Foundations of Quantum Mechanics, an Empiricist Approach Fundamental Theories of Physics An International Book Series on The Fundamental Theories of Physics: Their Clarification, Development and Application Editor: ALWYN VAN DER MERWE, University of Denver, U.S.A. EditorialAdvisoryBoard: JAMES T. CUSHING, University of Notre Dame, U.S.A. GIANCARLO GHIRARDI, University of Trieste, Italy LAWRENCE P. HORWITZ, Tel-Aviv University, Israel BRIAN D. JOSEPHSON, University of Cambridge, U.K. CLIVE KILMISTER, University of London, U.K. PEKKA J. LAHTI, University of Turku, Finland ASHER PERES, Israel Institute of Technology, Israel EDUARD PRUGOVECKI, University of Toronto, Canada TONY SUDBURY, University of York, U.K. HANS-JÜRGEN TREDER, Zentralinstitut für Astrophysik der Akademie der Wissenschaften, Germany Volume 127 Foundations of Quantum Mechanics, an Empiricist Approach by Willem M. de Muynck Eindhoven University of Technology, The Netherlands KLUWER ACADEMIC PUBLISHERS NEW YORK,BOSTON, DORDRECHT, LONDON, MOSCOW eBookISBN: 0-306-48047-6 Print ISBN: 1-4020-0932-1 ©2004 Kluwer Academic Publishers NewYork, Boston, Dordrecht, London, Moscow Print ©2002 Kluwer Academic Publishers Dordrecht All rights reserved No part of this eBook maybe reproducedor transmitted inanyform or byanymeans,electronic, mechanical, recording, or otherwise, without written consent from the Publisher Created in the United States of America Visit Kluwer Online at: http://kluweronline.com and Kluwer's eBookstoreat: http://ebooks.kluweronline.com For Yolanda, Catelijne and Sarah Contents Preface xix 1 Standard and generalized formalisms of quantum mechanics 1 1.1 Basic postulates of standard quantum mechanics 1 1.2 Some elements of quantum field theory 7 1.3 Simultaneous and joint measurement ofcompatible observables 9 1.3.1 Postulate of local commutativity 10 1.4 Mixtures 12 1.4.1 Density operators 12 1.4.2 Basic postulates for mixtures 14 1.4.3 Density operators as vectors in a linear space 16 1.5 Coupled systems 16 1.5.1 Subsystems 16 1.5.2 Correlation observables 18 1.5.3 Polar decomposition 19 1.5.4 Entangled states 21 1.6 Projection or reduction postulate 22 1.7 Uncertainty relations 26 1.7.1 Heisenberg inequality 26 1.7.2 Entropic uncertainty relations 28 1.8 Proposition calculus of standard quantum mechanics 31 1.8.1 Boolean lattice of classical propositions; objectivity 31 vii viii CONTENTS 1.8.2 Propositionsreferring toasinglequantummechanicalobservable 32 1.8.3 Two or more compatible observables 33 1.8.4 Incompatible standard observables 34 1.8.5 States on the lattice of propositions 37 1.9 Generalized quantum mechanical observables 40 1.9.1 Quantum mechanical observables as positive operator-valued measures 40 1.9.2 Joint measurement of generalized observables 43 1.9.3 Naimark’stheorem 47 1.9.4 Phase observables 50 1.10 Correspondence with classical mechanics 51 1.10.1 Ehrenfest’stheorem 52 1.10.2 Dirac quantization 53 1.10.3 Classical statistical mechanics 54 1.11 Phase space representations 57 1.11.1 Introduction 57 1.11.2 Wigner-Weyl representation 58 1.11.3 Husimi representation 61 1.11.4 representations 64 1.11.5 Relation to operator ordering 66 1.11.6 Wigner’s theorem 67 1.11.7 The Schrödinger equation in the representation 69 2 Empiricist and realist interpretations of quantum mechanics 73 2.1 Introduction 73 2.2 Empiricist interpretation of quantum mechanics 74 2.2.1 Logical positivism/empiricism and empiricist interpretation 77 2.3 Realist interpretation of quantum mechanics 82 2.4 Empiricist or realist interpretation: which one to choose? 87 CONTENTS ix 2.4.1 Logical positivism/empiricism and realist interpretation 87 2.4.2 The classical paradigm 88 2.4.3 Double role of the quantum mechanical observable 93 2.4.4 Interpretations of quantum field theory 95 2.4.5 Contextualistic-realist interpretation 101 2.5 Some consequences 106 2.5.1 Empiricist interpretation and generalized observables 106 2.5.2 Realist interpretation ofquantum mechanics, and hidden vari- ables 108 2.5.3 Interpretations and the classical limit 109 3 Quantum mechanical description of measurement, and the “mea- surement problem” 113 3.1 The (conventional) “measurement problem” 113 3.1.1 Schrödinger’s cat 113 3.1.2 Interpretations, the “measurement problem”, andthe problem of quantum mechanical measurement 115 3.1.3 Three tentative answers 117 3.2 Quantum mechanical description of the measurement process 121 3.2.1 Asimplifiedmodel 121 3.2.2 Von Neumann’s proofofconsistency ofprojection and unitary evolution 123 3.2.3 “Orthodox” solution to the “measurement problem” 125 3.2.4 Measurements of first and second kind 127 3.2.5 The “measurement problem” for measurements ofthe second kind 132 3.2.6 Conditional preparation 134 3.2.7 Quantum jumps 140 3.3 Quantum mechanical description of the measurement process and POVMs 146 x CONTENTS 3.3.1 Possibility of POVMs 146 3.3.2 Pointer observables and POVMs 149 3.3.3 Measurement and empiricist interpretation 151 3.3.4 Conditional preparation and generalized observables 152 3.3.5 Generalized vonNeumannprojection forgeneralizedobservables155 3.3.6 Measurement and information 157 3.4 Decoherence 160 3.4.1 Ergodicity 161 3.4.2 Environment-induced superselection 162 3.4.3 Spontaneous localization 163 3.4.4 General evolution equation for open systems 165 3.4.5 Critique of the decoherence solution to the problem of quan- tum measurement 166 4 The Copenhagen interpretation 171 4.1 Introduction 171 4.2 Completeness of quantum mechanics 173 4.2.1 Completeness in a wider sense 173 4.2.2 Completeness in a restricted sense 177 4.2.3 Entanglement of the ‘(in)completeness’ question with other issues; sources ofconfusion 182 4.3 The correspondence principle 190 4.3.1 Weak and strong forms ofthe correspondence principle 190 4.3.2 Strong form ofthe correspondence principle 191 4.3.3 Realism versus empiricism, and correspondence 193 4.3.4 Critique ofthe correspondence principle 195 4.4 Complementarity in a wider and in a restricted sense 197 4.5 ‘Thought experiments’ 200 4.5.1 Diffraction of particles through a slit 200 CONTENTS xi 4.5.2 The double-slit experiment 201 4.5.3 The microscope 206 4.6 Meaning ofthe ‘complementarity’ concept 207 4.6.1 Two meanings of‘to determine’ 208 4.6.2 Heisenberg’s disturbance theory ofmeasurement 212 4.6.3 Complementarity according to Bohr 214 4.6.4 Particle-wave duality 218 4.6.5 Parallel and circular complementarity 219 4.6.6 Complementarity and the projection postulate 220 4.6.7 Complementarity and consciousness 223 4.7 Critique ofthe complementarity principle 225 4.7.1 Einstein 225 4.7.2 Margenau 226 4.7.3 Ballentine 228 4.7.4 Recent developments 231 4.8 Complementarity and empiricist interpretation 235 5 The Einstein-Podolsky-Rosen problem 239 5.1 Introduction 239 5.2 Formulation of the EPR problem in terms of physical quantities 241 5.2.1 The EPR reasoning 241 5.2.2 Discussion ofthe EPR reasoning 243 5.3 Bohr’s answer to EPR 246 5.3.1 Criticisms of Bohr’s answer to EPR 250 5.4 Formulation of the EPR problem in terms of state vectors 255 5.4.1 The EPR reasoning in terms of state vectors 255 5.4.2 Discussion of the state vector approach to EPR 259 5.4.3 Modified EPR experiments 265