Interpretations of Probability (eBook)

Andrei Khrennikov (Autor)

eBook Download: PDF

2009 | 2nd rev. and ext. ed.
237 Seiten
De Gruyter (Verlag)
978-3-11-021319-5 (ISBN)

Infos

This is the first fundamental book devoted to non-Kolmogorov probability models. It provides a mathematical theory of negative probabilities, with numerous applications to quantum physics, information theory, complexity, biology and psychology. The book also presents an interesting model of cognitive information reality with flows of information probabilities, describing the process of thinking, social, and psychological phenomena.

Andrei Khrennikov, Växjö University, Sweden.

Autor

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Andrei Khrennikov, Växjö University, Sweden.

Inhalt

Preface: edition 2008 7
Preface: edition 1999 11
Contents 17
1. Foundations of probability theory 21
2. Quantum probabilities 57
3. Negative probabilities 105
4. p-adic probability theory 133
5. Tests for randomness for p-adic probability theory 160
6. Contextual probability and interference 177
7. Quantum-like representation 198
Bibliography 225
Index 235

Rezensionen

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"This book will be a useful source for graduate students and researchers."
In: Enseignement Mathematique 2/2009

Textauszug

1.4 Idealism (S. 39-40)

Empiricism is sometimes identified with idealism. By idealists viewpoint quantum systems have no objective properties at all. This approach immediately implies a death of realism (not only reality of the microworld, but also reality of macroworld which is composed of microsystems). However, in principle empiricism need not imply idealism. It is very well possible to believe in the objective existence of atoms and electrons without being committed to the thesis that this reality is described by the quantum mechanical formalism. It was Bohr’s position. Of course, one should not forget that Bohr’s position was time-dependent (views of everybody may change crucially during his life).

Early Bohr’s wrote about positions and momentums of quantum particles. He could be considered as f -realist: measurement devices change values of positions and momentums of quantum particles. However, from the very beginning he claimed that quantum mechanics is complete. One could not expect creation of new (more fundamental) theory which would provide an access to e.g. positions and momentums (at least simultaneously).

This is a result of uncontrollable exchange of momentum between a particle and a measurement device. Late Bohr did not write more about properties of quantum systems, but solely about results of measurements. Thus he moved from the camp of f -realists and he became pure empiricist.

1.5 Comparing realism and empiricism

The realist philosophy is very attractive for scientists working in classical physics. However, we shall see that the realist viewpoint induces some problems, e.g. Einstein– Podolsky–Rosen paradox [58] in the foundations of quantum physics. The empiricists approach seems to be free of such problems.

However, empiricism is not so attractive as the philosophic basis for the investigation of reality. If we even do not keep to idealism – not deny existence of objective reality (which is independent to our observations), then by the empiricists ideology we still have to assume that the quantum formalism describes not objective reality of microworld, but reality of equipment in our laboratories. In principle, the empiricists ideology were not be so bad, if it would not deny (following Bohr) a possibility to go beyond quantum mechanics, i.e., to create of a new more fundamental theory for which quantum mechanics would play the role of an approximation.

However, such thoughts were totally forbidden in Bohr’s kingdom. Nowadays some leading empiricists, e.g. already mentioned De Muynck as well as De Baere, see [52, 53], and Ballentine [21], do not exclude a possibility of construction of a subquantum model with mentioned features. We point out that in his early works, e.g. the fundamental paper [20], Ballentine presented the realist position, but later he kept to the empiricist one [21].

Produkt-Details

• Erscheint lt. Verlag: 26.2.2009
• Verlagsort: Berlin/Boston
• Sprache: englisch
• Themenwelt: Mathematik / Informatik ► Mathematik ► Allgemeines / Lexika
• Themenwelt: Naturwissenschaften ► Physik / Astronomie
• Themenwelt: Technik
• Schlagworte: Gruppentheorie • Kolmogorov Model • Kolmogorov-System • p-adic Groups • Quantenphysik • Quantum Physics • stochastics • Stochastics; p-adic Groups; Quantum Physics; Kolmogorov Model • Stochastik
• ISBN-10: 3-11-021319-2 / 3110213192
• ISBN-13: 978-3-11-021319-5 / 9783110213195