Probability and randomness: quantum versus classical
"Creating a rigorous mathematical theory of randomness is far from being complete, even in the classical case. Interrelation of Classical and Quantum Randomness rectifies this and introduces mathematical formalisms of classical and quantum probability and randomness with brief discussion of the...
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| Format: | Elektronisch E-Book |
| Sprache: | English |
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Covent Garden, London
Imperial College Press
[2016]
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| Schlagworte: | |
| Online-Zugang: | UBW01 Volltext Inhaltsverzeichnis Inhaltsverzeichnis |
| Zusammenfassung: | "Creating a rigorous mathematical theory of randomness is far from being complete, even in the classical case. Interrelation of Classical and Quantum Randomness rectifies this and introduces mathematical formalisms of classical and quantum probability and randomness with brief discussion of their interrelation and interpretational and foundational issues. The book presents the essentials of classical approaches to randomness, enlightens their successes and problems, and then proceeds to essentials of quantum randomness. Its wide-ranging and comprehensive scope makes it suitable for researchers in mathematical physics, probability and statistics at any level"... |
| Beschreibung: | Includes bibliographical references and index |
| Beschreibung: | 1 Online-Ressource (xvi, 282 Seiten) |
| ISBN: | 1783267968 9781783267972 9781783267989 |
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| adam_text | INTERRELATION OF CLASSICAL AND QUANTUM RANDOMNESS
/ KHRENNIKOV, A. IU.YYQ(ANDREI IUREVICH)YYD1958-YYEAUTHOR
: 2016
TABLE OF CONTENTS / INHALTSVERZEICHNIS
FOUNDATIONS OF PROBABILITY
RANDOMNESS
SUPPLEMENTARY NOTES ON MEASURE-THEORETIC AND FREQUENCY
APPROACHES
INTRODUCTION TO QUANTUM FORMALISM
QUANTUM AND CONTEXTUAL PROBABILITY
INTERPRETATIONS OF QUANTUM MECHANICS AND PROBABILITY
RANDOMNESS: QUANTUM VERSUS CLASSICAL
PROBABILISTIC STRUCTURE OF BELL S ARGUMENT
QUANTUM PROBABILITY OUTSIDE OF PHYSICS: FROM MOLECULAR BIOLOGY TO
COGNITION
DIESES SCHRIFTSTCK WURDE MASCHINELL ERZEUGT.
Titel: Probability and randomness
Autor: Khrennikov, Andrei Yu
Jahr: 2016
Contents
Preface vii
1. Foundations of Probability 1
1.1 Interpretation Problem in Quantum Mechanics and
Classical Probability Theory..................................4
1.2 Kolmogorov Axiomatics of Probability Theory..............6
1.2.1 Events as sets and probability as measure on a
family of sets representing events....................6
1.2.2 The role of countable-additivity (cr-additivity) . . 8
1.2.3 Probability space ....................................10
1.3 Elementary Properties of Probability Measure..............10
1.3.1 Consequences of finite-additivity....................11
1.3.2 Bell s inequality in Wigner s form..................13
1.3.3 Monotonicity of probability..........................14
1.4 Random Variables ............................................14
1.5 Conditional Probability; Independence; Repeatability ... 17
1.6 Formula of Total Probability ................................18
1.7 Law of Large Numbers........................................19
1.8 Kolmogorov s Interpretation of Probability..................21
1.9 Random Vectors; Existence of Joint Probability
Distribution....................................................22
1.9.1 Marginal probability..................................22
1.9.2 Prom Boole and Vorob ev to Bell....................23
1.9.3 No-signaling in quantum physics....................25
1.9.4 Kolmogorov theorem about existence of stochastic
processes..............................................28
xi
xii Probability and Randomness: Quantum versus Classical
1.10 Frequency (von Mises) Theory of Probability..............29
1.11 Subjective Interpretation of Probability....................36
1.12 Gnedenko s Viewpoint on Subjective Probability and
Bayesian Inference............................................43
1.13 Cournot s Principle............................................44
2. Randomness 47
2.1 Random Sequences..................................48
2.1.1 Approach of von Mises: randomness as
unpredictability......................................50
2.1.2 Laplace-Ville-Martin-Löf: randomness as typicality 53
2.2 Kolmogorov: Randomness as Complexity ..................54
2.3 Kolmogorov-Chaitin Randomness............................56
2.4 Randomness: Concluding Remarks..........................58
3. Supplementary Notes on Measure-theoretic and
Frequency Approaches 61
3.1 Extension of Probability Measure............................61
3.1.1 Lebesgue measure on the real line..................61
3.1.2 Outer and inner probabilities, Lebesgue
measurability..........................................62
3.2 Complete Probability..........................................65
3.3 Von Mises Views..............................................67
3.3.1 Problem of verification ..............................67
3.3.2 Jordan measurability ................................69
3.4 Role of the Axiom of Choice in the Measure Theory ... 70
3.5 Possible Generalizations of Probability Theory ............71
3.5.1 Negative probabilities................................72
3.5.2 On generalizations of the frequency theory of
probability............................................74
3.5.3 p-adic probability....................................75
3.6 Quantum Theory: No Statistical Stabilization for
Hidden Variables? ............................................77
4. Introduction to Quantum Formalism 79
4.1 Quantum States................................................79
4.2 First Steps Towards Quantum Measurement Theory ... 84
4.2.1 Projection measurements............................85
Contents xiii
4.2.2 Projection postulate for pure states................87
4.3 Conditional Probabilities......................................88
4.4 Quantum Logic................................................91
4.5 Atomic Instruments ..........................................92
4.6 Symmetric Informationally Complete Quantum
Instruments....................................................93
4.7 Schrödinger and von Neumann Equations..................94
4.8 Compound Systems............................................95
4.9 Dirac s Symbolic Notations ..................................98
4.10 Quantum Bits..................................................99
4.11 Entanglement..................................................100
4.12 General Theory of Quantum Instruments....................101
4.12.1 Davis-Levis instruments..............................102
4.12.2 Complete positivity..................................104
5. Quantum and Contextual Probability 107
5.1 Probabilistic Structure of Two-Slit Experiment............110
5.2 Quantum versus Classical Interference......................113
5.2.1 Quantum waves?......................................114
5.2.2 Prequantum classical statistical field theory .... 115
5.3 Formula of Total Probability with Interference Term ... 120
5.3.1 Context-conditioning ................................120
5.3.2 Contextual analog of the two-slit experiment . . . 123
5.3.3 Non-Kolmogorovean probability models............125
5.3.4 Trigonometric and hyperbolic interference..........126
5.4 Constructive Wave Function Approach......................127
5.4.1 Inverse Born s rule problem..........................127
5.4.2 Quantum-like representation algorithm............130
5.4.3 Double stochasticity..................................133
5.4.4 Supplementary observables..........................134
5.4.5 Symmetrically conditioned observables ............136
5.4.6 Non-doubly stochastic matrices of transition
probabilities..........................................137
5.5 Contextual Probabilistic Description of Measurements . . 137
5.5.1 Contexts, observables, and measurements.....137
5.5.2 Contextual probabilistic model......................139
5.5.3 Probabilistic compatibility (noncontextuality) . . 142
5.6 Quantum Formula of Total Probability......................146
5.6.1 Interference of von Neumann-Lüders observables . 147
xiv Probability and Randomness: Quantum versus Classical
5.6.2 Interference of positive operator valued measures . 150
6. Interpretations of Quantum Mechanics and Probability 155
6.1 Classification of Interpretations..............................155
6.1.1 Realism and reality..................................155
6.1.2 Epistemic and ontic description....................158
6.1.3 Individual and statistical interpretations ..........160
6.1.4 Subquantum models and models with hidden
variables..............................................161
6.1.5 Nonlocality............................................162
6.2 Interpretations of Probability and Quantum State.....162
6.3 Orthodox Copenhagen Interpretation........................163
6.4 Von Neumann s Interpretation................................167
6.5 Zeilinger-Brukner Information Interpretation ..............168
6.6 Copenhagen-Göttingen Interpretation: From Bohr and
Pauli to Plotnitsky............................................174
6.7 Quantum Bayesianism - QBism..............................177
6.7.1 QBism childhood in Växjö..........................177
6.7.2 Quantum theory is about evaluation of expecta-
tions for the content of personal experience .... 179
6.7.3 QBism as a probability update machinery..........180
6.7.4 Agents constrained by Born s rule..................185
6.7.5 QBism challenge: Born rule or Hilbert space
formalism?............................................187
6.7.6 QBism and Copenhagen interpretation?............188
6.8 Interpretations in the Spirit of Einstein......................189
6.9 Växjö Interpretation..........................................191
6.10 Projection Postulate: von Neumann and Lüders Versions . 194
7. Randomness: Quantum Versus Classical 199
7.1 Irreducible Quantum Randomness ..........................199
7.2 Lawless Universe? Digital Philosophy? ..........202
7.3 Unpredictability and Indeterminism ............205
8. Probabilistic Structure of Bell s Argument 211
8.1 CHSH-inequality in Kolmogorov Probability Theory . . . 214
8.2 Bell-test: Conditional Compatibility of Observables .... 215
8.2.1 Random choice of settings ..........................217
Contents xv
8.2.2 Construction of Kolmogorov probability space . . 218
8.2.3 Validity of CHSH-inequality for correlations taking
into account randomness of selection of experimen-
tal settings............................................220
8.2.4 Quantum correlations as conditional classical
correlations............................................221
8.2.5 Violation of the CHSH-inequality for classical
conditional correlations..............................223
8.3 Statistics: Data from Incompatible Contexts................224
8.3.1 Medical studies........................................224
8.3.2 Cognition and psychology..............225
8.3.3 Consistent histories..................................226
8.3.4 Hidden variables......................................226
8.4 Contextuality of Bell s Test from the Viewpoint of Quan-
tum Measurement Theory....................................227
8.5 Inter-relation of Observations on a Compound System and
its Subsystems ................................................229
8.5.1 Averages..............................................229
8.5.2 Correlations ..........................................230
8.5.3 Towards proper quantum formalization of Bell s
experiment............................................231
8.6 Quantum Conditional Correlations..........................232
8.7 Classical Probabilistic Realization of Random Numbers
Certified by Bell s Theorem ..................................234
9. Quantum Probability Outside of Physics: from Molecular
Biology to Cognition 237
9.1 Quantum Information Biology................................238
9.2 Inter-relation of Quantum Bio-physics and Information
Biology ........................................................240
9.3 Prom Information Physics to Information Biology.....242
9.3.1 Operational approach................................242
9.3.2 Free will problem ....................................242
9.3.3 Bohmian mechanics on information spaces and
mental phenomena....................................243
9.3.4 Information interpretation is biology friendly . . . 245
9.4 Nonclassical Probability? Yes! But, Why Quantum? . . . 245
Appendix A Växjö Interpretation-2002 249
xvi
Probability and Randomness: Quantum versus Classical
A.l Contextual Statistical Realist Interpretation of Physical
Theories........................................................249
A. 2 Citation with Comments......................................257
A.3 On Romantic Interpretation of Quantum Mechanics . . . 258
Appendix B Analogy between non-Kolmogorovian
Probability and non-Euclidean Geometry 261
Bibliography 263
Index
279
|
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| spelling | Khrennikov, Andrei 1958- (DE-588)128568410 aut Probability and randomness quantum versus classical Andrei Khrennikov (Linnaeus University, Sweden) Covent Garden, London Imperial College Press [2016] Singapore Distributed by World Scientific Publishing Co. Pte. Ltd © 2016 1 Online-Ressource (xvi, 282 Seiten) txt rdacontent c rdamedia cr rdacarrier Includes bibliographical references and index "Creating a rigorous mathematical theory of randomness is far from being complete, even in the classical case. Interrelation of Classical and Quantum Randomness rectifies this and introduces mathematical formalisms of classical and quantum probability and randomness with brief discussion of their interrelation and interpretational and foundational issues. The book presents the essentials of classical approaches to randomness, enlightens their successes and problems, and then proceeds to essentials of quantum randomness. Its wide-ranging and comprehensive scope makes it suitable for researchers in mathematical physics, probability and statistics at any level"... Mathematische Physik Quantentheorie Probabilities Quantum theory Mathematical physics Erscheint auch als Druckausgabe 978-1-78326-796-5 http://www.worldscientific.com/worldscibooks/10.1142/P1036#t=toc Verlag URL des Erstveröffentlichers Volltext LoC Fremddatenuebernahme application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=028987020&sequence=000001&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis HBZ Datenaustausch application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=028987020&sequence=000003&line_number=0002&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis |
| spellingShingle | Khrennikov, Andrei 1958- Probability and randomness quantum versus classical Mathematische Physik Quantentheorie Probabilities Quantum theory Mathematical physics |
| title | Probability and randomness quantum versus classical |
| title_auth | Probability and randomness quantum versus classical |
| title_exact_search | Probability and randomness quantum versus classical |
| title_full | Probability and randomness quantum versus classical Andrei Khrennikov (Linnaeus University, Sweden) |
| title_fullStr | Probability and randomness quantum versus classical Andrei Khrennikov (Linnaeus University, Sweden) |
| title_full_unstemmed | Probability and randomness quantum versus classical Andrei Khrennikov (Linnaeus University, Sweden) |
| title_short | Probability and randomness |
| title_sort | probability and randomness quantum versus classical |
| title_sub | quantum versus classical |
| topic | Mathematische Physik Quantentheorie Probabilities Quantum theory Mathematical physics |
| topic_facet | Mathematische Physik Quantentheorie Probabilities Quantum theory Mathematical physics |
| url | http://www.worldscientific.com/worldscibooks/10.1142/P1036#t=toc http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=028987020&sequence=000001&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=028987020&sequence=000003&line_number=0002&func_code=DB_RECORDS&service_type=MEDIA |
| work_keys_str_mv | AT khrennikovandrei probabilityandrandomnessquantumversusclassical |