Compendium of Quantum Physics

Daniel GreenbergerKlaus HentschelFriedel WeinertEditors

Compendiumof Quantum Physics

Concepts, Experiments, Historyand Philosophy

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Editors

Daniel GreenbergerDepartment of PhysicsThe City College of New York138th St. & Convent Ave.New York NY 10031USAgreenbgr@sci.ccny.cuny.edudansnzy@nyc.rr.com

Friedel WeinertDepartment of Social Sciencesand HumanitiesUniversity of BradfordBradford BD7 1DPUKf.weinert@bradford.ac.uk

Klaus HentschelUniversity of StuttgartSection for the History of Science& TechnologyKeplerstr. 17D-70174 StuttgartGermanyklaus.hentschel@po.hi.uni-stuttgart.de

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Preface

Since its inception in the early part of the twentieth century, quantum physics hasfascinated the academic world, its students, and even the general public. In fact, it is– or has become – a highly interdisciplinary field. On a topic such as “the physics ofthe atom” the disciplines of physics, philosophy, and history of science interconnectin a remarkable way, and to an extent that is revealed in this volume for the firsttime. This compendium brings together some 90 researchers, who have authoredapproximately 185 articles on all aspects of quantum theory. The project is trulyinternational and interdisciplinary because it is a compilation of contributions byhistorians of science, philosophers, and physicists, all interested in particular aspectsof quantum physics. A glance at the biographies at the end of the volume revealsauthor affiliations in no fewer than twenty countries: Australia, Austria, Belgium,Canada, Denmark, Finland, France, Germany, Greece, Italy, Israel, the Netherlands,New Zealand, Norway, Poland, Portugal, Spain, Switzerland, the United Kingdomand the United States. Indeed, the authors are not only international, they are alsointernationally renowned – with three Physics Nobel Prize laureates among them.

The basic idea and motivation behind the compendium is indicated in its subtitle,namely, to describe in concise and accessible form the essential concepts and exper-iments as well as the history and philosophy of quantum physics. The length of thecontributions varies according to the topic, and all texts are written by recognizedexperts in the respective fields. The need for such a compendium was originallyperceived by one of the editors (FW), who later discovered that many physicistsshared this view. Due to the interdisciplinary nature of this endeavor, it would havebeen impossible to realize it without the expertise and active participation of a pro-fessional physicist (DG) and a historian of science (KH). We should not forget,however, that it was brought to life by the numerous contributions of the manyauthors from around the world, who generously offered their time and expertise towrite their respective articles. The contributions appear in alphabetical order by title,and include many cross-references, as well as selected references to the literature.The volume includes a short English–French–German lexicon of common terms inquantum physics. This will be especially helpful to anyone interested in exploring

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historical documents on quantum physics, the theory of which was developed side-by-side in these three cultures and languages.

The editors would like to thank Brigitte Falkenburg and Peter Mittelstaedt fortheir initial work on the project. Angela Lahee (at Springer publishers) deserves ourgratitude for her unwavering support and patience during the four years it has takento turn the idea for this compendium into reality.

January 2009 Dan GreenbergerKlaus HentschelFriedel Weinert

Contents

Alphabetical Compendium

Aharonov–Bohm Effect . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1

Aharonov–Casher Effect . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3

Algebraic Quantum Mechanics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6

Angular Momentum . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10

Anyons . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10

Aspect Experiment . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14

Asymptotic Freedom . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18

Atomic Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18

Atomic Models, J.J. Thomson’s “Plum Pudding” Model . . . . . . . . . . . . . . . . . . . . 18

Atomic Models, Nagaoka’s Saturnian Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22

Bell’s Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24

Berry’s Phase . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31

Black Body . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36

Black-Body Radiation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39

Bohm Interpretation of Quantum Mechanics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43

Bohmian Mechanics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47

Bohm’s Approach to the EPR Paradox . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55

Bohr’s Atomic Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58

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Bohr–Kramers–Slater Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62

Born Rule and its Interpretation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64

Bose–Einstein Condensation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71

Bose–Einstein Statistics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74

Bremsstrahlung . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 78

Brownian Motion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81

Bub–Clifton Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 84

Casimir Effect . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 87

Cathode Rays . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 89

Causal Inference and EPR . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93

Clauser-Horne-Shimony-Holt (CHSH) – Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . 96

Cluster States . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 96

Coherent States . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 106

Color Charge Degree of Freedom in Particle Physics . . . . . . . . . . . . . . . . . . . . . . . . 109

Complementarity Principle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111

Complex-Conjugate Number . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 114

Compton Experiment (or Compton Effect) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 115

Consistent Histories . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 117

Copenhagen Interpretation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 122

Correlations in Quantum Mechanics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 122

Correspondence Principle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 125

Counterfactuals in Quantum Mechanics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 132

Covariance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 136

CPT Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 138

Creation and Annihilation Operators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 139

Creation and Detection of Entanglement . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 145

Davisson–Germer Experiment . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 150

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De Broglie Wavelength (λ = h/p) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 152

Decay . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 154

Decoherence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 155

Degeneracy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 159

Delayed-Choice Experiments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 161

Density Matrix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 166

Density Operator . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 166

Diffeomorphism Invariance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 170

Dirac Equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 171

Dirac Notation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 172

Double-Slit Experiment (or Two-Slit Experiment) . . . . . . . . . . . . . . . . . . . . . . . . . . . 174

Effect . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 179

Ehrenfest Theorems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 180

Eigenstates, Eigenvalues. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 182

Einstein Locality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 182

Electron Interferometry .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 188

Electrons . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 195

Ensembles in Quantum Mechanics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 199

Entanglement . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 201

Entanglement Purification and Distillation. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 202

Entropy of Entanglement . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 205

EPR-Problem (Einstein-Podolsky-Rosen Problem) . . . . . . . . . . . . . . . . . . . . . . . . . . 209

Errors and Paradoxes in Quantum Mechanics. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 211

Exclusion Principle (or Pauli Exclusion Principle) . . . . . . . . . . . . . . . . . . . . . . . . . . . 220

Experimental Observation of Decoherence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 223

Fermi–Dirac Statistics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 230

Feynman Diagrams . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 235

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Fine-Structure Constant . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 239

Franck–Hertz Experiment . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 241

Functional Integration; Path Integrals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 243

Gauge Symmetry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 248

Generalizations of Quantum Statistics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 255

GHZ (Greenberger–Horne–Zeilinger) Theorem and GHZStates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 258

Gleason’s Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 263

Grover’s Algorithm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 266

GRW Theory (Ghirardi, Rimini, Weber Model of QuantumMechanics) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 266

Hamiltonian Operator. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 271

Hardy Paradox . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 275

Heisenberg Microscope . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 279

Heisenberg Picture . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 280

Heisenberg Uncertainty Relation (Indeterminacy Relations) . . . . . . . . . . . . . . . 281

Hermitian Operator . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 284

Hidden Variables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 284

Hidden-Variables Models of Quantum Mechanics(Noncontextual and Contextual) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 287

Hilbert Space . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 291

Holism in Quantum Mechanics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 295

Identity of Quanta . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 299

Identity Operator . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 304

Ignorance Interpretation of Quantum Mechanics . . . . . . . . . . . . . . . . . . . . . . . . . . . 305

Indeterminacy Relations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 306

Indeterminism and Determinism in Quantum Mechanics . . . . . . . . . . . . . . . . . . . 307

Indistinguishability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 311

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Interaction-Free Measurements (Elitzur–Vaidman, EV IFM). . . . . . . . . . . . . . . 317

Interpretations of Quantum Mechanics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 322

Invariance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 322

Ithaca Interpretation of Quantum Mechanics. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 325

jj-Coupling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 327

Kaluza–Klein Theory. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 328

Kochen–Specker Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 331

Lande’s g-factor and g-formula . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 336

Large-Angle Scattering . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 337

Light Quantum . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 339

Locality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 347

Loopholes in Experiments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 348

Luders Rule . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 356

Mach–Zehnder Interferometer . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 359

Magnetic Resonance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 359

Many Worlds Interpretation of Quantum Mechanics . . . . . . . . . . . . . . . . . . . . . . . 363

Matrix Mechanics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 368

Matter Waves . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 371

Measurement Problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 373

Measurement Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 374

Mesoscopic Quantum Phenomena . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 379

Metaphysics of Quantum Mechanics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 384

Mixed State . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 389

Mixing and Oscillations of Particles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 390

Modal Interpretations of Quantum Mechanics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 394

Neutron Interferometry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 402

No-Cloning Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 404

xii Contents

Nonlocality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 405

Nuclear Fission . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 411

Nuclear Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 414

Objectification . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 417

Objective Quantum Probabilities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 420

Observable . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 425

One- and Two-Photon Interference . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 428

Operational Quantum Mechanics, Quantum Axiomaticsand Quantum Structures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 434

Operator . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 440

Orthodox Interpretation of Quantum Mechanics . . . . . . . . . . . . . . . . . . . . . . . . . . . . 444

Orthonormal Basis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 447

Parity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 450

Particle Physics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 455

Particle Tracks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 460

Parton Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 465

Paschen–Back Effect . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 468

Pauli Exclusion Principle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 470

Pauli Spin Matrices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 470

Photoelectric Effect . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 472

Photon . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 476

Pilot Waves . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 476

Planck’s Constant h . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 478

POVM (Positive Operator Value Measure) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 480

Probabilistic Interpretation of Quantum Mechanics . . . . . . . . . . . . . . . . . . . . . . . . 485

Probability in Quantum Mechanics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 492

Projection . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 497

Contents xiii

Projection Postulate . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 499

Propensities in Quantum Mechanics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 502

Protective Measurements. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 505

Pure States . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 508

Quantization (First, Second) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 509

Quantization (Systematic) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 510

Quantum Chaos . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 514

Quantum Chemistry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 518

Quantum Chromodynamics (QCD) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 524

Quantum Communication . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 527

Quantum Computation. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 533

Quantum Electrodynamics (QED) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 539

Quantum Entropy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 543

Quantum Eraser . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 546

Quantum Field Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 549

Quantum Gravity (General) and Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 565

Quantum Hall Effect . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 572

Quantum Interrogation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 591

Quantum Jump Experiments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 595

Quantum Jumps . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 599

Quantum Logic . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 601

Quantum Mechanics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 605

Quantum Numbers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 605

Quantum State Diffusion Theory (QSD) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 608

Quantum State Reconstruction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 609

Quantum Statistics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 611

Quantum Theory, 1914–1922 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 613

xiv Contents

Quantum Theory, Crisis Period 1923–Early 1925 . . . . . . . . . . . . . . . . . . . . . . . . . . . 613

Quantum Theory, Early Period (1900–1913) .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 617

Quantum Zeno Effect . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 622

Quarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 626

Quasi-Classical Limit. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 626

Radioactive Decay Law (Rutherford–Soddy) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 630

Relativistic Quantum Mechanics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 632

Renormalization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 637

Rigged Hilbert Spaces in Quantum Physics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 640

Rigged Hilbert Spaces for the Dirac Formalismof Quantum Mechanics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 651

Rigged Hilbert Spaces and Time Asymmetric Quantum Theory . . . . . . . . . . . . 660

Russell–Saunders Coupling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 671

Rutherford Atom . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 671

Scattering Experiments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 676

Schrodinger Equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 681

Schrodinger’s Cat . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 685

Schrodinger Picture . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 689

Selection Rules . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 690

Self-Adjoint Operator . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 692

Semi-classical Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 697

Shor’s Algorithm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 702

Solitons . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 702

Sommerfeld School . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 716

Specific Heats . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 719

Spectral Decomposition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 721

Spectroscopy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 721

Contents xv

Spin . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 726

Spin Echo . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 731

Spin Statistics Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 733

Squeezed States . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 736

Standard Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 738

Stark Effect . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 738

States in Quantum Mechanics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 742

States, Pure and Mixed, and Their Representations . . . . . . . . . . . . . . . . . . . . . . . . . 744

State Operator . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 746

Statistical Operator . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 746

Stern–Gerlach Experiment. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 746

Superconductivity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 750

Superfluidity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 758

Superluminal Communication in Quantum Mechanics . . . . . . . . . . . . . . . . . . . . . 766

Superposition Principle (Coherent and IncoherentSuperposition) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 769

Superselection Rules. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 771

Symmetry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 779

Time in Quantum Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 786

Trace . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 793

Transactional Interpretation of Quantum Mechanics . . . . . . . . . . . . . . . . . . . . . . . 795

Tunneling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 799

Two-State Vector Formalism . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 802

Uncertainty Principle, Indetermincay Relations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 807

Unitary Operator . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 807

Vector Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 810

Wave Function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 812

xvi Contents

Wave Function Collapse . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 813

Wave Mechanics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 822

Wave Packet . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 828

Wave-Particle Duality: Some History . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 830

Wave-Particle Duality: A Modern View . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 835

Weak Value and Weak Measurements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 840

Werner States . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 843

Which-Way or Welcher-Weg-Experiments .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 845

Wigner Distribution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 851

Wigner’s Friend . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 854

X-Rays . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 859

Zeeman Effect . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 862

Zero-Point Energy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 864

English/German/French Lexicon of Terms . . . . . . . . . . . . . . . . . .867

Selected Resources for Historical Studies . . . . . . . . . . . . . . . . . . . .869

The Contributors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .871

AAharonov–Bohm Effect

Holger Lyre

The Aharonov–Bohm effect (for short: AB effect) is, quite generally, a non-localeffect in which a physical object travels along a closed loop through a gauge field-free region and thereby undergoes a physical change. As such, the AB effect can bedescribed as a holonomy. Its paradigmatic realization became widely known afterAharonov and Bohm’s 1959 paper – with forerunners by Weiss [1] and Ehrenbergand Siday [2]. Aharonov and Bohm [3] consider the following scenario: A splitelectron beam passes around a solenoid in which a magnetic field is confined. Theregion outside the solenoid is field-free, but nevertheless a shift in the interferencepattern on a screen behind the solenoid can be observed upon alteration of the mag-netic field. The schematic experimental setting can be grasped from the followingfigure:

e− beam� �

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screen

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solenoid

The phase shift can be calculated from the loop integral over the potential,which – due to Stokes’ theorem – relates to the magnetic flux

�χ = q

∮C

A dr = q

∫S

B ds = q Φmag. (1)

Convincing arguments can be given that the effect is no artifact of some impropershielding of the fields involved. On the one hand, the magnetic field can perfectly beconfined by the usage of toroidal magnets [15], the unavoidable penetration of thequantum � wave function into the solenoid, on the other hand, is not known to becorrelated to any scaling of the effect with the quality of the solenoid’s shielding.

While the above experimental setting is called the magnetic AB effect, it is alsopossible to consider the electric pendant where the phase of the wave function

D. Greenberger et al. (eds.), Compendium of Quantum Physics: Concepts, Experiments, 1History and Philosophy, c© Springer-Verlag Berlin Heidelberg 2009

2 Aharonov–Bohm Effect

depends upon varying the electric potential for two paths of a particle travellingthrough regions free of an electric field. Moreover, Aharonov and Casher [4] de-scribed a dual to the AB effect, called the � Aharonov–Casher effect, where a phaseshift in the interference of the magnetic moment in an electric field is considered.

The discovery of the AB effect has caused a flood of publications both about thetheoretical nature of the effect as well as about the various experimental realizations.Much of the relevant material is covered in Peshkin and Tonomura [14]. The theo-retical debate can basically be centered around the questions, whether and in whichsense the AB effect is of (1) quantum, (2) topological, and (3) non-local nature.

1. Contrary to a widely held view in the literature, the point can be made thatthe AB effect is not of a genuine quantum nature, since there exist classical gravi-tational AB effects as well ([5]; [6]; [7]). A simple case is the geometry of a conewhere the curvature is flat everywhere except at the apex (which may be smoothed).Parallel transport on a loop enclosing the apex leads to a holonomy. Also, the secondclock effect in Weylian spacetime can be construed as an AB analogue, as Brownand Pooley [8] have pointed out. In Weylian spacetime, a clock travelling on a loopthrough a field free region enclosing a non-vanishing electromagnetic field under-goes a shift. It has been shown that the AB effect can be generalized to any SU(N)gauge theory ([9]; [10]).

2. The AB effect does not depend on the particular path as long as the regionof the non-vanishing gauge field strength is enclosed. It is therefore no instanceof the � Berry phase, which is a path-dependent geometrical quantum phase. Itdoes depend on the topology of the configuration space of the considered physicalobject (in case of the electric AB effect this space is homeomorphic to a circle).Nevertheless, the AB effect can still be distinguished from topological effects withingauge theories such as monopoles or instantons, where the topological nature canbe described as non-trivial mappings from the gauge group into the configurationspace (this incidentally also applies to the magnetic AB effect, but generally not toSU(N) or gravitational AB effects).

3. It is obvious that the AB effect is in some sense non-local. A closer inspectiondepends directly on the question about the genuine entities involved, and this ques-tion has been in the focus of the philosophy of physics literature. In the magneticAB effect, the electron wave function does not directly interact with the confinedmagnetic field, but since the vector gauge potential outside the solenoid is non-zero,it is a common view to consider the AB effect as a proof for the reality of the gaugepotential. This, however, renders real entities gauge-dependent. Healey [11] there-fore argues for the holonomy itself as the genuine gauge theoretic entity. In boththe potential and the holonomy interpretation the AB effect is non-local in the sensethat it is non-separable, since properties of the whole – the holonomy – do not su-pervene on properties of the parts. As a third possibility even an interpretation solelyin terms of field strengths can be given at the expense of violating the principle oflocal action. The case can be made that this is an instance of ontological underde-termination, where only the gauge group structure is invariant (and, hence, a case infavour of structural realism [12]).

Aharonov–Casher Effect 3

ARemarkably, van Kampen [13] has argued that the AB effect is in fact instan-

taneous, but that this cannot be directly observed since the instantaneous actionof the magnetic effect is accordingly cancelled by the electric AB effect. � AlsoBerry’s Phase.

Primary Literature

1. P. Weiss: On the Hamilton-Jacobi Theory and Quantization of a Dynamical Continuum. Proc.Roy. Soc. Lond. A 169, 102–19 (1938)

2. W. Ehrenberg, R.E. Siday: The Refractive Index in Electron Optics and the Principles ofDynamics. Proc. Phys. Soc. Lond. B 62, 8–21 (1949)

3. Y. Aharonov, D. Bohm: Significance of Electromagnetic Potentials in the Quantum Theory.Phys. Rev. 115(3), 485–491(1959)

4. Y. Aharonov, A. Casher: Topological Quantum Effects for Neutral Particles. Phys. Rev. Lett.53, 319–321(1984)

5. J.-S. Dowker: A Gravitational Aharonov–Bohm Effect. Nuovo. Cim. 52B(1), 129–135 (1967)6. J. Anandan: Interference, Gravity and Gauge Fields. Nuovo. Cim 53A(2), 221–249 (1979)7. J. Stachel: Globally Stationary but Locally Static Space-times: A Gravitational Analog of the

Aharonov–Bohm Effect. Phys. Rev. D 26(6), 1281–1290 (1982)8. H.R. Brown, O. Pooley: The origin of the spacetime metric: Bell’s ‘Lorentzian pedagogy’ and

its significance in general relativity. In C. Callender and N. Huggett, editors. Physics meetsPhilosophy at the Planck Scale. (Cambridge University Press, Cambridge 2001)

9. T.T. Wu, C.N. Yang: Concept of Nonintegrable Phase Factors and Global Formulation of GaugeFields. Phys. Rev. D 12(12), 3845–3857 (1975)

10. C.N. Yang: Integral Formalism for Gauge Fields. Phys. Rev. Lett. 33(7), 445–447(1974)11. R. Healey: On the Reality of Gauge Potentials. Phil. Sci. 68(4), 432–455 (2001)12. H. Lyre: Holism and Structuralism in U(1) Gauge Theory. Stud. Hist. Phil. Mod. Phys. 35(4),

643–670 (2004)13. N.G. van Kampen: Can the Aharonov-Bohm Effect Transmit Signals Faster than Light? Phys.

Lett. A 106(1), 5–6 (1984)

Secondary Literature

14. M.A. Peshkin, A. Tonomura: The Aharonov-Bohm Effect. (Lecture Notes in Physics 340.Springer, Berlin 2001)

15. A. Tonomura: The Quantum World Unveiled by Electron Waves. (World Scientific, Singapore1998)

Aharonov–Casher Effect

Daniel Rohrlich

In 1984, 25 years after the prediction of the � Aharonov–Bohm (AB) effect,Aharonov and Casher [1] predicted a “dual” effect. In both effects, a particle is

4 Aharonov–Casher Effect

excluded from a tubular region of space, but otherwise no force acts on it. Yet itacquires a measurable quantum phase that depends on what is inside the tube ofspace from which it is excluded. In the AB effect, the particle is charged and thetube contains a magnetic flux. In the Aharonov–Casher (AC) effect, the particle isneutral, but has a magnetic moment, and the tube contains a line of charge. Experi-ments in neutron [2], vortex [3], atom [4], and electron [5] interferometry bear outthe prediction of Aharonov and Casher. Here we briefly explain the logic of the ACeffect and how it is dual to the AB effect.

We begin with a two-dimensional version of the AB effect. Figure 1 shows anelectron moving in a plane, and also a “fluxon”, i.e. a small region of magneticflux (pointing out of the plane) from which the electron is excluded. In Fig. 1 thefluxon is in a quantum � superposition of two positions, and the electron diffractsaround one of the positions but not the other. Initially, the fluxon and electron are ina product state |Ψin〉:

|Ψin〉 = 1

2(|f1〉 + |f2〉)⊗ (|e1〉 + |e2〉),

where |f1〉 and |f2〉 represent the two fluxon wave packets and |e1〉 and |e2〉 repre-sent the two electron wave packets. After the electron passes the fluxon, their state|Ψfin〉 is not a product state; the relative phase between |e1〉 and |e2〉 depends on thefluxon position:

|Ψfin〉 = 1

2|f1〉 ⊗ (|e1〉 + |e2〉)+ 1

2|f2〉 ⊗ (|e1〉 + eiφAB |e2〉).

Fig. 1 An electron and afluxon, each in a superpositionof two wave packets; theelectron wave packets encloseonly one of the fluxon wavepackets

Aharonov–Casher Effect 5

AHere φAB is the Aharonov–Bohm phase, and |f2〉 represents the fluxon positionedbetween the two electron � wave packets. Now if we always measure the position ofthe fluxon and the relative phase of the electron, we discover the Aharonov–Bohmeffect: the electron acquires the relative phase φAB if and only if the fluxon liesbetween the two electron paths. But we can rewrite |Ψfin〉 as follows:

|Ψfin〉 = 1

2(|f1〉 + |f2〉)⊗ |e1〉 + 1

2(|f1〉 + eiφAB |f2〉)⊗ |e2〉.

This rewriting implies that if we always measure the relative phase of the fluxon andthe position of the electron, we discover an effect that is analogous to the Aharonov–Bohm effect: the fluxon acquires the relative phase φAB if and only if the electronpasses between the two fluxon wave packets. Indeed, the effects are equivalent: wecan choose a reference frame in which the fluxon passes by the stationary electron.Then we find the same relative phase whether the electron paths enclose the fluxonor the fluxon paths enclose the electron.

In two dimensions, the two effects are equivalent, but there are two inequivalentways to go from two to three dimensions while preserving the topology (of pathsof one particle that enclose the other): either the electron remains a particle and thefluxon becomes a tube of flux, or the fluxon remains a particle (a neutral particlewith a magnetic moment) and the electron becomes a tube of charge. These twoinequivalent ways correspond to the AB and AC effects, respectively. They are notequivalent but dual, i.e. equivalent up to interchange of electric charge and magneticflux.

In the AB effect, the electron does not cross through a magnetic field; in the ACeffect, the neutral particle does cross through an electric field. However, there is noforce on either particle. The proof [6] is surprisingly subtle and holds only if the lineof charge is straight and parallel to the magnetic moment of the neutral particle [8].Hence only for such a line of charge are the AB and AC effects dual.

Duality has another derivation. To derive their effect, Aharonov and Casher [1]first obtained the nonrelativistic Lagrangian for a neutral particle of magnetic mo-ment μ interacting with a particle of charge e. In Gaussian units, it is

L = 1

2mv2 + 1

2MV 2 + e

cA(r− R) · (v− V),

where M,R,V and m, r, v are the mass, position and velocity of the neutral andcharged particle, respectively, and the vector potential A(r− R) is

A(r− R) = μ× (r− R)|r− R|3 .

Note L is invariant under respective interchange of r, v and R,V. Thus L is thesame whether an electron interacts with a line of magnetic moments (AB effect) ora magnetic moment interacts with a line of electrons (AC effect). However, if webegin with the AC effect and replace the magnetic moment with an electron, and all

6 Algebraic Quantum Mechanics

the electrons with the original magnetic moment, we end up with magnetic momentsthat all point in the same direction, i.e. with a straight line of magnetic flux. Hencethe original line of electrons must have been straight. We see intuitively that theeffects are dual only for a straight line of charge.1

Primary Literature

1. Y. Aharonov, A. Casher: Topological quantum effects for neutral particles. Phys. Rev. Lett. 53,319–21 (1984).

2. A. Cimmino, G. I. Opat, A. G. Klein, H. Kaiser, S. A. Werner, M. Arif, R. Clothier: Observationof the topological Aharonov–Casher phase shift by neutron interferometry. Phys. Rev. Lett. 63,380–83 (1989).

3. W. J. Elion, J. J. Wachters, L. L. Sohn, J. D. Mooij: Observation of the Aharonov–Casher effectfor vortices in Josephson-junction arrays. Phys. Rev. Lett. 71, 2311–314 (1993).

4. K. Sangster, E. A. Hinds, S. M. Barnett, E. Riis: Measurement of the Aharonov–Casher phasein an atomic system. Phys. Rev. Lett. 71, 3641–3644 (1993); S. Yanagimachi, M. Kajiro,M. Machiya, A. Morinaga: Direct measurement of the Aharonov–Casher phase and tensorStark polarizability using a calcium atomic polarization interferometer. Phys. Rev. A65, 042104(2002).

5. M. Konig et al.: Direct observation of the Aharonov–Casher Phase. Phys. Rev. Lett. 96, 076804(2006).

6. Y. Aharonov, P. Pearle, L. Vaidman: Comment on Proposed Aharonov–Casher effect: Anotherexample of an Aharonov–Bohm effect arising from a classical lag. Phys. Rev. A37, 4052–055(1988).

Secondary Literature

7. For a review, see L. Vaidman: Torque and force on a magnetic dipole. Am. J. Phys. 58, 978–83(1990).

1 I thank Prof. Aharonov for a conversation on this point.

Algebraic Quantum Mechanics

N.P. Landsman

Algebraic quantum mechanics is an abstraction and generalization of the � Hilbertspace formulation of quantum mechanics due to von Neumann [5]. In fact, von Neu-mann himself played a major role in developing the algebraic approach. Firstly, hisjoint paper [3] with Jordan and Wigner was one of the first attempts to go beyondHilbert space (though it is now mainly of historical value). Secondly, he foundedthe mathematical theory of operator algebras in a magnificent series of papers [4, 6].

Algebraic Quantum Mechanics 7

AAlthough his own attempts to apply this theory to quantum mechanics were unsuc-cessful [18], the operator algebras that he introduced (which are now aptly calledvon Neumann algebras) still play a central role in the algebraic approach to quantumtheory. Another class of operator algebras, now called C∗-algebras, introduced byGelfand and Naimark [1], is of similar importance in algebraic quantum mechanicsand quantum field theory. Authoritative references for the theory of C∗-algebras andvon Neumann algebras are [14] and [21]. Major contributions to algebraic quantumtheory were also made by Segal [7, 8] and Haag and his collaborators [2, 13].

The need to go beyond Hilbert space initially arose in attempts at a mathemati-cally rigorous theory of systems with an infinite number of degrees of freedom, bothin quantum statistical mechanics [9, 12, 13, 19, 20, 22] and in quantum field theory[2, 13, 20]. These remain active fields of study. More recently, the algebraic ap-proach has also been applied to � quantum chemistry [17], to the quantization and� quasi-classical limit of finite-dimensional systems [15, 16], and to the philosophyof physics [10, 11, 16].

Besides its mathematical rigour, an important advantage of the algebraic ap-proach is that it enables one to incorporate � Superselection Rules. Indeed, it wasa fundamental insight of Haag that the superselection sectors of a quantum systemcorrespond to (unitarily) inequivalent representations of its algebra of � observ-ables (see below). As shown in the references just cited, in quantum field theorysuch representations (and hence the corresponding superselection sectors) are typ-ically labeled by charges, whereas in quantum statistical mechanics they describedifferent thermodynamic phases of the system. In chemistry, the chirality of certainmolecules can be understood as a superselection rule. The algebraic approach alsoleads to a transparent description of situations where � locality and/or � entangle-ment play a role [11, 13].

The notion of a C∗-algebra is basic in algebraic quantum theory. This is a com-plex algebra A that is complete in a norm ‖ · ‖ satisfying ‖ab‖ � ‖a‖ ‖b‖ for alla, b ∈ A, and has an involution a �→ a∗ such that ‖a∗a‖ = ‖a‖2. A quantum systemis then supposed to be modeled by a C∗-algebra whose self-adjoint elements (i.e.a∗ = a) form the observables of the system. Of course, further structure than theC∗-algebraic one alone is needed to describe the system completely, such as a time-evolution or (in the case of quantum field theory) a description of the localization ofeach observable [13].

A basic example of a C∗-algebra is the algebra Mn of all complex n×n matrices,which describes an n-level system. Also, one may take A = B(H), the algebra ofall bounded operators on an infinite-dimensional Hilbert space H , equipped withthe usual operator norm and adjoint. By the Gelfand–Naimark theorem [1], anyC∗-algebra is isomorphic to a norm-closed self-adjoint subalgebra of B(H), forsome Hilbert space H . Another key example is A = C0(X), the space of all con-tinuous complex-valued functions on a (locally compact Hausdorff) space X thatvanish at infinity (in the sense that for every ε > 0 there is a compact subsetK ⊂ X such that |f (x)| < ε for all x /∈ K), equipped with the supremum norm‖f ‖∞ := supx∈X |f (x)|, and involution given by (pointwise) complex conjugation.By the Gelfand–Naimark lemma [1], any commutative C∗-algebra is isomorphic to

8 Algebraic Quantum Mechanics

C0(X) for some locally compact Hausdorff space X. The algebra of observables ofa classical system can often be modeled as a commutative C∗-algebra.

A von Neumann algebra M is a special kind of C∗-algebra, namely one thatis concretely given on some Hilbert space, i.e. M ⊂ B(H), and is equal to itsown bicommutant: (M ′)′ = M (where M ′ consists of all bounded operators on H

that commute with every element of M). For example, B(H) is always a von Neu-mann algebra. Whereas C∗-algebras are usually considered in their norm-topology,a von Neumann algebra in addition carries a second interesting topology, called theσ -weak topology, in which its is complete as well. In this topology, one has conver-gence an → a if Tr ρ(an−a)→ 0 for each density matrix ρ on H . Unlike a generalC∗-algebra (which may not have any nontrivial projections at all), a von Neumannalgebra is generated by its projections (i.e. its elements p satisfying p2 = p∗ = p).It is often said, quite rightly, that C∗-algebras describe “non-commutative topol-ogy” whereas von Neumann algebra form the domain of “non-commutative measuretheory”.

In the algebraic framework the notion of a state is defined in a different way fromwhat one is used to in quantum mechanics. An (algebraic) state on a C∗-algebra A isa linear functional ρ:A → C that is positive in that ρ(a∗a) � 0 for all a ∈ A andnormalized in that ρ(1) = 1, where 1 is the unit element ofA (providedA has a unit;if not, an equivalent requirement given positivity is ‖ρ‖ = 1). If A is a von Neumannalgebra, the same definition applies, but one has the finer notion of a normal state,which by definition is continuous in the σ -weak topology (a state is automaticallycontinuous in the norm topology). If A = B(H), then a fundamental theorem of vonNeumann [5] states that each normal state ρ on A is given by a � density matrixρ on H , so that ρ(a) = Tr ρa for each a ∈ A. (If H is infinite-dimensional, thenB(H) also possesses states that are not normal. For example, if H = L2(R) theDirac eigenstates |x〉 of the position operator are well known not to exist as vectorsin H , but it turns out that they do define non-normal states on B(H).) On this basis,algebraic states are interpreted in the same way as states in the usual formalism, inthat the number ρ(a) is taken to be the expectation value of the observable a in thestate ρ (this is essentially the � Born rule).

The notions of pure and mixed states can be defined in a general way now.Namely, a state ρ : A → C is said to be pure when a decomposition ρ =λω + (1 − λ)σ for some λ ∈ (0, 1) and two states ω and σ is possible only ifω = σ = ρ. Otherwise, ρ is called mixed, in which case it evidently does havea nontrivial decomposition. It then turns out that a normal pure state on B(H) isnecessarily of the form ψ(a) = (Ψ, aΨ ) for some unit vector Ψ ∈ H ; of course,the state ρ defined by a density matrix ρ that is not a one-dimensional projectionis mixed. Thus one recovers the usual notion of pure and mixed states from thealgebraic formalism.

In the algebraic approach, however, states play a role that has no counterpart inthe usual formalism of quantum mechanics. Namely, each state ρ on a C∗-algebraA defines a representation πρ of A on a Hilbert space Hρ by means of the so-called GNS-construction (after Gelfand, Naimark and Segal [1, 7]). First, assumethat ρ is faithful in that ρ(a∗a) > 0 for all nonzero a ∈ A. It follows that (a, b) :=

Algebraic Quantum Mechanics 9

Aρ(a∗b) defines a positive definite sesquilinear form on A; the completion of A in thecorresponding norm is a Hilbert space denoted by Hρ . By construction, it containsA as a dense subspace. For each a ∈ A, define an operator πρ(a) on A by πρ(a)b :=ab, where b ∈ A. It easily follows that πρ(a) is bounded, so that it may be extendedby continuity to all of Hρ . One then checks that πρ : A → B(Hρ) is linear andsatisfies πρ(a1a2) = πρ(a1)πρ(a2) and πρ(a

∗) = πρ(a)∗. This means that πρ is a

representation of A on Hρ . If ρ is not faithful, the same construction applies withone additional step: since the sesquilinear form is merely positive semidefinite, onehas to take the quotient of A by the kernel Nρ of the form (i.e. the collection of allc ∈ A for which ρ(c∗c) = 0), and construct the Hilbert space Hρ as the completionof A/Nρ .

As in group theory, one has a notion of unitary (in)equivalence of representationsof C∗-algebras. As already mentioned, this provides a mathematical explanation forthe phenomenon of superselection rules, an insight that remains one of the mostimportant achievements of algebraic quantum theory to date. See also � operationalquantum mechanics; relativistic quantum mechanics.

Primary Literature

1. I.M. Gelfand & M.A. Naimark: On the imbedding of normed rings into the ring of operators inHilbert space. Mat. Sbornik 12, 197–213 (1943)

2. R. Haag & D. Kastler: An algebraic approach to quantum field theory. J. Math. Phys. 7,848–861 (1964)

3. P. Jordan, J. von Neumann & E. Wigner: On an algebraic generalization of the quantum me-chanical formalism. Ann. Math. 35, 29–64 (1934)

4. F.J. Murray & J. von Neumann: On rings of operators I, II, IV. Ann. Math. 37, 116–229 (1936),Trans. Amer. Math. Soc. 41, 208–248 (1937), Ann. Math. 44, 716–808 (1943)

5. J. von Neumann: Mathematische Grundlagen der Quantenmechanik (Springer, Berlin 1932).English translation: Mathematical Foundations of Quantum Mechanics (Princeton UniversityPress, Berlin 1955)

6. J. von Neumann: On rings of operators III, V. Ann. Math. 41, 94–161 (1940), ibid. 50, 401–485(1949)

7. I.E. Segal: Irreducible representations of operator algebras. Bull. Amer. Math. Soc. 61, 69–105(1947)

8. I.E. Segal: Postulates for general quantum mechanics. Ann. Math. 48, 930–948 (1947)

Secondary Literature

9. O. Bratteli & D.W. Robinson: Operator Algebras and Quantum Statistical Mechanics. Vol 1:C* and W*-algebras, Symmetry Groups, Decomposition of States; Vol. 2: Equilibrium States,Models in Quantum Statistical Mechanics, Second Edition (Springer, Heidelberg 1996, 2003)

10. J. Butterfield & J. Earman (ed.): Handbook of the Philosophy of Science Vol. 2: Philosophy ofPhysics (North-Holland, Elsevier, Amsterdam 2007)

11. R. Clifton, J. Butterfield & H. Halvorson: Quantum Entanglements - Selected Papers (OxfordUniversity Press, Oxford 2004)

12. G.G. Emch: Mathematical and Conceptual Foundations of 20th-Century Physics (North-Holland, Amsterdam 1984)

10 Anyons

13. R. Haag: Local Quantum Physics: Fields, Particles, Algebras (Springer, Heidelberg 1992)14. R.V. Kadison & J.R. Ringrose: Fundamentals of the Theory of Operator Algebras. Vol. 1:

Elementary Theory; Vol. 2: Advanced Theory (Academic, New York 1983, 1986)15. N.P. Landsman: Mathematical Topics Between Classical and Quantum Mechanics (Springer,

New York 1998)16. N.P. Landsman: Between classical and quantum, in Handbook of the Philosophy of Science

Vol. 2: Philosophy of Physics, ed. by J. Butterfield and J. Earman, pp. 417–554 (North-Holland,Elsevier, Amsterdam 2007)

17. H. Primas: Chemistry, Quantum Mechanics and Reductionism, Second Edition (Springer,Berlin 1983)

18. M. Redei: Why John von Neumann did not like the Hilbert space formalism of quantum me-chanics (and what he liked instead). Stud. Hist. Phil. Mod. Phys. 27, 493–510 (1996).

19. G.L. Sewell: Quantum Mechanics and its Emergent Macrophysics (Princeton University Press,Princeton 2002)

20. F. Strocchi: Elements of Quantum mechanics of Infinite Systems (World Scientific, Singapore1985)

21. M. Takesaki: Theory of Operator Algebras. Vols. I-III. (Springer, New York 2003)22. W. Thirring: Quantum Mathematical Physics: Atoms, Molecules and Large Systems, Second

Edition (Springer, New York 2002)

Angular Momentum

� See Spin; Stern–Gerlach experiment; Vector model.

Anyons

Jon Magne Leinaas

Quantum mechanics gives a unique characterization of elementary particles as be-ing either bosons or fermions. This property, referred to as the � quantum statisticsof the particles, follows from a simple symmetry argument, where the � wave func-tions of a system of identical particles are restricted to be either symmetric (bosons)or antisymmetric (fermions) under permutation of particle coordinates. For twospinless particles, this symmetry is expressed through a sign factor which is as-sociated with the switching of positions

ψ(r1, r2) = ±ψ(r2, r1) , (1)

with + for bosons and − for fermions. From the symmetry constraint, when ap-plied to a many-particle system, the statistical distributions of particles over singleparticle states can be derived, and the completely different collective behaviour ofsystems like � electrons (fermions) and photons (bosons) (� light quantum) can beunderstood.

Anyons 11

AThe restriction to two possible kinds of quantum statistics, represented by the

sign factor in (1), seems almost obvious. On one hand the permutation of parti-cle coordinates has no physical significance when the particles are identical, whichmeans that the wave function can change at most by a complex phase factor eiθ .On the other hand a double permutation seems to make no change at all, which fur-ther restricts the phase factor to a sign ±1. This is the standard argument used intextbooks like [14].

However, there is a loophole to this argument, as pointed out by J.M. Leinaas andJ. Myrheim in 1976 [1]. If the dimension of space is reduced from three to two theconstraint on the phase factor is lifted and a continuum of possibilities appears thatinterpolates between the boson and fermion cases. In [1] these unconventional typesof quantum statistics were found by analysis of the wave functions defined on themany-particle configuration space. Other approaches by G.A. Goldin, R. Menikoff,and D.H. Sharp [2] and by F. Wilczek [3] lead to similar results, and Wilczek in-troduced the name anyon for these new types of particles. As a precursor to thisdiscussion M.G.G. Laidlaw and C.M. DeWitt had already shown that a path integraldescription applied to systems of identical particles reproduces standard results, butonly in a space of dimensions higher than two [4].

The difference between continuous interchange of positions in two and three di-mensions can readily be demonstrated, as illustrated in Fig. 1a. In two dimensionsa two-particle interchange path comes with an orientation, and as a consequence aright-handed path and its inverse, a left-handed path, may be associated with dif-ferent (inverse) phase factors. In three and higher dimensions there is no intrinsicdifference between orientations of a path, since a right-handed path can be continu-ously changed to a left-handed one by a rotation in the extra dimension. Therefore,in dimensions higher than two the exchange phase factor has to be equal to its in-verse, and is consequently restricted to ±1. This explains why anyons are possiblein two but not in three dimensions. Since the statistics angle θ in the exchange fac-tor eiθ is a free parameter, there is a different type of anyon for each value of θ . For

time

eiθ e−iθ

a b

Fig. 1 Switching positions in two dimensions. (a) The difference between right-handed and left-handed interchange may give rise to quantum phase factors e±iθ that are different from ±1.(b) When many particles switch positions the collection of continuous particle paths can be viewedas forming a braid and the associated phase factor can be viewed as a representing an element ofthe braid group

12 Anyons

systems with more than two particles the different paths define more complicatedpatterns (Fig. 1b), which are generally known as braids, and in this view of quan-tum statistics the corresponding braid group is therefore more fundamental than thepermutation group. The generalized types of quantum statistics characterized by theparameter θ is often referred to as fractional statistics or braiding statistics.

Since anyons can only exist in two dimensions, elementary particles in the worldof three space dimensions are still restricted to be either fermions or bosons. But incondensed matter physics the creation of quasi-twodimensional systems is possible,and in such systems anyons may emerge. They are excitations of the quantum sys-tem with sharply defined particle properties, generally known as quasiparticles.The presence of anyons in such systems is not only a theoretical possibility, aswas realized after the discovery of the fractional � quantum Hall effect in 1982.This effect is due to the formation of a two-dimensional, incompressible electronfluid in a strong magnetic field, and the anyon character of the quasiparticles inthis system was demonstrated quite convincingly in theoretical studies [5, 6]. Al-though theoretical developments have given further support to this idea, a directexperimental evidence has been lacking. However, experiments performed by V.J.Goldman and his group in 2005, with studies on interference effects in tunnellingcurrents, have given clear indications for the presence of excitations with fractionalstatistics [7].

The discovery of the fractional quantum Hall effect and the subsequent de-velopment of ideas of anyon superconductivity [15] gave a boost in interest foranyons, which later on has been followed up by ideas of anyons in other typesof systems with exotic quantum properties. One of these ideas applies to rotatingatomic � Bose-Einstein condensation, where theoretical studies have lead to pre-dictions that at sufficiently high angular velocities a transition of the condensate toa bosonic analogy of a quantum Hall state will occur, and in this new quantum stateanyon excitations should exist [8].

Topology is an important element in the description of anyons, since the focusis on continuous paths rather than simply on permutations of particle coordinates[1]. This focus on topology and on braids places the theory of anyons into a widercontext of modern physics. Thus, anyons form a natural part of an approach tothe physics of exotic condensed matter systems known as topologically orderedsystems, where the two-dimensional electron gas of the quantum Hall system is aspecial realization [9]. The braid formulation also opens for generalizations in theform of non-abelian anyons. In this extension of the anyon theory, the phase factorassociated with the interchange of two anyon positions is replaced by non-abelianunitary operations (or matrices). This is an extension of the simple identical particlepicture of anyons, since new degrees of freedom are introduced which in a sense areshared by the participants in the braid. In the rich physics of the quantum Hall effectthere are indications that such nonabelions may indeed exist [10], and theoreticalideas of exploiting such objects in the form of topological � quantum computation[11] have gained much interest.

The topological aspects are important for the description of anyons, but at thesame time they create problems for the study of many-anyon systems. Even if no

Anyons 13

Aadditional interaction is present such systems can be studied in detail only when theparticle number is small. There are also limitations to the application of standardmany-particle methods. For these reasons the physics of many-anyon systems isonly partly understood. One approach to the many-anyon problem is to trade thenon-trivial braiding symmetry for a compensating statistics interaction [1], which isa two-body interaction that is sensitive to the braiding of particles, but is independentof distance. The same type of statistics transformation has also been used in fieldtheory descriptions of the fractional quantum Hall effect, where the fundamentalelectron field is changed by a statistics transmutation into an effective bosonic fieldof the system [12].

Even if anyons, as usually defined, are particles restricted to two dimensions,there are related many-particle effects in one dimension. The interchange of parti-cle positions cannot be viewed in the same way, since particles in one dimensioncannot switch place in a continuous way without actually passing through eachother. Nevertheless there are special kinds of interactions that can be interpreted asrepresenting unconventional types of quantum statistics also in one dimension [13].The name anyon is often applied also to these kinds of particles.

For further reading see [15] and [16].

Primary Literature

1. J.M. Leinaas, J. Myrheim: On the theory of identical particles. Il Nuovo Cimento B 37, 1(1977).

2. G.A. Goldin, R. Menikoff, D.H. Sharp: Representations of a local current algebra in nonsimplyconnected space and the Aharonov-Bohm effect. J. Math. Phys. 22, 1664 (1981).

3. F. Wilczek: Magnetic flux, angular momentum and statistics. Phys. Rev. Lett. 48, 1144 (1982).4. M.G.G. Laidlaw, C.M. DeWitt: Feynman integrals for systems of indistinguishable particles.

Phys. Rev. D. 3, 1375 (1971).5. B.I. Halperin: Statistics of quasiparticles and the hierarchy of fractional quantized Hall states.

Phys. Rev. Lett. 52, 1583 (1984).6. D. Arovas, J.R. Schrieffer, F. Wilzcek: Fractional statistics and the quantum Hall effect. Phys.

Rev. Lett. 53, 722 (1984).7. F.E. Camino, W. Zhou, V.J. Goldman: Realization of a Laughlin quasiparticle interferometer:

Observation of fractional statistics. Phys. Rev. B 72, 075342 (2005).8. N.K. Wilkin, J.M.F. Gunn: Condensation of composite bosons in a rotating BEC. Phys. Rev.

Lett. 84, 6 (2000).9. X.G. Wen: Topological orders in rigid states. Int. J. Mod. Phys. B 4, 239 (1990).

10. G. Moore, N. Read: Nonabelions in the fractional quantum Hall effect. Nucl. Phys. B. 360, 362(1991).

11. A.Yu. Kitaev: Fault-tolerant quantum computation by anyons. Ann. Phys. (N.Y.), 303, 2 (2003).12. S.C. Zhang, T.H. Hansson, S. Kivelson: Effective-field-theory model for the fractional quantum

Hall effect. Phys. Rev. Lett. 62, 82 (1989).13. J.M. Leinaas, J. Myrheim: Intermediate statistics for vortices in superfluid films. Phys. Rev. B

37, 9286 (1988).

14 Aspect Experiment

Secondary Literature

14. L.D. Landau, L.M. Lifshitz: Quantum Mechanics: Non-Relativistic Theory (Elsevier Science,Amsterdam, Third Edition 1977).

15. F. Wilczek: Fractional Statistics and Anyon Superconductivity (World Scientific, Singapore,1990).

16. A. Khare: Fractional Statistics and Quantum Theory (World Scientific, Singapore, SecondEdition 2005).

Aspect Experiment

A.J. Leggett

In 1965, John S. Bell proved a celebrated theorem [1] which essentially states thatno theory belonging to the class of “objective local theories” (OLT’s) can reproducethe experimental predictions of quantum mechanics for a situation in which two cor-related particles are detected at mutually distant stations (� Bell’s Theorem). A fewyears later Clauser et al. [2] extended the theorem so as to make possible an experi-ment which would in principle unambiguously discriminate between the predictionsof the class of OLT’s and those of quantum mechanics, and the first experiment ofthis type was carried out by Freedman and Clauser [3] in 1972. This experiment,and (with one exception) others performed in the next few years confirmed the pre-dictions of quantum mechanics. However, they did not definitively rule out the classof OLT’s, because of a number of “loopholes” (� Loopholes in Experiments). Ofthese various loopholes, probably the most worrying was the “locality loophole”:a crucial ingredient in the definition of an OLT is the postulate that the outcomeof a measurement at (e.g.) station 2 cannot depend on the nature of the measure-ment at the distant station 1 (i.e., on the experimenter’s choice of which of two ormore mutually incompatible measurements to perform). If the space-time intervalbetween the “event” of the choice of measurement at station 1 and that of the out-come of the measurement at station 2 were spacelike, then violation of the postulateunder the conditions of the experiment would imply, at least prima facie, a viola-tion of the principles of special relativity, so that most physicists would have a greatdeal of confidence in the postulate. Unfortunately, in the experiments mentioned, thechoice of which variable to measure was made in setting up the apparatus (polariz-ers, etc.) in a particular configuration, a process which obviously precedes the actualmeasurements by a time of the order of hours; since the spatial separation betweenthe stations was only of the order of a few meters, it is clear that the events of choiceat 1 and measurement at 2 fail to meet the condition of spacelike separation by manyorders of magnitude, and the possibility is left open that information concerning thesetting (choice) at station 1 has been transmitted (subluminally) to station 2 and

Aspect Experiment 15

Aaffected the outcome of the measurement there. While such a hypothesis certainlyseems bizarre within the framework of currently accepted physics, the question ofthe viability or not of the class of OLT’s is so fundamental an issue that one cannotafford to neglect it completely.

In this situation it becomes highly desirable, as emphasized by Bell in his orig-inal paper, to perform an experiment in which the choice of what to measure atstation 1 is made “at the last moment”, so that there is no time for informationabout this choice to be transmitted (subluminally or luminally) to station 2 beforethe outcome of the measurement there is realized. Of course, whether or not thiscondition is fulfilled in any given experiment depends crucially on exactly at whatstage the “realization” of a specific outcome is taken to occur, and this questionimmediately gets us into the fundamental problem of measurement in quantum me-chanics (� Measurement Theory); however, most discussions of the incompatibilityof OLT’s and quantum theory in the literature have been content to assume that therealization occurs no later than the first irreversible processes taking place in themacroscopic measuring device.(For example, in a typical photomultiplier it is as-sumed to take place when the photon hits the cathode and ejects the first electron,since in practice any processes taking place thereafter are irreversible). Althoughthis assumption is certainly questionable, for the sake of definiteness it will be madeuntil further notice.

The first experiment to attempt to evade the locality loophole was that of Aspectet al. [4] in 1982, and subsequent experiments which continue this approach areoften referred to as “Aspect-type”. In some sense these experiments are a sub-classof the more general category of “delayed-choice” experiments � Delayed-ChoiceExperiment), but they have a special significance in their role of attempting to ex-clude the class of OLT’s. In the original experiment [4], the distance between thedetection stations is about 12 m, corresponding to a transit time for light of 40 nsec.At each station, the “switch” which decides which of the two alternative measure-ments to make is an acousto-optical device; in each case two electro-acousticaltransducers, driven in phase, create ultrasonic standing waves in a slab of waterthrough which the relevant photon must pass, with a period of about 25 MHz (thefrequency is different for the two stations). The periodic density variation in thewave acts as a diffraction grating: If a given photon � wave packet (length intime ∼5 nsec) arrives at (say) station 1 when the wave has a node (i.e., the densityand hence dielectric constant of the water is uniform) it is transmitted rectilinearlythrough the slab and enters a polarizer set in direction a; if on the other hand it ar-rives at an antinode (periodic density variation) it undergoes Bragg diffraction and isdirected into a polarizer set at a’. (See Fig. 1). Photons (� light quantum) incident atintermediate phases of the wave are deflected into neither polarizer and thus missedin the counting. The period of switching between the alternative choices (a quarterperiod of the transducers) is about 10 nsec., short compared to the transit time oflight between the stations. To the extent, then, that one can regard the switching asa “random” process, the locality loophole is blocked. The data obtained in ref. [4]violate the OLT predictions by 5 standard deviations.