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1 Introduction
E. Joos ........................................................ 1
2 Basic Concepts and Their Interpretation
H. D. Zeh ........................................ ............. 7
2.1 The Phenomenon of Decoherence ............................. 7
2.1.1 Superpositions ................. ............. .. ... 7
2.1.2 Superselection Rules ................................... 11
2.1.3 Decoherence by "Measurements" ......................... 13
2.2 Observables as a Derived Concept ........................... 17
2.3 The Measurement Problem .................................. 21
2.4 Density Matrix, Coarse Graining, and "Events" ................ 33
2.5 Conclusions .................................... ........ 40
3 Decoherence Through Interaction with the Environment
E. Joos ................................................. 41
3.1 The General Mechanisms of Decoherence ...................... 47
3.1.1 Dynamics of Quantum Correlations ..................... 47
3.1.2 Scattering Processes ................................... 55
3.1.3 Environment-Induced Superselection Rules ............... 57
3.2 Localization of Objects .................................... 62
3.2.1 Localization Through Ideal Measurements ................. 64
3.2.1.1 Spatial Decoherence ............................ 64
3.2.1.2 Equation of Motion ............ ........... 70
3.2.1.3 Decohering Wave Packets ................ ...... 75
3.2.1.4 More General Recoil-Free Decoherence .............. 78
3.2.2 Decoherence and Dissipation ............................ 79
3.2.2.1 Quantum Brownian Motion ................. ... 79
3.2.2.2 Ehrenfest Theorems ........................... 87
3.2.2.3 Decoherence Versus Friction .......................89
3.2.3 Wigner Function Description of Decoherence .............. 90
3.2.4 Molecular Structure .................................. 99
3.2.5 Decoherence in the Brain .............................. 107
3.3 Dynamical Consequences ............................. ... 109
3.3.1 The Quantum Zeno Effect ............................. 110
3.3.1.1 Phenomenological Description ..... ............ 111
3.3.1.2 An Experimental Test ...... ..... ......... 118
3.3.1.3 Models for the Quantum Zeno Effect . ........... 122
3.3.2 Master Equations ........................... ........ . 126
3.3.2.1 Pauli Equation ................... ... . .. 126
3.3.2.2 Lindblad's Form of Master Equations ............... 133
3.3.3 Dynamical Stability of States . ................ 135
3.3.3.1 Sensitivity to the Presence of an Environment ...... 136
3.3.3.2 Decoherence and Quantum Computation ......... 145
3.3.3.3 Quantum Nondemolition Measurements ............ 147
3.3.4 Decoherence and Quantum Chaos . ... ......... 149
3.3.4.1 Classical Versus Quantum Chaos................... 149
3.3.4.2 Example: The Kicked Rotator ..................... 152
3.3.4.3 Quantum (?) Chaos in the Solar System .. ..... 156
3.3.4.4 Decoherence Through Chaotic Environments ........ 159
3.4 Interpretational Issues .................. ................. 161
3.4.1 Null Measurements, Histories and Quantum Jumps......... 161
3.4.2 Quantum "Information" and Teleportation . ............... 172
3.4.3 True, False, and Fake Decoherence . ................. . 175
4 Decoherence in Quantum Field Theory
and Quantum Gravity
C. Kiefer..... ......... ............... ......................... 181
4.1 Decoherence in Quantum Electrodynamics ..................... 182
4.1.1 "Measurement" of Charges by Fields ..................... 182
4.1.2 "Measurement" of Electromagnetic Fields by Charges ...... 188
4.2 Decoherence and the Gravitational Field ...................... 192
4.2.1 Emergence of Classical Spacetime ........................ 192
4.2.2 The Formalism of Quantum Cosmology ................... 196
4.2.3 Decoherence in Quantum Cosmology ..................... 199
4.2.4 Classicality of Primordial Fluctuations in the Early Universe 209
4.2.5 Black Holes, Wormholes, and String Theory ............... 218
5 Consistent Histories and Decoherence
C. Kiefer..... ................. . ..... .......... ............. 227
5.1 Influence Functional and Its Application
to Quantum Brownian Motion ............................... 229
5.2 Definition and Properties of Consistent Histories ............... 238
5.3 Reduced Density Matrix and Decoherence ..................... 247
5.4 Consistent Histories, Arrow of Time, and Quantum Gravity...... 251
6 Superselection Rules and Symmetries
D. Giulini.................. ............................. 259
6.1 States, Observables, and Superselection Rules .................. 261
6.1.1 Spaces of States ........................................ 262
6.1.2 Spaces of Observables .................................. 267
6.1.3 Superselection Rules ................................. 275
6.2 Symmetries and Superselection Rules ......................... 278
6.2.1 Symmetries .......... .. .................... ..... 279
6.2.2 Superselection Rules from Symmetries .................... 284
6.2.3 An Example: The Univalence Superselection Rule .......... 285
6.2.4 Discussion and Caveats ................... ......... 287
6.3 Physical Symmetries Versus Gauge Transformations .......... 289
6.3.1 Configuration Spaces and Spaces of State ................. 289
6.3.2 Symmetries and Redundant State Spaces ................. 292
6.3.3 Symmetries, Redundancies, and Superselection Rules ....... 297
6.4 Superselection Rules in Field Theory ......................... 303
6.4.1 Charge and Asymptotic Flux-Distribution in QED ......... 304
6.4.2 Poincare Charges in General Relativity .................. 310
6.4.3 Decoherence and Charge Superselection Rules ............ 312
7 Open Quantum Systems
J. Kupsch ...................................................... 317
7.1 Reduced Dynamics ................................... ... 319
7.2 Projection Methods ......................................... 321
7.3 Generalized Master Equations ................................ 327
7.4 Markov Approximation and Semigroups ....................... 330
7.5 Quantum Stochastic Processes ............................... 333
7.6 Induced Superselection Sectors ............................... 339
7.6.1 General Considerations ................................. 339
7.6.2 Hamiltonian Models of Decoherence ...................... 341
7.6.2.1 The Araki-Zurek Models .......................... 343
7.6.2.2 Particle Coupled to a Massless Boson Field.......... 345
7.6.2.3 Models with Scattering ........................ 347
7.6.2.4 Heisenberg Picture ................... 348
7.7 Mathematical Supplement ...................................350
7.7.1 Spaces of Linear Operators .............................. 350
7.7.2 Complete Positivity ................................... 354
7.7.3 Entropy Inequalities ........... ............... . 355
8 Stochastic Collapse Models
I.-O. Stamatescu ............................................... 357
8.1 The Question of State Vector Reduction ....................... 357
8.1.1 Two Points of View .................................... 357
8.1.2 Decoherence, Collapse, Measurement ..................... 358
8.1.3 Various Approaches to Collapse ......................... 363
8.2 Spontaneous Collapse Models ............................... 369
8.2.1 The Dynamical Collapse Hypothesis ...................... 369
8.2.2 Spontaneous Localization by a Jump Process .............. 370
8.2.3 Continuous Spontaneous Localization ..................... 373
8.3 Spontaneous Localization, Quantum State Diffusion
and Decoherence ........................................... 378
9 Related Concepts and Methods
H. D. Zeh ................... ............. .................. 383
9.1 Phase Averaging in Ensembles ("Dephasing") .................. 383
9.2 Ergodicity and Irreversible Amplification ...................... 385
9.3 Dressing of States ...................... ............ ..... 387
9.4 Symmetry Breaking and Collective Motion ..................... 388
Al Equation of Motion of a Mass Point
E. Joos ............................................ . 395
Ax Solutions for the Equation of Motion
E. Joos................... .. ... ........................... 399
A2 1 Gaussian Density Matrices ..... ..... ......... ..... 399
A2 2 Green Functions ....................................... . 402
A2 3 Some Derived Quantities ...... .......................... 403
As Elementary Properties of Composite Systems
in Quantum Mechanics
D . G iulini ................... ....................... 407
Az Quantum Correlations
I.-O . Stam atescu ................................................415
A5 Hamiltonian Formulation of Quantum Mechanics
D. Giulini ...................................................... 419
A6 Galilean Symmetry
of Non-Relativistic Quantum Mechanics
D. Giulini.......................... .. ........ 425
A7 Stochastic Processes
I.-O. Stamatescu...... ....... ......... ............... 433
A7 1 Random Variables and Distributions .......................... 433
A7 2 Markov Chains ... .................................. .... 434
A7 3 Stochastic Processes ......................... 436
A7 4 The Fokker-Planck Equation ................................. 437
A7 5 Stochastic Differential Equations ............................. 439
References ..................................... .............. 445
Library of Congress Subject Headings for this publication: Quantum theory, Quantum field theory, Coherent states