Compendium of quantum physics; concepts, experiments, history and philosophy

Neu-Algebraic Quantum Mechanics 7A Although his own attempts to apply this theory to quantum mechanics were cessful [18], the operator algebras that he introduced which are now aptly cal

Compendium of Quantum Physics

Daniel Greenberger

Department of Physics

The City College of New York

138th St & Convent Ave

& TechnologyKeplerstr 17D-70174 StuttgartGermanyklaus.hentschel@po.hi.uni-stuttgart.de

ISBN 978-3-540-70622-9 e-ISBN 978-3-540-70626-7

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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 interconnect

in 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 aspects

of 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

to turn the idea for this compendium into reality.

Klaus HentschelFriedel Weinert

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

viii Contents

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

Contents ix

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

x Contents

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 GHZ States 258

Gleason’s Theorem 263

Grover’s Algorithm 266

GRW Theory (Ghirardi, Rimini, Weber Model of Quantum Mechanics) 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

Contents xi

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

Land´e’s g-factor and g-formula 336

Large-Angle Scattering 337

Light Quantum 339

Locality 347

Loopholes in Experiments 348

L ¨uders 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 Axiomatics and 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 Formalism of Quantum Mechanics 651

Rigged Hilbert Spaces and Time Asymmetric Quantum Theory 660

Russell–Saunders Coupling 671

Rutherford Atom 671

Scattering Experiments 676

Schr¨odinger Equation 681

Schr¨odinger’s Cat 685

Schr¨odinger 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 Incoherent Superposition) 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

ZZZZZZZZZ

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 region

of the non-vanishing gauge field strength is enclosed It is therefore no instance

of 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 can

be 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 magnetic

AB 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 solely

in 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])

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 meets

Philosophy 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 Gauge

Fields 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),

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 meets

Philosophy 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 Gauge

Fields 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),

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 superpositionof two positions, and the electron diffractsaround one of the positions but not the other Initially, the fluxon and electron are in

a 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 the

fluxon position:

|Ψfin = 1

2|f1 ⊗ (|e1 + |e2) + 1

2|f2 ⊗ (|e1 + eiφAB|e2).

Fig 1 An electron and a

fluxon, each in a superposition

of two wave packets; the

electron wave packets enclose

only one of the fluxon wave

packets

Aharonov–Casher Effect 5

A

Here φABis the Aharonov–Bohm phase, and|f2 represents the fluxon positioned

between 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–Bohm

effect: 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:

passes 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 fluxon

or 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 paths

of 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 not

equivalent but dual, i.e equivalent up to interchange of electric charge and magnetic

flux

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 line

of 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-mentμ interacting with a particle of charge e In Gaussian units, it is

Note L is invariant under respective interchange of r, v and R, V Thus L is the

same whether an electron interacts with a line of magnetic moments (AB effect) or

a 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: Observation

of 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 effect for 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 phase

in 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 tensor Stark polarizability using a calcium atomic polarization interferometer Phys Rev A65, 042104 (2002).

5 M K¨onig 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: Another example 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 Hilbert

spaceformulation of quantum mechanics due to von Neumann [5] In fact, von 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]

Neu-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: Observation

of 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 effect for 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 phase

in 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 tensor Stark polarizability using a calcium atomic polarization interferometer Phys Rev A65, 042104 (2002).

5 M K¨onig 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: Another example 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 Hilbert

spaceformulation of quantum mechanics due to von Neumann [5] In fact, von 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]

Neu-Algebraic Quantum Mechanics 7

A

Although his own attempts to apply this theory to quantum mechanics were 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 quantum

unsuc-theory Another class of operator algebras, now called C∗-algebras, introduced byGelfand and Naimark [1], is of similar importance in algebraic quantum mechanics

and 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 cally rigorous theory of systems with an infinite number of degrees of freedom, both

mathemati-in quantum statistical mechanics [9, 12, 13, 19, 20, 22] and mathemati-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 limitof finite-dimensional systems [15, 16], and to the philosophy

mentplay 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 all

a, b ∈ A, and has an involution a → a∗such thata∗a  = a2 A quantum system

is 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 the

C∗-algebraic one alone is needed to describe the system completely, such as a evolution or (in the case of quantum field theory) a description of the localization ofeach observable [13]

time-A basic example of a C∗-algebra is the algebra M

n of all complex n ×n matrices,

which describes an n-level system Also, one may take A = B(H ), the algebra of

all bounded operators on an infinite-dimensional Hilbert space H , equipped with

the usual operator norm and adjoint By the Gelfand–Naimark theorem [1], any

C∗-algebra is isomorphic to a norm-closed self-adjoint subalgebra of B(H ), for

some 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 that vanish at infinity (in the sense that for every ε > 0 there is a compact subset

K ⊂ 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 of

a 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 that

is concretely given on some Hilbert space, i.e M ⊂ B(H ), and is equal to its

own 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 mann algebra Whereas C∗-algebras are usually considered in their norm-topology,

Neu-a von NeumNeu-ann Neu-algebrNeu-a in Neu-addition cNeu-arries Neu-a second interesting topology, cNeu-alled the

σ-weak topology, in which its is complete as well In this topology, one has

conver-gence a n → a if Tr ˆρ(a n −a) → 0 for each density matrix ˆρ on H Unlike a general

C∗-algebra (which may not have any nontrivial projections at all), a von Neumann

algebra 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 ogy” whereas von Neumann algebra form the domain of “non-commutative measuretheory”

topol-In the algebraic framework the notion of a state is defined in a different way from

what one is used to in quantum mechanics An (algebraic) state on a C∗-algebra A is

a linear functional ρ : A → C that is positive in that ρ(a∗a)  0 for all a ∈ A and

normalized in that ρ(1) = 1, where 1 is the unit element of A (provided A has a unit;

if not, an equivalent requirement given positivity isρ = 1) If A is a von Neumann

algebra, 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 automatically continuous in the norm topology) If A = B(H ), then a fundamental theorem of von

Neumann [5] states that each normal state ρ on A is given by a density matrix

ˆρ on H , so that ρ(a) = Tr ˆρa for each a ∈ A (If H is infinite-dimensional, then

B(H ) also possesses states that are not normal For example, if H = L2( R) the

Dirac eigenstates|x of the position operator are well known not to exist as vectors

in 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, in

that the number ρ(a) is taken to be the expectation value of the observable a in the state 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 if

ω = σ = ρ Otherwise, ρ is called mixed, in which case it evidently does have

a nontrivial decomposition It then turns out that a normal pure state on B(H ) is necessarily of the form ψ(a) = (Ψ, aΨ ) for some unit vector Ψ ∈ H ; of course,

the state ρ defined by a density matrix ˆρ that is not a one-dimensional projection

is 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 in

the usual formalism of quantum mechanics Namely, each state ρ on a C∗-algebra

A defines a representation π ρ of A on a Hilbert space H ρ by means of the called GNS-construction (after Gelfand, Naimark and Segal [1, 7]) First, assume

so-that ρ is faithful in so-that ρ(a∗a) > 0 for all nonzero a ∈ A It follows that (a, b) :=

Algebraic Quantum Mechanics 9

A

ρ(a∗b) defines a positive definite sesquilinear form on A; the completion of A in the

corresponding norm is a Hilbert space denoted by H ρ By construction, it contains

A as a dense subspace For each a ∈ A, define an operator π ρ (a) on A by π ρ (a)b:=

ab , where b ∈ A It easily follows that π ρ (a)is bounded, so that it may be extended

by continuity to all of H ρ One then checks that π ρ : A → B(H ρ )is linear and

satisfies π ρ (a1a2) = π ρ (a1)π ρ (a2) and π ρ (a∗) = π ρ (a)∗ This means that π

ρ is a

representation of A on H ρ If ρ is not faithful, the same construction applies with

one additional step: since the sesquilinear form is merely positive semidefinite, one

has to take the quotient of A by the kernel N ρ of the form (i.e the collection of all

c ∈ A for which ρ(c∗c) = 0), and construct the Hilbert space H ρas the completion

of A/N ρ

As in group theory, one has a notion of unitary (in)equivalence of representations

of 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 operational

quantum mechanics; relativistic quantum mechanics

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 University

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 of

Physics (North-Holland, Elsevier, Amsterdam 2007)

11 R Clifton, J Butterfield & H Halvorson: Quantum Entanglements - Selected Papers (Oxford

University 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, Singapore

1985)

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 statistics

of the particles, follows from a simple symmetry argument, where the wave

func-tionsof 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

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

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, Singapore

1985)

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 statistics

of the particles, follows from a simple symmetry argument, where the wave

func-tionsof 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

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

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, Singapore

1985)

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 statistics

of the particles, follows from a simple symmetry argument, where the wave

func-tionsof 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

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

A

The restriction to two possible kinds of quantum statistics, represented by thesign factor in (1), seems almost obvious On one hand the permutation of parti-cle coordinates has no physical significance when the particles are identical, which

means that the wave function can change at most by a complex phase factor e iθ

On the other hand a double permutation seems to make no change at all, which ther restricts the phase factor to a sign±1 This is the standard argument used in

fur-textbooks like [14]

However, there is a loophole to this argument, as pointed out by J.M Leinaas and

J 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 types

of 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 this

discussion 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 mensions can readily be demonstrated, as illustrated in Fig 1a In two dimensions

di-a two-pdi-article interchdi-ange pdi-ath comes with di-an orientdi-ation, di-and di-as di-a consequence di-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 verse, and is consequently restricted to±1 This explains why anyons are possible

in-in two but not in-in three dimensions Sin-ince the statistics angle θ in-in the exchange

fac-tor eiθ is a free parameter, there is a different type of anyon for each value of θ For

time

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 viewed

as forming a braid and the associated phase factor can be viewed as a representing an element of

the 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 the

permutation group The generalized types of quantum statistics characterized by the

parameter θ is often referred to as fractional statistics or braiding statistics.

Since anyons can only exist in two dimensions, elementary particles in the world

of 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, as

was realized after the discovery of the fractional  quantum Hall effectin 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 for

anyons, which later on has been followed up by ideas of anyons in other types

of 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 to

a 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 focus

is 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 to

the physics of exotic condensed matter systems known as topologically ordered

systems, where the two-dimensional electron gas of the quantum Hall system is a

special realization [9] The braid formulation also opens for generalizations in theform of non-abelian anyons In this extension of the anyon theory, the phase factor

associated with the interchange of two anyon positions is replaced by non-abelian unitary operations (or matrices) This is an extension of the simple identical particle

picture 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 effect

there are indications that such nonabelions may indeed exist [10], and theoretical ideas 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

A

additional 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 the

non-trivial braiding symmetry for a compensating statistics interaction [1], which is

a two-body interaction that is sensitive to the braiding of particles, but is independent

of distance The same type of statistics transformation has also been used in fieldtheory descriptions of the fractional quantum Hall effect, where the fundamental

electron field is changed by a statistics transmutation into an effective bosonic field

of 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]

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,

In 1965, John S Bell proved a celebrated theorem [1] which essentially states that

no 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 predictions

of 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 class

of 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 outcome

of a measurement at (e.g.) station 2 cannot depend on the nature of the 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 choice

measure-at 1 and measurement measure-at 2 fail to meet the condition of spacelike separmeasure-ation 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

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,

In 1965, John S Bell proved a celebrated theorem [1] which essentially states that

no 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 predictions

of 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 class

of 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 outcome

of a measurement at (e.g.) station 2 cannot depend on the nature of the 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 choice

measure-at 1 and measurement measure-at 2 fail to meet the condition of spacelike separmeasure-ation 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

A

affected 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 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 incompatibility

orig-of 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 Aspect

et 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-class

of the more general category of “delayed-choice” experiments Delayed-Choice

Experiment), but they have a special significance in their role of attempting to 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

ex-At each station, the “switch” which decides which of the two alternative 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 density

measure-and 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 missed

in 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 as

a “random” process, the locality loophole is blocked The data obtained in ref [4]violate the OLT predictions by 5 standard deviations

Pa’

Ultrasonic transducer

Fig 1 Schema of switching devices in Aspect experiment Pa (Pa )are polarisers with

transmis-sion axis a (a) When a photon arrives at time on ultrasonic cycle when density of H2 O is constant,

it is directed into Pa; (b) if it arrives at a maximum of the standing wave, into Pa

Is the switching in fact a truly random process? On the one hand, since the ducer pairs are driven by different generators at different frequencies, there is nocorrelation between the choices made at the two stations, and as we have seen notime for information about the choice itself to be transmitted between them On theother hand, since the driving at each station separately is periodic, a sufficientlydetermined advocate of OLT’s might argue that station 2 has the information to pre-dict what the setting at station 1 will be at a given time in the future and to makearrangements accordingly (and of course vice versa) Thus, while the experiment

trans-of ref [4] is clearly a major advance on the original Freedman-Clauser one, noteveryone was convinced that it had definitively blocked the locality loophole

Of the various Aspect-type experiments performed subsequently to 1982, bly the most notable is that of Weihs et al [5] This experiment used a much longerbaseline, around 400 m, and the choice of measurement was made by a quantum ran-dom number generator (QRNG), with a total switching time of less than 100 nsec

proba-Aspect Experiment 17

A

A further feature of this experiment, unique up to now among the whole class of

“Bell’s theorem” experiments, is that instead of being channelled to a central cidence counter the detection outcomes are recorded in situ and compared, with thehelp of accurate timing, only hours or days later (so that, coming back to the ques-tion of the time of “realization”, its postponement until the time of comparison,which is not totally implausible in other experiments, would in this case seemdistinctly unnatural) The duration of the registration process was such that it iscompleted well within the signal transit time The data obtained are consistent withthe predictions of quantum mechanics and violate those of the class of OLT’s by 30standard deviations

coin-One further experiment which has some significance in the present context is that

of Tittel et al [6] Although there was no in-situ recording, this is otherwise similar

in spirit to that of ref [5], with an even longer base-line (10 km); the difference isthat the role of the QRNG which controls the choice of measurement is played bythe measured photon itself (it impinges on a beam splitter where the output beamscorrespond to different choices) Once more good agreement with the predictions ofquantum mechanics is obtained

In the light of these experiments, any attempt to continue to exploit the localityloophole to defend a theory of the OLT class would have either to deny that theQRNG’s used work in a genuinely random way, or postpone the realization pro-cess for at least 1.3 microsec after the photon enters the photomultiplier (the signaltransit time in the experiment of Weihs et al.) A truly definitive blocking of thisloophole would presumably require that the detection be directly conducted by twohuman observers with a spatial separation such that the signal transit time exceedshuman reaction times, a few hundred milliseconds (i.e., a separation of several tens

of thousand kilometers) Given the extraordinary progress made in quantum munication in recent years, this goal may not be indefinitely far in the future In themeantime, a small step in this direction might be taken by repeating the experiment

com-of Weihs et al with inspection com-of the outcomes by independent human observersbefore they are correlated, something which was not done in ref [5].1

Primary Literature

1 J.S Bell: On the Einstein-Podolsky-Rosen paradox Physics 1, 195 (1964)

2 J.F Clauser, M.A Horne, A Shimony and R.A Holt: Proposed experiment to test local

hidden-variable theories Phys Rev Lett 23, 880 (1969)

3 S.J Freedman, J.F Clauser: Experimental test of local hidden-variable theories Phys Rev Lett.

28, 938 (1972)

4 A Aspect, J Dalibard, G Roger: Experimental test of Bell’s inequalities using time-varying

analysers Phys Rev Lett 49, 1804 (1982)

5 G Weihs, T.Jennewein, C.Simon, H.Weinfurter and A.Zeilinger: Violation of Bell’s inequality

under strict Einstein locality conditions Phys Rev Lett 81, 5039 (1998)

1 This work was supported by the National Science Foundation under grant no 21568.

NSF-EIA-01-18 Atomic Models, J.J Thomson’s “Plum Pudding” Model

6 W Tittel, J Brendel, H Zbinden and N Gisin: Violation of Bell inequalities by photons more

than 10 km apart Phys Rev Lett 81, 3563 (1998)

See also: Bohr’s Atomic Model; Rutherford Atom

Atomic Models, J.J Thomson’s

“Plum Pudding” Model

Klaus Hentschel

In 1897, Joseph John Thomson (1856–1940) had announced the discovery of a puscle Others soon called it electron, despite Thomson’s stubborn preference forhis original term, borrowed from Robert Boyle (1627–91) to denote any particle-like structure Very soon afterwards, Thomson began to think about how to explainthe periodicity of properties of the chemical elements in terms of these negativelycharged corpuscles as atomic constituents Chemical properties would thus have todepend on the number and constellations of these corpuscles inside the atom Theywould have to have stable positions in it, bound by electrostatic and possibly kineticforces Because under normal conditions chemical atoms are electrically neutral,the total electric charge of all these negatively charged electrons had to be com-pensated for by an equal amount of positive charge For Thomson it was natural to

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

6 W Tittel, J Brendel, H Zbinden and N Gisin: Violation of Bell inequalities by photons more

than 10 km apart Phys Rev Lett 81, 3563 (1998)

See also: Bohr’s Atomic Model; Rutherford Atom

Atomic Models, J.J Thomson’s

“Plum Pudding” Model

Klaus Hentschel

In 1897, Joseph John Thomson (1856–1940) had announced the discovery of a puscle Others soon called it electron, despite Thomson’s stubborn preference forhis original term, borrowed from Robert Boyle (1627–91) to denote any particle-like structure Very soon afterwards, Thomson began to think about how to explainthe periodicity of properties of the chemical elements in terms of these negativelycharged corpuscles as atomic constituents Chemical properties would thus have todepend on the number and constellations of these corpuscles inside the atom Theywould have to have stable positions in it, bound by electrostatic and possibly kineticforces Because under normal conditions chemical atoms are electrically neutral,the total electric charge of all these negatively charged electrons had to be com-pensated for by an equal amount of positive charge For Thomson it was natural to

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

6 W Tittel, J Brendel, H Zbinden and N Gisin: Violation of Bell inequalities by photons more

than 10 km apart Phys Rev Lett 81, 3563 (1998)

See also: Bohr’s Atomic Model; Rutherford Atom

Atomic Models, J.J Thomson’s

“Plum Pudding” Model

Klaus Hentschel

In 1897, Joseph John Thomson (1856–1940) had announced the discovery of a puscle Others soon called it electron, despite Thomson’s stubborn preference forhis original term, borrowed from Robert Boyle (1627–91) to denote any particle-like structure Very soon afterwards, Thomson began to think about how to explainthe periodicity of properties of the chemical elements in terms of these negativelycharged corpuscles as atomic constituents Chemical properties would thus have todepend on the number and constellations of these corpuscles inside the atom Theywould have to have stable positions in it, bound by electrostatic and possibly kineticforces Because under normal conditions chemical atoms are electrically neutral,the total electric charge of all these negatively charged electrons had to be com-pensated for by an equal amount of positive charge For Thomson it was natural to

New York 1998)

16 N.P Landsman: Between classical and quantum, in Handbook of the Philosophy of Science... predictions

of 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)... predictions

of 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)