantum physics play an extremely stimulating role in the development of quantum probability mathematics. However, the original appearance of this mathematical apparatus in the framework of a special physical theory— quantum mechanics—created a barrier (in any event, a psychological barrier) on the way to generalizations and applications of quantum probability calculus outside quantum physics. The quantum mechanical origin of this mathematical formalism induced the impression that the quantum probabilistic behavior can only be found in very special (often regarded as even mystical) systems, namely, quantum particles and fields.1 1
“Quantum mechanics is magic,” Daniel Greenberger; “ Everything we call real is made of things
that cannot be regarded as real,” Niels Bohr; “Those who are not shocked when they first come across quantum theory cannot possibly have understood it,” Niels Bohr; “If you are not completely
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Preface
The aim of this book is to demystify quantum probability.2 The only possibility to do this is to derive the quantum probability calculus w
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