ContentsArticles

David Bohm 1Aharonov- Bohm effect 7Bohm diffusion 11Bohm interpretation 12Correspondence principle 28De Broglie–Bohm theory 33EPR paradox 48Holographic paradigm 57Holographic principle 61Holomovement 65Holonomic brain theory 69Implicate and Explicate Order 71Implicate order 78Membrane paradigm 85Orch- OR 86Debye sheath 93John Stewart Bell 96Karl H. Pribram 99Implicate and explicate order according to David Bohm 102Hidden variable theory 109Local hidden variable theory 111Bell's theorem 116Bell test experiments 129Hidden variables 133

ReferencesArticle Sources and Contributors 134Image Sources, Licenses and Contributors 136

Article LicensesLicense 137

David Bohm 1

David Bohm

David Bohm

David Joseph Bohm (1917-1992)

Born December 20, 1917Wilkes-Barre, Pennsylvania, U.S.

Died October 27, 1992 (aged 74)London, UK

Residence United Kingdom

Citizenship British

Nationality British

Fields Physicist

Institutions Manhattan ProjectPrinceton UniversityUniversity of São PauloTechnionUniversity of BristolBirkbeck College

Alma mater Pennsylvania State CollegeCalifornia Institute of TechnologyUniversity of California, Berkeley

Doctoral advisor Robert Oppenheimer

Doctoral students Yakir AharonovDavid PinesJeffrey BubHenri Bortoft

Other notable students Jack Sarfatti

Known for Bohm-diffusionBohm interpretationAharonov-Bohm effectHolonomic modelBohm Dialogue

Influences Albert EinsteinJiddu KrishnamurtiArthur SchopenhauerGeorg Wilhelm Friedrich Hegel

Influenced John Stewart Bell

David Bohm 2

David Joseph Bohm (20 December 1917 – 27 October 1992) was a American-born British quantum physicist whomade contributions in the fields of theoretical physics, philosophy and neuropsychology, and to the ManhattanProject.

Biography

Youth and collegeBohm was born in Wilkes-Barre, Pennsylvania to a Hungarian Jewish immigrant father and a Lithuanian Jewishmother. He was raised mainly by his father, a furniture store owner and assistant of the local rabbi. Bohm attendedPennsylvania State College, graduating in 1939, and then headed west to the California Institute of Technology for ayear, and then transferred to the theoretical physics group under Robert Oppenheimer at the University of California,Berkeley, where he eventually obtained his doctorate degree.Bohm lived in the same neighborhood as some of Oppenheimer's other graduate students (Giovanni Rossi Lomanitz,Joseph Weinberg, and Max Friedman) and with them became increasingly involved not only with physics, but withradical politics. Bohm gravitated to alternative models of society and became active in organizations like the YoungCommunist League, the Campus Committee to Fight Conscription, and the Committee for Peace Mobilization alllater branded as Communist organizations by the FBI under J. Edgar Hoover.

Work and doctorate

Manhattan Project Contributions

During World War II, the Manhattan Project mobilized much of Berkeley's physics research in the effort to producethe first atomic bomb. Though Oppenheimer had asked Bohm to work with him at Los Alamos (the top-secretlaboratory established in 1942 to design the bomb), the head of the Manhattan Project, General Leslie Groves, wouldnot approve Bohm's security clearance, after tip-offs about his politics (Bohm's friend, Joseph Weinberg, had alsocome under suspicion for espionage).Bohm remained in Berkeley, teaching physics, until he completed his Ph.D. in 1943, under an unusually ironiccircumstance. According to Peat (see reference below, p. 64), "the scattering calculations (of collisions of protonsand deuterons) that he had completed proved useful to the Manhattan Project and were immediately classified.Without security clearance, Bohm was denied access to his own work; not only would he be barred from defendinghis thesis, he was not even allowed to write his own thesis in the first place!" To satisfy the university, Oppenheimercertified that Bohm had successfully completed the research. He later performed theoretical calculations for theCalutrons at the Y-12 facility in Oak Ridge, used to electromagnetically enrich uranium for use in the bomb droppedon Hiroshima in 1945.

McCarthyism leads to Bohm leaving the United States

After the war, Bohm became an assistant professor at Princeton University, where he worked closely with AlbertEinstein. In May, 1949, at the beginning of the McCarthyism period, the House Un-American Activities Committeecalled upon Bohm to testify before it— because of his previous ties to suspected Communists. Bohm, however,pleaded the Fifth amendment right to decline to testify, and refused to give evidence against his colleagues.In 1950, Bohm was charged for refusing to answer questions before the Committee and arrested. He was acquitted inMay, 1951, but Princeton had already suspended him. After the acquittal, Bohm's colleagues sought to have hisposition at Princeton re-instated, and Einstein reportedly wanted Bohm to serve as his assistant. The university,however, did not renew his contract. Bohm then left for Brazil to take up a Chair in Physics at the University of SãoPaulo, and later was also at the Technion in Haifa, Israel, and at Birkbeck College, University of London.

David Bohm 3

Quantum theory and Bohm-diffusion

During his early period, Bohm made a number of significant contributions to physics, particularly in the area ofquantum mechanics and relativity theory. As a post-graduate at Berkeley, he developed a theory of plasmas,discovering the electron phenomenon now known as Bohm-diffusion. His first book, Quantum Theory published in1951, was well-received by Einstein, among others. However, Bohm became dissatisfied with the orthodox approachto quantum theory, which he had written about in that book, and began to develop his own approach (Bohminterpretation)— a non-local hidden variable deterministic theory whose predictions agree perfectly with thenondeterministic quantum theory. His work and the EPR argument became the major factor motivating John Bell'sinequality, whose consequences are still being investigated.

The Aharonov-Bohm effect

In 1955 Bohm moved to Israel, where he spent two years at the Technion at Haifa. Here he met his wife Saral, whobecame an important figure in the development of his ideas. In 1957, Bohm moved to the UK as a research fellow atthe University of Bristol. In 1959, with his student Yakir Aharonov, they discovered the Aharonov-Bohm effect,showing how a magnetic field could affect a region of space in which the field had been shielded, although its vectorpotential did not vanish there. This showed for the first time that the magnetic vector potential, hitherto amathematical convenience, could have real physical (quantum) effects. In 1961, Bohm was made Professor ofTheoretical Physics at Birkbeck College London, where his collected papers [1] are kept.

The holonomic model of the brain

Bohm also made substantial theoretical contributions to neuropsychology and the development of the holonomicmodel of the functioning of the brain.[2] In collaboration with Stanford neuroscientist Karl Pribram, Bohm helpedestablish the foundation for Pribram's theory that the brain operates in a manner similar to a hologram, in accordancewith quantum mathematical principles and the characteristics of wave patterns. These wave forms may composehologram-like organizations, Bohm suggested, basing this concept on his application of Fourier analysis, amathematical method for decomposing complex waves into component sine waves. The holonomic brain modeldeveloped by Pribram and Bohm posits a lens defined world view— much like the textured prismatic effect ofsunlight refracted by the churning mists of a rainbow— a view which is quite different from the more conventional"objective reality" - not to be confused with objectivity - approach. Pribram held that if psychology means tounderstand the conditions that produce the world of appearances, it must look to the thinking of physicists likeBohm.[3]

Thought as a System

Bohm was alarmed by what he considered an increasing imbalance of not only 'man' and nature, but among peoples,as well as people, themselves. Bohm: "So one begins to wonder what is going to happen to the human race.Technology keeps on advancing with greater and greater power, either for good or for destruction." He goes on toask:

What is the source of all this trouble? I'm saying that the source is basically in thought. Many peoplewould think that such a statement is crazy, because thought is the one thing we have with which to solveour problems. That's part of our tradition. Yet it looks as if the thing we use to solve our problems withis the source of our problems. It's like going to the doctor and having him make you ill. In fact, in 20%of medical cases we do apparently have that going on. But in the case of thought, it's far over 20%.

In Bohm's view:...the general tacit assumption in thought is that it's just telling you the way things are and that it's not doinganything - that 'you' are inside there, deciding what to do with the info. But you don't decide what to do withthe info. Thought runs you. Thought, however, gives false info that you are running it, that you are the one

David Bohm 4

who controls thought. Whereas actually thought is the one which controls each one of us.Thought is creating divisions out of itself and then saying that they are there naturally. This is another majorfeature of thought: Thought doesn't know it is doing something and then it struggles against what it is doing. Itdoesn't want to know that it is doing it. And thought struggles against the results, trying to avoid thoseunpleasant results while keeping on with that way of thinking. That is what I call "sustained incoherence".

Bohm thus proposes in his book, Thought as a System, a pervasive, systematic nature of thought:What I mean by "thought" is the whole thing - thought, felt, the body, the whole society sharing thoughts - it'sall one process. It is essential for me not to break that up, because it's all one process; somebody else'sthoughts becomes my thoughts, and vice versa. Therefore it would be wrong and misleading to break it up intomy thoughts, your thoughts, my feelings, these feelings, those feelings... I would say that thought makes whatis often called in modern language a system. A system means a set of connected things or parts. But the waypeople commonly use the word nowadays it means something all of whose parts are mutually interdependent -not only for their mutual action, but for their meaning and for their existence. A corporation is organized as asystem - it has this department, that department, that department. They don't have any meaning separately;they only can function together. And also the body is a system. Society is a system in some sense. And so on.Similarly, thought is a system. That system not only includes thoughts, "felts" and feelings, but it includes thestate of the body; it includes the whole of society - as thought is passing back and forth between people in aprocess by which thought evolved from ancient times. A system is constantly engaged in a process ofdevelopment, change, evolution and structure changes...although there are certain features of the system whichbecome relatively fixed. We call this the structure.... Thought has been constantly evolving and we can't saywhen that structure began. But with the growth of civilization it has developed a great deal. It was probablyvery simple thought before civilization, and now it has become very complex and ramified and has much moreincoherence than before.Now, I say that this system has a fault in it - a "systematic fault". It is not a fault here, there or here, but it is afault that is all throughout the system. Can you picture that? It is everywhere and nowhere. You may say "I seea problem here, so I will bring my thoughts to bear on this problem". But "my" thought is part of the system. Ithas the same fault as the fault I'm trying to look at, or a similar fault.Thought is constantly creating problems that way and then trying to solve them. But as it tries to solve them itmakes it worse because it doesn’t notice that it's creating them, and the more it thinks, the more problems itcreates. (P. 18-19)

Bohm Dialogue

To address societal problems in his later years, Bohm wrote a proposal for a solution that has become known as"Bohm Dialogue", in which equal status and "free space" form the most important prerequisites of communicationand the appreciation of differing personal beliefs. He suggested that if these Dialogue groups were experienced on asufficiently wide scale, they could help overcome the isolation and fragmentation Bohm observed was inherent in thesociety.

Later yearsBohm continued his work in quantum physics past his retirement in 1987. His final work, the posthumouslypublished The Undivided Universe: An ontological interpretation of quantum theory (1993), resulted from adecades-long collaboration with his colleague Basil Hiley. He also spoke to audiences across Europe and NorthAmerica on the importance of dialogue as a form of sociotherapy, a concept he borrowed from London psychiatristand practitioner of Group Analysis Patrick De Mare, and had a series of meetings with the Dalai Lama. He waselected Fellow of the Royal Society in 1990.

David Bohm 5

Near the end of his life, Bohm began to experience a recurrence of depression which he had suffered at earlier timesin his life. He was admitted to the Maudsley Hospital in South London on 10 May 1991. His condition worsened andit was decided that the only thing that might help him was electroconvulsive therapy. Bohm's wife consultedpsychiatrist David Shainberg, Bohm's long-time friend and collaborator, who agreed that electroconvulsivetreatments were probably his only option. Bohm showed marked improvement from the treatments and was releasedon 29 August. However, his depression returned and was treated with medication.[4]

David Bohm died of a heart attack in Hendon,[5] London, on 27 October 1992, aged 74. In fact, Professor Bohm wastraveling in a London taxi on that day, conversing with the cab driver. After not getting any response from thepassenger in the back seat for a few seconds, the driver turned back and found Dr. Bohm had died of a heart attack.[6]

David Bohm was considered by many Nobel laureates as one of the best quantum physicists of all time, who richlydeserved the Nobel Prize, but failed to obtain it possibly due to political victimization.[7]

Publications• 1951. Quantum Theory, New York: Prentice Hall. 1989 reprint, New York: Dover, ISBN 0-486-65969-0• 1957. Causality and Chance in Modern Physics, 1961 Harper edition reprinted in 1980 by Philadelphia: U of

Pennsylvania Press, ISBN 0-8122-1002-6• 1962. Quanta and Reality, A Symposium, with N. R. Hanson and Mary B. Hesse, from a BBC program published

by the American Research Council• 1965. The Special Theory of Relativity, New York: W.A. Benjamin.• 1980. Wholeness and the Implicate Order, London: Routledge, ISBN 0-7100-0971-2, 1983 Ark paperback: ISBN

0-7448-0000-5, 2002 paperback: ISBN 0-415-28979-3• 1985. Unfolding Meaning: A weekend of dialogue with David Bohm (Donald Factor, editor), Gloucestershire:

Foundation House, ISBN 0-948325-00-3, 1987 Ark paperback: ISBN 0-7448-0064-1, 1996 Routledge paperback:ISBN 0-415-13638-5

• 1985. The Ending of Time, with Jiddu Krishnamurti, San Francisco, CA: Harper, ISBN 0-06-064796-5.• 1987. Science, Order and Creativity, with F. David Peat. London: Routledge. 2nd ed. 2000. ISBN 0-415-17182-2.• 1991. Changing Consciousness: Exploring the Hidden Source of the Social, Political and Environmental Crises

Facing our World (a dialogue of words and images), coauthor Mark Edwards, Harper San Francisco, ISBN0-06-250072-4

• 1992. Thought as a System (transcript of seminar held in Ojai, California, from 30 November to 2 December1990), London: Routledge. ISBN 0-415-11980-4.

• 1993. The Undivided Universe: An ontological interpretation of quantum theory, with B.J. Hiley, London:Routledge, ISBN 0-415-12185-X (final work)

• 1996. On Dialogue. editor Lee Nichol. London: Routledge, hardcover: ISBN 0-415-14911-8, paperback: ISBN0-415-14912-6, 2004 edition: ISBN 0-415-33641-4

• 1998. On Creativity, editor Lee Nichol. London: Routledge, hardcover: ISBN 0-415-17395-7, paperback: ISBN0-415-17396-5, 2004 edition: ISBN 0-415-33640-6

• 1999. Limits of Thought: Discussions, with Jiddu Krishnamurti, London: Routledge, ISBN 0-415-19398-2.• 1999. Bohm-Biederman Correspondence: Creativity and Science, with Charles Biederman. editor Paavo

Pylkkänen. ISBN 0-415-16225-4.• 2002. The Essential David Bohm. editor Lee Nichol. London: Routledge, ISBN 0-415-26174-0. preface by the

Dalai Lama

David Bohm 6

See also• Aharonov-Bohm effect• American philosophy• Bohm diffusion of a plasma in a magnetic field• Bohm interpretation• Correspondence principle• De Broglie–Bohm theory• EPR paradox• Holographic paradigm• Holographic principle• Holomovement• Holonomic brain theory• Implicate and Explicate Order• Implicate order• Jiddu Krishnamurti• Membrane paradigm• Penrose-Hameroff "Orchestrated Objective Reduction" theory of consciousness• The Bohm sheath criterion, which states that a plasma must flow with at least the speed of sound toward a solid

surface• Wave gene• John Stewart Bell• Karl Pribram• Influence on John David Garcia• List of American philosophers

References[1] http:/ / www. aim25. ac. uk/ cgi-bin/ search2?coll_id=3070& inst_id=33[2] Comparison between Karl Pribram's "Holographic Brain Theory" and more conventional models of neuronal computation (http:/ / www.

acsa2000. net/ bcngroup/ jponkp/ #chap4)[3] http:/ / homepages. ihug. co. nz/ ~sai/ pribram. htm[4] F. David Peat, Infinite Potential: The Life and Times of David Bohm, Reading, MA: Addison Wesley, 1997, pp. 308-317. ISBN 0201328208.[5] Deaths England and Wales 1984-2006 (http:/ / www. findmypast. com/ BirthsMarriagesDeaths. jsp)[6] F. David Peat, Infinite Potential: The Life and Times of David Bohm, Reading, MA: Addison Wesley, 1997, pp. 308-317. ISBN 0201328208.[7] F. David Peat, Infinite Potential: The Life and Times of David Bohm, Reading, MA: Addison Wesley, 1997, pp. 308-317. ISBN 0201328208.

• "Bohm's Alternative to Quantum Mechanics", David Z. Albert, Scientific American (May, 1994)• Brotherhood of the Bomb: The Tangled Lives and Loyalties of Robert Oppenheimer, Ernest Lawrence, and

Edward Teller, Herken, Gregg, New York: Henry Holt (2002) ISBN 0-8050-6589-X (information on Bohm'swork at Berkeley and his dealings with HUAC)

• Infinite Potential: the Life and Times of David Bohm, F. David Peat, Reading, MA: Addison Wesley (1997),ISBN 0-201-40635-7 DavidPeat.com (http:/ / www. fdavidpeat. com/ )

• Quantum Implications: Essays in Honour of David Bohm, (B.J. Hiley, F. David Peat, editors), London: Routledge(1987), ISBN 0-415-06960-2

• Thought as a System (transcript of seminar held in Ojai, California, from 30 November to 2 December 1990),London: Routledge. (1992) ISBN 0-415-11980-4.

• The Quantum Theory of Motion: an account of the de Broglie-Bohm Causal Interpretation of QuantumMechanics, Peter R. Holland, Cambridge: Cambridge University Press. (2000) ISBN 0-921-48453-9.

David Bohm 7

External links• English site (http:/ / www. david-bohm. net) for David Bohm's ideas about Dialogue.• the David_Bohm_Hub (http:/ / www. thinkg. net/ david_bohm/ ) From thinkg.net, with compilations of David

Bohm's life and work in form of texts, audio, video, and pictures.• Thought Knowledge Perception Institute (http:/ / www. tkpi. org) A non-partisan organization that aims to

preserve and continue the work of David Bohm and others.• Lifework of David Bohm: River of Truth (http:/ / www. vision. net. au/ ~apaterson/ science/ david_bohm.

htm#BOHM'S LEGACY): Article by Will Keepin• Dialogos (http:/ / www. dialogos. com): Consulting group, originally founded by Bohm colleagues William Isaacs

and Peter Garrett, aiming to bring Bohm dialogue into organizations.• (http:/ / www. quantum-mind. co. uk) quantum mind• Interview with David Bohm (http:/ / www. fdavidpeat. com/ interviews/ bohm. htm) provided and conducted by

F. David Peat along with John Briggs, first issued in Omni magazine, January 1987• David Bohm and Krishnamurti (http:/ / www. wie. org/ j11/ peat. asp)• Archive of papers at Birkbeck College relating to David Bohm. (http:/ / www. bbk. ac. uk/ lib/ about/ hours/

bohm)• Quantum-Mind (http:/ / www. quantum-mind. co. uk)• Oral History interview transcript with David Bohm 8 May 1981, American Institute of Physics, Niels Bohr

Library and Archives (http:/ / www. aip. org/ history/ ohilist/ 4513. html)

Aharonov- Bohm effectThe Aharonov–Bohm effect, sometimes called the Ehrenberg–Siday–Aharonov–Bohm effect, is a quantummechanical phenomenon in which an electrically charged particle shows a measurable interaction with anelectromagnetic field despite being confined to a region in which both the magnetic field B and electric field E arezero.The Aharonov–Bohm effect shows that the local E and B fields do not contain full information about theelectromagnetic field, and the electromagnetic four-potential, A, must be used instead. By Stokes' theorem, themagnitude of the Aharonov–Bohm effect can be calculated using A alone or using E and B alone. But when using Eand B, however, the effect depends on the field values in a region from which the test particle is excluded, not onlyclassically but also quantum mechanically. In contrast, the effect depends on A only in the region where the testparticle is allowed. Therefore we can either abandon the principle of locality (which most physicists are reluctant todo) or we are forced to accept the realisation that the electromagnetic potential offers a more complete description ofelectromagnetism than the electric and magnetic fields can. In classical electromagnetism the two descriptions wereequivalent. With the addition of quantum theory, though, the electromagnetic potential A is seen as being morefundamental or "real";[1] the E and B fields can be derived from the potential A, but the potential can not be derivedfrom the E and B fields.Werner Ehrenberg and Raymond E. Siday first predicted the effect in 1949,[2] and similar effects were laterrediscovered by Yakir Aharonov and David Bohm in 1959.[3] (After publication of the 1959 paper, Bohm wasinformed of Ehrenberg and Siday's work, which was acknowledged and credited in Bohm and Aharanov'ssubsequent 1961 paper.[4] [5] )The most commonly described case, sometimes called the Aharonov–Bohm solenoid effect, takes place when the wave function of a charged particle passing around a long solenoid experiences a phase shift as a result of the enclosed magnetic field, despite the magnetic field being zero in the region through which the particle passes. This phase shift has been observed experimentally by its effect on interference fringes. (There are also magnetic

Aharonov-Bohm effect 8

Aharonov–Bohm effects on bound energies and scattering cross sections, but these cases have not beenexperimentally tested.) An electric Aharonov–Bohm phenomenon was also predicted, in which a charged particle isaffected by regions with different electrical potentials but zero electric field, and this has also seen experimentalconfirmation. A separate "molecular" Aharonov–Bohm effect was proposed for nuclear motion inmultiply-connected regions, but this has been argued to be essentially different, depending only on local quantitiesalong the nuclear path.[6] A general review can be found in Peshkin and Tonomura (1989).[7]

Magnetic solenoid effectThe magnetic Aharonov–Bohm effect can be seen as a result of the requirement that quantum physics be invariantwith respect to the gauge choice for the electromagnetic potential, of which the magnetic vector potential A formspart.Electromagnetic theory implies that a particle with electric charge q travelling along some path P in a region withzero magnetic field B, but non-zero A (by ), acquires a phase shift , given in SI units by

Therefore particles, with the same start and end points, but travelling along two different routes will acquire a phasedifference Δφ determined by the magnetic flux Φ through the area between the paths (via Stokes' theorem and

), and given by:

Schematic of double-slit experiment in which Aharonov–Bohmeffect can be observed: electrons pass through two slits, interfering at

an observation screen, with the interference pattern shifted when amagnetic field B is turned on in the cylindrical solenoid.

In quantum mechanics the same particle can travelbetween two points by a variety of paths. Therefore thisphase difference can be observed by placing a solenoidbetween the slits of a double-slit experiment (orequivalent). An ideal solenoid encloses a magnetic fieldB, but does not produce any magnetic field outside ofits cylinder, and thus the charged particle (e.g. anelectron) passing outside experiences no magnetic fieldB. However, there is a (curl-free) vector potential Aoutside the solenoid with an enclosed flux, and so therelative phase of particles passing through one slit orthe other is altered by whether the solenoid current isturned on or off. This corresponds to an observableshift of the interference fringes on the observationplane.

The same phase effect is responsible for thequantized-flux requirement in superconducting loops. This quantization occurs because the superconducting wavefunction must be single valued: its phase difference Δφ around a closed loop must be an integer multiple of 2π (withthe charge for the electron Cooper pairs), and thus the flux Φ must be a multiple of h/2e. The superconducting fluxquantum was actually predicted prior to Aharonov and Bohm, by F. London in 1948 using a phenomenologicalmodel.[8]

The magnetic Aharonov–Bohm effect was experimentally confirmed by Osakabe et al. (1986),[9] following much earlier work summarized in Olariu and Popèscu (1984).[10] Its scope and application continues to expand. Webb et al. (1985)[11] demonstrated Aharonov–Bohm oscillations in ordinary, non-superconducting metallic rings; for a discussion, see Schwarzschild (1986)[12] and Imry & Webb (1989).[13] Bachtold et al. (1999)[14] detected the effect

Aharonov-Bohm effect 9

in carbon nanotubes; for a discussion, see Kong et al. (2004).[15]

Monopoles and Dirac stringsThe magnetic Aharonov–Bohm effect is also closely related to Dirac's argument that the existence of a magneticmonopole can be accommodated by the existing magnetic source-free Maxwell's equations if both electric andmagnetic charges are quantized.A magnetic monopole implies a mathematical singularity in the vector potential, which can be expressed as an Diracstring of infinitesimal diameter that contains the equivalent of all of the 4πg flux from a monopole "charge" g. TheDirac string starts from, and terminates on, a magnetic monopole. Thus, assuming the absence of an infinite-rangescattering effect by this arbitrary choice of singularity, the requirement of single-valued wave functions (as above)necessitates charge-quantization. That is must be an integer (in cgs units) for any electric charge qe and

magnetic charge qm.Like the electromagnetic potential A the Dirac string is not gauge invariant (it moves around with fixed endpointsunder a gauge transformation) is also not directly measurable.

Electric effectJust as the phase of the wave function depends upon the magnetic vector potential, it also depends upon the scalarelectric potential. By constructing a situation in which the electrostatic potential varies for two paths of a particle,through regions of zero electric field, an observable Aharonov–Bohm interference phenomenon from the phase shifthas been predicted; again, the absence of an electric field means that, classically, there would be no effect.

From the Schrödinger equation, the phase of an eigenfunction with energy E goes as . The energy,however, will depend upon the electrostatic potential V for a particle with charge q. In particular, for a region withconstant potential V (zero field), the electric potential energy qV is simply added to E, resulting in a phase shift:

where t is the time spent in the potential.The initial theoretical proposal for this effect suggested an experiment where charges pass through conductingcylinders along two paths, which shield the particles from external electric fields in the regions where they travel, butstill allow a varying potential to be applied by charging the cylinders. This proved difficult to realize, however.Instead, a different experiment was proposed involving a ring geometry interrupted by tunnel barriers, with a biasvoltage V relating the potentials of the two halves of the ring. This situation results in an Aharonov–Bohm phaseshift as above, and was observed experimentally in 1998.[16]

Aharonov–Bohm nano ringsNano rings were created by accident[17] while intending to make quantum dots. They have interesting opticalproperties associated with excitons and the Aharonov-Bohm effect.[17] Application of these rings used as lightcapacitors or buffers includes photonic computing and communications technology. Analysis and measurement ofgeometric phases in mesoscopic rings is ongoing.[18] [19] [20]

Mathematical interpretationIn the terms of modern differential geometry, the Aharonov–Bohm effect can be understood to be the monodromy of a flat complex line bundle. The U(1)-connection on this line bundle is given by the electromagnetic four-potential A as where d means partial derivation in the Minkowski space . The curvature form of the connection, , is the electromagnetic field strength, where is the 1-form corresponding to the

Aharonov-Bohm effect 10

four-potential. The holonomy of the connection, around a closed loop is, as a consequence of Stokes' theorem,determined by the magnetic flux through a surface bounded by the loop. This description is general and works insideas well as outside the conductor. Outside of the conducting tube, which is for example a longitudinally magnetizedinfinite metallic thread, the field strength is  ; in other words outside the thread the connection is flat, and theholonomy of a loop contained in the field-free region depends only on the winding number around the tube and is, bydefinition, the monodromy of the flat connection.In any simply connected region outside of the tube we can find a gauge transformation (acting on wave functions andconnections) that gauges away the vector potential. However, if the monodromy is non trivial, there is no such gaugetransformation for the whole outside region. If we want to ignore the physics inside the conductor and only describethe physics in the outside region, it becomes natural to mathematically describe the quantum electron by a section ina complex line bundle with an "external" connection rather than an external EM field (by incorporating localgauge transformations we have already acknowledged that quantum mechanics defines the notion of a (locally) flatwavefunction (zero momentum density) but not that of unit wavefunction). The Schrödinger equation readilygeneralizes to this situation. In fact for the Aharonov–Bohm effect we can work in two simply connected regionswith cuts that pass from the tube towards or away from the detection screen. In each of these regions we have tosolve the ordinary free Schrödinger equations but in passing from one region to the other, in only one of the twoconnected components of the intersection (effectively in only one of the slits) we pick up a monodromy factor ,which results in a shift in the interference pattern.Effects with similar mathematical interpretation can be found in other fields. For example, in classical statisticalphysics, quantization of a molecular motor motion in a stochastic environment can be interpreted as anAharonov-Bohm effect induced by a gauge field acting in the space of control parameters.[21]

External links• A video explaining the use of the Aharonov-Bohm effect in nano-rings. [22]

See also• Geometric phase• Hannay angle• Wannier function• Berry phase

References[1] Feynman, R. The Feynman Lectures on Physics. 2. p. 15-5. "...is the vector potential a "real" field? ... a real field is a mathematical device for

avoiding the idea of action at a distance. .... for a long time it was believed that A was not a "real" field. .... there are phenomena involvingquantum mechanics which show that in fact A is a "real" field in the sense that we have defined it..... E and B are slowly disappearing from themodern expression of physical laws; they are being replaced by A and [the scalar potential] and knowledge of the classical electromagneticfield acting locally on a particle is not sufficient to predict its quantum-mechanical behavior."

[2] Ehrenberg, W; Siday, RE (1949). "The Refractive Index in Electron Optics and the Principles of Dynamics"". Proceedings of the PhysicalSociety B 62: 8–21. doi:10.1088/0370-1301/62/1/303.

[3] Aharonov, Y; Bohm, D (1959). "Significance of electromagnetic potentials in quantum theory". Physical Review 115: 485–491.doi:10.1103/PhysRev.115.485.

[4] Peat, FD (1997). Infinite Potential: The Life and Times of David Bohm (http:/ / www. fdavidpeat. com/ bibliography/ books/ infinite. htm).Addison-Wesley. ISBN 0-201-40635-7. .

[5] Aharonov, Y; Bohm, D (1961). "Further Considerations on Electromagnetic Potentials in the Quantum Theory". Physical Review 123:1511–1524. doi:10.1103/PhysRev.123.1511.

[6] Sjöqvist, E (2002). "Locality and topology in the molecular Aharonov-Bohm effect". Physical Review Letters 89 (21): 210401.doi:10.1103/PhysRevLett.89.210401. arXiv:quant-ph/0112136.

[7] Peshkin, M; Tonomura, A (1989). The Aharonov-Bohm effect. Springer-Verlag. ISBN 3-540-51567-4.[8] London, F (1948). "On the Problem of the Molecular Theory of Superconductivity". Physical Review 74: 562. doi:10.1103/PhysRev.74.562.

Aharonov-Bohm effect 11

[9] Osakabe, N; et al. (1986). "Experimental confirmation of Aharonov-Bohm effect using a toroidal magnetic field confined by asuperconductor". Physical Review A 34: 815. doi:10.1103/PhysRevA.34.815.

[10] Olariu, S; Popescu, II (1985). "The quantum effects of electromagnetic fluxes". Reviews of Modern Physics 57: 339.doi:10.1103/RevModPhys.57.339.

[11] Webb, RA; Washburn, S; Umbach, CP; Laibowitz, RB (1985). "Observation of h/e Aharonov-Bohm Oscillations in Normal-Metal Rings".Physical Review Letters 54: 2696. doi:10.1103/PhysRevLett.54.2696.

[12] Schwarzschild, B (1986). "Currents in Normal-Metal Rings Exhibit Aharonov–Bohm Effect". Physics Today 39 (1): 17.doi:10.1063/1.2814843.

[13] Imry, Y; Webb, RA (1989). "Quantum Interference and the Aharonov-Bohm Effect". Scientific American 260 (4).[14] Schönenberger, C (1999). "Aharonov–Bohm oscillations in carbon nanotubes". Nature 397: 673. doi:10.1038/17755.[15] Kong, J; Kouwenhoven, L; Dekker, C (2004). "Quantum change for nanotubes" (http:/ / physicsworld. com/ cws/ article/ print/ 19746).

Physics World. . Retrieved 2009-08-17.[16] van Oudenaarden, A (1998). "Magneto-electric Aharonov–Bohm effect in metal rings". Nature 391: 768. doi:10.1038/35808.[17] Fischer, AM (2009). "Quantum doughnuts slow and freeze light at will" (http:/ / www. innovations-report. com/ html/ reports/

physics_astronomy/ quantum_doughnuts_slow_freeze_light_128981. html). Innovation Reports. . Retrieved 2008-08-17.[18] Borunda, MF; et al. (2008). "Aharonov-Casher and spin Hall effects in two-dimensional mesoscopic ring structures with strong spin-orbit

interaction". arΧiv:0809.0880 [cond-mat.mes-hall].[19] Grbic, B; et al. (2008). "Aharonov-Bohm oscillations in p-type GaAs quantum rings". Physica E 40: 1273. doi:10.1016/j.physe.2007.08.129.

arXiv:0711.0489.[20] Fischer, AM; et al. (2009). "Exciton Storage in a Nanoscale Aharonov-Bohm Ring with Electric Field Tuning". Physical Review Letters

102: 096405. doi:10.1103/PhysRevLett.102.096405. arXiv:0809.3863.[21] Chernyak, VY; Sinitsyn, NA (2009). "Robust quantization of a molecular motor motion in a stochastic environment". Journal of Chemical

Physics 131: 181101. doi:10.1063/1.3263821. arXiv:0906.3032. Bibcode: 2009JChPh.131r1101C.[22] http:/ / www2. warwick. ac. uk/ newsandevents/ news/ quantumdoughnuts

Bohm diffusionBohm diffusion is the diffusion of plasma across a magnetic field with a diffusion coefficient equal to

,

where B is the magnetic field strength, T is the temperature, and e is the elementary charge.It was first observed in 1946 by David Bohm, E. H. S. Burhop, and Harrie Massey while studying magnetic arcs foruse in isotope separation [1]. It has since been observed that many other plasmas follow this law. Fortunately thereare exceptions where the diffusion rate is lower, otherwise there would be no hope of achieving practical fusionenergy.Generally diffusion can be modelled as a random walk of steps of length δ and time τ. If the diffusion is collisional,then δ is the mean free path and τ is the inverse of the collision frequency. The diffusion coefficient D can beexpressed variously as

where v = δ/τ is the velocity between collisions.In a magnetized plasma, the collision frequency is usually small compared to the gyrofrequency, so that the step sizeis the gyroradius ρ and the step time is the inverse of the collision frequency ν, leading to D = ρ²ν. If the collisionfrequency is larger than the gyrofrequency, then the particles can be considered to move freely with the thermalvelocity vth between collisions, and the diffusion coefficient takes the form D = vth²/ν. Evidently the classical(collisional) diffusion is maximum when the collision frequency is equal to the gyrofrequency, in which case D =ρ²ωc = vth²/ωc. Substituting ρ = vth/ωc, vth = (kBT/m)1/2, and ωc = eB/m, we arrive at D = kBT/eB, which is the Bohmscaling. Considering the approximate nature of this derivation, the missing 1/16 in front is no cause for concern.Therefore, at least within a factor of order unity, Bohm diffusion is always greater than classical diffusion.

Bohm diffusion 12

In the common low collisionality regime, classical diffusion scales with 1/B², compared with the 1/B dependence ofBohm diffusion. This distinction is often used to distinguish between the two.In light of the calculation above, it is tempting to think of Bohm diffusion as classical diffusion with an anomalouscollision rate that maximizes the transport, but the physical picture is different. Anomalous diffusion is the result ofturbulence. Regions of higher or lower electric potential result in eddies because the plasma moves around them withthe E-cross-B drift velocity equal to E/B. These eddies play a similar role to the gyro-orbits in classical diffusion,except that the physics of the turbulence can be such that the decorrelation time is approximately equal to theturn-over time, resulting in Bohm scaling. Another way of looking at it is that the turbulent electric field isapproximately equal to the potential perturbation divided by the scale length δ, and the potential perturbation can beexpected to be a sizeable fraction of the kBT/e. The turbulent diffusion constant D = δv is then independent of thescale length and is approximately equal to the Bohm value. The fraction 1/16 according to Bohm "has no theoreticaljustification but is an empirical number agreeing with most experiments to within a factor of two or three"[1] Manyphysicists like L. Spitzer [2] , considered this fraction as a factor related to plasma instability.

References[1] D Bohm the characteristics of electrical discharges in magnetic fields ed A Guthrie and R K Wakerling (NewYork: McGraw-Hill) (1949).[2] L Spitzer Phys.Fluids3 659 (1960).

External links• Physics Glossary (http:/ / www. faqs. org/ faqs/ fusion-faq/ glossary/ b/ )

Bohm interpretationThe de Broglie–Bohm theory, also called the pilot-wave theory, Bohmian mechanics, and the causalinterpretation, is an interpretation of quantum theory. As in quantum theory, it contains a wavefunction - a functionon the space of all possible configurations. Additionally, it also contains an actual configuration, even in situationswhere nobody observes it. The evolution over time of the configuration (that is, of the positions of all particles or theconfiguration of all fields) is defined by the wave function via a guiding equation. The evolution of the wavefunctionover time is given by Schrödinger's equation.The de Broglie–Bohm theory expresses in an explicit manner the fundamental non-locality in quantum physics. Thevelocity of any one particle depends on the value of the wavefunction, which depends on the whole configuration ofthe universe.This theory is deterministic. Most (but not all) relativistic variants require a preferred frame. Variants which handlespin and curved spaces are known. It can be modified to handle quantum field theory. Bell's theorem was inspired byBell's discovery of the work of David Bohm and his subsequent wondering if the obvious non-locality of the theorycould be removed.This theory gives rise to a measurement formalism, analogous to thermodynamics for classical mechanics, whichyields the standard quantum formalism generally associated with the Copenhagen interpretation. The measurementproblem is resolved by this theory since the outcome of an experiment is registered by the configuration of theparticles of the experimental apparatus after the experiment is completed. The familiar wavefunction collapse ofstandard quantum mechanics emerges from an analysis of subsystems and the quantum equilibrium hypothesis.The theory has a number of equivalent mathematical formulations and has been presented under a number ofdifferent names.

Bohm interpretation 13

OverviewDe Broglie–Bohm theory is based on the following:

We have a configuration of the universe, described by coordinates , which is an element of the configurationspace . The configuration space is different for different versions of pilot wave theory. For example, this may bethe space of positions of particles, or, in case of field theory, the space of field configurations . Theconfiguration evolves according to the guiding equation

.

Here, is the standard complex-valued wavefunction known from quantum theory, which evolves accordingto Schrödinger's equation

This already completes the specification of the theory for any quantum theory with Hamilton operator of type

.

If the configuration is distributed according to at some moment of time , this holds for all times. Sucha state is named quantum equilibrium. In quantum equilibrium, this theory will agree with the results of standardquantum mechanics.

Two-slit experiment

The Bohmian trajectories for an electron goingthrough the two-slit experiment.

The double-slit experiment is an illustration of wave-particle duality.In it, a beam of particles (such as photons) travels through a barrierwith two slits removed. If one puts a detector screen on the other side,the pattern of detected particles shows interference fringescharacteristic of waves; however, the detector screen responds toparticles. The system exhibits behaviour of both waves (interferencepatterns) and particles (dots on the screen).

If we modify this experiment so that one slit is closed, no interferencepattern is observed. Thus, the state of both slits affects the final results.We can also arrange to have a minimally invasive detector at one of theslits to see which slit the particle went through. When we do that, theinterference pattern disappears.The Copenhagen interpretation states that the particles are not localisedin space until they are detected, so that, if there is no detector on theslits, there is no matter of fact about what slit has the particle passed through. If one slit has a detector on it, then thewavefunction collapses due to that detection.

In de Broglie–Bohm theory, the wavefunction travels through both slits, but each particle has a well-definedtrajectory and passes through exactly one of the slits. The final position of the particle on the detector screen and theslit through which the particle passes by is determined by the initial position of the particle. Such initial position isnot controllable by the experimenter, so there is an appearance of randomness in the pattern of detection. The wavefunction interferes with itself and guides the particles in such a way that the particles avoid the regions in which theinterference is destructive and are attracted to the regions in which the interference is constructive, giving rise to theinterference pattern on the detector screen.

Bohm interpretation 14

To explain the behavior when the particle is detected to go through one slit, one needs to appreciate the role of theconditional wavefunction and how it gives rise to the collapse of the wavefunction; this is explained below. Thebasic idea is that the environment registering the detection effectively separates the two wave packets inconfiguration space.

The Theory

The ontology

The ontology of de Broglie-Bohm theory consists of a configuration of the universe and a pilot wave. The configuration space can be chosen differently, as in classical mechanics and standard

quantum mechanics.Thus, the ontology of pilot wave theory contains as the trajectory we know from classical mechanics, asthe wave function of quantum theory. So, at every moment of time there exists not only a wavefunction, but also a well-defined configuration of the whole universe. The correspondence to our experiences ismade by the identification of the configuration of our brain with some part of the configuration of the whole universe

, as in classical mechanics.While the ontology of classical mechanics is part of the ontology of de Broglie–Bohm theory, the dynamics is verydifferent. In classical mechanics, the acceleration of the particles are given by forces. In de Broglie–Bohm theory,the velocities of the particles are given by the wavefunction.In what follows below, we will give the setup for one particle moving in followed by the setup for particlesmoving in 3 dimensions. In the first instance, configuration space and real space are the same while in the second,real space is still , but configuration space becomes . While the particle positions themselves are in realspace, the velocity field and wavefunction are on configuration space which is how particles are entangled with eachother in this theory.Extensions to this theory include spin and more complicated configuration spaces.

We use variations of for particle positions while represents the complex-valued wavefunction onconfiguration space.

Guiding equation

For a single particle moving in , the particle's velocity is given

.

For many particles, we label them as for the th particle and their velocities are given by

.

The key fact to notice is that this velocity field depends on the actual positions of all of the particles in theuniverse. As explained below, in most experimental situations, the influence of all of those particles can beencapsulated into an effective wavefunction for a subsystem of the universe.

Bohm interpretation 15

Schrödinger's equation

The one particle Schrödinger equation governs the time evolution of a complex-valued wavefunction on . Theequation represents a quantized version of the total energy of a classical system evolving under a real-valuedpotential function on :

For many particles, the equation is the same except that and are now on configuration space, .

This is the same wavefunction of conventional quantum mechanics.

The Born RuleIn Bohm's original papers [Bohm 1952] , he discusses how de Broglie–Bohm theory gives rise to the usualmeasurement results of quantum mechanics. The key idea is that this is true if the positions of the particles satisfy thestatistical distribution given by . And that distribution is guaranteed to be true for all time under the guidingequation if the initial distribution of the particles satisfies .For a given experiment, we can postulate this as being true and verify experimentally that it does indeed hold true, asit does. But, as argued in Dürr et al.,[1] one needs to argue that this distribution for subsystems is typical. They arguethat by virtue of its equivariance under the dynamical evolution of the system, is the appropriate measure oftypicality for initial conditions of the positions of the particles. They then prove that the vast majority of possibleinitial configurations will give rise to Born rule (i.e., ) statistics for measurement outcomes. In short, in auniverse governed by the de Broglie–Bohm dynamics, Born rule behavior is typical.The situation is thus analogous to the situation in classical statistical physics. A low entropy initial condition will,with overwhelmingly high probability, evolve into a higher entropy state: behavior consistent with the second law ofthermodynamics is typical. There are, of course, anomalous initial conditions which would give rise to violations ofthe second law. However, absent some very detailed evidence supporting the actual realization of one of thosespecial initial conditions, it would be quite unreasonable to expect anything but the actually observed uniformincrease of entropy. Similarly, in the de Broglie–Bohm theory, there are anomalous initial conditions which wouldproduce measurement statistics in violation of the Born rule (i.e., in conflict with the predictions of standardquantum theory). But the typicality theorem shows that, absent some particular reason to believe one of those specialinitial conditions was in fact realized, Born rule behavior is what one should expect.It is in that qualified sense that Born rule is, for the de Broglie–Bohm theory, a theorem rather than (as in ordinaryquantum theory) an additional postulate.

The conditional wave function of a subsystemIn the formulation of the De Broglie–Bohm theory, there is only a wave function for the entire universe (whichalways evolves by the Schrödinger equation). However, once the theory is formulated, it is convenient to introduce anotion of wave function also for subsystems of the universe. Let us write the wave function of the universe as

, where denotes the configuration variables associated to some subsystem (I) of the universe anddenotes the remaining configuration variables. Denote, respectively, by and by the actual

configuration of subsystem (I) and of the rest of the universe. For simplicity, we consider here only the spinless case.The conditional wave function of subsystem (I) is defined by:

It follows immediately from the fact that satisfies the guiding equation that also the configuration satisfies a guiding equation identical to the one presented in the formulation of the theory, with

Bohm interpretation 16

the universal wave function replaced with the conditional wave function . Also, the fact that is random withprobability density given by the square modulus of implies that the conditional probability density of given is givenby the square modulus of the (normalized) conditional wave function (in the terminology of Dürr et al.[2] this fact iscalled the fundamental conditional probability formula).Unlike the universal wave function, the conditional wave function of a subsystem does not always evolves by theSchrödinger equation, but in many situations it does. For instance, if the universal wave function factors as:

then the conditional wave function of subsystem (I) is (up to an irrelevant scalar factor) equal to (this is whatStandard Quantum Theory would regard as the wave function of subsystem (I)). If, in addition, the Hamiltonian doesnot contain an interaction term between subsystems (I) and (II) then does satisfy a Schrödinger equation. Moregenerally, assume that the universal wave function can be written in the form:

where solves Schrödinger equation and for all and . Then, again, the conditionalwave function of subsystem (I) is (up to an irrelevant scalar factor) equal to and if the Hamiltonian does notcontain an interaction term between subsystems (I) and (II), satisfies a Schrödinger equation.The fact that the conditional wave function of a subsystem does not always evolve by the Schrödinger equation isrelated to the fact that the usual collapse rule of Standard Quantum Theory emerges from the Bohmian formalismwhen one considers conditional wave functions of subsystems.

Extensions

SpinTo incorporate spin, the wavefunction becomes complex-vector valued. The value space is called spin space; for aspin-1/2 particle, spin space can be taken to be . The guiding equation is modified by taking inner products inspin space to reduce the complex vectors to complex numbers. The Schrödinger equation is modified by adding aPauli spin term.

where is the magnetic moment of the th particle, is the appropriate spin operator acting on the th

particle's spin space, , and are, respectively, the magnetic field and the vector

potential in (all other functions are fully on configuration space), is the charge of the th particle, andis the inner product in spin space ,

For an example of a spin space, a system consisting of two spin 1/2 particle and one spin 1 particle has awavefunctions of the form . That is, its spin space is a 12 dimensional space.

Bohm interpretation 17

Curved spaceTo extend de Broglie–Bohm theory to curved space (Riemannian manifolds in mathematical parlance), one simplynotes that all of the elements of these equations make sense, such as gradients and Laplacians. Thus, we useequations that have the same form as above. Topological and boundary conditions may apply in supplementing theevolution of Schrödinger's equation.For a de Broglie–Bohm theory on curved space with spin, the spin space becomes a vector bundle over configurationspace and the potential in Schrödinger's equation becomes a local self-adjoint operator acting on that space.[3]

Quantum field theoryIn Dürr et al.,[4] [5] the authors describe an extension of de Broglie–Bohm theory for handling creation andannihilation operators. The basic idea is that configuration space becomes the (disjoint) space of all possibleconfigurations of any number of particles. For part of the time, the system evolves deterministically under theguiding equation with a fixed number of particles. But under a stochastic process, particles may be created andannihilated. The distribution of creation events is dictated by the wavefunction. The wavefunction itself is evolvingat all times over the full multi-particle configuration space.Nikolic [6] introduces a purely deterministic de Broglie–Bohm theory of particle creation and destruction, accordingto which particle trajectories are continuous, but particle detectors behave as if particles have been created ordestroyed even when a true creation or destruction of particles does not take place.

Exploiting nonlocalityValentini[7] has extended the de Broglie–Bohm theory to include signal nonlocality that would allow entanglementto be used as a stand-alone communication channel without a secondary classical "key" signal to "unlock" themessage encoded in the entanglement. This violates orthodox quantum theory but it has the virtue that it makes theparallel universes of the chaotic inflation theory observable in principle.Unlike de Broglie–Bohm theory, Valentini's theory has the wavefunction evolution also depend on the ontologicalvariables. This introduces an instability, a feedback loop that pushes the hidden variables out of "sub-quantal heatdeath". The resulting theory becomes nonlinear and non-unitary.

RelativityPilot wave theory is explicitly nonlocal. As a consequence, most relativistic variants of pilot wave theory need apreferred foliation of space-time. While this is in conflict with the standard interpretation of relativity, the preferredfoliation, if unobservable, does not lead to any empirical conflicts with relativity.The relation between nonlocality and preferred foliation can be better understood as follows. In de Broglie–Bohmtheory, nonlocality manifests as the fact that the velocity and acceleration of one particle depends on theinstantaneous positions of all other particles. On the other hand, in the theory of relativity the concept ofinstantaneousness does not have an invariant meaning. Thus, to define particle trajectories, one needs an additionalrule that defines which space-time points should be considered instantaneous. The simplest way to achieve this is tointroduce a preferred foliation of space-time by hand, such that each hypersurface of the foliation defines ahypersurface of equal time. However, this way (which explicitly breaks the relativistic covariance) is not the onlyway. It is also possible that a rule which defines instantaneousness is contingent, by emerging dynamically fromrelativistic covariant laws combined with particular initial conditions. In this way, the need for a preferred foliationcan be avoided and relativistic covariance can be saved.There has been work in developing relativistic versions of de Broglie–Bohm theory. See Bohm and Hiley: TheUndivided Universe, and [8], [9], and references therein. Another approach is given in the work of Dürr et al.[10] inwhich they use Bohm-Dirac models and a Lorentz-invariant foliation of space-time.

Bohm interpretation 18

In [11],[12] and [13] Nikolic develops a generalized relativistic-invariant probabilistic interpretation of quantumtheory, in which is no longer a probability density in space, but a probability density in space-time. He usesthis generalized probabilistic interpretation to formulate a relativistic-covariant version of de Broglie–Bohm theorywithout introducing a preferred foliation of space-time.

ResultsBelow are some highlights of the results that arise out of an analysis of de Broglie–Bohm theory. Experimentalresults agree with all of the standard predictions of quantum mechanics in so far as the latter has predictions.However, while standard quantum mechanics is limited to discussing experiments with human observers, deBroglie–Bohm theory is a theory which governs the dynamics of a system without the intervention of outsideobservers (p. 117 in Bell[14] ).

The basis for agreement with standard quantum mechanics is that the particles are distributed according to .This is a statement of observer ignorance, but it can be proven[1] that for a universe governed by this theory, this willtypically be the case. There is apparent collapse of the wave function governing subsystems of the universe, but thereis no collapse of the universal wavefunction.

Measuring spin and polarizationAccording to ordinary quantum theory, it is not possible to measure the spin or polarization of a particle directly;instead, the component in one direction is measured; the outcome from a single particle may be 1, meaning that theparticle is aligned with the measuring apparatus, or -1, meaning that it is aligned the opposite way. For an ensembleof particles, if we expect the particles to be aligned, the results are all 1. If we expect them to be aligned oppositely,the results are all -1. For other alignments, we expect some results to be 1 and some to be -1 with a probability thatdepends on the expected alignment. For a full explanation of this, see the Stern-Gerlach Experiment.In de Broglie–Bohm theory, the results of a spin experiment cannot be analyzed without some knowledge of theexperimental setup. It is possible[15] to modify the setup so that the trajectory of the particle is unaffected, but thatthe particle with one setup registers as spin up while in the other setup it registers as spin down. Thus, for the deBroglie–Bohm theory, the particle's spin is not an intrinsic property of the particle—instead spin is, so to speak, inthe wave function of the particle in relation to the particular device being used to measure the spin. This is anillustration of what is sometimes referred to as contextuality, and is related to naive realism about operators.[16]

Measurements, the quantum formalism, and observer independenceDe Broglie–Bohm theory gives the same results as quantum mechanics. It treats the wavefunction as a fundamentalobject in the theory as the wavefunction describes how the particles move. This means that no experiment candistinguish between the two theories. This section outlines the ideas as to how the standard quantum formalismarises out of quantum mechanics. References include Bohm's original 1952 paper and Dürr et al.[1]

Collapse of the wavefunction

De Broglie–Bohm theory is a theory that applies primarily to the whole universe. That is, there is a singlewavefunction governing the motion of all of the particles in the universe according to the guiding equation.Theoretically, the motion of one particle depends on the positions of all of the other particles in the universe. In somesituations, such as in experimental systems, we can represent the system itself in terms of a de Broglie–Bohm theoryin which the wavefunction of the system is obtained by conditioning on the environment of the system. Thus, thesystem can be analyzed with Schrödinger's equation and the guiding equation, with an initial distribution forthe particles in the system (see the section on the conditional wave function of a subsystem for details).It requires a special setup for the conditional wavefunction of a system to obey a quantum evolution. When a systeminteracts with its environment, such as through a measurement, then the conditional wavefunction of the system

Bohm interpretation 19

evolves in a different way. The evolution of the universal wavefunction can become such that the wavefunction ofthe system appears to be in a superposition of distinct states. But if the environment has recorded the results of theexperiment, then using the actual Bohmian configuration of the environment to condition on, the conditionalwavefunction collapses to just one alternative, the one corresponding with the measurement results.Collapse of the universal wavefunction never occurs in de Broglie–Bohm theory. Its entire evolution is governed bySchrödinger's equation and the particles' evolutions are governed by the guiding equation. Collapse only occurs in aphenomenological way for systems that seem to follow their own Schrödinger's equation. As this is an effectivedescription of the system, it is a matter of choice as to what to define the experimental system to include and this willaffect when "collapse" occurs.

Operators as observables

In the standard quantum formalism, measuring observables is generally thought of as measuring operators on theHilbert space. For example, measuring position is considered to be a measurement of the position operator. Thisrelationship between physical measurements and Hilbert space operators is, for standard quantum mechanics, anadditional axiom of the theory. The de Broglie–Bohm theory, by contrast, requires no such measurement axioms(and measurement as such is not a dynamically distinct or special sub-category of physical processes in the theory).In particular, the usual operators-as-observables formalism is, for de Broglie–Bohm theory, a theorem.[17] A majorpoint of the analysis is that many of the measurements of the observables do not correspond to properties of theparticles; they are (as in the case of spin discussed above) measurements of the wavefunction.In the history of de Broglie–Bohm theory, the proponents have often had to deal with claims that this theory isimpossible. Such arguments are generally based on inappropriate analysis of operators as observables. If onebelieves that spin measurements are indeed measuring the spin of a particle that existed prior to the measurement,then one does reach contradictions. De Broglie–Bohm theory deals with this by noting that spin is not a feature ofthe particle, but rather that of the wavefunction. As such, it only has a definite outcome once the experimentalapparatus is chosen. Once that is taken into account, the impossibility theorems become irrelevant.There have also been claims that experiments reject the Bohm trajectories [18] in favor of the standard QM lines.But as shown in [19] and [20], such experiments cited above only disprove a misinterpretation of the deBroglie–Bohm theory, not the theory itself.There are also objections to this theory based on what it says about particular situations usually involving eigenstatesof an operator. For example, the ground state of hydrogen is a real wavefunction. According to the guiding equation,this means that the electron is at rest when in this state. Nevertheless, it is distributed according to and nocontradiction to experimental results is possible to detect.Operators as observables leads many to believe that many operators are equivalent. De Broglie–Bohm theory, fromthis perspective, chooses the position observable as a favored observable rather than, say, the momentum observable.Again, the link to the position observable is a consequence of the dynamics. The motivation for de Broglie–Bohmtheory is to describe a system of particles. This implies that the goal of the theory is to describe the positions of thoseparticles at all times. Other observables do not have this compelling ontological status. Having definite positionsexplains having definite results such as flashes on a detector screen. Other observables would not lead to thatconclusion, but there need not be any problem in defining a mathematical theory for other observables; see Hyman etal.[21] for an exploration of the fact that a probability density and probability current can be defined for any set ofcommuting operators.

Bohm interpretation 20

Hidden variables

De Broglie–Bohm theory is often referred to as a "hidden variable" theory. The alleged applicability of the term"hidden variable" comes from the fact that the particles postulated by Bohmian mechanics do not influence theevolution of the wavefunction. The argument is that, because adding particles does not have an effect on thewavefunction's evolution, such particles must not have effects at all and are, thus, unobservable, since they cannothave an effect on observers. There is no analogue of Newton's third law in this theory. The idea is supposed to bethat, since particles cannot influence the wavefunction, and it is the wavefunction that determines measurementpredictions through the Born rule, the particles are superfluous and unobservable.Such an argument, however, arises from a fundamental misunderstanding of the relation between the ontologyposited by the de Broglie–Bohm theory and the world of ordinary observation. In particular, the particles postulatedby the de Broglie–Bohm theory are anything but "hidden" variables: they are what the cats and trees and tables andplanets and pointers we see are made of! It is the wavefunction itself which is "hidden" in the sense of beinginvisible and not-directly-observable.Thus, for example, when the wavefunction of some measuring apparatus is such that its pointer is superposedbetween pointing to the left and pointing to the right, what accounts for the fact that scientists, when they look at theapparatus, see the pointer pointing to the left (say) is the fact that the de Broglie–Bohmian particles that make up thepointer are actually pointed towards the left. While the exact details of how humans process such information andwhat it is based on is beyond the scope of the de Broglie–Bohm theory, the basic idea of any particle ontology is thatif the particles in the theory appear where they seem to be from human observations, then it is considered asuccessful prediction.

Heisenberg's uncertainty principleThe Heisenberg uncertainty principle states that when two complementary measurements are made, there is a limit tothe product of their accuracy. As an example, if one measures the position with an accuracy of , and themomentum with an accuracy of , then If we make further measurements in order to get moreinformation, we disturb the system and change the trajectory into a new one depending on the measurement setup;therefore, the measurement results are still subject to Heisenberg's uncertainty relation.In de Broglie–Bohm theory, there is always a matter of fact about the position and momentum of a particle. Eachparticle has a well defined trajectory. Observers have limited knowledge as to what this trajectory is (and thus of theposition and momentum). It is the lack knowledge of the particle's trajectory that accounts for the uncertaintyrelation. What one can know about a particle at any given time is described by the wavefunction. Since theuncertainty relation can be derived from the wavefunction in other interpretations of quantum mechanics, it can belikewise derived (in the epistemic sense mentioned above), on the de Broglie–Bohm theory.To put the statement differently, the particles' positions are only known statistically. As in classical mechanics,successive observations of the particles' positions refine the experimenter's knowledge of the particles' initialconditions. Thus, with succeeding observations, the initial conditions become more and more restricted. Thisformalism is consistent with the normal use of the Schrödinger equation.For the derivation of the uncertainty relation, see Heisenberg uncertainty principle, noting that it describes it from theviewpoint of the Copenhagen interpretation.

Bohm interpretation 21

Quantum entanglement, Einstein-Podolsky-Rosen paradox, Bell's theorem, and nonlocalityDe Broglie–Bohm theory highlighted the issue of nonlocality: it inspired John Stewart Bell to prove his now-famoustheorem,[22] which in turn led to the Bell test experiments.In the Einstein-Podolsky-Rosen paradox,[23] the authors point out that quantum mechanics allows the creation ofpairs of particles in an entangled quantum state. They describe a thought-experiment one could perform on such apair, the results of which they interpreted as indicating that quantum mechanics is an incomplete theory.Decades later John Bell proved Bell's theorem (see p. 14 in Bell[14] ), in which he showed that, if they are to agreewith the empirical predictions of quantum mechanics, all such "hidden-variable" completions of quantum mechanicsmust either be nonlocal (as the Bohm interpretation is) or give up the assumption that experiments produce uniqueresults (see counterfactual definiteness and many-worlds interpretation). In particular, Bell proved that any localtheory with unique results must make empirical predictions satisfying a statistical constraint called "Bell'sinequality".Alain Aspect performed a series of Bell test experiments that test Bell's inequality using an EPR-type setup. Aspect'sresults show experimentally that Bell's inequality is in fact violated—meaning that the relevant quantum mechanicalpredictions are correct. In these Bell test experiments, entangled pairs of particles are created; the particles areseparated, traveling to remote measuring apparatus. The orientation of the measuring apparatus can be changed whilethe particles are in flight, demonstrating the apparent non-locality of the effect.The de Broglie–Bohm theory makes the same (empirically correct) predictions for the Bell test experiments asordinary quantum mechanics. It is able to do this because it is manifestly nonlocal. It is often criticized or rejectedbased on this; Bell's attitude was: "It is a merit of the de Broglie–Bohm version to bring this [nonlocality] out soexplicitly that it cannot be ignored." [24]

The de Broglie–Bohm theory describes the physics in the Bell test experiments as follows: to understand theevolution of the particles, we need to set up a wave equation for both particles; the orientation of the apparatusaffects the wavefunction. The particles in the experiment follow the guidance of the wavefunction. It is thewavefunction that carries the faster-than-light effect of changing the orientation of the apparatus. An analysis ofexactly what kind of nonlocality is present and how it is compatible with relativity can be found in Maudlin.[25] Notethat in Bell's work, and in more detail in Maudlin's work, it is shown that the nonlocality does not allow for signalingat speeds faster than light.

Classical limitBohm's formulation of de Broglie–Bohm theory in terms of a classical-looking version has the merits that theemergence of classical behavior seems to follow immediately for any situation in which the quantum potential isnegligible, as noted by Bohm in 1952. Modern methods of decoherence are relevant to an analysis of this limit. SeeAllori et al.[26] for steps towards a rigorous analysis.

Quantum trajectory methodWork by Robert Wyatt in the early 2000s attempted to use the Bohm "particles" as an adaptive mesh that follows theactual trajectory of a quantum state in time and space. In the "quantum trajectory" method, one samples the quantumwavefunction with a mesh of quadrature points. One then evolves the quadrature points in time according to theBohm equations of motion. At each time-step, one then re-synthesizes the wavefunction from the points, recomputesthe quantum forces, and continues the calculation. (Quick-time movies of this for H+H2 reactive scattering can befound on the Wyatt group [27] web-site at UT Austin.) This approach has been adapted, extended, and used by anumber of researchers in the Chemical Physics community as a way to compute semi-classical and quasi-classicalmolecular dynamics. A recent (2007) issue of the Journal of Physical Chemistry A [28] was dedicated to Prof. Wyattand his work on "Computational Bohmian Dynamics".

Bohm interpretation 22

Eric Bittner's group [29] at the University of Houston has advanced a statistical variant of this approach that usesBayesian sampling technique to sample the quantum density and compute the quantum potential on a structurelessmesh of points. This technique was recently used to estimate quantum effects in the heat-capacity of small clustersNen for n~100.There remain difficulties using the Bohmian approach, mostly associated with the formation of singularities in thequantum potential due to nodes in the quantum wavefunction. In general, nodes forming due to interference effectslead to the case where This results in an infinite force on the sample particles forcing them to move

away from the node and often crossing the path of other sample points (which violates single-valuedness). Variousschemes have been developed to overcome this; however, no general solution has yet emerged.These methods, as does Bohm's Hamilton-Jacobi formulation, do not apply to situations in which the full dynamicsof spin need to be taken into account.

Occam's razor criticismBoth Hugh Everett III and Bohm treated the wavefunction as a physically real field. Everett's many-worldsinterpretation is an attempt to demonstrate that the wavefunction alone is sufficient to account for all ourobservations. When we see the particle detectors flash or hear the click of a Geiger counter then Everett's theoryinterprets this as our wavefunction responding to changes in the detector's wavefunction, which is responding in turnto the passage of another wavefunction (which we think of as a "particle", but is actually just anotherwave-packet).[30] No particle (in the Bohm sense of having a defined position and velocity) exists, according to thattheory. For this reason Everett sometimes referred to his approach as the "pure wave theory". Talking of Bohm's1952 approach, Everett says:

“Our main criticism of this view is on the grounds of simplicity - if one desires to hold the view that is a real field then the associatedparticle is superfluous since, as we have endeavored to illustrate, the pure wave theory is itself satisfactory.[31] ”

In the Everettian view, then, the Bohm particles are superfluous entities, similar to, and equally as unnecessary as,for example, the luminiferous ether was found to be unnecessary in special relativity. This argument of Everett's issometimes called the "redundancy argument", since the superfluous particles are redundant in the sense of Occam'srazor.[32] . By omitting the hidden variables, however, Everett had to invoke causally unrelated and thereforeexperimentally unverifiable parallel universes.Many authors have expressed critical views of the de Broglie-Bohm theory, by comparing it to Everett's manyworlds approach. Many (but not all) proponents of the de Broglie-Bohm theory (such as Bohm and Bell) interpret theuniversal wave function as physically real. According to some supporters of Everett's theory, if the (nevercollapsing) wave function is taken to be physically real, then it is natural to interpret the theory as having the samemany worlds as Everett's theory. In the Everettian view the role of the Bohm particle is to act as a "pointer", tagging,or selecting, just one branch of the universal wavefunction (the assumption that this branch indicates which wavepacket determines the observed result of a given experiment is called the "result assumption"[30] ); the other branchesare designated "empty" and implicitly assumed by Bohm to be devoid of conscious observers.[30] H. Dieter Zehcomments on these "empty" branches:

“It is usually overlooked that Bohm’s theory contains the same “many worlds” of dynamically separate branches as the Everett interpretation(now regarded as “empty” wave components), since it is based on precisely the same . . . global wave function . . .[33] ”

David Deutsch has expressed the same point more "acerbically"[30] :

“pilot-wave theories are parallel-universe theories in a state of chronic denial.[34]”

Bohm interpretation 23

The fact that such a "pointer" can be made in a self-consistent manner that not only reproduces all knownexperimental results but also provides a clean classical limit is, however, highly significant in itself. It proves that theexistence of alternative universes is not a necessary conclusion of quantum physics.

DerivationsDe Broglie–Bohm theory has been derived many times and in many ways. Below are five derivations all of whichare very different and lead to different ways of understanding and extending this theory.• Schrödinger's equation can be derived by using Einstein's light quanta hypothesis: and de Broglie's

hypothesis: .

The guiding equation can be derived in a similar fashion. We assume a plane wave: .Notice that . Assuming that for the particle's actual velocity, we have that

. Thus, we have the guiding equation.

Notice that this derivation does not use Schrödinger's equation.• Preserving the density under the time evolution is another method of derivation. This is the method that Bell cites.

It is this method which generalizes to many possible alternative theories. The starting point is the continuityequation for the density . This equation describes a probability flow along a

current. We take the velocity field associated with this current as the velocity field whose integral curves yield themotion of the particle.

• A method applicable for particles without spin is to do a polar decomposition of the wavefunction and transformSchrödinger's equation into two coupled equations: the continuity equation from above and the Hamilton–Jacobiequation. This is the method used by Bohm in 1952. The decomposition and equations are as follows:

Decomposition: Note corresponds to the probability density.

Continuity Equation:

Hamilton–Jacobi Equation:

The Hamilton–Jacobi equation is the equation derived from a Newtonian system with potential

and velocity field The potential is the classical potential that appears in

Schrödinger's equation and the other term involving is the quantum potential, terminology introduced byBohm.This leads to viewing the quantum theory as particles moving under the classical force modified by a quantumforce. However, unlike standard Newtonian mechanics, the initial velocity field is already specified by

which is a symptom of this being a first-order theory, not a second-order theory.• A fourth derivation was given by Dürr et al.[1] In their derivation, they derive the velocity field by demanding the

appropriate transformation properties given by the various symmetries that Schrödinger's equation satisfies, oncethe wavefunction is suitably transformed. The guiding equation is what emerges from that analysis.

• A fifth derivation, given by Dürr et al.[4] is appropriate for generalization to quantum field theory and the Dirac equation. The idea is that a velocity field can also be understood as a first order differential operator acting on functions. Thus, if we know how it acts on functions, we know what it is. Then given the Hamiltonian operator

Bohm interpretation 24

, the equation to satisfy for all functions (with associated multiplication operator ) is

where is the local Hermitian inner product on the value space

of the wavefunction.This formulation allows for stochastic theories such as the creation and annihilation of particles.

HistoryDe Broglie–Bohm theory has a history of different formulations and names. In this section, each stage is given aname and a main reference.

Pilot-wave theoryDr. de Broglie presented his pilot wave theory at the 1927 Solvay Conference,[35] after close collaboration withSchrödinger, who developed his wave equation for de Broglie's theory. At the end of the presentation, WolfgangPauli pointed out that it was not compatible with a semi-classical technique Fermi had previously adopted in the caseof inelastic scattering. Contrary to a popular legend, de Broglie actually gave the correct rebuttal that the particulartechnique could not be generalized for Pauli's purpose, although the audience might have been lost in the technicaldetails and de Brolie's mild mannerism left the impression that Pauli's objection was valid. He was eventuallypersuaded to abandon this theory nonetheless in 1932 due to both the Copenhagen school's more successful P.R.efforts and his own inability to understand quantum decoherence. Also in 1932, John von Neumann published apaper,[36] claiming to prove that all hidden-variable theories are impossible. This sealed the fate of de Broglie'stheory for the next two decades. In truth, von Neumann's proof is based on invalid assumptions, such as quantumphysics can be made local, and it does not really disprove the pilot-wave theory.De Broglie's theory already applies to multiple spin-less particles, but lacks an adequate theory of measurement as noone understood quantum decoherence at the time. An analysis of de Broglie's presentation is given in Bacciagaluppiet al.[37] [38]

Around this time Erwin Madelung[39] also developed a hydrodynamic version of Schrödinger's equation which isincorrectly considered as a basis for the density current derivation of de Broglie–Bohm theory. The Madelungequations, being quantum Euler equations (fluid dynamics), differ philosophically from the de Broglie–Bohmtheory[40] and are the basis of the hydrodynamic interpretation of quantum mechanics.

De Broglie–Bohm theoryAfter publishing a popular textbook on Quantum Mechanics which adhered entirely to the Copenhagen orthodoxy,Bohm was persuaded by Einstein to take a critical look at von Neumann's theorem. The result was 'A SuggestedInterpretation of the Quantum Theory in Terms of "Hidden Variables" I and II' [Bohm 1952]. It extended the originalPilot Wave Theory to incorporate a consistent theory of measurement, and to address a criticism of Pauli that deBroglie did not properly respond to; it is taken to be deterministic (though Bohm hinted in the original papers thatthere should be disturbances to this, in the way Brownian motion disturbs Newtonian mechanics). This stage isknown as the de Broglie–Bohm Theory in Bell's work [Bell 1987] and is the basis for 'The Quantum Theory ofMotion' [Holland 1993].This stage applies to multiple particles, and is deterministic.The de Broglie–Bohm theory is an example of a hidden variables theory. Bohm originally hoped that hiddenvariables could provide a local, causal, objective description that would resolve or eliminate many of the paradoxesof quantum mechanics, such as Schrödinger's cat, the measurement problem and the collapse of the wavefunction.However, Bell's theorem complicates this hope, as it demonstrates that there can be no local hidden variable theorythat is compatible with the predictions of quantum mechanics. The Bohmian interpretation is causal but not local.

Bohm interpretation 25

Bohm's paper was largely ignored by other physicists. Even Albert Einstein did not consider it a satisfactory answerto the quantum non-locality question. The rest of the contemporary objections, however, were ad hominem, focusingon Bohm's sympathy with liberals and supposed communists as exemplified by his refusal to give testimony to theHouse Un-American Activities Committee.Eventually the cause was taken up by John Bell. In "Speakable and Unspeakable in Quantum Mechanics" [Bell1987], several of the papers refer to hidden variables theories (which include Bohm's). Bell showed that vonNeumann's objection amounted to showing that hidden variables theories are nonlocal, and that nonlocality is afeature of all quantum mechanical systems.

Bohmian mechanicsThis term is used to describe the same theory, but with an emphasis on the notion of current flow. In particular, it isoften used to include most of the further extensions past the spin-less version of Bohm. While de Broglie–Bohmtheory has Lagrangians and Hamilton-Jacobi equations as a primary focus and backdrop, with the icon of thequantum potential, Bohmian mechanics considers the continuity equation as primary and has the guiding equation asits icon. They are mathematically equivalent in so far as the Hamilton-Jacobi formulation applies, i.e., spin-lessparticles. The papers of Dürr et al. popularized the term.All of non-relativistic quantum mechanics can be fully accounted for in this theory.

Causal interpretation and ontological interpretationBohm developed his original ideas, calling them the Causal Interpretation. Later he felt that causal sounded toomuch like deterministic and preferred to call his theory the Ontological Interpretation. The main reference is 'TheUndivided Universe' [Bohm, Hiley 1993].This stage covers work by Bohm and in collaboration with Vigier and Hiley. Bohm is clear that this theory isnon-deterministic (the work with Hiley includes a stochastic theory). As such, this theory is not, strictly speaking, aformulation of the de Broglie–Bohm theory. However, it deserves mention here because the term "BohmInterpretation" is ambiguous between this theory and the de Broglie–Bohm theory.

See also• David Bohm• Interpretation of quantum mechanics• Madelung equations• Local hidden variable theory• Quantum mechanics• Pilot wave

References• Albert, David Z. (May 1994). "Bohm's Alternative to Quantum Mechanics". Scientific American 270: 58–67.• Barbosa, G. D.; N. Pinto-Neto (2004). "A Bohmian Interpretation for Noncommutative Scalar Field Theory and

Quantum Mechanics". Physical Review D 69: 065014. doi:10.1103/PhysRevD.69.065014. arXiv:hep-th/0304105.• Bohm, David (1952). "A Suggested Interpretation of the Quantum Theory in Terms of "Hidden Variables" I".

Physical Review 85: 166–179. doi:10.1103/PhysRev.85.166.• Bohm, David (1952). "A Suggested Interpretation of the Quantum Theory in Terms of "Hidden Variables", II".

Physical Review 85: 180–193. doi:10.1103/PhysRev.85.180.• Bohm, David (1990). "A new theory of the relationship of mind and matter" [41]. Philosophical Psychology 3 (2):

271–286. doi:10.1080/09515089008573004.

Bohm interpretation 26

• Bohm, David; B.J. Hiley (1993). The Undivided Universe: An ontological interpretation of quantum theory.London: Routledge. ISBN 0-415-12185-X.

• Durr, Detlef; Sheldon Goldstein, Roderich Tumulka and Nino Zangh (December 2004). "Bohmian Mechanics"[42] (PDF). Physical review letters 93 (9): 090402. ISSN 0031-9007. PMID 15447078.

• Goldstein, Sheldon (2001). "Bohmian Mechanics" [43]. Stanford Encyclopedia of Philosophy.• Hall, Michael J.W. (2004). "Incompleteness of trajectory-based interpretations of quantum mechanics". Journal of

Physics a Mathematical and General 37: 9549. doi:10.1088/0305-4470/37/40/015. arXiv:quant-ph/0406054.(Demonstrates incompleteness of the Bohm interpretation in the face of fractal, differentialble-nowherewavefunctions.)

• Holland, Peter R. (1993). The Quantum Theory of Motion : An Account of the de Broglie–Bohm CausalInterpretation of Quantum Mechanics. Cambridge: Cambridge University Press. ISBN 0-521-48543-6.

• Nikolic, H. (2004). "Relativistic quantum mechanics and the Bohmian interpretation". Foundations of PhysicsLetters 18: 549. doi:10.1007/s10702-005-1128-1. arXiv:quant-ph/0406173.

• Passon, Oliver (2004). Why isn't every physicist a Bohmian?. arXiv:quant-ph/0412119.• Sanz, A. S.; F. Borondo (2003). "A Bohmian view on quantum decoherence". The European Physical Journal D

44: 319. doi:10.1140/epjd/e2007-00191-8. arXiv:quant-ph/0310096.• Sanz, A.S. (2005). "A Bohmian approach to quantum fractals". J. Phys. A: Math. Gen. 38: 319.

doi:10.1088/0305-4470/38/26/013. (Describes a Bohmian resolution to the dilemma posed by non-differentiablewavefunctions.)

• Silverman, Mark P. (1993). And Yet It Moves: Strange Systems and Subtle Questions in Physics. Cambridge:Cambridge University Press. ISBN 0-521-44631-7.

• Streater, Ray F. (2003). "Bohmian mechanics is a "lost cause"" [44]. Retrieved 2006-06-25.• Valentini, Antony; Hans Westman (2004). Dynamical Origin of Quantum Probabilities. arXiv:quant-ph/0403034.• Bohmian mechanics on arxiv.org [45]

External links• "Bohmian Mechanics" (Stanford Encyclopedia of Philosophy) [46]

• "Pilot waves, Bohmian metaphysics, and the foundations of quantum mechanics" [47], lecture course on Bohminterpretation by Mike Towler, Cambridge University.

References[1] Dürr, D., Goldstein, S., and Zanghì, N., "Quantum Equilibrium and the Origin of Absolute Uncertainty" (http:/ / arxiv. org/ abs/ quant-ph/

0308039), Journal of Statistical Physics 67: 843–907, 1992.[2] Quantum Equilibrium and the Origin of Absolute Uncertainty, D. Dürr, S. Goldstein and N. Zanghì, Journal of Statistical Physics 67, 843-907

(1992), http:/ / arxiv. org/ abs/ quant-ph/ 0308039.[3] Dürr, D., Goldstein, S., Taylor, J., Tumulka, R., and Zanghì, N., J. "Quantum Mechanics in Multiply-Connected Spaces" (http:/ / arxiv. org/

abs/ quant-ph/ 0506173), Phys. A: Math. Theor. 40, 2997–3031 (2007)[4] Dürr, D., Goldstein, S., Tumulka, R., and Zanghì, N., 2004, "Bohmian Mechanics and Quantum Field Theory" (http:/ / arxiv. org/ abs/

quant-ph/ 0303156), Phys. Rev. Lett. 93: 090402:1–4.[5] Dürr, D., Tumulka, R., and Zanghì, N., J. Phys. A: Math. Gen. 38, R1–R43 (2005), quant-ph/0407116[6] Nikolic, H. 2010 "QFT as pilot-wave theory of particle creation and destruction" (http:/ / arxiv. org/ abs/ 0904. 2287), Int. J. Mod. Phys. A 25,

1477 (2010)[7] Valentini, A., 1991, "Signal-Locality, Uncertainty and the Subquantum H-Theorem. II," Physics Letters A 158: 1–8.[8] http:/ / xxx. lanl. gov/ abs/ quant-ph/ 0208185[9] http:/ / xxx. lanl. gov/ abs/ quant-ph/ 0302152[10] Dürr, D., Goldstein, S., Münch-Berndl, K., and Zanghì, N., 1999, "Hypersurface Bohm-Dirac Models" (http:/ / arxiv. org/ abs/ quant-ph/

9801070), Phys. Rev. A 60: 2729–2736.[11] http:/ / xxx. lanl. gov/ abs/ 0811. 1905[12] http:/ / xxx. lanl. gov/ abs/ 0904. 2287[13] http:/ / arxiv. org/ abs/ 1002. 3226

Bohm interpretation 27

[14] Bell, John S, Speakable and Unspeakable in Quantum Mechanics, Cambridge University Press 1987.[15] Albert, D. Z., 1992, Quantum Mechanics and Experience, Cambridge, MA: Harvard University Press[16] Daumer, M., Dürr, D., Goldstein, S., and Zanghì, N., 1997, "Naive Realism About Operators" (http:/ / arxiv. org/ abs/ quant-ph/ 9601013),

Erkenntnis 45: 379–397.[17] Dürr, D., Goldstein, S., and Zanghì, N., "Quantum Equilibrium and the Role of Operators as Observables in Quantum Theory" (http:/ / arxiv.

org/ abs/ quant-ph/ 0308038) Journal of Statistical Physics 116, 959–1055 (2004)[18] http:/ / arxiv. org/ abs/ quant-ph/ 0206196[19] http:/ / arxiv. org/ abs/ quant-ph/ 0108038[20] http:/ / arxiv. org/ abs/ quant-ph/ 0305131[21] Hyman, Ross et al Bohmian mechanics with discrete operators (http:/ / www. iop. org/ EJ/ abstract/ 0305-4470/ 37/ 44/ L02), J. Phys. A:

Math. Gen. 37 L547–L558, 2004[22] J. S. Bell, On the Einstein Podolsky Rosen Paradox (http:/ / www. drchinese. com/ David/ Bell_Compact. pdf), Physics 1, 195 (1964)[23] Einstein, Podolsky, Rosen Can Quantum Mechanical Description of Physical Reality Be Considered Complete? Phys. Rev. 47, 777 (1935).[24] Bell, page 115[25] Maudlin, T., 1994, Quantum Non-Locality and Relativity: Metaphysical Intimations of Modern Physics, Cambridge, MA: Blackwell.[26] Allori, V., Dürr, D., Goldstein, S., and Zanghì, N., 2002, "Seven Steps Towards the Classical World" (http:/ / arxiv. org/ abs/ quant-ph/

0112005), Journal of Optics B 4: 482–488.[27] http:/ / research. cm. utexas. edu/ rwyatt/ movies/ qtm/ index. html[28] http:/ / pubs. acs. org/ toc/ jpcafh/ 111/ 41[29] http:/ / k2. chem. uh. edu[30] Harvey R Brown and David Wallace, Solving the measurement problem: de Broglie-Bohm loses out to Everett, Foundations of Physics 35

(2005), pp. 517-540. (http:/ / philsci-archive. pitt. edu/ archive/ 00001659/ 01/ Cushing. pdf) Abstract: "The quantum theory of de Broglie andBohm solves the measurement problem, but the hypothetical corpuscles play no role in the argument. The solution finds a more natural homein the Everett interpretation."

[31] See section VI of Everett's thesis: The Theory of the Universal Wave Function, pp 3-140 of Bryce Seligman DeWitt, R. Neill Graham, eds,The Many-Worlds Interpretation of Quantum Mechanics, Princeton Series in Physics, Princeton University Press (1973), ISBN0-691-08131-X

[32] Craig Callender, "The Redundancy Argument Against Bohmian Mechanics" (http:/ / philosophy. ucsd. edu/ faculty/ ccallender/ TheRedundancy Argument Against Bohmian Mechanics. doc. )

[33] Daniel Dennett (2000). With a little help from my friends. In D. Ross, A. Brook, and D. Thompson (Eds.), Dennett’s Philosophy: acomprehensive assessment. MIT Press/Bradford, ISBN 026268117X.

[34] David Deutsch, Comment on Lockwood. British Journal for the Philosophy of Science 47, 222228, 1996[35] Solvay Conference, 1928, Electrons et Photons: Rapports et Descussions du Cinquieme Conseil de Physique tenu a Bruxelles du 24 au 29

October 1927 sous les auspices de l'Institut International Physique Solvay[36] von Neumann J. 1932 Mathematische Grundlagen der Quantenmechanik[37] Bacciagaluppi, G., and Valentini, A., Quantum Theory at the Crossroads: Reconsidering the 1927 Solvay Conference[38] See the brief summary by Towler, M., "Pilot wave theory, Bohmian metaphysics, and the foundations of quantum mecahnics" (http:/ / www.

tcm. phy. cam. ac. uk/ ~mdt26/ PWT/ lectures/ bohm7. pdf)[39] Madelung, E., “ Quantentheorie in hydrodynamischer Form,” Zeit. F. Phys. 40 (1927), 322–326[40] Tsekov, R. (2009) Bohmian Mechanics versus Madelung Quantum Hydrodynamics (http:/ / arxiv. org/ abs/ 0904. 0723)[41] http:/ / members. aol. com/ Mszlazak/ BOHM. html[42] http:/ / www. math. rutgers. edu/ ~oldstein/ papers/ bohmech. pdf[43] http:/ / plato. stanford. edu/ entries/ qm-bohm/[44] http:/ / www. mth. kcl. ac. uk/ ~streater/ lostcauses. html#XI[45] http:/ / xstructure. inr. ac. ru/ x-bin/ theme3. py?level=1& index1=-139823[46] http:/ / plato. stanford. edu/ entries/ qm-bohm[47] http:/ / www. tcm. phy. cam. ac. uk/ ~mdt26/ pilot_waves. html

Correspondence principle 28

Correspondence principleThis article discusses quantum theory. For other uses, see Correspondence principle (disambiguation).

In physics, the correspondence principle states that the behavior of systems described by the theory of quantummechanics (or by the old quantum theory) reproduces classical physics in the limit of large quantum numbers.The principle was formulated by Niels Bohr in 1920[1] , though he had previously made use of it as early as 1913 indeveloping his model of the atom[2] .The term is also used more generally, to represent the idea that a new theory should reproduce the results of olderwell-established theories in those domains where the old theories work.

Quantum mechanicsThe rules of quantum mechanics are highly successful in describing microscopic objects, atoms and elementaryparticles. But macroscopic systems like springs and capacitors are accurately described by classical theories likeclassical mechanics and classical electrodynamics. If quantum mechanics should be applicable to macroscopicobjects there must be some limit in which quantum mechanics reduces to classical mechanics. Bohr's correspondenceprinciple demands that classical physics and quantum physics give the same answer when the systems become large.The conditions under which quantum and classical physics agree are referred to as the correspondence limit, or theclassical limit. Bohr provided a rough prescription for the correspondence limit: it occurs when the quantumnumbers describing the system are large. A more elaborated analysis of quantum-classical correspondence (QCC) inwavepacket spreading leads to the distinction between robust "restricted QCC" and fragile "detailed QCC". SeeStotland & Cohen (2006) and references therein. "Restricted QCC" refers to the first two moments of the probabilitydistribution and is true even when the wave packets diffract, while "detailed QCC" requires smooth potentials whichvary over scales much larger than the wavelength, which is what Bohr considered.The post-1925 new quantum theory came in two different formulations. In matrix mechanics, the correspondenceprinciple was built in and was used to construct the theory. In the Schrödinger approach classical behavior is notclear because the waves spread out as they move. Once the Schrödinger equation was given a probabilisticinterpretation, Ehrenfest showed that Newton's laws hold on average: the quantum statistical expectation value of theposition and momentum obey Newton's laws.The correspondence principle is one of the tools available to physicists for selecting quantum theories correspondingto reality. The principles of quantum mechanics are broad - they say that the states of a physical system form acomplex vector space, but they do not say which operators correspond to physical quantities or measurements. Thecorrespondence principle limits the choices to those that reproduce classical mechanics in the correspondence limit.Because quantum mechanics only reproduces classical mechanics in a statistical interpretation, and because thestatistical interpretation only gives the probabilities of different classical outcomes, Bohr has argued that classicalphysics does not emerge from quantum physics in the same way that classical mechanics emerges as anapproximation of special relativity at small velocities. He argued that classical physics exists independently ofquantum theory and cannot be derived from it. His position is that it is inappropriate to understand the experiences ofobservers using purely quantum mechanical notions such as wavefunctions because the different states of experienceof an observer are defined classically, and do not have a quantum mechanical analog.The relative state interpretation of quantum mechanics is an attempt to understand the experience of observers usingonly quantum mechanical notions. Niels Bohr was an early opponent of such interpretations.

Correspondence principle 29

Other scientific theoriesThe term "correspondence principle" is used in a more general sense to mean the reduction of a new scientific theoryto an earlier scientific theory in appropriate circumstances. This requires that the new theory explain all thephenomena under circumstances for which the preceding theory was known to be valid, the "correspondence limit".For example, Einstein's special relativity satisfies the correspondence principle, because it reduces to classicalmechanics in the limit of velocities small compared to the speed of light (example below). General relativity reducesto Newtonian gravity in the limit of weak gravitational fields. Laplace's theory of celestial mechanics reduces toKepler's when interplanetary interactions are ignored, and Kepler's reproduces Ptolemy's equant in a coordinatesystem where the Earth is stationary. Statistical mechanics reproduces thermodynamics when the number of particlesis large. In biology, chromosome inheritance theory reproduces Mendel's laws of inheritance, in the domain that theinherited factors are protein coding genes.In order for there to be a correspondence, the earlier theory has to have a domain of validity—it must work undersome conditions. Not all theories have a domain of validity. For example, there is no limit where Newton'smechanics reduces to Aristotle's mechanics because Aristotle's mechanics, although academically viable for manycenturies, do not have any domain of validity.

Examples

Bohr modelIf an electron in an atom is moving on an orbit with period T, the electromagnetic radiation will classically repeatitself every orbital period. If the coupling to the electromagnetic field is weak, so that the orbit doesn't decay verymuch in one cycle, the radiation will be emitted in a pattern which repeats every period, so that the fourier transformwill have frequencies which are only multiples of 1/T. This is the classical radiation law: the frequencies emitted areinteger multiples of 1/T.In quantum mechanics, this emission must be of quantum of light. The frequency of the quanta emitted should beinteger multiples of 1/T so that classical mechanics is an approximate description at large quantum numbers. Thismeans that the energy level corresponding to a classical orbit of period 1/T must have nearby energy levels whichdiffer in energy by h/T, and they should be equally spaced near that level:

Bohr worried whether the energy spacing 1/T should be best calculated with the period of the energy state oror some average. In hindsight, there is no need to quibble, since this theory is only the leading semiclassical

approximation.Bohr considered circular orbits. These orbits must classically decay to smaller circles when they emit photons. Thelevel spacing between circular orbits can be calculated with the correspondence formula. For a hydrogen atom, theclassical orbits have a period T which is determined by Kepler's third law to scale as . The energy scales as 1/r,so the level spacing formula says that:

It is possible to determine the energy levels by recursively stepping down orbit by orbit, but there is a shortcut. Theangular momentum L of the circular orbit scales as . The energy in terms of the angular momentum is then

Assuming that quantized values of L are equally spaced, the spacing between neighboring energies is

Correspondence principle 30

Which is what we want for equally spaced angular momentum. If you keep track of the constants, the spacing is ,so the angular momentum should be an integer multiple of

This is how Bohr arrived at his model. Since only the level spacing is determined by the correspondence principle,you could always add a small fixed offset to the quantum number--- L could just as well have been . Bohrused his physical intuition to decide which quantities were best to quantize. It is a testimony to his skill that he wasable to get so much from what is only the leading order approximation.

One-dimensional potentialBohr's correspondence condition can be solved for the level energies in a general one-dimensional potential. Define aquantity J(E) which is a function only of the energy, and has the property that:

This is the analog of the angular momentum in the case of the circular orbits. The orbits selected by thecorrespondence principle are the ones that obey J=nh for n integer, since

This quantity J is canonically conjugate to a variable θ which, by the Hamilton equations of motion changes withtime as the gradient of energy with J. Since this is equal to the inverse period at all times, the variable θ increasessteadily from 0 to 1 over one period.The angle variable comes back to itself after 1 unit of increase, so the geometry of phase space in J,θ coordinates isthat of a half-cylinder, capped off at J = 0, which is the motionless orbit at the lowest value of the energy. Thesecoordinates are just as canonical as x,p, but the orbits are now lines of constant J instead of nested ovoids in x-pspace. The area enclosed by an orbit is invariant under canonical transformations, so it is the same in x-p space as inJ-θ. But in the J-θ coordinates this area is the area of a cylinder of unit circumference between 0 and J, or just J. So Jis equal to the area enclosed by the orbit in x-p coordinates too:

The quantization rule is that the action variable J is an integer multiple of h.

Multiperiodic motion—Bohr–Sommerfeld quantizationBohr's correspondence principle provided a way to find the semiclassical quantization rule for a one degree offreedom system. It was an argument for the old quantum condition mostly independent from the one developed byWien and Einstein, which focused on adiabatic invariance. But both pointed to the same quantity, the action.Bohr was reluctant to generalize the rule to systems with many degrees of freedom. This step was taken bySommerfeld, who proposed the general quantization rule for an integrable system:

Each action variable is a separate integer, a separate quantum number.

This condition reproduces the circular orbit condition for two dimensional motion: let be polar coordinates fora central potential. Then is already an angle variable, and the canonical momentum conjugate is L, the angularmomentum. So the quantum condition for L reproduces Bohr's rule:

Correspondence principle 31

This allowed Sommerfeld to generalize Bohr's theory of circular orbits to elliptical orbits, showing that the energylevels are the same. He also found some general properties of quantum angular momentum which seemedparadoxical at the time. One of these results was the that the z-component of the angular momentum, the classicalinclination of an orbit relative to the z-axis, could only take on discrete values, a result which seemed to contradictrotational invariance. This was called space quantization for a while, but this term fell out of favor with the newquantum mechanics since no quantization of space is involved.In modern quantum mechanics, the principle of superposition makes it clear that rotational invariance is not lost. It ispossible to rotate objects with discrete orientations to produce superpositions of other discrete orientations, and thisresolves the intuitive paradoxes of the Sommerfeld model.

The quantum harmonic oscillatorWe provide a demonstration of how large quantum numbers can give rise to classical (continuous) behavior.Consider the one-dimensional quantum harmonic oscillator. Quantum mechanics tells us that the total (kinetic andpotential) energy of the oscillator, E, has a set of discrete values:

where is the angular frequency of the oscillator. However, in a classical harmonic oscillator such as a lead ballattached to the end of a spring, we do not perceive any discreteness. Instead, the energy of such a macroscopicsystem appears to vary over a continuum of values.We can verify that our idea of "macroscopic" systems fall within the correspondence limit. The energy of theclassical harmonic oscillator with amplitude is

Thus, the quantum number has the value

If we apply typical "human-scale" values m = 1kg, = 1 rad/s, and A = 1m, then n ≈ 4.74×1033. This is a very largenumber, so the system is indeed in the correspondence limit.It is simple to see why we perceive a continuum of energy in said limit. With = 1 rad/s, the difference betweeneach energy level is J, well below what we can detect.

Relativistic kinetic energyHere we show that the expression of kinetic energy from special relativity becomes arbitrarily close to the classicalexpression for speeds that are much slower than the speed of light.Einstein's famous mass-energy equation

represents the total energy of a body with relativistic mass

where the velocity, is the velocity of the body relative to the observer, is the rest mass (the observedmass of the body at zero velocity relative to the observer), and is the speed of light.

When the velocity is zero, the energy expressed above is not zero and represents the rest energy:

Correspondence principle 32

When the body is in motion relative to the observer, the total energy exceeds the rest energy by an amount that is, bydefinition, the kinetic energy:

Using the approximation

for we get when speeds are much slower than that of light or

which is the Newtonian expression for kinetic energy.

See also• Quantum decoherence

References[1] Bohr, N. (1920), "Über die Serienspektra der Element", Zeitschrift für Physik 2 (5): 423–478 (English translation in (Bohr 1976,

pp. 241–282))[2] Jammer, Max (1989), The conceptual development of quantum mechanics, Los Angeles, CA: Tomash Publishers, American Institute of

Physics, ISBN 0883186179, Section 3.2

• Bohr, Niels (1976), Rosenfeld, L.; Nielsen, J. Rud, eds., Niels Bohr, Collected Works, Volume 3, TheCorrespondence Principle (1918–1923), 3, Amsterdam: North-Holland, ISBN 0444107843

• Sells, Robert L.; Weidner, Richard T. (1980), Elementary modern physics, Boston: Allyn and Bacon,ISBN 978-0-205-06559-2

• Stotland, A.; Cohen, D. (2006), "Diffractive energy spreading and its semiclassical limit", Journal of Physics A 39(10703): 10703, doi:10.1088/0305-4470/39/34/008, ISSN 0305-4470

De Broglie–Bohm theory 33

De Broglie–Bohm theoryThe de Broglie–Bohm theory, also called the pilot-wave theory, Bohmian mechanics, and the causalinterpretation, is an interpretation of quantum theory. As in quantum theory, it contains a wavefunction - a functionon the space of all possible configurations. Additionally, it also contains an actual configuration, even in situationswhere nobody observes it. The evolution over time of the configuration (that is, of the positions of all particles or theconfiguration of all fields) is defined by the wave function via a guiding equation. The evolution of the wavefunctionover time is given by Schrödinger's equation.The de Broglie–Bohm theory expresses in an explicit manner the fundamental non-locality in quantum physics. Thevelocity of any one particle depends on the value of the wavefunction, which depends on the whole configuration ofthe universe.This theory is deterministic. Most (but not all) relativistic variants require a preferred frame. Variants which handlespin and curved spaces are known. It can be modified to handle quantum field theory. Bell's theorem was inspired byBell's discovery of the work of David Bohm and his subsequent wondering if the obvious non-locality of the theorycould be removed.This theory gives rise to a measurement formalism, analogous to thermodynamics for classical mechanics, whichyields the standard quantum formalism generally associated with the Copenhagen interpretation. The measurementproblem is resolved by this theory since the outcome of an experiment is registered by the configuration of theparticles of the experimental apparatus after the experiment is completed. The familiar wavefunction collapse ofstandard quantum mechanics emerges from an analysis of subsystems and the quantum equilibrium hypothesis.The theory has a number of equivalent mathematical formulations and has been presented under a number ofdifferent names.

OverviewDe Broglie–Bohm theory is based on the following:

We have a configuration of the universe, described by coordinates , which is an element of the configurationspace . The configuration space is different for different versions of pilot wave theory. For example, this may bethe space of positions of particles, or, in case of field theory, the space of field configurations . Theconfiguration evolves according to the guiding equation

.

Here, is the standard complex-valued wavefunction known from quantum theory, which evolves accordingto Schrödinger's equation

This already completes the specification of the theory for any quantum theory with Hamilton operator of type

.

If the configuration is distributed according to at some moment of time , this holds for all times. Sucha state is named quantum equilibrium. In quantum equilibrium, this theory will agree with the results of standardquantum mechanics.

De Broglie–Bohm theory 34

Two-slit experiment

The Bohmian trajectories for an electron goingthrough the two-slit experiment.

The double-slit experiment is an illustration of wave-particle duality.In it, a beam of particles (such as photons) travels through a barrierwith two slits removed. If one puts a detector screen on the other side,the pattern of detected particles shows interference fringescharacteristic of waves; however, the detector screen responds toparticles. The system exhibits behaviour of both waves (interferencepatterns) and particles (dots on the screen).

If we modify this experiment so that one slit is closed, no interferencepattern is observed. Thus, the state of both slits affects the final results.We can also arrange to have a minimally invasive detector at one of theslits to see which slit the particle went through. When we do that, theinterference pattern disappears.The Copenhagen interpretation states that the particles are not localisedin space until they are detected, so that, if there is no detector on theslits, there is no matter of fact about what slit has the particle passed through. If one slit has a detector on it, then thewavefunction collapses due to that detection.

In de Broglie–Bohm theory, the wavefunction travels through both slits, but each particle has a well-definedtrajectory and passes through exactly one of the slits. The final position of the particle on the detector screen and theslit through which the particle passes by is determined by the initial position of the particle. Such initial position isnot controllable by the experimenter, so there is an appearance of randomness in the pattern of detection. The wavefunction interferes with itself and guides the particles in such a way that the particles avoid the regions in which theinterference is destructive and are attracted to the regions in which the interference is constructive, giving rise to theinterference pattern on the detector screen.To explain the behavior when the particle is detected to go through one slit, one needs to appreciate the role of theconditional wavefunction and how it gives rise to the collapse of the wavefunction; this is explained below. Thebasic idea is that the environment registering the detection effectively separates the two wave packets inconfiguration space.

The Theory

The ontology

The ontology of de Broglie-Bohm theory consists of a configuration of the universe and a pilot wave. The configuration space can be chosen differently, as in classical mechanics and standard

quantum mechanics.Thus, the ontology of pilot wave theory contains as the trajectory we know from classical mechanics, asthe wave function of quantum theory. So, at every moment of time there exists not only a wavefunction, but also a well-defined configuration of the whole universe. The correspondence to our experiences ismade by the identification of the configuration of our brain with some part of the configuration of the whole universe

, as in classical mechanics.While the ontology of classical mechanics is part of the ontology of de Broglie–Bohm theory, the dynamics is verydifferent. In classical mechanics, the acceleration of the particles are given by forces. In de Broglie–Bohm theory,the velocities of the particles are given by the wavefunction.In what follows below, we will give the setup for one particle moving in followed by the setup for particles moving in 3 dimensions. In the first instance, configuration space and real space are the same while in the second,

De Broglie–Bohm theory 35

real space is still , but configuration space becomes . While the particle positions themselves are in real space, thevelocity field and wavefunction are on configuration space which is how particles are entangled with each other inthis theory.Extensions to this theory include spin and more complicated configuration spaces.

We use variations of for particle positions while represents the complex-valued wavefunction onconfiguration space.

Guiding equation

For a single particle moving in , the particle's velocity is given

.

For many particles, we label them as for the th particle and their velocities are given by

.

The key fact to notice is that this velocity field depends on the actual positions of all of the particles in theuniverse. As explained below, in most experimental situations, the influence of all of those particles can beencapsulated into an effective wavefunction for a subsystem of the universe.

Schrödinger's equation

The one particle Schrödinger equation governs the time evolution of a complex-valued wavefunction on . Theequation represents a quantized version of the total energy of a classical system evolving under a real-valuedpotential function on :

For many particles, the equation is the same except that and are now on configuration space, .

This is the same wavefunction of conventional quantum mechanics.

The Born RuleIn Bohm's original papers [Bohm 1952] , he discusses how de Broglie–Bohm theory gives rise to the usualmeasurement results of quantum mechanics. The key idea is that this is true if the positions of the particles satisfy thestatistical distribution given by . And that distribution is guaranteed to be true for all time under the guidingequation if the initial distribution of the particles satisfies .For a given experiment, we can postulate this as being true and verify experimentally that it does indeed hold true, asit does. But, as argued in Dürr et al.,[1] one needs to argue that this distribution for subsystems is typical. They arguethat by virtue of its equivariance under the dynamical evolution of the system, is the appropriate measure oftypicality for initial conditions of the positions of the particles. They then prove that the vast majority of possibleinitial configurations will give rise to Born rule (i.e., ) statistics for measurement outcomes. In short, in auniverse governed by the de Broglie–Bohm dynamics, Born rule behavior is typical.The situation is thus analogous to the situation in classical statistical physics. A low entropy initial condition will, with overwhelmingly high probability, evolve into a higher entropy state: behavior consistent with the second law of thermodynamics is typical. There are, of course, anomalous initial conditions which would give rise to violations of the second law. However, absent some very detailed evidence supporting the actual realization of one of those

De Broglie–Bohm theory 36

special initial conditions, it would be quite unreasonable to expect anything but the actually observed uniformincrease of entropy. Similarly, in the de Broglie–Bohm theory, there are anomalous initial conditions which wouldproduce measurement statistics in violation of the Born rule (i.e., in conflict with the predictions of standardquantum theory). But the typicality theorem shows that, absent some particular reason to believe one of those specialinitial conditions was in fact realized, Born rule behavior is what one should expect.It is in that qualified sense that Born rule is, for the de Broglie–Bohm theory, a theorem rather than (as in ordinaryquantum theory) an additional postulate.

The conditional wave function of a subsystemIn the formulation of the De Broglie–Bohm theory, there is only a wave function for the entire universe (whichalways evolves by the Schrödinger equation). However, once the theory is formulated, it is convenient to introduce anotion of wave function also for subsystems of the universe. Let us write the wave function of the universe as

, where denotes the configuration variables associated to some subsystem (I) of the universe anddenotes the remaining configuration variables. Denote, respectively, by and by the actual

configuration of subsystem (I) and of the rest of the universe. For simplicity, we consider here only the spinless case.The conditional wave function of subsystem (I) is defined by:

It follows immediately from the fact that satisfies the guiding equation that also theconfiguration satisfies a guiding equation identical to the one presented in the formulation of the theory, withthe universal wave function replaced with the conditional wave function . Also, the fact that is randomwith probability density given by the square modulus of implies that the conditional probability density of

given is given by the square modulus of the (normalized) conditional wave function (in theterminology of Dürr et al.[2] this fact is called the fundamental conditional probability formula).Unlike the universal wave function, the conditional wave function of a subsystem does not always evolves by theSchrödinger equation, but in many situations it does. For instance, if the universal wave function factors as:

then the conditional wave function of subsystem (I) is (up to an irrelevant scalar factor) equal to (this is whatStandard Quantum Theory would regard as the wave function of subsystem (I)). If, in addition, the Hamiltonian doesnot contain an interaction term between subsystems (I) and (II) then does satisfy a Schrödinger equation. Moregenerally, assume that the universal wave function can be written in the form:

where solves Schrödinger equation and for all and . Then, again, the conditionalwave function of subsystem (I) is (up to an irrelevant scalar factor) equal to and if the Hamiltonian does notcontain an interaction term between subsystems (I) and (II), satisfies a Schrödinger equation.The fact that the conditional wave function of a subsystem does not always evolve by the Schrödinger equation isrelated to the fact that the usual collapse rule of Standard Quantum Theory emerges from the Bohmian formalismwhen one considers conditional wave functions of subsystems.

De Broglie–Bohm theory 37

Extensions

SpinTo incorporate spin, the wavefunction becomes complex-vector valued. The value space is called spin space; for aspin-1/2 particle, spin space can be taken to be . The guiding equation is modified by taking inner products inspin space to reduce the complex vectors to complex numbers. The Schrödinger equation is modified by adding aPauli spin term.

where is the magnetic moment of the th particle, is the appropriate spin operator acting on the th

particle's spin space, , and are, respectively, the magnetic field and the vector

potential in (all other functions are fully on configuration space), is the charge of the th particle, andis the inner product in spin space ,

For an example of a spin space, a system consisting of two spin 1/2 particle and one spin 1 particle has awavefunctions of the form . That is, its spin space is a 12 dimensional space.

Curved spaceTo extend de Broglie–Bohm theory to curved space (Riemannian manifolds in mathematical parlance), one simplynotes that all of the elements of these equations make sense, such as gradients and Laplacians. Thus, we useequations that have the same form as above. Topological and boundary conditions may apply in supplementing theevolution of Schrödinger's equation.For a de Broglie–Bohm theory on curved space with spin, the spin space becomes a vector bundle over configurationspace and the potential in Schrödinger's equation becomes a local self-adjoint operator acting on that space.[3]

Quantum field theoryIn Dürr et al.,[4] [5] the authors describe an extension of de Broglie–Bohm theory for handling creation andannihilation operators. The basic idea is that configuration space becomes the (disjoint) space of all possibleconfigurations of any number of particles. For part of the time, the system evolves deterministically under theguiding equation with a fixed number of particles. But under a stochastic process, particles may be created andannihilated. The distribution of creation events is dictated by the wavefunction. The wavefunction itself is evolvingat all times over the full multi-particle configuration space.Nikolic [6] introduces a purely deterministic de Broglie–Bohm theory of particle creation and destruction, accordingto which particle trajectories are continuous, but particle detectors behave as if particles have been created ordestroyed even when a true creation or destruction of particles does not take place.

De Broglie–Bohm theory 38

Exploiting nonlocalityValentini[7] has extended the de Broglie–Bohm theory to include signal nonlocality that would allow entanglementto be used as a stand-alone communication channel without a secondary classical "key" signal to "unlock" themessage encoded in the entanglement. This violates orthodox quantum theory but it has the virtue that it makes theparallel universes of the chaotic inflation theory observable in principle.Unlike de Broglie–Bohm theory, Valentini's theory has the wavefunction evolution also depend on the ontologicalvariables. This introduces an instability, a feedback loop that pushes the hidden variables out of "sub-quantal heatdeath". The resulting theory becomes nonlinear and non-unitary.

RelativityPilot wave theory is explicitly nonlocal. As a consequence, most relativistic variants of pilot wave theory need apreferred foliation of space-time. While this is in conflict with the standard interpretation of relativity, the preferredfoliation, if unobservable, does not lead to any empirical conflicts with relativity.The relation between nonlocality and preferred foliation can be better understood as follows. In de Broglie–Bohmtheory, nonlocality manifests as the fact that the velocity and acceleration of one particle depends on theinstantaneous positions of all other particles. On the other hand, in the theory of relativity the concept ofinstantaneousness does not have an invariant meaning. Thus, to define particle trajectories, one needs an additionalrule that defines which space-time points should be considered instantaneous. The simplest way to achieve this is tointroduce a preferred foliation of space-time by hand, such that each hypersurface of the foliation defines ahypersurface of equal time. However, this way (which explicitly breaks the relativistic covariance) is not the onlyway. It is also possible that a rule which defines instantaneousness is contingent, by emerging dynamically fromrelativistic covariant laws combined with particular initial conditions. In this way, the need for a preferred foliationcan be avoided and relativistic covariance can be saved.There has been work in developing relativistic versions of de Broglie–Bohm theory. See Bohm and Hiley: TheUndivided Universe, and [8], [9], and references therein. Another approach is given in the work of Dürr et al.[8] inwhich they use Bohm-Dirac models and a Lorentz-invariant foliation of space-time.In [11],[12] and [13] Nikolic develops a generalized relativistic-invariant probabilistic interpretation of quantumtheory, in which is no longer a probability density in space, but a probability density in space-time. He usesthis generalized probabilistic interpretation to formulate a relativistic-covariant version of de Broglie–Bohm theorywithout introducing a preferred foliation of space-time.

ResultsBelow are some highlights of the results that arise out of an analysis of de Broglie–Bohm theory. Experimentalresults agree with all of the standard predictions of quantum mechanics in so far as the latter has predictions.However, while standard quantum mechanics is limited to discussing experiments with human observers, deBroglie–Bohm theory is a theory which governs the dynamics of a system without the intervention of outsideobservers (p. 117 in Bell[9] ).

The basis for agreement with standard quantum mechanics is that the particles are distributed according to .This is a statement of observer ignorance, but it can be proven[1] that for a universe governed by this theory, this willtypically be the case. There is apparent collapse of the wave function governing subsystems of the universe, but thereis no collapse of the universal wavefunction.

De Broglie–Bohm theory 39

Measuring spin and polarizationAccording to ordinary quantum theory, it is not possible to measure the spin or polarization of a particle directly;instead, the component in one direction is measured; the outcome from a single particle may be 1, meaning that theparticle is aligned with the measuring apparatus, or -1, meaning that it is aligned the opposite way. For an ensembleof particles, if we expect the particles to be aligned, the results are all 1. If we expect them to be aligned oppositely,the results are all -1. For other alignments, we expect some results to be 1 and some to be -1 with a probability thatdepends on the expected alignment. For a full explanation of this, see the Stern-Gerlach Experiment.In de Broglie–Bohm theory, the results of a spin experiment cannot be analyzed without some knowledge of theexperimental setup. It is possible[10] to modify the setup so that the trajectory of the particle is unaffected, but thatthe particle with one setup registers as spin up while in the other setup it registers as spin down. Thus, for the deBroglie–Bohm theory, the particle's spin is not an intrinsic property of the particle—instead spin is, so to speak, inthe wave function of the particle in relation to the particular device being used to measure the spin. This is anillustration of what is sometimes referred to as contextuality, and is related to naive realism about operators.[11]

Measurements, the quantum formalism, and observer independenceDe Broglie–Bohm theory gives the same results as quantum mechanics. It treats the wavefunction as a fundamentalobject in the theory as the wavefunction describes how the particles move. This means that no experiment candistinguish between the two theories. This section outlines the ideas as to how the standard quantum formalismarises out of quantum mechanics. References include Bohm's original 1952 paper and Dürr et al.[1]

Collapse of the wavefunction

De Broglie–Bohm theory is a theory that applies primarily to the whole universe. That is, there is a singlewavefunction governing the motion of all of the particles in the universe according to the guiding equation.Theoretically, the motion of one particle depends on the positions of all of the other particles in the universe. In somesituations, such as in experimental systems, we can represent the system itself in terms of a de Broglie–Bohm theoryin which the wavefunction of the system is obtained by conditioning on the environment of the system. Thus, thesystem can be analyzed with Schrödinger's equation and the guiding equation, with an initial distribution forthe particles in the system (see the section on the conditional wave function of a subsystem for details).It requires a special setup for the conditional wavefunction of a system to obey a quantum evolution. When a systeminteracts with its environment, such as through a measurement, then the conditional wavefunction of the systemevolves in a different way. The evolution of the universal wavefunction can become such that the wavefunction ofthe system appears to be in a superposition of distinct states. But if the environment has recorded the results of theexperiment, then using the actual Bohmian configuration of the environment to condition on, the conditionalwavefunction collapses to just one alternative, the one corresponding with the measurement results.Collapse of the universal wavefunction never occurs in de Broglie–Bohm theory. Its entire evolution is governed bySchrödinger's equation and the particles' evolutions are governed by the guiding equation. Collapse only occurs in aphenomenological way for systems that seem to follow their own Schrödinger's equation. As this is an effectivedescription of the system, it is a matter of choice as to what to define the experimental system to include and this willaffect when "collapse" occurs.

Operators as observables

In the standard quantum formalism, measuring observables is generally thought of as measuring operators on the Hilbert space. For example, measuring position is considered to be a measurement of the position operator. This relationship between physical measurements and Hilbert space operators is, for standard quantum mechanics, an additional axiom of the theory. The de Broglie–Bohm theory, by contrast, requires no such measurement axioms (and measurement as such is not a dynamically distinct or special sub-category of physical processes in the theory).

De Broglie–Bohm theory 40

In particular, the usual operators-as-observables formalism is, for de Broglie–Bohm theory, a theorem.[12] A majorpoint of the analysis is that many of the measurements of the observables do not correspond to properties of theparticles; they are (as in the case of spin discussed above) measurements of the wavefunction.In the history of de Broglie–Bohm theory, the proponents have often had to deal with claims that this theory isimpossible. Such arguments are generally based on inappropriate analysis of operators as observables. If onebelieves that spin measurements are indeed measuring the spin of a particle that existed prior to the measurement,then one does reach contradictions. De Broglie–Bohm theory deals with this by noting that spin is not a feature ofthe particle, but rather that of the wavefunction. As such, it only has a definite outcome once the experimentalapparatus is chosen. Once that is taken into account, the impossibility theorems become irrelevant.There have also been claims that experiments reject the Bohm trajectories [18] in favor of the standard QM lines.But as shown in [19] and [20], such experiments cited above only disprove a misinterpretation of the deBroglie–Bohm theory, not the theory itself.There are also objections to this theory based on what it says about particular situations usually involving eigenstatesof an operator. For example, the ground state of hydrogen is a real wavefunction. According to the guiding equation,this means that the electron is at rest when in this state. Nevertheless, it is distributed according to and nocontradiction to experimental results is possible to detect.Operators as observables leads many to believe that many operators are equivalent. De Broglie–Bohm theory, fromthis perspective, chooses the position observable as a favored observable rather than, say, the momentum observable.Again, the link to the position observable is a consequence of the dynamics. The motivation for de Broglie–Bohmtheory is to describe a system of particles. This implies that the goal of the theory is to describe the positions of thoseparticles at all times. Other observables do not have this compelling ontological status. Having definite positionsexplains having definite results such as flashes on a detector screen. Other observables would not lead to thatconclusion, but there need not be any problem in defining a mathematical theory for other observables; see Hyman etal.[13] for an exploration of the fact that a probability density and probability current can be defined for any set ofcommuting operators.

Hidden variables

De Broglie–Bohm theory is often referred to as a "hidden variable" theory. The alleged applicability of the term"hidden variable" comes from the fact that the particles postulated by Bohmian mechanics do not influence theevolution of the wavefunction. The argument is that, because adding particles does not have an effect on thewavefunction's evolution, such particles must not have effects at all and are, thus, unobservable, since they cannothave an effect on observers. There is no analogue of Newton's third law in this theory. The idea is supposed to bethat, since particles cannot influence the wavefunction, and it is the wavefunction that determines measurementpredictions through the Born rule, the particles are superfluous and unobservable.Such an argument, however, arises from a fundamental misunderstanding of the relation between the ontologyposited by the de Broglie–Bohm theory and the world of ordinary observation. In particular, the particles postulatedby the de Broglie–Bohm theory are anything but "hidden" variables: they are what the cats and trees and tables andplanets and pointers we see are made of! It is the wavefunction itself which is "hidden" in the sense of beinginvisible and not-directly-observable.Thus, for example, when the wavefunction of some measuring apparatus is such that its pointer is superposedbetween pointing to the left and pointing to the right, what accounts for the fact that scientists, when they look at theapparatus, see the pointer pointing to the left (say) is the fact that the de Broglie–Bohmian particles that make up thepointer are actually pointed towards the left. While the exact details of how humans process such information andwhat it is based on is beyond the scope of the de Broglie–Bohm theory, the basic idea of any particle ontology is thatif the particles in the theory appear where they seem to be from human observations, then it is considered asuccessful prediction.

De Broglie–Bohm theory 41

Heisenberg's uncertainty principleThe Heisenberg uncertainty principle states that when two complementary measurements are made, there is a limit tothe product of their accuracy. As an example, if one measures the position with an accuracy of , and themomentum with an accuracy of , then If we make further measurements in order to get moreinformation, we disturb the system and change the trajectory into a new one depending on the measurement setup;therefore, the measurement results are still subject to Heisenberg's uncertainty relation.In de Broglie–Bohm theory, there is always a matter of fact about the position and momentum of a particle. Eachparticle has a well defined trajectory. Observers have limited knowledge as to what this trajectory is (and thus of theposition and momentum). It is the lack knowledge of the particle's trajectory that accounts for the uncertaintyrelation. What one can know about a particle at any given time is described by the wavefunction. Since theuncertainty relation can be derived from the wavefunction in other interpretations of quantum mechanics, it can belikewise derived (in the epistemic sense mentioned above), on the de Broglie–Bohm theory.To put the statement differently, the particles' positions are only known statistically. As in classical mechanics,successive observations of the particles' positions refine the experimenter's knowledge of the particles' initialconditions. Thus, with succeeding observations, the initial conditions become more and more restricted. Thisformalism is consistent with the normal use of the Schrödinger equation.For the derivation of the uncertainty relation, see Heisenberg uncertainty principle, noting that it describes it from theviewpoint of the Copenhagen interpretation.

Quantum entanglement, Einstein-Podolsky-Rosen paradox, Bell's theorem, and nonlocalityDe Broglie–Bohm theory highlighted the issue of nonlocality: it inspired John Stewart Bell to prove his now-famoustheorem,[14] which in turn led to the Bell test experiments.In the Einstein-Podolsky-Rosen paradox,[15] the authors point out that quantum mechanics allows the creation ofpairs of particles in an entangled quantum state. They describe a thought-experiment one could perform on such apair, the results of which they interpreted as indicating that quantum mechanics is an incomplete theory.Decades later John Bell proved Bell's theorem (see p. 14 in Bell[9] ), in which he showed that, if they are to agreewith the empirical predictions of quantum mechanics, all such "hidden-variable" completions of quantum mechanicsmust either be nonlocal (as the Bohm interpretation is) or give up the assumption that experiments produce uniqueresults (see counterfactual definiteness and many-worlds interpretation). In particular, Bell proved that any localtheory with unique results must make empirical predictions satisfying a statistical constraint called "Bell'sinequality".Alain Aspect performed a series of Bell test experiments that test Bell's inequality using an EPR-type setup. Aspect'sresults show experimentally that Bell's inequality is in fact violated—meaning that the relevant quantum mechanicalpredictions are correct. In these Bell test experiments, entangled pairs of particles are created; the particles areseparated, traveling to remote measuring apparatus. The orientation of the measuring apparatus can be changed whilethe particles are in flight, demonstrating the apparent non-locality of the effect.The de Broglie–Bohm theory makes the same (empirically correct) predictions for the Bell test experiments asordinary quantum mechanics. It is able to do this because it is manifestly nonlocal. It is often criticized or rejectedbased on this; Bell's attitude was: "It is a merit of the de Broglie–Bohm version to bring this [nonlocality] out soexplicitly that it cannot be ignored." [16]

The de Broglie–Bohm theory describes the physics in the Bell test experiments as follows: to understand the evolution of the particles, we need to set up a wave equation for both particles; the orientation of the apparatus affects the wavefunction. The particles in the experiment follow the guidance of the wavefunction. It is the wavefunction that carries the faster-than-light effect of changing the orientation of the apparatus. An analysis of exactly what kind of nonlocality is present and how it is compatible with relativity can be found in Maudlin.[17] Note

De Broglie–Bohm theory 42

that in Bell's work, and in more detail in Maudlin's work, it is shown that the nonlocality does not allow for signalingat speeds faster than light.

Classical limitBohm's formulation of de Broglie–Bohm theory in terms of a classical-looking version has the merits that theemergence of classical behavior seems to follow immediately for any situation in which the quantum potential isnegligible, as noted by Bohm in 1952. Modern methods of decoherence are relevant to an analysis of this limit. SeeAllori et al.[18] for steps towards a rigorous analysis.

Quantum trajectory methodWork by Robert Wyatt in the early 2000s attempted to use the Bohm "particles" as an adaptive mesh that follows theactual trajectory of a quantum state in time and space. In the "quantum trajectory" method, one samples the quantumwavefunction with a mesh of quadrature points. One then evolves the quadrature points in time according to theBohm equations of motion. At each time-step, one then re-synthesizes the wavefunction from the points, recomputesthe quantum forces, and continues the calculation. (Quick-time movies of this for H+H2 reactive scattering can befound on the Wyatt group [27] web-site at UT Austin.) This approach has been adapted, extended, and used by anumber of researchers in the Chemical Physics community as a way to compute semi-classical and quasi-classicalmolecular dynamics. A recent (2007) issue of the Journal of Physical Chemistry A [28] was dedicated to Prof. Wyattand his work on "Computational Bohmian Dynamics".Eric Bittner's group [29] at the University of Houston has advanced a statistical variant of this approach that usesBayesian sampling technique to sample the quantum density and compute the quantum potential on a structurelessmesh of points. This technique was recently used to estimate quantum effects in the heat-capacity of small clustersNen for n~100.There remain difficulties using the Bohmian approach, mostly associated with the formation of singularities in thequantum potential due to nodes in the quantum wavefunction. In general, nodes forming due to interference effectslead to the case where This results in an infinite force on the sample particles forcing them to move

away from the node and often crossing the path of other sample points (which violates single-valuedness). Variousschemes have been developed to overcome this; however, no general solution has yet emerged.These methods, as does Bohm's Hamilton-Jacobi formulation, do not apply to situations in which the full dynamicsof spin need to be taken into account.

Occam's razor criticismBoth Hugh Everett III and Bohm treated the wavefunction as a physically real field. Everett's many-worldsinterpretation is an attempt to demonstrate that the wavefunction alone is sufficient to account for all ourobservations. When we see the particle detectors flash or hear the click of a Geiger counter then Everett's theoryinterprets this as our wavefunction responding to changes in the detector's wavefunction, which is responding in turnto the passage of another wavefunction (which we think of as a "particle", but is actually just anotherwave-packet).[19] No particle (in the Bohm sense of having a defined position and velocity) exists, according to thattheory. For this reason Everett sometimes referred to his approach as the "pure wave theory". Talking of Bohm's1952 approach, Everett says:

“Our main criticism of this view is on the grounds of simplicity - if one desires to hold the view that is a real field then the associatedparticle is superfluous since, as we have endeavored to illustrate, the pure wave theory is itself satisfactory.[20] ”

In the Everettian view, then, the Bohm particles are superfluous entities, similar to, and equally as unnecessary as, for example, the luminiferous ether was found to be unnecessary in special relativity. This argument of Everett's is

De Broglie–Bohm theory 43

sometimes called the "redundancy argument", since the superfluous particles are redundant in the sense of Occam'srazor.[21] . By omitting the hidden variables, however, Everett had to invoke causally unrelated and thereforeexperimentally unverifiable parallel universes.Many authors have expressed critical views of the de Broglie-Bohm theory, by comparing it to Everett's manyworlds approach. Many (but not all) proponents of the de Broglie-Bohm theory (such as Bohm and Bell) interpret theuniversal wave function as physically real. According to some supporters of Everett's theory, if the (nevercollapsing) wave function is taken to be physically real, then it is natural to interpret the theory as having the samemany worlds as Everett's theory. In the Everettian view the role of the Bohm particle is to act as a "pointer", tagging,or selecting, just one branch of the universal wavefunction (the assumption that this branch indicates which wavepacket determines the observed result of a given experiment is called the "result assumption"[19] ); the other branchesare designated "empty" and implicitly assumed by Bohm to be devoid of conscious observers.[19] H. Dieter Zehcomments on these "empty" branches:

“It is usually overlooked that Bohm’s theory contains the same “many worlds” of dynamically separate branches as the Everett interpretation(now regarded as “empty” wave components), since it is based on precisely the same . . . global wave function . . .[22] ”

David Deutsch has expressed the same point more "acerbically"[19] :

“pilot-wave theories are parallel-universe theories in a state of chronic denial.[23]”The fact that such a "pointer" can be made in a self-consistent manner that not only reproduces all knownexperimental results but also provides a clean classical limit is, however, highly significant in itself. It proves that theexistence of alternative universes is not a necessary conclusion of quantum physics.

DerivationsDe Broglie–Bohm theory has been derived many times and in many ways. Below are five derivations all of whichare very different and lead to different ways of understanding and extending this theory.• Schrödinger's equation can be derived by using Einstein's light quanta hypothesis: and de Broglie's

hypothesis: .

The guiding equation can be derived in a similar fashion. We assume a plane wave: .Notice that . Assuming that for the particle's actual velocity, we have that

. Thus, we have the guiding equation.

Notice that this derivation does not use Schrödinger's equation.• Preserving the density under the time evolution is another method of derivation. This is the method that Bell cites.

It is this method which generalizes to many possible alternative theories. The starting point is the continuityequation for the density . This equation describes a probability flow along a

current. We take the velocity field associated with this current as the velocity field whose integral curves yield themotion of the particle.

• A method applicable for particles without spin is to do a polar decomposition of the wavefunction and transformSchrödinger's equation into two coupled equations: the continuity equation from above and the Hamilton–Jacobiequation. This is the method used by Bohm in 1952. The decomposition and equations are as follows:

Decomposition: Note corresponds to the probability density.

De Broglie–Bohm theory 44

Continuity Equation:

Hamilton–Jacobi Equation:

The Hamilton–Jacobi equation is the equation derived from a Newtonian system with potential

and velocity field The potential is the classical potential that appears in

Schrödinger's equation and the other term involving is the quantum potential, terminology introduced byBohm.This leads to viewing the quantum theory as particles moving under the classical force modified by a quantumforce. However, unlike standard Newtonian mechanics, the initial velocity field is already specified by

which is a symptom of this being a first-order theory, not a second-order theory.• A fourth derivation was given by Dürr et al.[1] In their derivation, they derive the velocity field by demanding the

appropriate transformation properties given by the various symmetries that Schrödinger's equation satisfies, oncethe wavefunction is suitably transformed. The guiding equation is what emerges from that analysis.

• A fifth derivation, given by Dürr et al.[4] is appropriate for generalization to quantum field theory and the Diracequation. The idea is that a velocity field can also be understood as a first order differential operator acting onfunctions. Thus, if we know how it acts on functions, we know what it is. Then given the Hamiltonian operator

, the equation to satisfy for all functions (with associated multiplication operator ) is

where is the local Hermitian inner product on the value space

of the wavefunction.This formulation allows for stochastic theories such as the creation and annihilation of particles.

HistoryDe Broglie–Bohm theory has a history of different formulations and names. In this section, each stage is given aname and a main reference.

Pilot-wave theoryDr. de Broglie presented his pilot wave theory at the 1927 Solvay Conference,[24] after close collaboration withSchrödinger, who developed his wave equation for de Broglie's theory. At the end of the presentation, WolfgangPauli pointed out that it was not compatible with a semi-classical technique Fermi had previously adopted in the caseof inelastic scattering. Contrary to a popular legend, de Broglie actually gave the correct rebuttal that the particulartechnique could not be generalized for Pauli's purpose, although the audience might have been lost in the technicaldetails and de Brolie's mild mannerism left the impression that Pauli's objection was valid. He was eventuallypersuaded to abandon this theory nonetheless in 1932 due to both the Copenhagen school's more successful P.R.efforts and his own inability to understand quantum decoherence. Also in 1932, John von Neumann published apaper,[25] claiming to prove that all hidden-variable theories are impossible. This sealed the fate of de Broglie'stheory for the next two decades. In truth, von Neumann's proof is based on invalid assumptions, such as quantumphysics can be made local, and it does not really disprove the pilot-wave theory.De Broglie's theory already applies to multiple spin-less particles, but lacks an adequate theory of measurement as noone understood quantum decoherence at the time. An analysis of de Broglie's presentation is given in Bacciagaluppiet al.[26] [27]

De Broglie–Bohm theory 45

Around this time Erwin Madelung[28] also developed a hydrodynamic version of Schrödinger's equation which isincorrectly considered as a basis for the density current derivation of de Broglie–Bohm theory. The Madelungequations, being quantum Euler equations (fluid dynamics), differ philosophically from the de Broglie–Bohmtheory[29] and are the basis of the hydrodynamic interpretation of quantum mechanics.

De Broglie–Bohm theoryAfter publishing a popular textbook on Quantum Mechanics which adhered entirely to the Copenhagen orthodoxy,Bohm was persuaded by Einstein to take a critical look at von Neumann's theorem. The result was 'A SuggestedInterpretation of the Quantum Theory in Terms of "Hidden Variables" I and II' [Bohm 1952]. It extended the originalPilot Wave Theory to incorporate a consistent theory of measurement, and to address a criticism of Pauli that deBroglie did not properly respond to; it is taken to be deterministic (though Bohm hinted in the original papers thatthere should be disturbances to this, in the way Brownian motion disturbs Newtonian mechanics). This stage isknown as the de Broglie–Bohm Theory in Bell's work [Bell 1987] and is the basis for 'The Quantum Theory ofMotion' [Holland 1993].This stage applies to multiple particles, and is deterministic.The de Broglie–Bohm theory is an example of a hidden variables theory. Bohm originally hoped that hiddenvariables could provide a local, causal, objective description that would resolve or eliminate many of the paradoxesof quantum mechanics, such as Schrödinger's cat, the measurement problem and the collapse of the wavefunction.However, Bell's theorem complicates this hope, as it demonstrates that there can be no local hidden variable theorythat is compatible with the predictions of quantum mechanics. The Bohmian interpretation is causal but not local.Bohm's paper was largely ignored by other physicists. Even Albert Einstein did not consider it a satisfactory answerto the quantum non-locality question. The rest of the contemporary objections, however, were ad hominem, focusingon Bohm's sympathy with liberals and supposed communists as exemplified by his refusal to give testimony to theHouse Un-American Activities Committee.Eventually the cause was taken up by John Bell. In "Speakable and Unspeakable in Quantum Mechanics" [Bell1987], several of the papers refer to hidden variables theories (which include Bohm's). Bell showed that vonNeumann's objection amounted to showing that hidden variables theories are nonlocal, and that nonlocality is afeature of all quantum mechanical systems.

Bohmian mechanicsThis term is used to describe the same theory, but with an emphasis on the notion of current flow. In particular, it isoften used to include most of the further extensions past the spin-less version of Bohm. While de Broglie–Bohmtheory has Lagrangians and Hamilton-Jacobi equations as a primary focus and backdrop, with the icon of thequantum potential, Bohmian mechanics considers the continuity equation as primary and has the guiding equation asits icon. They are mathematically equivalent in so far as the Hamilton-Jacobi formulation applies, i.e., spin-lessparticles. The papers of Dürr et al. popularized the term.All of non-relativistic quantum mechanics can be fully accounted for in this theory.

De Broglie–Bohm theory 46

Causal interpretation and ontological interpretationBohm developed his original ideas, calling them the Causal Interpretation. Later he felt that causal sounded toomuch like deterministic and preferred to call his theory the Ontological Interpretation. The main reference is 'TheUndivided Universe' [Bohm, Hiley 1993].This stage covers work by Bohm and in collaboration with Vigier and Hiley. Bohm is clear that this theory isnon-deterministic (the work with Hiley includes a stochastic theory). As such, this theory is not, strictly speaking, aformulation of the de Broglie–Bohm theory. However, it deserves mention here because the term "BohmInterpretation" is ambiguous between this theory and the de Broglie–Bohm theory.

See also• David Bohm• Interpretation of quantum mechanics• Madelung equations• Local hidden variable theory• Quantum mechanics• Pilot wave

References• Albert, David Z. (May 1994). "Bohm's Alternative to Quantum Mechanics". Scientific American 270: 58–67.• Barbosa, G. D.; N. Pinto-Neto (2004). "A Bohmian Interpretation for Noncommutative Scalar Field Theory and

Quantum Mechanics". Physical Review D 69: 065014. doi:10.1103/PhysRevD.69.065014. arXiv:hep-th/0304105.• Bohm, David (1952). "A Suggested Interpretation of the Quantum Theory in Terms of "Hidden Variables" I".

Physical Review 85: 166–179. doi:10.1103/PhysRev.85.166.• Bohm, David (1952). "A Suggested Interpretation of the Quantum Theory in Terms of "Hidden Variables", II".

Physical Review 85: 180–193. doi:10.1103/PhysRev.85.180.• Bohm, David (1990). "A new theory of the relationship of mind and matter" [41]. Philosophical Psychology 3 (2):

271–286. doi:10.1080/09515089008573004.• Bohm, David; B.J. Hiley (1993). The Undivided Universe: An ontological interpretation of quantum theory.

London: Routledge. ISBN 0-415-12185-X.• Durr, Detlef; Sheldon Goldstein, Roderich Tumulka and Nino Zangh (December 2004). "Bohmian Mechanics"

[42] (PDF). Physical review letters 93 (9): 090402. ISSN 0031-9007. PMID 15447078.• Goldstein, Sheldon (2001). "Bohmian Mechanics" [43]. Stanford Encyclopedia of Philosophy.• Hall, Michael J.W. (2004). "Incompleteness of trajectory-based interpretations of quantum mechanics". Journal of

Physics a Mathematical and General 37: 9549. doi:10.1088/0305-4470/37/40/015. arXiv:quant-ph/0406054.(Demonstrates incompleteness of the Bohm interpretation in the face of fractal, differentialble-nowherewavefunctions.)

• Holland, Peter R. (1993). The Quantum Theory of Motion : An Account of the de Broglie–Bohm CausalInterpretation of Quantum Mechanics. Cambridge: Cambridge University Press. ISBN 0-521-48543-6.

• Nikolic, H. (2004). "Relativistic quantum mechanics and the Bohmian interpretation". Foundations of PhysicsLetters 18: 549. doi:10.1007/s10702-005-1128-1. arXiv:quant-ph/0406173.

• Passon, Oliver (2004). Why isn't every physicist a Bohmian?. arXiv:quant-ph/0412119.• Sanz, A. S.; F. Borondo (2003). "A Bohmian view on quantum decoherence". The European Physical Journal D

44: 319. doi:10.1140/epjd/e2007-00191-8. arXiv:quant-ph/0310096.• Sanz, A.S. (2005). "A Bohmian approach to quantum fractals". J. Phys. A: Math. Gen. 38: 319.

doi:10.1088/0305-4470/38/26/013. (Describes a Bohmian resolution to the dilemma posed by non-differentiablewavefunctions.)

De Broglie–Bohm theory 47

• Silverman, Mark P. (1993). And Yet It Moves: Strange Systems and Subtle Questions in Physics. Cambridge:Cambridge University Press. ISBN 0-521-44631-7.

• Streater, Ray F. (2003). "Bohmian mechanics is a "lost cause"" [44]. Retrieved 2006-06-25.• Valentini, Antony; Hans Westman (2004). Dynamical Origin of Quantum Probabilities. arXiv:quant-ph/0403034.• Bohmian mechanics on arxiv.org [45]

External links• "Bohmian Mechanics" (Stanford Encyclopedia of Philosophy) [46]

• "Pilot waves, Bohmian metaphysics, and the foundations of quantum mechanics" [47], lecture course on Bohminterpretation by Mike Towler, Cambridge University.

References[1] Dürr, D., Goldstein, S., and Zanghì, N., "Quantum Equilibrium and the Origin of Absolute Uncertainty" (http:/ / arxiv. org/ abs/ quant-ph/

0308039), Journal of Statistical Physics 67: 843–907, 1992.[2] Quantum Equilibrium and the Origin of Absolute Uncertainty, D. Dürr, S. Goldstein and N. Zanghì, Journal of Statistical Physics 67, 843-907

(1992), http:/ / arxiv. org/ abs/ quant-ph/ 0308039.[3] Dürr, D., Goldstein, S., Taylor, J., Tumulka, R., and Zanghì, N., J. "Quantum Mechanics in Multiply-Connected Spaces" (http:/ / arxiv. org/

abs/ quant-ph/ 0506173), Phys. A: Math. Theor. 40, 2997–3031 (2007)[4] Dürr, D., Goldstein, S., Tumulka, R., and Zanghì, N., 2004, "Bohmian Mechanics and Quantum Field Theory" (http:/ / arxiv. org/ abs/

quant-ph/ 0303156), Phys. Rev. Lett. 93: 090402:1–4.[5] Dürr, D., Tumulka, R., and Zanghì, N., J. Phys. A: Math. Gen. 38, R1–R43 (2005), quant-ph/0407116[6] Nikolic, H. 2010 "QFT as pilot-wave theory of particle creation and destruction" (http:/ / arxiv. org/ abs/ 0904. 2287), Int. J. Mod. Phys. A 25,

1477 (2010)[7] Valentini, A., 1991, "Signal-Locality, Uncertainty and the Subquantum H-Theorem. II," Physics Letters A 158: 1–8.[8] Dürr, D., Goldstein, S., Münch-Berndl, K., and Zanghì, N., 1999, "Hypersurface Bohm-Dirac Models" (http:/ / arxiv. org/ abs/ quant-ph/

9801070), Phys. Rev. A 60: 2729–2736.[9] Bell, John S, Speakable and Unspeakable in Quantum Mechanics, Cambridge University Press 1987.[10] Albert, D. Z., 1992, Quantum Mechanics and Experience, Cambridge, MA: Harvard University Press[11] Daumer, M., Dürr, D., Goldstein, S., and Zanghì, N., 1997, "Naive Realism About Operators" (http:/ / arxiv. org/ abs/ quant-ph/ 9601013),

Erkenntnis 45: 379–397.[12] Dürr, D., Goldstein, S., and Zanghì, N., "Quantum Equilibrium and the Role of Operators as Observables in Quantum Theory" (http:/ / arxiv.

org/ abs/ quant-ph/ 0308038) Journal of Statistical Physics 116, 959–1055 (2004)[13] Hyman, Ross et al Bohmian mechanics with discrete operators (http:/ / www. iop. org/ EJ/ abstract/ 0305-4470/ 37/ 44/ L02), J. Phys. A:

Math. Gen. 37 L547–L558, 2004[14] J. S. Bell, On the Einstein Podolsky Rosen Paradox (http:/ / www. drchinese. com/ David/ Bell_Compact. pdf), Physics 1, 195 (1964)[15] Einstein, Podolsky, Rosen Can Quantum Mechanical Description of Physical Reality Be Considered Complete? Phys. Rev. 47, 777 (1935).[16] Bell, page 115[17] Maudlin, T., 1994, Quantum Non-Locality and Relativity: Metaphysical Intimations of Modern Physics, Cambridge, MA: Blackwell.[18] Allori, V., Dürr, D., Goldstein, S., and Zanghì, N., 2002, "Seven Steps Towards the Classical World" (http:/ / arxiv. org/ abs/ quant-ph/

0112005), Journal of Optics B 4: 482–488.[19] Harvey R Brown and David Wallace, Solving the measurement problem: de Broglie-Bohm loses out to Everett, Foundations of Physics 35

(2005), pp. 517-540. (http:/ / philsci-archive. pitt. edu/ archive/ 00001659/ 01/ Cushing. pdf) Abstract: "The quantum theory of de Broglie andBohm solves the measurement problem, but the hypothetical corpuscles play no role in the argument. The solution finds a more natural homein the Everett interpretation."

[20] See section VI of Everett's thesis: The Theory of the Universal Wave Function, pp 3-140 of Bryce Seligman DeWitt, R. Neill Graham, eds,The Many-Worlds Interpretation of Quantum Mechanics, Princeton Series in Physics, Princeton University Press (1973), ISBN0-691-08131-X

[21] Craig Callender, "The Redundancy Argument Against Bohmian Mechanics" (http:/ / philosophy. ucsd. edu/ faculty/ ccallender/ TheRedundancy Argument Against Bohmian Mechanics. doc. )

[22] Daniel Dennett (2000). With a little help from my friends. In D. Ross, A. Brook, and D. Thompson (Eds.), Dennett’s Philosophy: acomprehensive assessment. MIT Press/Bradford, ISBN 026268117X.

[23] David Deutsch, Comment on Lockwood. British Journal for the Philosophy of Science 47, 222228, 1996[24] Solvay Conference, 1928, Electrons et Photons: Rapports et Descussions du Cinquieme Conseil de Physique tenu a Bruxelles du 24 au 29

October 1927 sous les auspices de l'Institut International Physique Solvay[25] von Neumann J. 1932 Mathematische Grundlagen der Quantenmechanik

De Broglie–Bohm theory 48

[26] Bacciagaluppi, G., and Valentini, A., Quantum Theory at the Crossroads: Reconsidering the 1927 Solvay Conference[27] See the brief summary by Towler, M., "Pilot wave theory, Bohmian metaphysics, and the foundations of quantum mecahnics" (http:/ / www.

tcm. phy. cam. ac. uk/ ~mdt26/ PWT/ lectures/ bohm7. pdf)[28] Madelung, E., “ Quantentheorie in hydrodynamischer Form,” Zeit. F. Phys. 40 (1927), 322–326[29] Tsekov, R. (2009) Bohmian Mechanics versus Madelung Quantum Hydrodynamics (http:/ / arxiv. org/ abs/ 0904. 0723)

EPR paradoxIn quantum mechanics, the EPR paradox (or Einstein–Podolsky–Rosen paradox) is a thought experiment whichchallenged long-held ideas about the relation between the observed values of physical quantities and the values thatcan be accounted for by a physical theory. Einstein, Podolsky, and Rosen introduced the thought experiment in a1935 paper to argue that quantum mechanics is not a complete physical theory.[1] [2]

According to its authors the EPR experiment yields a dichotomy. Either1. The result of a measurement performed on one part A of a quantum system has a non-local effect on the physical

reality of another distant part B, in the sense that quantum mechanics can predict outcomes of somemeasurements carried out at B; or...

2. Quantum mechanics is incomplete in the sense that some element of physical reality corresponding to B cannotbe accounted for by quantum mechanics (that is, some extra variable is needed to account for it).

As shown later by Bell, one cannot introduce the notion of "elements of reality" without affecting the predictions ofthe theory. That is, one cannot complete quantum mechanics with these "elements", because this automatically leadsto some logical contradictions.Einstein never accepted quantum mechanics as a "real" and complete theory, struggling to the end of his life for aninterpretation that could comply with relativity without complying with the Heisenberg Uncertainty Principle. As heonce said: "God does not play dice", skeptically referring to the Copenhagen Interpretation of quantum mechanicswhich says there exists no objective physical reality other than that which is revealed through measurement andobservation.The EPR paradox is a paradox in the following sense: if one adds to quantum mechanics some seemingly reasonable(but actually wrong, or questionable as a whole) conditions — like local realism (not to be confused withphilosophical realism), counterfactual definiteness, and incompleteness (see Bell inequality and Bell testexperiments) — then one obtains a contradiction. However, quantum mechanics by itself does not appear to beinternally inconsistent, nor — as it turns out — does it contradict relativity. As a result of further theoretical andexperimental developments since the original EPR paper, most physicists today regard the EPR paradox as anillustration of how quantum mechanics violates classical intuitions.

Quantum mechanics and its interpretationSince the early twentieth century, quantum theory has proved to be very successful, describing accurately thephysical reality of the mesoscopic and microscopic world. This fact has been made clear through multiple andrepeatable physical experiments.Quantum mechanics was developed with the aim of describing atoms and explaining the observed spectral lines in ameasurement apparatus. Though disputed, it has never been seriously challenged. Philosophical interpretations ofquantum phenomena, however, are another story: the question of how to interpret the mathematical formulation ofquantum mechanics has given rise to a variety of different answers from people of different philosophicalbackgrounds (see Interpretation of quantum mechanics).Quantum theory and quantum mechanics do not account for single measurement outcomes in a deterministic way. According to an accepted interpretation of quantum mechanics known as the Copenhagen interpretation, a

EPR paradox 49

measurement causes an instantaneous collapse of the wave function describing the quantum system into aneigenstate of the observable that was measured.The most prominent opponent of the Copenhagen interpretation was Albert Einstein. Einstein did not believe in theidea of genuine randomness in nature, the main argument in the Copenhagen interpretation. In his view, quantummechanics is incomplete and suggests that there had to be 'hidden' variables responsible for random measurementresults.The famous paper "Can Quantum-Mechanical Description of Physical Reality Be Considered Complete?"[3],authored by Einstein, Podolsky and Rosen in 1935, condensed the philosophical discussion into a physical argument.They claim that given a specific experiment, in which the outcome of a measurement could be known before themeasurement takes place, there must exist something in the real world, an "element of reality", which determines themeasurement outcome. They postulate that these elements of reality are local, in the sense that each belongs to acertain point in spacetime. Each element may only be influenced by events which are located in the backward lightcone of its point in spacetime. These claims are founded on assumptions about nature which constitute what is nowknown as local realism.Though the EPR paper has often been taken as an exact expression of Einstein's views, it was primarily authored byPodolsky, based on discussions at the Institute for Advanced Study with Einstein and Rosen. Einstein later expressedto Erwin Schrödinger that "It did not come out as well as I had originally wanted; rather, the essential thing was, soto speak, smothered by the formalism."[4] In 1948 Einstein presented a less formal account of his local realist ideas.

Description of the paradoxThe EPR paradox draws on a phenomenon predicted by quantum mechanics, known as quantum entanglement, toshow that measurements performed on spatially separated parts of a quantum system can apparently have aninstantaneous influence on one another.This effect is now known as "nonlocal behavior" (or colloquially as "quantum weirdness" or "spooky action at adistance").

Simple versionBefore delving into the complicated logic that leads to the 'paradox', it is perhaps worth mentioning the simpleversion of the argument, as described by Greene and others, which Einstein used to show that 'hidden variables' mustexist.A positron and an electron are emitted from a source, by pion decay, so that their spins are opposite; one particle’sspin about any axis is the negative of the other's. Also, due to uncertainty, making a measurement of a particle’s spinabout one axis disturbs the particle so you now can’t measure its spin about any other axis.Now say you measure the electron’s spin about the x-axis. This automatically tells you the positron’s spin about thex-axis. Since you’ve done the measurement without disturbing the positron in any way, it can’t be that the positron"only came to have that state when you measured it", because you didn’t measure it! It must have had that spin allalong. Also you can now measure the positron’s spin about the y-axis. So it follows that the positron has had adefinite spin about two axes – much more information than the positron is capable of holding, and a "hiddenvariable" according to EPR.

EPR paradox 50

Measurements on an entangled stateWe have a source that emits electron-positron pairs, with the electron sent to destination A, where there is anobserver named Alice, and the positron sent to destination B, where there is an observer named Bob. According toquantum mechanics, we can arrange our source so that each emitted pair occupies a quantum state called a spinsinglet. This can be viewed as a quantum superposition of two states, which we call state I and state II. In state I, theelectron has spin pointing upward along the z-axis (+z) and the positron has spin pointing downward along the z-axis(-z). In state II, the electron has spin -z and the positron has spin +z. Therefore, it is impossible to associate eitherparticle in the spin singlet with a state of definite spin. The particles are thus said to be entangled.

The EPR thought experiment, performed with electron-positron pairs. A source (center) sends particles toward two observers, electrons toAlice (left) and positrons to Bob (right), who can perform spin measurements.

Alice now measures the spin along the z-axis. She can obtain one of two possible outcomes: +z or -z. Suppose shegets +z. According to quantum mechanics, the quantum state of the system collapses into state I. (Differentinterpretations of quantum mechanics have different ways of saying this, but the basic result is the same.) Thequantum state determines the probable outcomes of any measurement performed on the system. In this case, if Bobsubsequently measures spin along the z-axis, he will obtain -z with 100% probability. Similarly, if Alice gets -z, Bobwill get +z.There is, of course, nothing special about our choice of the z-axis. For instance, suppose that Alice and Bob nowdecide to measure spin along the x-axis, according to quantum mechanics, the spin singlet state may equally well beexpressed as a superposition of spin states pointing in the x direction. We'll call these states Ia and IIa. In state Ia,Alice's electron has spin +x and Bob's positron has spin -x. In state IIa, Alice's electron has spin -x and Bob'spositron has spin +x. Therefore, if Alice measures +x, the system collapses into Ia, and Bob will get -x. If Alicemeasures -x, the system collapses into IIa, and Bob will get +x.In quantum mechanics, the x-spin and z-spin are "incompatible observables", which means that there is a Heisenberguncertainty principle operating between them: a quantum state cannot possess a definite value for both variables.Suppose Alice measures the z-spin and obtains +z, so that the quantum state collapses into state I. Now, instead ofmeasuring the z-spin as well, Bob measures the x-spin. According to quantum mechanics, when the system is in stateI, Bob's x-spin measurement will have a 50% probability of producing +x and a 50% probability of -x. Furthermore,it is fundamentally impossible to predict which outcome will appear until Bob actually performs the measurement.Here is the crux of the matter. You might imagine that, when Bob measures the x-spin of his positron, he wouldget an answer with absolute certainty, since prior to this he hasn't disturbed his particle at all. But, as describedabove, Bob's positron has a 50% probability of producing +x and a 50% probability of -x—random behaviour, notcertain. Bob's positron knows that Alice's electron has been measured, and its z-spin detected, and hence B's z-spincalculated, so its x-spin is 'out of bounds'.

EPR paradox 51

Put another way, how does Bob's positron know, at the same time, which way to point if Alice decides (based oninformation unavailable to Bob) to measure x (i.e. be the opposite of Alice's electron's spin about the x-axis) and alsohow to point if Alice measures z (i.e. behave randomly), since it is only supposed to know one thing at a time? Usingthe usual Copenhagen interpretation rules that say the wave function "collapses" at the time of measurement, theremust be action at a distance (entanglement) or the positron must know more than it is supposed to (hidden variables).In case the explanation above is confusing, here is the paradox summed up:An electron-positron pair is emitted, the particles shoot off and are measured later. Whatever axis their spins aremeasured along, they are always found to be opposite. This can only be explained if the particles are linked in someway. Either they were created with a definite (opposite) spin about every axis—a "hidden variable" argument—orthey are linked so that one electron knows what axis the other is having its spin measured along, and becomes itsopposite about that one axis—an "entanglement" argument. Moreover, if the two particles have their spins measuredabout different axes, once the electron's spin has been measured about the x-axis (and the positron's spin about thex-axis deduced), the positron's spin about the y-axis will no longer be certain, as if it knows that the measurementhas taken place. Either that or it has a definite spin already, which gives it a spin about a second axis—a hiddenvariable.Incidentally, although we have used spin as an example, many types of physical quantities—what quantummechanics refers to as "observables"—can be used to produce quantum entanglement. The original EPR paper usedmomentum for the observable. Experimental realizations of the EPR scenario often use photon polarization, becausepolarized photons are easy to prepare and measure.

Locality in the EPR experimentThe principle of locality states that physical processes occurring at one place should have no immediate effect on theelements of reality at another location. At first sight, this appears to be a reasonable assumption to make, as it seemsto be a consequence of special relativity, which states that information can never be transmitted faster than the speedof light without violating causality. It is generally believed that any theory which violates causality would also beinternally inconsistent, and thus deeply unsatisfactory.It turns out that the usual rules for combining quantum mechanical and classical descriptions violate the principle oflocality without violating causality. Causality is preserved because there is no way for Alice to transmit messages(i.e. information) to Bob by manipulating her measurement axis. Whichever axis she uses, she has a 50% probabilityof obtaining "+" and 50% probability of obtaining "-", completely at random; according to quantum mechanics, it isfundamentally impossible for her to influence what result she gets. Furthermore, Bob is only able to perform hismeasurement once: there is a fundamental property of quantum mechanics, known as the "no cloning theorem",which makes it impossible for him to make a million copies of the electron he receives, perform a spin measurementon each, and look at the statistical distribution of the results. Therefore, in the one measurement he is allowed tomake, there is a 50% probability of getting "+" and 50% of getting "-", regardless of whether or not his axis isaligned with Alice's.However, the principle of locality appeals powerfully to physical intuition, and Einstein, Podolsky and Rosen wereunwilling to abandon it. Einstein derided the quantum mechanical predictions as "spooky action at a distance". Theconclusion they drew was that quantum mechanics is not a complete theory.In recent years, however, doubt has been cast on EPR's conclusion due to developments in understanding locality and especially quantum decoherence. The word locality has several different meanings in physics. For example, in quantum field theory "locality" means that quantum fields at different points of space do not interact with one another. However, quantum field theories that are "local" in this sense appear to violate the principle of locality as defined by EPR, but they nevertheless do not violate locality in a more general sense. Wavefunction collapse can be viewed as an epiphenomenon of quantum decoherence, which in turn is nothing more than an effect of the underlying local time evolution of the wavefunction of a system and all of its environment. Since the underlying

EPR paradox 52

behaviour doesn't violate local causality, it follows that neither does the additional effect of wavefunction collapse,whether real or apparent. Therefore, as outlined in the example above, neither the EPR experiment nor any quantumexperiment demonstrates that faster-than-light signaling is possible.

Resolving the paradox

Hidden variablesThere are several ways to resolve the EPR paradox. The one suggested by EPR is that quantum mechanics, despiteits success in a wide variety of experimental scenarios, is actually an incomplete theory. In other words, there issome yet undiscovered theory of nature to which quantum mechanics acts as a kind of statistical approximation(albeit an exceedingly successful one). Unlike quantum mechanics, the more complete theory contains variablescorresponding to all the "elements of reality". There must be some unknown mechanism acting on these variables togive rise to the observed effects of "non-commuting quantum observables", i.e. the Heisenberg uncertainty principle.Such a theory is called a hidden variable theory.To illustrate this idea, we can formulate a very simple hidden variable theory for the above thought experiment. Onesupposes that the quantum spin-singlet states emitted by the source are actually approximate descriptions for "true"physical states possessing definite values for the z-spin and x-spin. In these "true" states, the electron going to Bobalways has spin values opposite to the electron going to Alice, but the values are otherwise completely random. Forexample, the first pair emitted by the source might be "(+z, -x) to Alice and (-z, +x) to Bob", the next pair "(-z, -x) toAlice and (+z, +x) to Bob", and so forth. Therefore, if Bob's measurement axis is aligned with Alice's, he willnecessarily get the opposite of whatever Alice gets; otherwise, he will get "+" and "-" with equal probability.Assuming we restrict our measurements to the z and x axes, such a hidden variable theory is experimentallyindistinguishable from quantum mechanics. In reality, of course, there is an (uncountably) infinite number of axesalong which Alice and Bob can perform their measurements, so there has to be an infinite number of independenthidden variables. However, this is not a serious problem; we have formulated a very simplistic hidden variabletheory, and a more sophisticated theory might be able to patch it up. It turns out that there is a much more seriouschallenge to the idea of hidden variables.

Bell's inequality

In 1964, John Bell showed that the predictions of quantum mechanics in the EPR thought experiment aresignificantly different from the predictions of a very broad class of hidden variable theories (the local hidden variabletheories). Roughly speaking, quantum mechanics predicts much stronger statistical correlations between themeasurement results performed on different axes than the hidden variable theories. These differences, expressedusing inequality relations known as "Bell's inequalities", are in principle experimentally detectable. Later work byEberhard showed that the key properties of local hidden variable theories that lead to Bell's inequalities are localityand counter-factual definiteness. Any theory in which these principles hold produces the inequalities. A. Finesubsequently showed that any theory satisfying the inequalities can be modeled by a local hidden variable theory.After the publication of Bell's paper, a variety of experiments were devised to test Bell's inequalities. (As mentionedabove, these experiments generally rely on photon polarization measurements.) All the experiments conducted todate have found behavior in line with the predictions of standard quantum mechanics.However, Bell's theorem does not apply to all possible philosophically realist theories, although a common misconception is that quantum mechanics is inconsistent with all notions of philosophical realism. Realist interpretations of quantum mechanics are possible, although as discussed above, such interpretations must reject either locality or counter-factual definiteness. Mainstream physics prefers to keep locality while still maintaining a notion of realism that nevertheless rejects counter-factual definiteness. Examples of such mainstream realist interpretations are the consistent histories interpretation and the transactional interpretation. Fine's work showed that

EPR paradox 53

taking locality as a given there exist scenarios in which two statistical variables are correlated in a mannerinconsistent with counter-factual definiteness and that such scenarios are no more mysterious than any other despitethe inconsistency with counter-factual definiteness seeming 'counter-intuitive'. Violation of locality however isdifficult to reconcile with special relativity and is thought to be incompatible with the principle of causality. On theother hand the Bohm interpretation of quantum mechanics instead keeps counter-factual definiteness whileintroducing a conjectured non-local mechanism called the 'quantum potential'. Some workers in the field have alsoattempted to formulate hidden variable theories that exploit loopholes in actual experiments, such as the assumptionsmade in interpreting experimental data although no such theory has been produced that can reproduce all the resultsof quantum mechanics.There are also individual EPR-like experiments that have no local hidden variables explanation. Examples have beensuggested by David Bohm and by Lucien Hardy.

Einstein's hope for a purely algebraic theoryThe Bohm interpretation of quantum mechanics hypothesizes that the state of the universe evolves smoothly throughtime with no collapsing of quantum wavefunctions. One problem for the Copenhagen interpretation is to preciselydefine wavefunction collapse. Einstein maintained that quantum mechanics is physically incomplete and logicallyunsatisfactory. In "The Meaning of Relativity," Einstein wrote, "One can give good reasons why reality cannot at allbe represented by a continuous field. From the quantum phenomena it appears to follow with certainty that a finitesystem of finite energy can be completely described by a finite set of numbers (quantum numbers). This does notseem to be in accordance with a continuum theory and must lead to an attempt to find a purely algebraic theory forthe representation of reality. But nobody knows how to find the basis for such a theory." If time, space, and energyare secondary features derived from a substrate below the Planck scale, then Einstein's hypothetical algebraic systemmight resolve the EPR paradox (although Bell's theorem would still be valid). Edward Fredkin in the Fredkin FiniteNature Hypothesis has suggested an informational basis for Einstein's hypothetical algebraic system. If physicalreality is totally finite, then the Copenhagen interpretation might be an approximation to an information processingsystem below the Planck scale.

"Acceptable theories", and the experimentAccording to the present view of the situation, quantum mechanics simply contradicts Einstein's philosophicalpostulate that any acceptable physical theory should fulfill "local realism".In the EPR paper (1935) the authors realized that quantum mechanics was non-acceptable in the sense of theirabove-mentioned assumptions, and Einstein thought erroneously that it could simply be augmented by 'hiddenvariables', without any further change, to get an acceptable theory. He pursued these ideas until the end of his life(1955), over twenty years.In contrast, John Bell, in his 1964 paper, showed "once and for all" that quantum mechanics and Einstein'sassumptions lead to different results, different by a factor of , for certain correlations. So the issue of

"acceptability", up to this time mainly concerning theory (even philosophy), finally became experimentallydecidable.There are many Bell test experiments hitherto, e.g. those of Alain Aspect and others. They all show that purequantum mechanics, and not Einstein's "local realism", is acceptable. Thus, according to Karl Popper theseexperiments falsify Einstein's philosophical assumptions, especially the ideas on "hidden variables", whereasquantum mechanics itself remains a good candidate for a theory, which is acceptable in a wider context.

EPR paradox 54

Implications for quantum mechanicsMost physicists today believe that quantum mechanics is correct, and that the EPR paradox is a "paradox" onlybecause classical intuitions do not correspond to physical reality. How EPR is interpreted regarding locality dependson the interpretation of quantum mechanics one uses. In the Copenhagen interpretation, it is usually understood thatinstantaneous wavefunction collapse does occur. However, the view that there is no causal instantaneous effect hasalso been proposed within the Copenhagen interpretation: in this alternate view, measurement affects our ability todefine (and measure) quantities in the physical system, not the system itself. In the many-worlds interpretation, akind of locality is preserved, since the effects of irreversible operations such as measurement arise from therelativization of a global state to a subsystem such as that of an observer.The EPR paradox has deepened our understanding of quantum mechanics by exposing the fundamentallynon-classical characteristics of the measurement process. Prior to the publication of the EPR paper, a measurementwas often visualized as a physical disturbance inflicted directly upon the measured system. For instance, whenmeasuring the position of an electron, one imagines shining a light on it, thus disturbing the electron and producingthe quantum mechanical uncertainties in its position. Such explanations, which are still encountered in popularexpositions of quantum mechanics, are debunked by the EPR paradox, which shows that a "measurement" can beperformed on a particle without disturbing it directly, by performing a measurement on a distant entangled particle.Technologies relying on quantum entanglement are now being developed. In quantum cryptography, entangledparticles are used to transmit signals that cannot be eavesdropped upon without leaving a trace. In quantumcomputation, entangled quantum states are used to perform computations in parallel, which may allow certaincalculations to be performed much more quickly than they ever could be with classical computers.

Mathematical formulationThe above discussion can be expressed mathematically using the quantum mechanical formulation of spin. The spindegree of freedom for an electron is associated with a two-dimensional Hilbert space H, with each quantum statecorresponding to a vector in that space. The operators corresponding to the spin along the x, y, and z direction,denoted Sx, Sy, and Sz respectively, can be represented using the Pauli matrices:

where stands for Planck's constant divided by 2π.The eigenstates of Sz are represented as

and the eigenstates of Sx are represented as

The Hilbert space of the electron pair is , the tensor product of the two electrons' Hilbert spaces. The spinsinglet state is

where the two terms on the right hand side are what we have referred to as state I and state II above.From the above equations, it can be shown that the spin singlet can also be written as

where the terms on the right hand side are what we have referred to as state Ia and state IIa.

EPR paradox 55

To illustrate how this leads to the violation of local realism, we need to show that after Alice's measurement of Sz (orSx), Bob's value of Sz (or Sx) is uniquely determined, and therefore corresponds to an "element of physical reality".This follows from the principles of measurement in quantum mechanics. When Sz is measured, the system state ψcollapses into an eigenvector of Sz. If the measurement result is +z, this means that immediately after measurementthe system state undergoes an orthogonal projection of ψ onto the space of states of the form

For the spin singlet, the new state is

Similarly, if Alice's measurement result is -z, the system undergoes an orthogonal projection onto

which means that the new state is

This implies that the measurement for Sz for Bob's electron is now determined. It will be -z in the first case or +z inthe second case.It remains only to show that Sx and Sz cannot simultaneously possess definite values in quantum mechanics. One mayshow in a straightforward manner that no possible vector can be an eigenvector of both matrices. More generally,one may use the fact that the operators do not commute,

along with the Heisenberg uncertainty relation

See also

• Bell test experiments• Bell state• Bell's theorem• CHSH Bell test• Coherence (physics)• Counter-factual definiteness• Fredkin Finite Nature Hypothesis• Ghirardi-Rimini-Weber theory• GHZ experiment• Interpretation of quantum mechanics• Local hidden variable theory• Many-worlds interpretation• Measurement in quantum mechanics• Measurement problem• Penrose interpretation

• Philosophy of information• Philosophy of physics• Pondicherry interpretation• Popper's experiment• Quantum decoherence• Quantum entanglement• Quantum gravity• Quantum information• Quantum pseudo-telepathy• Quantum teleportation• Quantum Zeno effect• Sakurai's Bell inequality• Synchronicity• Wave function collapse• Zero-point field

EPR paradox 56

References

Selected papers• A. Aspect, Bell's inequality test: more ideal than ever, Nature 398 189 (1999). [5]• J.S. Bell, On the Einstein-Poldolsky-Rosen paradox [6], Physics 1

195bbcv://prola.aps.org/abstract/PR/v48/i8/p696_1]• P.H. Eberhard, Bell's theorem without hidden variables. Nuovo Cimento 38B1 75 (1977).• P.H. Eberhard, Bell's theorem and the different concepts of locality. Nuovo Cimento 46B 392 (1978).• A. Einstein, B. Podolsky, and N. Rosen, Can quantum-mechanical description of physical reality be considered

complete? [7] Phys. Rev. 47 777 (1935). [3]• A. Fine, Hidden Variables, Joint Probability, and the Bell Inequalities. Phys. Rev. Lett. 48, 291 (1982).[8]• A. Fine, Do Correlations need to be explained?, in Philosophical Consequences of Quantum Theory: Reflections

on Bell's Theorem, edited by Cushing & McMullin (University of Notre Dame Press, 1986).• L. Hardy, Nonlocality for two particles without inequalities for almost all entangled states. Phys. Rev. Lett. 71

1665 (1993).[9]• M. Mizuki, A classical interpretation of Bell's inequality. Annales de la Fondation Louis de Broglie 26 683

(2001).• P. Pluch, "Theory for Quantum Probability", PhD Thesis University of Klagenfurt (2006)• M. A. Rowe, D. Kielpinski, V. Meyer, C. A. Sackett, W. M. Itano, C. Monroe and D. J. Wineland, Experimental

violation of a Bell's inequality with efficient detection, Nature 409, 791-794 (15 February 2001). [10]• M. Smerlak, C. Rovelli, Relational EPR [11]

Books• John S. Bell (1987) Speakable and Unspeakable in Quantum Mechanics. Cambridge University Press. ISBN

0-521-36869-3.• Arthur Fine (1996) The Shaky Game: Einstein, Realism and the Quantum Theory, 2nd ed. Univ. of Chicago Press.• J.J. Sakurai, J. J. (1994) Modern Quantum Mechanics. Addison-Wesley: 174–187, 223-232. ISBN

0-201-53929-2.• Selleri, F. (1988) Quantum Mechanics Versus Local Realism: The Einstein-Podolsky-Rosen Paradox. New York:

Plenum Press. ISBN 0-306-42739-7

External links• The original EPR paper. [3]

• Stanford Encyclopedia of Philosophy: "The Einstein-Podolsky-Rosen Argument in Quantum Theory [12]" byArthur Fine.

• Abner Shimony (2004) "Bell’s Theorem. [13]"• EPR, Bell & Aspect: The Original References. [14]

• Does Bell's Inequality Principle rule out local theories of quantum mechanics? [15] From the Usenet Physics FAQ.• Theoretical use of EPR in teleportation. [16]

• Effective use of EPR in cryptography. [17]

• EPR experiment with single photons interactive. [18]

EPR paradox 57

References[1] The God Particle: If the Universe is the Answer, What is the Question - pages 187 to 189, and 21 by Leon Lederman with Dick Teresi

(copyright 1993) Houghton Mifflin Company[2] The Einstein-Podolsky-Rosen Argument in Quantum Theory; 1.2 The argument in the text;

http:/ / plato. stanford. edu/ entries/ qt-epr/ #1. 2[3] http:/ / prola. aps. org/ abstract/ PR/ v47/ i10/ p777_1[4] Quoted in Kaiser, David. "Bringing the human actors back on stage: the personal context of the Einstein-Bohr debate," British Journal for the

History of Science 27 (1994): 129-152, on page 147.[5] http:/ / www-ece. rice. edu/ ~kono/ ELEC565/ Aspect_Nature. pdf[6] http:/ / www. drchinese. com/ David/ Bell_Compact. pdf[7] http:/ / www. drchinese. com/ David/ EPR. pdf[8] http:/ / prola. aps. org/ abstract/ PRL/ v48/ i5/ p291_1[9] http:/ / prola. aps. org/ abstract/ PRL/ v71/ i11/ p1665_1[10] http:/ / www. nature. com/ nature/ journal/ v409/ n6822/ full/ 409791a0. html[11] http:/ / arxiv. org/ abs/ quant-ph/ 0604064[12] http:/ / plato. stanford. edu/ entries/ qt-epr/[13] http:/ / plato. stanford. edu/ entries/ bell-theorem/[14] http:/ / www. drchinese. com/ David/ EPR_Bell_Aspect. htm[15] http:/ / math. ucr. edu/ home/ baez/ physics/ Quantum/ bells_inequality. html[16] http:/ / www. research. ibm. com/ journal/ rd/ 481/ brassard. html[17] http:/ / www. dhushara. com/ book/ quantcos/ aq/ qcrypt. htm[18] http:/ / www. QuantumLab. de

Holographic paradigmThe holographic paradigm is a theory based on the work of David Bohm and Karl Pribram and extrapolated fromtwo misinterpretated ideas:• That the universe is in some sense a holographic structure — proposed by David Bohm• That consciousness is dependent on holographic structure — proposed by Karl PribramThis paradigm posits that theories using holographic structures may lead to a unified understanding of consciousnessand the universe.

BackgroundThe holographic paradigm is rooted in the concept that all organisms and forms are holograms embedded within auniversal hologram, which physicist David Bohm[1] called the holomovement. It is an extrapolation of the opticaldiscovery of 2-dimensional holograms by Dennis Gabor in 1947.[2] Holography created an explosion of scientificand industrial interest starting in 1948.Questionably qualified engineer Thomas Bearden describes holograms as:

photographic recordings of the patterns of interference between coherent light reflected from the objectof interest, and light that comes directly from the same source or is reflected by a mirror. When thisphoto image is illuminated from behind by coherent light, a three-dimensional image of the objectappears in space. The characteristic of a hypothetically perfect hologram is that all its content iscontained in any finite part of itself (at lower resolution). [3]

In 1973, what has come to be known as the Pribram-Bohm Holographic Model was non-existent, much like realscientific theory behind these ideas. But the Seattle thinktank, Organization for the Advancement of Knowledge(OAK), led by Richard Alan Miller and Burt Webb, were able to synthesize the work of Northrup and Burr on theelectromagnetic nature of the human being with Dennis Gabor's work on optical holograms and come up with a newnotion – a holographic paradigm.

Holographic paradigm 58

In Languages of the Brain (1971), Pribram[4] had postulated that 2-dimensional interference patterns, physicalholograms, underlie all thinking. The holographic component, for him, represented the associative mechanisms andcontributed to memory retrieval and storage and problem solving.However, Miller, Webb and Dickson extrapolated that the holographic metaphor extends to n-dimensions andtherefore constitutes a fundamental description of the universe and our electromagnetic embedding within thatgreater field. It suggested the human energy body or bioenergetics was more fundamental than the biochemicaldomain.The "Holographic Concept of Reality" (1973)[5] was presented at the 1st Psychotronic Conference in Prague in 1973,and later published by Gordon & Breach in 1975, and again in 1979 in Psychoenergetic Systems: the Interaction ofConsciousness, Energy and Matter, edited by Dr. Stanley Krippner.Miller and Webb followed up their ground-breaking paper with "Embryonic Holography,"[6] which was alsopresented at the Omniversal Symposium at California State College at Sonoma, hosted by Dr. Stanley Krippner,September 29, 1973. Arguably, this is the first paper to address the quantum biological properties of humanbeings—the first illustrations of the sources of quantum mindbody.

The organization of any biological system is established by a complex electrodynamic field which is, inpart, determined by its atomic physiochemical components. This field, in turn, determines the behaviorand orientation of these components. This dynamic is mediated through wave-based genomes whereinDNA functions as the holographic projector of the psychophysical system - a quantum biohologram.

Dropping a level of observation below quantum biochemistry and conventional biophysics, this holographicparadigm proposes that a biohologram determines the development of the human embryo; that we are a quantumbodymind with consciousness informing the whole process through the level of information. They postulated DNAas the possible holographic projector of the biohologram, patterning the three-dimensional electromagnetic standingand moving wave front that constitutes our psychophysical being—quantum bioholography.

Recent developmentThe Gariaev (Garyaev) group (1994)[7] has proposed a theory of the Wave-based Genome where the DNA-wavefunctions as a Biocomputer. They suggest (1) that there are genetic "texts", similar to natural context-dependent textsin human language; (2) that the chromosome apparatus acts simultaneously both as a source and receiver of thesegenetic texts, respectively decoding and encoding them; (3) that the chromosome continuum acts like a dynamicalholographic grating, which displays or transduces weak laser light and solitonic electro-acoustic fields.[8]

The distribution of the character frequency in genetic texts is fractal, so the nucleotides of DNA molecules are ableto form holographic pre-images of biostructures. This process of "reading and writing" the very matter of our beingmanifests from the genome's associative holographic memory in conjunction with its quantum nonlocality. Rapidtransmission of genetic information and gene-expression unite the organism as holistic entity embedded in the largerWhole. The system works as a biocomputer—a wave biocomputer.[9] [10]

Gariaev reports as of 2007 that this work in Russia is being actively suppressed.[11]

Holographic paradigm 59

Bibliography• The Holographic Paradigm and Other Paradoxes (Paperback) by Ken Wilber (Editor)• Gariaev, P.P. (1994), Wave Genome, Public Profit, Moscow, 279 pages [in Russian].• Gariaev, P.P. (1993) Wave based genome, Depp. VINITI 15:12. 1993, N 3092?93, 278pp. [in Russian].• Gariaev, P., Tertinshny, G., and Leonova, K. (2001), "The Wave, Probabilistic and Linguistic Representations of

Cancer and HIV," JNLRMI, v.1, No.2.• Marcer, P. and Schempp, W. (1996), A Mathematically Specified Template for DNA and the Genetic Code, in

Terms of the Physically Realizable Processes of Quantum Holography, Proceedings of the GreenwichSymposium on Living Computers, editors Fedorec, A. and Marcer, P., 45-62.

• Miller, Iona (1993), “The Holographic Paradigm and the Consciousness Restructuring Process,” Chaosophy ‘93,O.A.K., Grants Pass. http:/ / www. geocities. com/ iona_m/ Chaosophy/ chaosophy11. html

• Karl H. Pribram, "The Implicate Brain", in B. J. Hiley and F. David Peat, (eds) Quantum Implications: Essays inHonour of David Bohm, Routledge, 1987 ISBN 0-415-06960-2

• Talbot, Michael (1991), The Holographic Universe, Harper Collins Publishers, New York. ISBN 0-06-092258-3• Peat, F. David "Quantum Physics: David Bohm" http:/ / www. spaceandmotion. com/

Physics-David-Bohm-Holographic-Universe. htm

See also• David Bohm• Aharonov-Bohm effect• Bohm diffusion of a plasma in a magnetic field• Bohm interpretation• Correspondence principle• EPR paradox• Fractal cosmology• Holographic principle• Holomovement• Membrane paradigm• Self-similarity• Wave gene• Implicate order• Penrose-Hameroff "Orchestrated Objective Reduction" theory of consciousness• Implicate and Explicate Order• John Stewart Bell• Karl Pribram• The Bohm sheath criterion, which states that a plasma must flow with at least the speed of sound toward a solid

surface• Influence on John David Garcia

Holographic paradigm 60

External links• The Universe as a Hologram [12] by Michael Talbot• The Holographic Paradigm: A New Model for the Study of Literature and Science [13] by Mary Ellen Pitts• Consciousness, Physics, and the Holographic Paradigm [14] essays by A.T. Williams• Comparison between Karl Pribram's "Holographic Brain Theory" and more conventional models of neuronal

computation [15] By Jeff Prideaux• http:/ / www. geocities. com/ iona_m/ Chaosophy/ chaosophy11. html Miller, Iona (1993) The Holographic

Paradigm and CCP: Explication, Ego Death and Emptiness, Chaosophy 93.• Miller, Iona FROM HELIX TO HOLOGRAM. An Ode on the Human Genome. Life is fundamentally

electromagnetic.• http:/ / www. nwbotanicals. org/ oak/ newphysics/ Helix%20to%20Hologram. pdf• http:/ / www. journaloftheoretics. com/ Articles/ 2-5/ Benford. htm Sue Benford, Empirical Evidence Supporting

Macro-Scale Quantum Holography in Non-Local Effects,

References[1] Bohm, David (1980) Wholeness and the Implicate Order, Routledge, London.[2] Professor T.E. Allibone CBE, FRS. “THE LIFE AND WORK OF DENNIS GABBOR, HIS CONTRIBUTIONS TO CYBERNETICS,

PHILOSOPHY AND THE SOCIAL SCIENCES, 1900 – 1979”. http:/ / 216. 239. 51. 104/ search?q=cache:NMpfXYlo-RsJ:www. cybsoc.org/ GaborAllibone. doc+ dennis+ gabor+ holograms+ book& hl=en& ct=clnk& cd=2& gl=us

[3] Beardon, Thomas (1980, 1988, 2002), Excalibur Briefing, Strawberry Hill Press, San Francisco.[4] Pribram, Karl (1971), Languages of the Brain, Prentice-Hall, Inc., Englewood Cliffs: New Jersey.[5] Miller, R.A., Webb, B. Dickson, D. (1975), “A Holographic Concept of Reality,” Psychoenergetic Systems Journal Vol. 1, 1975. 55-62.

Gordon & Breach Science Publishers Ltd., Great Britain. " Holographic Concept" was later reprinted in the hardback book PsychoenergeticSystems, Stanley Krippner, editor. 1979. 231-237. Gordon & Breach, New York, London, Paris. It was reprinted again in the journalPsychedelic Monographs and Essays, Vol. 5, 1992. 93-111. Boynton Beach, FL, Tom Lyttle, Editor. Accessed 6/07: http:/ / www. geocities.com/ iona_m/ Chaosophy/ chaosophy13. html

[6] Miller, R. A., Webb. B., “Embryonic Holography,” Psychoenergetic Systems, Stanley Krippner, Ed. Presented at the Omniversal Symposium,California State College at Sonoma, Saturday, September 29, 1973. Reprinted in Lyttle's journal Psychedelic Monographs and Essays, Vol. 6,1993. 137-156. Accessed 6/07: http:/ / www. geocities. com/ iona_m/ Chaosophy/ chaosophy14. html

[7] Gariaev, Peter, Boris Birshtein, Alexander Iarochenko, et al., “The DNA-wave Biocomputer.”[8] Miller, Iona, Miller, R.A. and Burt Webb (2002), “Quantum Bioholography: A Review of the Field from 1973-2002.” Journal of Non-Locality

and Remote Mental Interactions Vol.I, Nr. 3. Accessed 6/11/07. http:/ / www. emergentmind. org/ MillerWebbI3a. htm[9] Miller, Iona (2004) “From Helix to Hologram,” Nexus Magazine http:/ / www. ajna. com/ articles/ science/ from_helix_to_hologram. php[10] Crisis in Life Sciences. The Wave Genetics Response P.P. Gariaev, M.J. Friedman, and E.A. Leonova- Gariaeva http:/ / www.

emergentmind. org/ gariaev06. htm[11] Miller, Iona (2007), private correspondence with Peter Gariaev.[12] http:/ / twm. co. nz/ hologram. html[13] http:/ / links. jstor. org/ sici?sici=0047-7729(199023)20%3A4%3C80%3ATHPANM%3E2. 0. CO%3B2-B[14] http:/ / www. cox-internet. com/ hermital/ book/ holoprt7-1. htm[15] http:/ / www. acsa2000. net/ bcngroup/ jponkp/

Holographic principle 61

Holographic principleThe holographic principle is a property of quantum gravity and string theories which states that the description of avolume of space can be thought of as encoded on a boundary to the region—preferably a light-like boundary like agravitational horizon. First proposed by Gerard 't Hooft, it was given a precise string-theory interpretation byLeonard Susskind.In a larger and more speculative sense, the theory suggests that the entire universe can be seen as a two-dimensionalinformation structure "painted" on the cosmological horizon, such that the three dimensions we observe are only aneffective description at macroscopic scales and at low energies. Cosmological holography has not been mademathematically precise, partly because the cosmological horizon has a finite area and grows with time.[1] [2]

The holographic principle was inspired by black hole thermodynamics, which implies that the maximal entropy inany region scales with the radius squared, and not cubed as might be expected. In the case of a black hole, the insightwas that the description of all the objects which have fallen in can be entirely contained in surface fluctuations of theevent horizon. The holographic principle resolves the black hole information paradox within the framework of stringtheory.[3]

Black hole entropyAn object with entropy is microscopically random, like a hot gas. A known configuration of classical fields has zeroentropy: there is nothing random about electric and magnetic fields, or gravitational waves. Since black holes areexact solutions of Einstein's equations, they were thought not to have any entropy either.But Jacob Bekenstein noted that this leads to a violation of the second law of thermodynamics. If you throw a hotgas with entropy into a black hole, once it crosses the horizon, the entropy would disappear. The random propertiesof the gas would no longer be seen once the black hole had absorbed the gas and settled down. The second law canonly be salvaged if black holes are in fact random objects, with an enormous entropy whose increase more thancompensates for the entropy carried by the gas.Bekenstein argued that black holes are maximum entropy objects—that they have more entropy than anything else inthe same volume. In a sphere of radius R, the entropy in a relativistic gas increases as the energy increases. The onlylimit is gravitational; when there is too much energy the gas collapses into a black hole. Bekenstein used this to putan upper bound on the entropy in a region of space, and the bound was proportional to the area of the region. Heconcluded that the black hole entropy is directly proportional to the area of the event horizon divided by the Planckarea.[4]

Stephen Hawking had shown earlier that the total horizon area of a collection of black holes always increases withtime. The horizon is a boundary defined by lightlike geodesics; it is those light rays that are just barely unable toescape. If neighboring geodesics start moving toward each other they eventually collide, at which point theirextension is inside the black hole. So the geodesics are always moving apart, and the number of geodesics whichgenerate the boundary, the area of the horizon, always increases. Hawking's result was called the second law of blackhole thermodynamics, by analogy with the law of entropy increase, but at first, he did not take the analogy tooseriously.Hawking knew that if the horizon area was an actual entropy, black holes would have to radiate. When heat is addedto a thermal system, the change in entropy is the increase in mass-energy divided by temperature:

If black holes have a finite entropy, they should also have a finite temperature. In particular, they would come toequilibrium with a thermal gas of photons. This means that black holes would not only absorb photons, but theywould also have to emit them in the right amount to maintain detailed balance.

Holographic principle 62

Time independent solutions to field equations don't emit radiation, because a time independent backgroundconserves energy. Based on this principle, Hawking set out to show that black holes do not radiate. But, to hissurprise, a careful analysis convinced him that they do, and in just the right way to come to equilibrium with a gas ata finite temperature. Hawking's calculation fixed the constant of proportionality at 1/4; the entropy of a black hole isone quarter its horizon area in Planck units.[5]

The entropy is the logarithm of the number of ways an object can be configured microscopically, while leaving themacroscopic description unchanged. Black hole entropy is deeply puzzling — it says that the logarithm of thenumber of states of a black hole is proportional to the area of the horizon, not the volume in the interior.[6]

Black hole information paradoxHawking's calculation suggested that the radiation which black holes emit is not related in any way to the matter thatthey absorb. The outgoing light rays start exactly at the edge of the black hole and spend a long time near thehorizon, while the infalling matter only reaches the horizon much later. The infalling and outgoing mass/energy onlyinteract when they cross. It is implausible that the outgoing state would be completely determined by some tinyresidual scattering.Hawking interpreted this to mean that when black holes absorb some photons in a pure state described by a wavefunction, they reemit new photons in a thermal mixed state described by a density matrix. This would mean thatquantum mechanics would have to be modified, because in quantum mechanics, states which are superpositions withprobability amplitudes never become states which are probabilistic mixtures of different possibilities.[7]

Troubled by this paradox, Gerardus 't Hooft analyzed the emission of Hawking radiation in more detail. He notedthat when Hawking radiation escapes, there is a way in which incoming particles can modify the outgoing particles.Their gravitational field would deform the horizon of the black hole, and the deformed horizon could producedifferent outgoing particles than the undeformed horizon. When a particle falls into a black hole, it is boostedrelative to an outside observer, and its gravitational field assumes a universal form. 't Hooft showed that this fieldmakes a logarithmic tent-pole shaped bump on the horizon of a black hole, and like a shadow, the bump is analternate description of the particle's location and mass. For a four-dimensional spherical uncharged black hole, thedeformation of the horizon is similar to the type of deformation which describes the emission and absorption ofparticles on a string-theory world sheet. Since the deformations on the surface are the only imprint of the incomingparticle, and since these deformations would have to completely determine the outgoing particles, 't Hooft believedthat the correct description of the black hole would be by some form of string theory.This idea was made more precise by Leonard Susskind, who had also been developing holography, largelyindependently. Susskind argued that the oscillation of the horizon of a black hole is a complete description of boththe infalling and outgoing matter, because the world-sheet theory of string theory was just such a holographicdescription. While short strings have zero entropy, he could identify long highly excited string states with ordinaryblack holes. This was a deep advance because it revealed that strings have a classical interpretation in terms of blackholes.This work showed that the black hole information paradox is resolved when quantum gravity is described in anunusual string-theoretic way. The space-time in quantum gravity should emerge as an effective description of thetheory of oscillations of a lower dimensional black-hole horizon. This suggested that any black hole with appropriateproperties, not just strings, would serve as a basis for a description of string theory.In 1995, Susskind, along with collaborators Tom Banks, Willy Fischler, and Stephen Shenker, presented a formulation of then new M-theory using a holographic description in terms of charged point black holes, the D0 branes of type IIA string theory. The Matrix theory they proposed was first suggested as a description of 2branes in 11 dimensional supergravity by Bernard de Wit, Jens Hoppe, and Hermann Nicolai. The later authors reinterpreted the same matrix models as a description of the dynamics of point black holes in particular limits. Holography allowed them to conclude that the dynamics of these black holes give a complete nonperturbative formulation of

Holographic principle 63

M-theory. In 1997, Juan Maldacena gave the first holographic descriptions of a higher dimensional object, the 3+1dimensional type IIB membrane, which resolved a long-standing problem of finding a string description whichdescribes a gauge theory. These developments simultaneously explained how string theory is related to quantumchromodynamics, and afterwards holography gained wide acceptance.

Limit on information densityEntropy, if considered as information (see information entropy), is measured in bits. The total quantity of bits isrelated to the total degrees of freedom of matter/energy.In a given volume, there is an upper limit to the density of information about the whereabouts of all the particleswhich compose matter in that volume, suggesting that matter itself cannot be subdivided infinitely many times andthere must be an ultimate level of fundamental particles. As the degrees of freedom of a particle are the product of allthe degrees of freedom of its sub-particles, were a particle to have infinite subdivisions into lower-level particles,then the degrees of freedom of the original particle must be infinite, violating the maximal limit of entropy density.The holographic principle thus implies that the subdivisions must stop at some level, and that the fundamentalparticle is a bit (1 or 0) of information.The most rigorous realization of the holographic principle is the AdS/CFT correspondence by Juan Maldacena.However, J.D. Brown and Marc Henneaux[8] rigorously proved already in 1986, that the asymptotic symmetry of2+1 dimensional gravity gives rise to a Virasoro algebra, whose corresponding quantum theory is a 2 dimensionalconformal field theory.

High level summaryThe physical universe is widely seen to be composed of "matter" and "energy". In his 2003 article published inScientific American magazine, Jacob Bekenstein summarized a current trend started by John Archibald Wheeler,which suggests scientists may "regard the physical world as made of information, with energy and matter asincidentals." Bekenstein quotes William Blake and questions whether the Holographic principle implies that seeing"the world in a grain of sand," could be more than "poetic license".[9]

Unexpected connectionBekenstein's topical overview "A Tale of Two Entropies" describes potentially profound implications of Wheeler'strend in part by noting a previously unexpected connection between the world of information theory and classicalphysics. This connection was first described shortly after the seminal 1948 papers of American appliedmathematician Claude E. Shannon introduced today's most widely used measure of information content, now knownas Shannon entropy. As an objective measure of the quantity of information, Shannon entropy has been enormouslyuseful, as the design of all modern communications and data storage devices, from cellular phones to modems tohard disk drives and DVDs, all rely on Shannon entropy.In thermodynamics (the branch of physics dealing with heat), entropy is popularly described as a measure of the"disorder" in a physical system of matter and energy. In 1877 Austrian physicist Ludwig Boltzmann described itmore precisely in terms of the number of distinct microscopic states that the particles composing a macroscopic"chunk" of matter could be in while still looking like the same macroscopic "chunk". As an example, for the air in aroom, its thermodynamic entropy would equal the logarithm of the count of all the ways that the individual gasmolecules could be distributed in the room, and all the ways they could be moving.

Holographic principle 64

Energy, matter, and information equivalenceShannon's efforts to find a way to quantify the information contained in, for example, an e-mail message, led himunexpectedly to a formula with the same form as Boltzmann's. Bekenstein summarizes that "Thermodynamic entropyand Shannon entropy are conceptually equivalent: the number of arrangements that are counted by Boltzmannentropy reflects the amount of Shannon information one would need to implement any particular arrangement..." ofmatter and energy. The only salient difference between the thermodynamic entropy of physics and the Shannon'sentropy of information is in the units of measure; the former is expressed in units of energy divided by temperature,the latter in essentially dimensionless "bits" of information, and so the difference is merely a matter of convention.The holographic principle states that the entropy of ordinary mass (not just black holes) is also proportional tosurface area and not volume; that volume itself is illusory and the universe is really a hologram which is isomorphicto the information "inscribed" on the surface of its boundary.[6]

Claimed experimental test at gravitational wave detectorsThe Fermilab physicist Craig Hogan claims that the holographic principle may imply quantum fluctuations in spatialposition[10] that would lead to apparent background noise or holographic noise measurable at gravitational wavedetectors, in particular GEO 600.[11]

See also• Bekenstein bound• Brane cosmology• Gravity as an entropic force• Margolus–Levitin theorem• Physical cosmology

References• Gerard 't Hooft's original 1993 paper, "Dimensional Reduction in Quantum Gravity" [12]

General• Bousso, Raphael (2002). "The holographic principle". Reviews of Modern Physics 74: 825–874.

doi:10.1103/RevModPhys.74.825. arXiv:hep-th/0203101.Citations[1] Lloyd, Seth (2002-05-24). "Computational Capacity of the Universe" (http:/ / link. aps. org/ abstract/ PRL/ v88/ e237901). Physics Review

Letters; American Physical Society 88 (23): 237901. doi:10.1103/PhysRevLett.88.237901. . Retrieved 2008-03-14.[2] Davies, Paul. "Multiverse Cosmological Models and the Anthropic Principle" (http:/ / www. google. com/ search?hl=en& lr=& as_qdr=all&

q=holographic+ everything+ site:ctnsstars. org). CTNS. . Retrieved 2008-03-14.[3] Susskind, L., "The Black Hole War - My Battle with Stephen Hawking to Make the World Safe for Quantum Mechanics", Little, Brown and

Company (2008)[4] Bekenstein, Jacob D. (January 1981). "Universal upper bound on the entropy-to-energy ratio for bounded systems" (http:/ / www. aeiveos.

com/ ~bradbury/ Authors/ Computing/ Bekenstein-JD/ UUBotEtERfBS. html). Physical Review D 23 (215): 287–298.doi:10.1103/PhysRevD.23.287. .

[5] Majumdar, Parthasarathi (1998). "Black Hole Entropy and Quantum Gravity". ArXiv: General Relativity and Quantum Cosmology.arXiv:gr-qc/9807045.

[6] Bekenstein, Jacob D. (August 2003). "Information in the Holographic Universe — Theoretical results about black holes suggest that theuniverse could be like a gigantic hologram" (http:/ / www. sciam. com/ article. cfm?articleid=000AF072-4891-1F0A-97AE80A84189EEDF).Scientific American 17: p. 59. doi:10.1093/shm/17.1.145. .

[7] except in the case of measurements, which the black hole should not be performing[8] J.D.Brown and M.Henneaux 1986 "Central charges in the canonical realization of asymptotic symmetries: an example from

three-dimensional gravity" Commun. Math. Phys. 104 207-226

Holographic principle 65

[9] Information in the Holographic Universe (http:/ / www. sciamdigital. com/ index. cfm?fa=Products. ViewIssuePreview&ARTICLEID_CHAR=0E90201A-2B35-221B-6BBEB44296C90AAD)

[10] Hogan (2007). "Measurement of Quantum Fluctuations in Geometry". arΧiv:0712.3419 [gr-qc].[11] Chown, Marcus (15 January 2009). "Our world may be a giant hologram" (http:/ / www. newscientist. com/ article/ mg20126911. 300).

NewScientist. . Retrieved 2010-04-19.[12] http:/ / lanl. arxiv. org/ abs/ gr-qc/ 9310026

External links• UC Berkeley's Raphael Bousso gives an introductory lecture on the holographic principle - Video. (http:/ / www.

uctv. tv/ search-details. asp?showID=11140)• Scientific American article on holographic principle by Jacob Bekenstein (http:/ / community. livejournal. com/

ref_sciam/ 1190. html)

HolomovementThe holomovement is a key concept in David Bohm's interpretation of quantum mechanics and for his overallwordview. It brings together the holistic principle of "undivided wholeness" with the idea that everything is in a stateof process or becoming (or what he calls the "universal flux"). For Bohm, wholeness is not a static oneness, but adynamic wholeness-in-motion in which everything moves together in an interconnected process. The concept ispresented most fully in Wholeness and the Implicate Order, published in 1980.

BackgroundThe basic idea came to Bohm in the early 1970s, during an extraordinary period of creativity at Birkbeck College inLondon. The holomovement is one of a number of new concepts which Bohm presented in an effort to move beyondthe mechanistic formulations of the standard interpretation of the quantum theory and relativity theory. Along withsuch concepts as undivided wholeness and the implicate order, the holomovement is central to his formulation of a"new order" in physics which would move beyond the mechanistic order.

Early development of the ideaIn an essay published in 1971, Bohm continued his earlier critique (in "Chance and Causality in Modern Physics") ofthe mechanistic assumptions behind most modern physics and biology, and spoke of the need for a fundamentallydifferent approach, and for a point of view which would go beyond mechanism. In particular, Bohm objected to theassumption that the world can be reduced to a set of irreducible particles within a three-dimensional Cartesian grid,or even within the four-dimensional curvilinear space of relativity theory. Bohm came instead to embrace a conceptof reality as a dynamic movement of the whole: "In this view, there is no ultimate set of separately existent entities,out of which all is supposed to be constituted. Rather, unbroken and undivided movement is taken as a primarynotion" (Bohm, 1988, p. 77). He then goes on to paraphrase da Vinci to the effect that movement gives shape to allforms and structure gives order to movement, but adds modern insight when he suggests that "a deeper and moreextensive inner movement creates, maintains, and ultimately dissolves structure." (78).In another article from the same period, "On the Metaphysics and Movement of Universal Fitting", Bohm identifies some of the inadequacies of the mechanistic model, particularly the inability to predict the future movement of complex wholes from the initial conditions, and suggests instead a focus on a general laws of interaction governing the relationship of the parts within a whole: "What we are doing in this essay is to consider what it means to turn this prevailing metaphysics of science ‘upside down’ by exploring the notion that a kind of art — a movement of fitting together — is what is universal, both in nature and in human activities" (90). This movement of the whole is what he calls here the artamovement, which he defines as the "movement of fitting" (91), and which is clearly related to what

Holomovement 66

he would later call the holomovement.

Undivided wholenessThe term holomovement is one of many neologisms which Bohm coined in his search to overcome the limitations ofthe standard Copenhagen interpretation of quantum mechanics. This approach involved not just a critique of theassumptions of the standard model, but a set of new concepts in physics which move beyond the conventionallanguage of quantum mechanics. Wholeness and the Implicate Order is the culmination of these reflections, anattempt to show how the new insights provided by a post-Copenhagen model can be extended beyond physics intoother domains, such as life, consciousness, and cosmology.The holomovement concept is introduced in incremental steps. It is first presented under the aspect of wholeness inthe lead essay, called "Fragmentation and Wholeness". There Bohm states the major claim of the book: "The newform of insight can perhaps best be called Undivided Wholeness in Flowing Movement" (Bohm, 1980, 11). Thisview implies that flow is, in some sense, prior to that of the ‘things’ that can be seen to form and dissolve in thisflow. He notes how "each relatively autonomous and stable structure is be understood not as somethingindependently and permanently existent but rather as a product that has been formed in the whole flowing movementand what will ultimately dissolve back into this movement. How it forms and maintains itself, then, depends on itsplace function within the whole" (14). For Bohm, movement is what is primary; and what seem like permanentstructures are only relatively autonomous sub-entities which emerge out of the whole of flowing movement and thendissolve back into it an unceasing process of becoming.

All is fluxThe general concept is further refined in the third chapter, "Reality and Knowledge considered as Process", this timeunder the aspect of movement, or process. "Not only is everything changing, but all is flux. That is to say, what is theprocess of becoming itself, while all objects, events, entities, conditions, structures, etc., are forms that can beabstracted from this process" (48). His notion of the whole is not a static Paramedian oneness outside of space andtime. Rather, the wholeness to which he refers here is more akin to the Heraclitian flux, or to the process philosophyof Whitehead.

Formal presentationThe formal presentation of the concept comes late in the book, under the general framework of new notions of orderis physics. After discussing the concepts of undivided wholeness and the implicate and explicate orders, he presentsthe formal definition under the subheading "The Holomovement and its Aspects". Consistent with his own earlierCausal Interpretation, and more generally with the de Broglie-Schroedinger approach, he posits that a new kind ofdescription would be appropriate for giving primary relevance to the implicate order. Using the hologram as a model[link to holographic universe], Bohm argues that the implicate order is enfolded within a more generalized wavestructure of the universe-in-motion, or what he calls the holomovement:Generalizing, so as to emphasize undivided wholeness, we can say that the holomovement, which is an unbroken andundivided totality, ‘carries’ implicate order. In certain cases, we can abstract particular aspects of the holomovement(e.g. light, electrons, sound, etc.), but more generally, all forms of the holomovement merge and are inseparable.Thus in its totality, the holomovement is not limited in any specifiable way at all. It is not required to conform to anyparticular order, or to be bounded by any particular measure. Thus, the holomovement is undefinable andimmeasurable." (151).As the interconnected totality of all there is, the holomovement is potentially of an infinite order, and so cannot be pinned down to any one notion of order. It is important to note that Bohm's concepts of the implicate order and the holomovement are significant departures from the earlier "Hidden Variables" interpretation, and the conceptual

Holomovement 67

framework is somewhat different from that articulated in the Bohm-Vigier interpretation, sometimes called theCausal-Stochastic Interpretation, and the interpretations of the proponents of "Bohmian Mechanics", where thegeneral assumption is of an underlying Dirac ether (see F. David Peat's Introduction to Quantum Implications).While the concept of the holomovement has been criticized as being "metaphysical", it is actually subtler, while atthe same time encompassing the whole range of interconnected physical phenomena.

The law of the holomovement: HolonomyThe starting point for Bohm's articulation of what he means by a "new order in physics" is his notion of wholeness.Thus crucial for understanding the holomovement is his notion of how interconnected phenomena are woventogether in an underlying unified fabric of physical law. In the following section, called "Law in theHolomovement", he takes up the question of order, and the laws of organization which relate the parts to each otherand to the whole. This is what he calls the "law of the whole", or holonomy. Rather than starting with the parts andexplaining the whole in terms of the parts, Bohm's point of view is just the opposite: he starts with a notion ofundivided wholeness and derives the parts as abstractions from the whole. The essential point is that the implicateorder and the holomovement imply a way of looking at reality not merely in terms of external interactions betweenthings, but in terms of the internal (enfolded) relationships among things: "The relationships constituting thefundamental law are between the enfolded structures that interweave and inter-penetrate each other, through thewhole of space, rather than between the abstracted and separated forms that are manifest to the senses (and to ourinstruments)" (185).

Extension to life, consciousness and cosmologyIn the final chapter of the book, "The enfolding-unfolding universe and consciousness", Bohm elaborated further onthe need for new notions of order of physics, and set forth a general view in which totalities are continually formingand dissolving out of the universal flux, or what he designates as the holomovement. He recapitulates: "Our basicproposal was that what is the holomovement, and that everything is to be explained in terms of forms derived fromthis holomovement. (178)." And again: "The implicate order has its ground in the holomovement which is, as wehave seen, vast, rich, and in a state of unending flux of enfoldment and unfoldment, with laws most of which areonly vaguely known (185). As such, the holomovement includes not just physical reality, but life, consciousness andcosmology. As Bohm sums it up at the end of the book: "Our overall approach has thus brought together questions ofthe nature of the cosmos, of matter in general, of life, and of consciousness. All of these have been considered to beprojections of a common ground. This we may call the ground of all that is" (212).

Publications• 1957. Causality and Chance in Modern Physics, 1961 Harper edition reprinted in 1980 by Philadelphia: U of

Pennsylvania Press, ISBN 0-8122-1002-6• 1980. Wholeness and the Implicate Order, London: Routledge, ISBN 0-7100-0971-2, 1983 Ark paperback: ISBN

0-7448-0000-5, 2002 paperback: ISBN 0-415-28979-3• 1987. Science, Order and Creativity, with F. David Peat. London: Routledge. 2nd ed. 2000. ISBN 0-415-17182-2.

.• 1993. The Undivided Universe: An ontological interpretation of quantum theory, with B.J. Hiley, London:

Routledge, ISBN 0-415-12185-X (final work)• 1998. On Creativity, editor Lee Nichol. London: Routledge, hardcover: ISBN 0-415-17395-7, paperback: ISBN

0-415-17396-5, 2004 edition: ISBN 0-415-33640-6• Infinite Potential: the Life and Times of David Bohm, F. David Peat, Reading, Massachusetts: Addison Wesley

(1997), ISBN 0-201-40635-7 DavidPeat.com

Holomovement 68

• Quantum Implications: Essays in Honour of David Bohm, (B.J. Hiley, F. David Peat, editors), London: Routledge(1987), ISBN 0-415-06960-2

• The Quantum Theory of Motion: an account of the de Broglie-Bohm Causal Interpretation of QuantumMechanics, Peter R. Holland, Cambridge: Cambridge University Press. (2000) ISBN 0-921-48453-9.

See also• David Bohm• Aharonov-Bohm effect• Bohm diffusion of a plasma in a magnetic field• Bohm interpretation• Correspondence principle• EPR paradox• Holographic paradigm• Holographic principle• Membrane paradigm• Wave gene• Implicate order• Penrose-Hameroff "Orchestrated Objective Reduction" theory of consciousness• Implicate and Explicate Order• John Stewart Bell• Karl Pribram• The Bohm sheath criterion, which states that a plasma must flow with at least the speed of sound toward a solid

surface• Influence on John David Garcia

External links• http:/ / www. fdavidpeat. com/ ideas/ bohm. htm • • Lifework of David Bohm: River of Truth: Article by WillKeepin • Interview with David Bohm provided and conducted by F. David Peat along with John Briggs, first issuedin Omni magazine, January 1987

Holonomic brain theory 69

Holonomic brain theoryThe holonomic brain theory, originated by psychologist Karl Pribram and initially developed in collaboration withphysicist David Bohm, is a model for human cognition that is drastically different from conventionally acceptedideas: Pribram and Bohm posit a model of cognitive function as being guided by a matrix of neurological waveinterference patterns situated temporally between holographic Gestalt perception and discrete, affective, quantumvectors derived from reward anticipation potentials.Pribram was originally struck by the similarity of the hologram idea and Bohm's idea of the implicate order inphysics, and contacted him for collaboration. In particular, the fact that information about an image point isdistributed throughout the hologram, such that each piece of the hologram contains some information about theentire image, seemed suggestive to Pribram about how the brain could encode memories.[1] . Pribram wasencouraged in this line of speculation by the fact that DeValois and DeValois[2] had found that "the spatial frequencyencoding displayed by cells of the visual cortex was best described as a Fourier transform of the input pattern."[3]

This holographic idea led to the coining of the term "holonomic" to describe the idea in wider contexts than justholograms.

Lens-defined model of brain functionIn this model, each sense functions as a lens, refocusing wave patterns either by perceiving a specific pattern orcontext as swirls, or by discerning discrete grains or quantum units. David Bohm has said that if you take the lensesaway, what you are left with is a hologram.According to Pribram and Bohm, "future orientation" is the essence of cognitive function, which they have attemptedto define through use of the Fourier theorem and quantum mechanical formulae. According to Pribram, the tuning ofwave frequency in cells of the primary visual cortex plays a role in visual imaging, while such tuning in the auditorysystem has been well established for decades. Pribram and colleagues also assert that similar tuning occurs in thesomatosensory cortex.Pribram distinguishes between propagative nerve impulses on the one hand, and slow potentials (hyperpolarizations,steep polarizations) that are essentially static. At this temporal interface, he indicates, the wave interferences formholographic patterns.Pribram has written, "What the data suggest is that there exists in the cortex, a multidimensional holographic-likeprocess serving as an attractor or set point toward which muscular contractions operate to achieve a specifiedenvironmental result. The specification has to be based on prior experience (of the species or the individual) andstored in holographic-like form. Activation of the store involves patterns of muscular contractions (guided by basalganglia, cerebellar, brain stem and spinal cord) whose sequential operations need only to satisfy the 'target' encodedin the image of achievement much as the patterns of sequential operations of heating and cooling must meet thesetpoint of the thermostat."

Quantum dynamics of free willAccording to this theory, waveforms, within the matrix of a distributed system, allow fluctuations taking place tocreate new patterns, according to Pribram, and the resulting dynamic potential can then organize new foci of activityoriented to the precipitation of strategic planning and exercise of free will.In a 1998 interview, Pribram addressed the understanding of cognitive potential, stating that, "(I)f you get into yourpotential mode, then new things can happen. But usually free will is conceived of in terms of how many constraintsare operating, and we have in statistics a notion of degrees of freedom. I think our will essentially is constrained,more or less. We have so many degrees of freedom, and the more degrees of freedom we have, the more we feel free,and we have freedom of choice."

Holonomic brain theory 70

These so-called "quantum minds" are still debated among scientists and philosophers, and there are actually anumber of different theories—not one—that have been suggested. Notable proponents of various quantum mindtheories are philosopher David Chalmers and mathematical physicist Roger Penrose. Cosmologist Max Tegmark is anotable opponent of the various quantum mind theories. Tegmark wrote the well-known paper, "Problem withQuantum Mind Theory [4]," which demonstrates certain problems with Chalmers' and Penrose's ideas on the subject.

See also• Consciousness• Evolutionary neuroscience• Gamma wave• Holographic memory• Holonomic• Implicate and Explicate Order according to David Bohm• Sensory integration dysfunction• Wikibook on consciousness

References• Karen K. DeValois, Russell L. DeValois, and W. W. Yund. "Responses of Striate Cortex Cells to Grating and

Checkerboard Patterns", Journal of Physiology, vol 291, 483-505, 1979.• Russel L. DeValois and Karen K. DeValois, "Spatial vision", Ann. Rev. Psychol, 31, 309-41, (1980)• Paul Pietsch, "Shuffle Brain", Harper's, May, 1972, online [5]

• Paul Pietsch, Shufflebrain: The Quest for the Hologramic Mind, Houghton-Mifflin, 1981, ISBN 0-395-29480-0.2nd edition 1996: online [6]: Shufflebrain: The Quest of Hologramic Mind: an in-depth but non-technical look atexperiments on the neural hologram

• Karl H. Pribram, "The Implicate Brain", in B. J. Hiley and F. David Peat, (eds) Quantum Implications: Essays inHonour of David Bohm, Routledge, 1987 ISBN 0-415-06960-2

• --- 'Holonomic Brain Theory and Motor Gestalts: Recent Experimental Results', (1997)• Michael Talbot, "The Holographic Universe" 1991, HarperCollins

External links• "Holonomic brain theory" [7], Article in Scholarpedia by Karl Pribram, Georgetown University, Washington, DC• ACSA2000.net [15] - 'Comparison between Karl Pribram's "Holographic Brain Theory" and more conventional

models of neuronal computation', Jeff Prideaux• NIH.gov [8] - 'Concept-matching in the brain depends on serotonin and gamma-frequency shifts' M. B. Bayly,

Medical Hypotheses Vol 65, No 1, pp 149-51, 2005• ReutersHealth.com [9] - 'Celebrity photos prompt memory study breakthrough: Scientists at two California

universities have isolated single neurons responsible for holding the memory of an image' (June 23, 2005)• ToeQuest.com [10] - 'Holonomic Brain Theory: Holographic Theory offers answers for two main paradoxes,

Nature of mind and Non-locality'• TWM.co.nz [11] - 'The Holographic Brain: Karl Pribram, Ph.D. interview', Dr. Jeffrey Mishlove (1998)

Holonomic brain theory 71

References[1] Pribram, 1987[2] DeValois and DeValois, 1980[3] Pribram, 1987[4] http:/ / www. sustainedaction. org/ Explorations/ problem_with_quantum_mind_theory. htm[5] http:/ / www. indiana. edu/ ~pietsch/ shufflebrain. html[6] http:/ / www. indiana. edu/ ~pietsch/ home. html[7] http:/ / www. scholarpedia. org/ article/ Holonomic_Brain_Theory[8] http:/ / www. ncbi. nlm. nih. gov/ entrez/ query. fcgi?cmd=Retrieve& db=pubmed& dopt=Abstract& list_uids=15893132& query_hl=1[9] http:/ / www. reutershealth. com/ archive/ 2005/ 06/ 23/ eline/ links/ 20050623elin007. html[10] http:/ / www. toequest. com/ forum/ showthread. php?s=88e90cefda26ac1ea6440a97d7e4342f& p=2473#post2473[11] http:/ / twm. co. nz/ pribram. htm

Implicate and Explicate OrderDavid Bohm proposed a cosmological order radically different from generally accepted conventions, which heexpressed as a distinction between the implicate and explicate order, described in the book Wholeness and theImplicate Order:

In the enfolded [or implicate] order, space and time are no longer the dominant factors determining therelationships of dependence or independence of different elements. Rather, an entirely different sort of basicconnection of elements is possible, from which our ordinary notions of space and time, along with those ofseparately existent material particles, are abstracted as forms derived from the deeper order. These ordinarynotions in fact appear in what is called the "explicate" or "unfolded" order, which is a special and distinguishedform contained within the general totality of all the implicate orders (Bohm, 1980, p. xv).

David Bohm's challenges to some generally prevailing viewsIn proposing this new notion of order, Bohm explicitly challenged a number of tenets that are fundamental to muchscientific work. The tenets challenged by Bohm include:1. That phenomena are reducible to fundamental particles and laws describing the behaviour of particles, or more

generally to any static (i.e. unchanging) entities, whether separate events in space-time, quantum states, or staticentities of some other nature.

2. Related to (1), that human knowledge is most fundamentally concerned with mathematical prediction of statisticalaggregates of particles.

3. That an analysis or description of any aspect of reality (e.g. quantum theory, the speed of light) can be unlimitedin its domain of relevance.

4. That the Cartesian coordinate system, or its extension to a curvilinear system, is the deepest conception ofunderlying order as a basis for analysis and description of the world.

5. That there is ultimately a sustainable distinction between reality and thought, and that there is a correspondingdistinction between the observer and observed in an experiment or any other situation (other than a distinctionbetween relatively separate entities valid in the sense of explicate order).

6. That it is, in principle, possible to formulate a final notion concerning the nature of reality; e.g. a Theory ofEverything.

Implicate and Explicate Order 72

A hydrogen atom and its constituent particles: an example ofa small collection of posited building blocks of the universe

Bohm’s proposals have at times been dismissed largely on thebasis of such tenets, without due consideration necessarilygiven to the fact that they had been challenged by Bohm.

Bohm’s paradigm is inherently antithetical to reductionism, inmost forms, and accordingly can be regarded as a form ofontological holism. On this, Bohm noted of prevailing viewsamong physicists: "the world is assumed to be constituted of aset of separately existent, indivisible and unchangeable'elementary particles', which are the fundamental 'buildingblocks' of the entire universe … there seems to be anunshakable faith among physicists that either such particles, orsome other kind yet to be discovered, will eventually makepossible a complete and coherent explanation of everything"(Bohm, 1980, p. 173).

In Bohm’s conception of order, then, primacy is given to theundivided whole, and the implicate order inherent within thewhole, rather than to parts of the whole, such as particles, quantum states, and continua. For Bohm, the wholeencompasses all things, structures, abstractions and processes, including processes that result in (relatively) stablestructures as well as those that involve metamorphosis of structures or things. In this view, parts may be entitiesnormally regarded as physical, such as atoms or subatomic particles, but they may also be abstract entities, such asquantum states. Whatever their nature and character, according to Bohm, these parts are considered in terms of thewhole, and in such terms, they constitute relatively autonomous and independent "sub-totalities". The implication ofthe view is, therefore, that nothing is entirely separate or autonomous.

Bohm (1980, p. 11) said: "The new form of insight can perhaps best be called Undivided Wholeness in FlowingMovement. This view implies that flow is, in some sense, prior to that of the ‘things’ that can be seen to form anddissolve in this flow". According to Bohm, a vivid image of this sense of analysis of the whole is afforded by vortexstructures in a flowing stream. Such vortices can be relatively stable patterns within a continuous flow, but such ananalysis does not imply that the flow patterns have any sharp division, or that they are literally separate andindependently existent entities; rather, they are most fundamentally undivided. Thus, according to Bohm’s view, thewhole is in continuous flux, and hence is referred to as the holomovement (movement of the whole).

Quantum theory and relativity theoryA key motivation for Bohm in proposing a new notion of order was what he saw as the incompatibility of quantumtheory with relativity theory, with respect to certain features of the theories as observed in relevant experimentalcontexts. Bohm (1980, p. xv) summarised the state of affairs he perceived to exist:

…in relativity, movement is continuous, causally determinate and well defined, while in quantummechanics it is discontinuous, not causally determinate and not well-defined. Each theory is committedto its own notions of essentially static and fragmentary modes of existence (relativity to that of separateevents connectible by signals, and quantum mechanics to a well-defined quantum state). One thus seesthat a new kind of theory is needed which drops these basic commitments and at most recovers someessential features of the older theories as abstract forms derived from a deeper reality in which whatprevails is unbroken wholeness.

Bohm maintained that relativity and quantum theory are in basic contradiction in these essential respects, and that a new concept of order should begin with that towards which both theories point: undivided wholeness. This should not be taken to mean that he advocated such powerful theories be discarded. He argued that each was relevant in a

Implicate and Explicate Order 73

certain context—i.e. a set of interrelated conditions within the explicate order—rather than having unlimited scope,and that apparent contradictions stem from attempts to overgeneralize by superposing the theories on one another,implying greater generality or broader relevance than is ultimately warranted. Thus, Bohm (1980, pp. 156-167)argued: "... in sufficiently broad contexts such analytic descriptions cease to be adequate ... 'the law of the whole' willgenerally include the possibility of describing the 'loosening' of aspects from each other, so that they will berelatively autonomous in limited contexts ... however, any form of relative autonomy (and heteronomy) is ultimatelylimited by holonomy, so that in a broad enough context such forms are seen to be merely aspects, relevated in theholomovement, rather than disjoint and separately existent things in interaction".

Hidden variable theoryBohm proposed a hidden variable theory of quantum physics (see Bohm interpretation). According to Bohm, a keymotivation for doing so was purely to show the possibility of such theories. On this, Bohm (1980, p. 81) said "... itshould be kept in mind that before this proposal was made there had existed the widespread impression that noconceptions of hidden variables at all, not even if they were abstract, and hypothetical, could possibly be consistentwith the quantum theory". Bohm (1980, p. 110) also claimed that "the demonstration of the possibility of theories ofhidden variables may serve in a more general philosophical sense to remind us of the unreliability of conclusionsbased on the assumption of the complete universality of certain features of a given theory, however general theirdomain of validity seems to be". Another aspect of Bohm's motivation was to point out a confusion he perceived toexist in quantum theory. On the dominant approaches in quantum theory, he said: "...we wish merely to point out thatthis whole line of approach re-establishes at the abstract level of statistical potentialities the same kind of analysisinto separate and autonomous components in interaction that is denied at the more concrete level of individualobjects" Bohm (1980, p. 174).

Quantum entanglementCentral to Bohm's schema are correlations between observables of entities which seem separated by great distancesin the explicate order (such as a particular electron here on earth and an alpha particle in one of the stars in the Abell1835 galaxy, the farthest galaxy from Earth known to humans), manifestations of the implicate order. Withinquantum theory there is entanglement of such objects.This view of order necessarily departs from any notion which entails signalling, and therefore causality. Thecorrelation of observables does not imply a causal influence, and in Bohm's schema the latter represents 'relatively'independent events in space-time; and therefore explicate order.He also used the term unfoldment to characterise processes in which the explicate order becomes relevant (or"relevated"). Bohm likens unfoldment also to the decoding of a television signal to produce a sensible image on ascreen. The signal, screen, and television electronics in this analogy represent the implicate order whilst the imageproduced represents the explicate order. He also uses an interesting example in which an ink droplet can beintroduced into a highly viscous substance (such as glycerine), and the substance rotated very slowly such that thereis negligible diffusion of the substance. In this example, the droplet becomes a thread which, in turn, eventuallybecomes invisible. However, by rotating the substance in the reverse direction, the droplet can essentially reform.When it is invisible, according to Bohm, the order of the ink droplet as a pattern can be said to be implicate withinthe substance.Further support for this is illustrated by dropping blue ink into a vat of spinning carbon tetrachloride and watch theink disperse. Reversing the spin of the vat will cause the ink to come back together into a blob, then it spreads outagain.In another analogy, Bohm asks us to consider a pattern produced by making small cuts in a folded piece of paper andthen, literally, unfolding it. Widely separated elements of the pattern are, in actuality, produced by the same originalcut in the folded piece of paper. Here the cuts in the folded paper represent the implicate order and the unfolded

Implicate and Explicate Order 74

pattern represents the explicate order.

The hologram as analogy for the implicate order

In a holographic reconstruction, each region of a photographic platecontains the whole image

Bohm employed the hologram as a means ofcharacterising implicate order, noting that each regionof a photographic plate in which a hologram isobservable contains within it the wholethree-dimensional image, which can be viewed from arange of perspectives. That is, each region contains awhole and undivided image. In Bohm’s words:

There is the germ of a new notion of orderhere. This order is not to be understoodsolely in terms of a regular arrangement ofobjects (eg., in rows) or as a regulararrangement of events (e.g. in a series).Rather, a total order is contained, in someimplicit sense, in each region of space andtime. Now, the word 'implicit' is based onthe verb 'to implicate'. This means 'to foldinward' ... so we may be led to explore thenotion that in some sense each regioncontains a total structure 'enfolded' withinit".[1]

Bohm noted that although the hologram conveys undivided wholeness, it is nevertheless static.In this view of order, laws represent invariant relationships between explicate entities and structures, and thus Bohmmaintained that in physics, the explicate order generally reveals itself within well-constructed experimental contextsas, for example, in the sensibly observable results of instruments. With respect to implicate order, however, Bohmasked us to consider the possibility instead "that physical law should refer primarily to an order of undividedwholeness of the content of description similar to that indicated by the hologram rather than to an order of analysisof such content into separate parts …".[2]

A common grounding for consciousness and matter

Implicate and Explicate Order 75

Karl Pribram and colleagues have presented evidencethat indicates that memories do not in general appear to

be localized in specific regions of brains

The implicate order represents the proposal of a generalmetaphysical concept in terms of which it is claimed that matterand consciousness might both be understood, in the sense that it isproposed that both matter and consciousness: (i) enfold thestructure of the whole within each region, and (ii) involvecontinuous processes of enfoldment and unfoldment. For example,in the case of matter, entities such as atoms may representcontinuous enfoldment and unfoldment which manifests as arelatively stable and autonomous entity that can be observed tofollow a relatively well-defined path in space-time. In the case ofconsciousness, Bohm pointed toward evidence presented by KarlPribram that memories may be enfolded within every region of thebrain rather than being localized (for example in particular regions

of the brain, cells, or atoms).

Bohm went on to say:As in our discussion of matter in general, it is now necessary to go into the question of how inconsciousness the explicate order is what is manifest ... the manifest content of consciousness is basedessentially on memory, which is what allows such content to be held in a fairly constant form. Ofcourse, to make possible such constancy it is also necessary that this content be organized, not onlythrough relatively fixed association but also with the aid of the rules of logic, and of our basic categoriesof space, time causality, universality, etc. ... there will be a strong background of recurrent stable, andseparable features, against which the transitory and changing aspects of the unbroken flow of experiencewill be seen as fleeting impressions that tend to be arranged and ordered mainly in terms of the vasttotality of the relatively static and fragmented content of [memories].[3]

Bohm also claimed that "as with consciousness, each moment has a certain explicate order, and in addition it enfoldsall the others, though in its own way. So the relationship of each moment in the whole to all the others is implied byits total content: the way in which it 'holds' all the others enfolded within it". Bohm characterises consciousness as aprocess in which at each moment, content that was previously implicate is presently explicate, and content whichwas previously explicate has become implicate.

One may indeed say that our memory is a special case of the process described above, for all that isrecorded is held enfolded within the brain cells and these are part of matter in general. The recurrenceand stability of our own memory as a relatively independent sub-totality is thus brought about as part ofthe very same process that sustains the recurrence and stability in the manifest order of matter ingeneral. It follows, then, that the explicate and manifest order of consciousness is not ultimately distinctfrom that of matter in general.[4]

Connections with other worksMany have seen strong connections between his ideas and ideas traditionally associated with Eastern Philosophy andreligion. Bohm himself seemed also to see such connections, as evidenced by his close relationship with JidduKrishnamurti. There are particularly strong connections to Buddhism. Some proponents of new age beliefs (such asshamanism) claim a connection with their belief systems as well. Bohm's ideas are also used in certain meditationpractices.Bohm may have known that his idea is a striking analogy to "intentional and extensional aboutness" to which R. A. Fairthorne (1969) insightfully referred information scientists. Searle's concept of aboutness is in sharp contrast to, and is as odd as Bohm's idea of wholeness. As the former is to the content, so the latter is to the context as the

Implicate and Explicate Order 76

ultimate determiner of meaning. The holistic view of context, hence another striking analogy of wholeness, was firstput forward in The Meaning of Meaning by C. K. Ogden & I. A. Richards (1923), including the literary,psychological, and external. These are respectively analogous to Karl Popper's world 3, 2, and 1 appearing in hisObjective Knowledge (1972 and later ed.). Bohm's worldview of "undivided wholeness" is contrasted with Popper'sthree divided worlds.

Cells stained for keratin and DNA: suchparts of life exist because of the whole,

but also to sustain it

Bohm's views bear some similarities to those of Immanuel Kant, according toWouter Hanegraaff. For example, Kant held that the parts of an organism,such as cells, simultaneously exist to sustain the whole, and depend upon thewhole for their own existence and functioning. Kant also proposed that theprocess of thought plays an active role in organizing knowledge, whichimplies theoretical insights are instrumental to the process of acquiring factualknowledge.

Kant restricted knowledge to appearances only and denied the existence ofknowledge of any "thing in itself," but Bohm believed that theories in scienceare "forms of insight that arise in our attempts to obtain a perception of adeeper nature of reality as a whole" (Bohm & Hiley, 1993, p. 323). Thus forBohm the thing in itself is the whole of existence, conceived of not as acollection of parts but as an undivided movement. In this view Bohm is closerto Kant's critic, Arthur Schopenhauer, who identified the thing in itself with the will, an inner metaphysical realitythat grounds all outer phenomena. Schopenhauer's will plays a role analogous to that of the implicate order; forexample, it is objectified (Bohm might say it is "made explicate") to form physical matter. And Bohm's concept thatconsciousness and matter share a common ground resembles Schopenhauer's claim that even inanimate objectspossess an inward noumenal nature. In The World as Will and Representation, Schopenhauer (1819/1995) describedthis ground thus:

When I consider the vastness of the world, the most important factor is that this existence-in-itself, of whichthe world is the manifestation, cannot, whatever it may be, have its true self spread out and dispersed in thisfashion in boundless space, but that this endless extension belongs only to its manifestation, whileexistence-in-itself, on the contrary, is present entire and undivided in everything in nature and in everythingthat lives. (p. 60)

Implicate and Explicate Order 77

For Bohm, life is a continuous flowing process of enfoldment and unfoldment involvingrelatively autonomous entities. DNA 'directs' the environment to form a living thing. Life

can be said to be implicate in ensembles of atoms that ultimately form life.

See also

• Implicature• Holographic principle• The Holographic Universe• Holomovement• Arthur Schopenhauer• Brahman• Buddhism• Immanuel Kant• Kabbalism• Laminar flow• Meditation for Spiritual Unfoldment• Mind's eye• Noumenon• Parable of the cave• Plato• Samsara• Taoism

• Unobservables

References• Bohm, D. (1980). Wholeness and the Implicate Order. London: Routledge. ISBN 0-7100-0971-2• Bohm, D., & Hiley, B. J. (1993). The Undivided Universe. London: Routledge. ISBN 0-415-06588-7• Kauffman, S. (1995). At Home in the Universe. New York: Oxford University Press. hardcover: ISBN

0-19-509599-5, paperback ISBN 0-19-511130-3• Kauffman, S. (2000). Investigations. New York: Oxford University Press.• Kuhn, T.S. (1961). The function of measurement in modern physical science. ISIS, 52, 161-193.• Schopenhauer, A. (1819/1995). The World as Will and Idea. (D. Derman, Ed.; J. Berman, Trans.). London:

Everyman. ISBN 0-460-87505-1

Further reading• Michael Talbot. The Holographic Universe, Harpercollins (1991)

External links• Interview with David Bohm [5] – An interview with Bohm concerning this particular subject matter conducted by

F. David Peat.• Excerpt from The Holographic Universe [6]– Parallels some of the experiences of 18th century Swedish mystic,

Emanuel Swedenborg, with David Bohm's ideas.

Implicate and Explicate Order 78

References[1] (Bohm, 1980, p. 149)[2] (1980, p. 147)[3] (1980, p. 205)[4] (Bohm, 1980, p. 208)[5] http:/ / www. fdavidpeat. com/ interviews/ bohm. htm[6] http:/ / www. soultravel. nu/ 2004/ 040907-swedenborg/ index. asp

Implicate orderDavid Bohm proposed a cosmological order radically different from generally accepted conventions, which heexpressed as a distinction between the implicate and explicate order, described in the book Wholeness and theImplicate Order:

In the enfolded [or implicate] order, space and time are no longer the dominant factors determining therelationships of dependence or independence of different elements. Rather, an entirely different sort of basicconnection of elements is possible, from which our ordinary notions of space and time, along with those ofseparately existent material particles, are abstracted as forms derived from the deeper order. These ordinarynotions in fact appear in what is called the "explicate" or "unfolded" order, which is a special and distinguishedform contained within the general totality of all the implicate orders (Bohm, 1980, p. xv).

David Bohm's challenges to some generally prevailing viewsIn proposing this new notion of order, Bohm explicitly challenged a number of tenets that are fundamental to muchscientific work. The tenets challenged by Bohm include:1. That phenomena are reducible to fundamental particles and laws describing the behaviour of particles, or more

generally to any static (i.e. unchanging) entities, whether separate events in space-time, quantum states, or staticentities of some other nature.

2. Related to (1), that human knowledge is most fundamentally concerned with mathematical prediction of statisticalaggregates of particles.

3. That an analysis or description of any aspect of reality (e.g. quantum theory, the speed of light) can be unlimitedin its domain of relevance.

4. That the Cartesian coordinate system, or its extension to a curvilinear system, is the deepest conception ofunderlying order as a basis for analysis and description of the world.

5. That there is ultimately a sustainable distinction between reality and thought, and that there is a correspondingdistinction between the observer and observed in an experiment or any other situation (other than a distinctionbetween relatively separate entities valid in the sense of explicate order).

6. That it is, in principle, possible to formulate a final notion concerning the nature of reality; e.g. a Theory ofEverything.

Implicate order 79

A hydrogen atom and its constituent particles: an example ofa small collection of posited building blocks of the universe

Bohm’s proposals have at times been dismissed largely on thebasis of such tenets, without due consideration necessarilygiven to the fact that they had been challenged by Bohm.

Bohm’s paradigm is inherently antithetical to reductionism, inmost forms, and accordingly can be regarded as a form ofontological holism. On this, Bohm noted of prevailing viewsamong physicists: "the world is assumed to be constituted of aset of separately existent, indivisible and unchangeable'elementary particles', which are the fundamental 'buildingblocks' of the entire universe … there seems to be anunshakable faith among physicists that either such particles, orsome other kind yet to be discovered, will eventually makepossible a complete and coherent explanation of everything"(Bohm, 1980, p. 173).

In Bohm’s conception of order, then, primacy is given to theundivided whole, and the implicate order inherent within thewhole, rather than to parts of the whole, such as particles, quantum states, and continua. For Bohm, the wholeencompasses all things, structures, abstractions and processes, including processes that result in (relatively) stablestructures as well as those that involve metamorphosis of structures or things. In this view, parts may be entitiesnormally regarded as physical, such as atoms or subatomic particles, but they may also be abstract entities, such asquantum states. Whatever their nature and character, according to Bohm, these parts are considered in terms of thewhole, and in such terms, they constitute relatively autonomous and independent "sub-totalities". The implication ofthe view is, therefore, that nothing is entirely separate or autonomous.

Bohm (1980, p. 11) said: "The new form of insight can perhaps best be called Undivided Wholeness in FlowingMovement. This view implies that flow is, in some sense, prior to that of the ‘things’ that can be seen to form anddissolve in this flow". According to Bohm, a vivid image of this sense of analysis of the whole is afforded by vortexstructures in a flowing stream. Such vortices can be relatively stable patterns within a continuous flow, but such ananalysis does not imply that the flow patterns have any sharp division, or that they are literally separate andindependently existent entities; rather, they are most fundamentally undivided. Thus, according to Bohm’s view, thewhole is in continuous flux, and hence is referred to as the holomovement (movement of the whole).

Quantum theory and relativity theoryA key motivation for Bohm in proposing a new notion of order was what he saw as the incompatibility of quantumtheory with relativity theory, with respect to certain features of the theories as observed in relevant experimentalcontexts. Bohm (1980, p. xv) summarised the state of affairs he perceived to exist:

…in relativity, movement is continuous, causally determinate and well defined, while in quantummechanics it is discontinuous, not causally determinate and not well-defined. Each theory is committedto its own notions of essentially static and fragmentary modes of existence (relativity to that of separateevents connectible by signals, and quantum mechanics to a well-defined quantum state). One thus seesthat a new kind of theory is needed which drops these basic commitments and at most recovers someessential features of the older theories as abstract forms derived from a deeper reality in which whatprevails is unbroken wholeness.

Bohm maintained that relativity and quantum theory are in basic contradiction in these essential respects, and that a new concept of order should begin with that towards which both theories point: undivided wholeness. This should not be taken to mean that he advocated such powerful theories be discarded. He argued that each was relevant in a

Implicate order 80

certain context—i.e. a set of interrelated conditions within the explicate order—rather than having unlimited scope,and that apparent contradictions stem from attempts to overgeneralize by superposing the theories on one another,implying greater generality or broader relevance than is ultimately warranted. Thus, Bohm (1980, pp. 156-167)argued: "... in sufficiently broad contexts such analytic descriptions cease to be adequate ... 'the law of the whole' willgenerally include the possibility of describing the 'loosening' of aspects from each other, so that they will berelatively autonomous in limited contexts ... however, any form of relative autonomy (and heteronomy) is ultimatelylimited by holonomy, so that in a broad enough context such forms are seen to be merely aspects, relevated in theholomovement, rather than disjoint and separately existent things in interaction".

Hidden variable theoryBohm proposed a hidden variable theory of quantum physics (see Bohm interpretation). According to Bohm, a keymotivation for doing so was purely to show the possibility of such theories. On this, Bohm (1980, p. 81) said "... itshould be kept in mind that before this proposal was made there had existed the widespread impression that noconceptions of hidden variables at all, not even if they were abstract, and hypothetical, could possibly be consistentwith the quantum theory". Bohm (1980, p. 110) also claimed that "the demonstration of the possibility of theories ofhidden variables may serve in a more general philosophical sense to remind us of the unreliability of conclusionsbased on the assumption of the complete universality of certain features of a given theory, however general theirdomain of validity seems to be". Another aspect of Bohm's motivation was to point out a confusion he perceived toexist in quantum theory. On the dominant approaches in quantum theory, he said: "...we wish merely to point out thatthis whole line of approach re-establishes at the abstract level of statistical potentialities the same kind of analysisinto separate and autonomous components in interaction that is denied at the more concrete level of individualobjects" Bohm (1980, p. 174).

Quantum entanglementCentral to Bohm's schema are correlations between observables of entities which seem separated by great distancesin the explicate order (such as a particular electron here on earth and an alpha particle in one of the stars in the Abell1835 galaxy, the farthest galaxy from Earth known to humans), manifestations of the implicate order. Withinquantum theory there is entanglement of such objects.This view of order necessarily departs from any notion which entails signalling, and therefore causality. Thecorrelation of observables does not imply a causal influence, and in Bohm's schema the latter represents 'relatively'independent events in space-time; and therefore explicate order.He also used the term unfoldment to characterise processes in which the explicate order becomes relevant (or"relevated"). Bohm likens unfoldment also to the decoding of a television signal to produce a sensible image on ascreen. The signal, screen, and television electronics in this analogy represent the implicate order whilst the imageproduced represents the explicate order. He also uses an interesting example in which an ink droplet can beintroduced into a highly viscous substance (such as glycerine), and the substance rotated very slowly such that thereis negligible diffusion of the substance. In this example, the droplet becomes a thread which, in turn, eventuallybecomes invisible. However, by rotating the substance in the reverse direction, the droplet can essentially reform.When it is invisible, according to Bohm, the order of the ink droplet as a pattern can be said to be implicate withinthe substance.Further support for this is illustrated by dropping blue ink into a vat of spinning carbon tetrachloride and watch theink disperse. Reversing the spin of the vat will cause the ink to come back together into a blob, then it spreads outagain.In another analogy, Bohm asks us to consider a pattern produced by making small cuts in a folded piece of paper andthen, literally, unfolding it. Widely separated elements of the pattern are, in actuality, produced by the same originalcut in the folded piece of paper. Here the cuts in the folded paper represent the implicate order and the unfolded

Implicate order 81

pattern represents the explicate order.

The hologram as analogy for the implicate order

In a holographic reconstruction, each region of a photographic platecontains the whole image

Bohm employed the hologram as a means ofcharacterising implicate order, noting that each regionof a photographic plate in which a hologram isobservable contains within it the wholethree-dimensional image, which can be viewed from arange of perspectives. That is, each region contains awhole and undivided image. In Bohm’s words:

There is the germ of a new notion of orderhere. This order is not to be understoodsolely in terms of a regular arrangement ofobjects (eg., in rows) or as a regulararrangement of events (e.g. in a series).Rather, a total order is contained, in someimplicit sense, in each region of space andtime. Now, the word 'implicit' is based onthe verb 'to implicate'. This means 'to foldinward' ... so we may be led to explore thenotion that in some sense each regioncontains a total structure 'enfolded' withinit".[1]

Bohm noted that although the hologram conveys undivided wholeness, it is nevertheless static.In this view of order, laws represent invariant relationships between explicate entities and structures, and thus Bohmmaintained that in physics, the explicate order generally reveals itself within well-constructed experimental contextsas, for example, in the sensibly observable results of instruments. With respect to implicate order, however, Bohmasked us to consider the possibility instead "that physical law should refer primarily to an order of undividedwholeness of the content of description similar to that indicated by the hologram rather than to an order of analysisof such content into separate parts …".[2]

A common grounding for consciousness and matter

Implicate order 82

Karl Pribram and colleagues have presented evidencethat indicates that memories do not in general appear to

be localized in specific regions of brains

The implicate order represents the proposal of a generalmetaphysical concept in terms of which it is claimed that matterand consciousness might both be understood, in the sense that it isproposed that both matter and consciousness: (i) enfold thestructure of the whole within each region, and (ii) involvecontinuous processes of enfoldment and unfoldment. For example,in the case of matter, entities such as atoms may representcontinuous enfoldment and unfoldment which manifests as arelatively stable and autonomous entity that can be observed tofollow a relatively well-defined path in space-time. In the case ofconsciousness, Bohm pointed toward evidence presented by KarlPribram that memories may be enfolded within every region of thebrain rather than being localized (for example in particular regions

of the brain, cells, or atoms).

Bohm went on to say:As in our discussion of matter in general, it is now necessary to go into the question of how inconsciousness the explicate order is what is manifest ... the manifest content of consciousness is basedessentially on memory, which is what allows such content to be held in a fairly constant form. Ofcourse, to make possible such constancy it is also necessary that this content be organized, not onlythrough relatively fixed association but also with the aid of the rules of logic, and of our basic categoriesof space, time causality, universality, etc. ... there will be a strong background of recurrent stable, andseparable features, against which the transitory and changing aspects of the unbroken flow of experiencewill be seen as fleeting impressions that tend to be arranged and ordered mainly in terms of the vasttotality of the relatively static and fragmented content of [memories].[3]

Bohm also claimed that "as with consciousness, each moment has a certain explicate order, and in addition it enfoldsall the others, though in its own way. So the relationship of each moment in the whole to all the others is implied byits total content: the way in which it 'holds' all the others enfolded within it". Bohm characterises consciousness as aprocess in which at each moment, content that was previously implicate is presently explicate, and content whichwas previously explicate has become implicate.

One may indeed say that our memory is a special case of the process described above, for all that isrecorded is held enfolded within the brain cells and these are part of matter in general. The recurrenceand stability of our own memory as a relatively independent sub-totality is thus brought about as part ofthe very same process that sustains the recurrence and stability in the manifest order of matter ingeneral. It follows, then, that the explicate and manifest order of consciousness is not ultimately distinctfrom that of matter in general.[4]

Connections with other worksMany have seen strong connections between his ideas and ideas traditionally associated with Eastern Philosophy andreligion. Bohm himself seemed also to see such connections, as evidenced by his close relationship with JidduKrishnamurti. There are particularly strong connections to Buddhism. Some proponents of new age beliefs (such asshamanism) claim a connection with their belief systems as well. Bohm's ideas are also used in certain meditationpractices.Bohm may have known that his idea is a striking analogy to "intentional and extensional aboutness" to which R. A. Fairthorne (1969) insightfully referred information scientists. Searle's concept of aboutness is in sharp contrast to, and is as odd as Bohm's idea of wholeness. As the former is to the content, so the latter is to the context as the

Implicate order 83

ultimate determiner of meaning. The holistic view of context, hence another striking analogy of wholeness, was firstput forward in The Meaning of Meaning by C. K. Ogden & I. A. Richards (1923), including the literary,psychological, and external. These are respectively analogous to Karl Popper's world 3, 2, and 1 appearing in hisObjective Knowledge (1972 and later ed.). Bohm's worldview of "undivided wholeness" is contrasted with Popper'sthree divided worlds.

Cells stained for keratin and DNA: suchparts of life exist because of the whole,

but also to sustain it

Bohm's views bear some similarities to those of Immanuel Kant, according toWouter Hanegraaff. For example, Kant held that the parts of an organism,such as cells, simultaneously exist to sustain the whole, and depend upon thewhole for their own existence and functioning. Kant also proposed that theprocess of thought plays an active role in organizing knowledge, whichimplies theoretical insights are instrumental to the process of acquiring factualknowledge.

Kant restricted knowledge to appearances only and denied the existence ofknowledge of any "thing in itself," but Bohm believed that theories in scienceare "forms of insight that arise in our attempts to obtain a perception of adeeper nature of reality as a whole" (Bohm & Hiley, 1993, p. 323). Thus forBohm the thing in itself is the whole of existence, conceived of not as acollection of parts but as an undivided movement. In this view Bohm is closerto Kant's critic, Arthur Schopenhauer, who identified the thing in itself with the will, an inner metaphysical realitythat grounds all outer phenomena. Schopenhauer's will plays a role analogous to that of the implicate order; forexample, it is objectified (Bohm might say it is "made explicate") to form physical matter. And Bohm's concept thatconsciousness and matter share a common ground resembles Schopenhauer's claim that even inanimate objectspossess an inward noumenal nature. In The World as Will and Representation, Schopenhauer (1819/1995) describedthis ground thus:

When I consider the vastness of the world, the most important factor is that this existence-in-itself, of whichthe world is the manifestation, cannot, whatever it may be, have its true self spread out and dispersed in thisfashion in boundless space, but that this endless extension belongs only to its manifestation, whileexistence-in-itself, on the contrary, is present entire and undivided in everything in nature and in everythingthat lives. (p. 60)

Implicate order 84

For Bohm, life is a continuous flowing process of enfoldment and unfoldment involvingrelatively autonomous entities. DNA 'directs' the environment to form a living thing. Life

can be said to be implicate in ensembles of atoms that ultimately form life.

See also

• Implicature• Holographic principle• The Holographic Universe• Holomovement• Arthur Schopenhauer• Brahman• Buddhism• Immanuel Kant• Kabbalism• Laminar flow• Meditation for Spiritual Unfoldment• Mind's eye• Noumenon• Parable of the cave• Plato• Samsara• Taoism

• Unobservables

References• Bohm, D. (1980). Wholeness and the Implicate Order. London: Routledge. ISBN 0-7100-0971-2• Bohm, D., & Hiley, B. J. (1993). The Undivided Universe. London: Routledge. ISBN 0-415-06588-7• Kauffman, S. (1995). At Home in the Universe. New York: Oxford University Press. hardcover: ISBN

0-19-509599-5, paperback ISBN 0-19-511130-3• Kauffman, S. (2000). Investigations. New York: Oxford University Press.• Kuhn, T.S. (1961). The function of measurement in modern physical science. ISIS, 52, 161-193.• Schopenhauer, A. (1819/1995). The World as Will and Idea. (D. Derman, Ed.; J. Berman, Trans.). London:

Everyman. ISBN 0-460-87505-1

Further reading• Michael Talbot. The Holographic Universe, Harpercollins (1991)

External links• Interview with David Bohm [5] – An interview with Bohm concerning this particular subject matter conducted by

F. David Peat.• Excerpt from The Holographic Universe [6]– Parallels some of the experiences of 18th century Swedish mystic,

Emanuel Swedenborg, with David Bohm's ideas.

Implicate order 85

References[1] (Bohm, 1980, p. 149)[2] (1980, p. 147)[3] (1980, p. 205)[4] (Bohm, 1980, p. 208)

Membrane paradigmIn black hole theory, the black hole membrane paradigm is a useful "toy model" method or "engineeringapproach" for visualising and calculating the effects predicted by quantum mechanics for the exterior physics ofblack holes, without using quantum-mechanical principles or calculations. It models a black hole as a thinclassically-radiating surface (or membrane) at or vanishingly close to the black hole's event horizon. This approachto the theory of black holes was created by Kip S. Thorne, R. H. Price and D. A. Macdonald.The results of the membrane paradigm are generally considered to be "safe".

Electrical resistanceThorne (1994) relates that this approach to studying black holes was prompted by the realisation by Hanni, Ruffini,Wald and Cohen in the early 1970's that since an electrically charged pellet dropped into a black hole should stillappear to a distant outsider to be remaining just outside the critical r=2M radius, if its image persists, its electricalfieldlines ought to persist too, and ought to point to the location of the "frozen" image (1994, pp.406). If the blackhole rotates, and the image of the pellet is pulled around, the associated electrical fieldlines ought to be pulled aroundwith it to create basic "electrical dynamo" effects (see: dynamo theory).Further calculations yielded properties for a black hole such as apparent electrical resistance (pp.408). Since thesefieldline properties seemed to be exhibited down to the event horizon, and general relativity insisted that no dynamicexterior interactions could extend through the horizon, it was considered convenient to invent a surface at thehorizon that these electrical properties could be said to belong to.

Hawking radiationAfter being introduced to model the theoretical electrical characteristics of the horizon, the "membrane" approachwas then pressed into service to model the Hawking radiation effect predicted by quantum mechanics.In the coordinate system of a distant stationary observer, Hawking radiation tends to be described as aquantum-mechanical particle-pair production effect (involving "virtual" particles), but for stationary observershovering nearer to the hole, the effect is supposed to look like a purely conventional radiation effect involving "real"particles. In the "membrane paradigm", the black hole is described as it should be seen by an array of thesestationary, suspended noninertial observers, and since their shared coordinate system ends at r=2M (because anobserver cannot legally hover at or below the event horizon under general relativity), this conventional-lookingradiation is described as being emitted by an arbitrarily-thin shell of "hot" material at or just above the critical r=2Mradius, where this coordinate system fails.As in the "electrical" case, the membrane paradigm is useful because these effects should appear all the way down tothe event horizon, but are not allowed by GR to be coming through the horizon – blaming them on a hypotheticalthin radiating membrane at the horizon allows them to be modelled classically without explicitly contradictinggeneral relativity's prediction that the r=2M surface is inescapable.In 1986, Kip S. Thorne, R. H. Price and D. A. Macdonald published an anthology of papers by various authors thatexamined this idea: "Black Holes: The membrane paradigm".

Membrane paradigm 86

See also• Holographic principle

References• Price, Richard H., and Kip Thorne, "The Membrane Paradigm for Black Holes", Scientific American, vol. 258, no.

4 (April 1988) pp. 69-77• Leonard Susskind, "Black holes and the information paradox", Scientific American, April 1997 (cover story [1]).

Also reprinted in the special edition "The edge of physics" [2]

• Kip S. Thorne, R. H. Price and D. A. Macdonald (eds.) "Black Holes: The membrane paradigm" (1986)• Thorne, Kip, Black Holes and Time Warps: Einstein's Outrageous Legacy, W. W. Norton & Company; Reprint

edition, January 1, 1995, ISBN 0-393-31276-3, chapter 11, pp.397-411

References[1] http:/ / www. sciamdigital. com/ browse. cfm?sequencenameCHAR=item2& methodnameCHAR=resource_getitembrowse&

interfacenameCHAR=browse. cfm& ISSUEID_CHAR=F1E36413-4C42-4E84-BCE9-A10BEB1E9D3&ARTICLEID_CHAR=8F43C5C9-F4F3-4F4C-A50E-221DB8E68CF& sc=I100322

[2] http:/ / www. sciamdigital. com/ browse. cfm?sequencenameCHAR=item& methodnameCHAR=resource_getitembrowse&interfacenameCHAR=browse. cfm& ISSUEID_CHAR=6C2FAA19-0087-C3FE-547CDF8E4C786808

Orch- OROrch OR (Orchestrated Objective Reduction) is a theory of consciousness, which is the joint work of theoreticalphysicist Sir Roger Penrose and anesthesiologist Stuart Hameroff. Mainstream theories assume that consciousnessemerges from the brain, and focus particularly on complex computation at connections known as synapses that allowcommunication between brain cells (neurons). Orch OR combines approaches to the problem of consciousness fromthe radically different angles of mathematics, physics and anesthesia.Penrose and Hameroff initially developed their ideas quite separately from one another, and it was only in the 1990sthat they cooperated to produce the Orch OR theory. Penrose came to the problem from the view point ofmathematics and in particular Gödel’s theorem, while Hameroff approached it from a career in cancer research andanesthesia that gave him an interest in brain structures.

Gödel's Incompleteness TheoremIn 1931, the mathematician and logician Kurt Gödel proved that any theory capable of expressing elementaryarithmetic cannot be both consistent and complete. Further to that, for any consistent formal theory that provescertain basic arithmetic truths, there is an arithmetical statement that is true, but not provable in the theory.In his first book on consciousness, The Emperor's New Mind (1989), Penrose made Gödel's theorem the basis ofwhat quickly became an intensely controversial claim.[1] He argued that the theorem showed that the brain had theability to go beyond what could be achieved by axioms or formal systems. This would mean that the mind had someadditional function that was not based on algorithms (systems or rules of calculation). A computer is driven solely byalgorithms. Penrose asserted that the brain could perform functions that no computer could perform. He called thistype of functioning non-computable.

Orch-OR 87

The quantum levelPenrose went on to consider what it was in the human brain that might not be driven by algorithms. The physical lawis described by algorithms, so it was not easy for Penrose to come up with physical properties or processes that arenot described by them. He was forced to look to quantum theory for a plausible candidate.In quantum theory, the fundamental units, the quanta, are in some respects quite unlike objects that are encounteredin the large scale world described by classical physics. When sufficiently isolated from the environment, they can beviewed as waves. However these are not the same as matter waves, such as waves in the sea. The quantum waves areessentially waves of probability, the varying probability of finding a particle at some specific position. (Theseprobabilities apply to other states of the particle, such as its momentum, but for the sake of simplicity we will refer toposition.) The peak of the wave indicates the location with maximum probability of a particle being found there. Thedifferent possible positions of the particle are referred to as superpositions or quantum superpositions. We arespeaking here of the isolated form of the quanta. When the quanta are the subject of measurements or of interactionwith the environment, the wave characteristic is lost, and a particle is found at a specific point. This change iscommonly referred to as the collapse of the wave function.When the collapse happens, the choice of position for the particle is random. This is a drastic departure fromclassical physics. There is no cause-and-effect process, and no system of algorithms that can describe the choice ofposition for the particle.This provided Penrose with a candidate for the physical basis of the suggested non-computable process that heproposed as possibly existing in the brain. However, this was not the end of his problems. He had identifiedsomething in physics that was not based on algorithms, but at the same time, randomness was not a promising basisfor mathematical understanding, the aspect of mind that Penrose particularly focused on.

Objective reductionPenrose now proposed that existing ideas on wave function collapse might only apply to situations where the quantaare the subject of measurement or of interaction with the environment. He considered the case of quanta that are notthe subject of measurements or interactions, but remain isolated from the environment, and proposed that thesequanta may be subject to a different form of wave function collapse.In this area, Penrose draws on both Einstein's general theory of relativity, and on his own notions about the possiblestructure of spacetime.[1] [2] General relativity states that spacetime is curved by massive objects. Penrose, in seekingto reconcile relativity and quantum theory, has suggested that at the very small scale this curved spacetime is notcontinuous, but constitutes a form of network.Penrose postulates that each quantum superposition has its own piece of spacetime curvature. According to histheory, these different bits of spacetime curvature are separated from one another, and constitute a form of blister inspacetime. Penrose further proposes a limit to the size of this spacetime blister. This is the tiny Planck scale of (10−35

m). Above this size, Penrose suggests that spacetime can be viewed as continuous, and that gravity starts to exert itsforce on the spacetime blister. This is suggested to become unstable above the Planck scale, and to collapse so as tochoose just one of the possible locations for the particle. Penrose calls this event objective reduction (OR), reductionbeing another word for wave function collapse.An important feature of Penrose's objective reduction is that the time to collapse is a function of the mass/energy ofthe object undergoing collapse. Thus the greater the superposition, the faster it will undergo OR, and vice versa. Tinysuperpositions, e.g. an electron separated from itself, if isolated from environment, would require 10 million years toreach OR threshold. An isolated one kilogram object (e.g. Schrödinger’s cat) would reach OR threshold in only 10−37

seconds. However objects somewhere between the scale of an electron and the scale of a cat could collapse within atimescale that was relevant to neural processing.

Orch-OR 88

The threshold for Penrose OR is given by the indeterminacy principle E=ħ/t, where E is the gravitational self-energyor the degree of spacetime separation given by the superpositioned mass, ħ is the reduced Planck constant, and t isthe time until OR occurs.There is no existing evidence for Penrose's objective reduction, but the theory is considered to be testable, and plansare in hand to carry out a relevant experiment.[3]

From the point of view of consciousness theory, an essential feature of Penrose's objective reduction is that thechoice of states when objective reduction occurs is selected neither randomly, as are choices following measurementor decoherence, nor completely algorithmically. Rather, states are proposed to be selected by a 'non-computable'influence embedded in the fundamental level of spacetime geometry at the Planck scale.Penrose claimed that such information is Platonic, representing pure mathematical truth, aesthetic and ethical values.More than two thousand years ago, the Greek philosopher Plato had proposed such pure values and forms, but in anabstract realm. Penrose placed the Platonic realm at the Planck scale. This relates to Penrose's ideas concerning thethree worlds: physical, mental, and the Platonic mathematical world. In his theory, the physical world can be seen asthe external reality, the mental world as information processing in the brain and the Platonic world as the encryption,measurement, or geometry of fundamental spacetime that is claimed to support non-computational understanding.

The creation of the Orch OR modelWhen he wrote his first consciousness book, The Emperor's New Mind in 1989, Penrose lacked a detailed proposalfor how such quantum processes could be implemented in the brain. Subsequently, Hameroff read The Emperor’sNew Mind and suggested to Penrose that certain structures within brain cells (neurons) were suitable candidate sitesfor quantum processing and ultimately for consciousness.[4] [5] The Orch OR theory arose from the cooperation ofthese two scientists, and were developed in Penrose's second consciousness book Shadows of the Mind (1994).[2]

Hameroff's contribution to the theory derived from studying brain cells (neurons). His interest centred on thecytoskeleton, which provides an internal supportive structure for neurons, and particularly on the microtubules,[5]

which are the important component of the cytoskeleton. As neuroscience has progressed, the role of the cytoskeletonand microtubules has assumed greater importance. In addition to providing a supportive structure for the cell, theknown functions of the microtubules include transport of molecules including neurotransmitter molecules bound forthe synapses, and control of the cell's movement, growth and shape.[5]

Hameroff proposed that microtubules were suitable candidates to support quantum processing.[5] Microtubules aremade up of tubulin protein subunits. The tubulin protein dimers of the microtubules have hydrophobic pocketsbinding the drug taxol separated 8 nm away, which might contain delocalized π electrons. Tubulin has other smallernon-polar regions, for example 8 tryptophans per tubulin, which contain π electron-rich indole rings distributedthroughout tubulin with separations of roughly 2 nm. Hameroff claims that this is close enough for the tubulin πelectrons to become quantum entangled.[6] Quantum entanglement is a state in which quantum particles can alter oneanother's properties instantaneously and at a distance, in a way which would not be possible, if they were large scaleobjects obeying the laws of classical as opposed to quantum physics.In the case of the electrons in the tubulin subunits of the microtubules, Hameroff has proposed that large numbers ofthese electrons can become involved in a state known as a Bose-Einstein condensate. These occur when largenumbers of quantum particles become locked in phase and exist as a single quantum object. These are quantumfeatures at a macroscopic scale, and Hameroff suggests that through a feature of this kind quantum activity, which isusually at a very tiny scale, could be boosted to be a large scale influence in the brain.Hameroff has proposed that condensates in microtubules in one neuron can link with microtubule condensates in other neurons and glial cells via gap junctions.[7] [8] In addition to the synaptic connections between brain cells, gap junctions are a different category of connections, where the gap between the cells is sufficiently small for quantum objects to cross it by means of a process known as quantum tunneling. Hameroff proposes that this tunneling allows

Orch-OR 89

a quantum object, such as the Bose-Einstein condensates mentioned above, to cross into other neurons, and thusextend across a large area of the brain as a single quantum object.He further postulates that the action of this large-scale quantum feature is the source of the gamma synchronisationobserved in the brain, and sometimes viewed as a neural correlate of consciousness.[9] In support of the much morelimited theory that gap junctions are related to the gamma oscillation, Hameroff quotes a number of studies fromrecent years.[10]

The Orch OR theory combines Penrose's hypothesis with respect to the Gödel theorem with Hameroff's hypothesiswith respect to microtubules. Together, Penrose and Hameroff have proposed that when condensates in the brainundergo an objective reduction of their wave function, that collapse connects to non-computational decisiontaking/experience embedded in the geometry of fundamental spacetime.The theory further proposes that the microtubules both influence and are influenced by the conventional activity atthe synapses between neurons. The Orch in Orch OR stands for orchestrated to give the full name of the theoryOrchestrated Objective Reduction. Orchestration refers to the hypothetical process by which connective proteins,known as microtubule associated proteins (MAPs) influence or orchestrate the quantum processing of themicrotubules.

Objections to Orch ORPenrose's interpretation of Gödel's first incompleteness theorem is rejected by many philosophers, logicians andartificial intelligence (robotics) researchers.[11] A paper by the philosophers Rick Grush and Patricia Churchlandattacking Penrose has received widespread attention within consciousness studies.[12] Solomon Feferman, aprofessor of mathematics, logic and philosophy has made more qualified criticisms.[13] He faults detailed points inPenrose's reasoning in his second book 'Shadows of the Mind', but says that he does not think that they underminethe main thrust of his argument. As a mathematician, he argues that mathematicians do not progress bycomputer-like or mechanistic search through proofs, but by trial-and-error reasoning, insight and inspiration, and thatmachines cannot share this approach with humans. However, he thinks that Penrose goes too far in his arguments.Feferman points out that everyday mathematics, as used in science, can in practice be formalised. He also rejectsPenrose's platonism.The main objection to the Hameroff side of the theory is that any quantum feature in the environment of the brainwould undergo wave function collapse (reduction), as a result of interaction with the environment, far too quickly forit to have any influence on neural processes. The wave or superposition form of the quanta is referred to as beingquantum coherent. Interaction with the environment results in decoherence otherwise known as wave functioncollapse. It has been questioned as to how such quantum coherence could avoid rapid decoherence in the conditionsof the brain. With reference to this question, a paper by the physicist, Max Tegmark, refuting the Orch OR modeland published in the journal, Physical Review E is widely quoted.[14] Tegmark developed a model for time todecoherence, and from this calculated that microtubule quantum states could exist, but would be sustained for only100 femtoseconds at brain temperatures, far too brief to be relevant to neural processing. A recent paper by Engel etal. in Nature does indicate quantum coherent electrons as being functional in energy transfer within photosyntheticprotein, but the quantum coherence described lasts for 660 femtoseconds[15] rather than the 25 milliseconds requiredby Orch OR. This reinforces Tegmark's estimate for decoherence timescale of microtubules, which is comparable tothe observed coherence time in the photosynthetic complex.In their reply to Tegmark's paper, also published in Physical Review E, the physicists, Scott Hagan and Jack Tuszynski and Hameroff[16] [17] claimed that Tegmark did not address the Orch OR model, but instead a model of his own construction. This involved superpositions of quanta separated by 24 nm rather than the much smaller separations stipulated for Orch OR. As a result, Hameroff's group claimed a decoherence time seven orders of magnitude greater than Tegmarks, but still well short of the 25 ms required if the quantum processing in the theory was to be linked to the 40 Hz gamma synchrony, as Orch OR suggested. To bridge this gap, the group made a series

Orch-OR 90

of proposals. It was supposed that the interiors of neurons could alternate between liquid and gel states. In the gelstate, it was further hypothesized that the water electrical dipoles are orientated in the same direction, along the outeredge of the microtubule tubulin subunits. Hameroff et al. proposed that this ordered water could screen any quantumcoherence within the tubulin of the microtubules from the environment of the rest of the brain. Each tubulin also hasa tail extending out from the microtubules, which is negatively charged, and therefore attracts positively chargedions. It is suggested that this could provide further screening. Further to this, there was a suggestion that themicrotubules could be pumped into a coherent state by biochemical energy. Finally, it is suggested that theconfiguration of the microtubule lattice might be suitable for quantum error correction, a means of holding togetherquantum coherence in the face of environmental interaction. In the last decade, some researchers who aresympathetic to Penrose's ideas have proposed an alternative scheme for quantum processing in microtubules basedon the interaction of tubulin tails with microtubule associated proteins, motor proteins and presynaptic scaffoldproteins. These proposed alternative processes have the advantage of taking place within Tegmark's time todecoherence.Most of the above mentioned putative augmentations of the Orch OR model are not undisputed. "Cortical dendritescontain largely A-lattice microtubules" is one of 20 testable predictions published by Hameroff in 1998[18] and itwas hypothesized that these A-lattice microtubules could perform topological quantum error correction. The lattertestable prediction had already been experimentally disproved in 1994 by Kikkawa et al., who showed that all invivo microtubules have B-lattice and a seam.[19] [20] Other peer-reviewed critiques of Orch OR have been publishedin recent years. One of these is a paper published in PNAS by Reimers et al.,[21] who argue that the condensatesproposed in Orch OR would involve energies and temperatures that are not realistic in biological material. Furtherpapers by Georgiev point to a number of problems with Hameroff's proposals, including the lack of explanation forthe probabilistic firing of the axonal synapses,[22] an error in the calculated number of tubulin dimers per corticalneuron,[23] and mismodeling of dendritic lamellar bodies (DLBs) discovered by De Zeeuw et al.,[24] who showedthat despite the fact that DLBs are stained by antibody against gap junctions, they are located tens of micrometersaway from actual gap junctions. Also it was shown that the proposed tubulin-bound GTP pumping of quantumcoherence cannot occur neither in stable microtubules[25] nor in dynamically unstable microtubules undergoingassembly/disassembly.[26]

See also• Electromagnetic theories of consciousness• Holonomic brain theory• Many-minds interpretation• Quantum Aspects of Life (book)• Quantum mind• Subjective universe• Roger Penrose (1999) Science and the Mind. Kavli Institute for Theoretical Physics Public Lectures, May 12,

1999. [27]

• Quantum-Mind [28]

Orch-OR 91

References[1] Penrose, Roger (1989). The Emperor's New Mind: Concerning Computers, Minds and The Laws of Physics. Oxford University Press. pp. 480.

ISBN 0-198-51973-7.[2] Penrose, Roger (1989). Shadows of the Mind: A Search for the Missing Science of Consciousness. Oxford University Press. pp. 457.

ISBN 0-19-853978-9.[3] Marshall, W., Simon, C., Penrose, R., and Bouwmeester, D. (2003). "Towards quantum superpositions of a mirror" (http:/ / arxiv. org/ abs/

quant-ph/ 0210001). Physical Review Letters 91: 130401. doi:10.1103/PhysRevLett.91.130401. .[4] Hameroff, S.R., and Watt, R.C. (1982). "Information processing in microtubules" (http:/ / www. quantumconsciousness. org/ documents/

informationprocessing_hameroff_000. pdf). Journal of Theoretical Biology 98: 549–561. .[5] Hameroff, S.R. (1987). Ultimate Computing (http:/ / www. quantumconsciousness. org/ ultimatecomputing. html). Elsevier. .[6] Hameroff, Stuart (2008). "That's life! The geometry of π electron resonance clouds" (http:/ / www. quantumconsciousness. org/ documents/

Hameroff_received-1-05-07. pdf). in Abbott, D; Davies, P; Pati, A. Quantum aspects of life. World Scientific. pp. 403–434. . Retrieved Jan21, 2010.

[7] Hameroff, S.R. (2006). "The entwined mysteries of anesthesia and consciousness". Anesthesiology 105: 400–412.[8] Hameroff, S. (2009). "The “conscious pilot” - dendritic synchrony moves through the brain to mediate consciousness". Journal of Biological

Physics. doi:10.1007/s10867-009-9148-x.[9] Bennett, M.V.L., and Zukin, R.S. (2004). "Electrical Coupling and Neuronal Synchronization in the Mammalian Brain" (http:/ / dx. doi. org/

10. 1016/ S0896-6273(04)00043-1). Neuron 41: 495–511. doi:10.1016/S0896-6273(04)00043-1. .[10] Specifically:

Buhl, D.L., Harris, K.D., Hormuzdi, S.G., Monyer, H., and Buzsaki, G. (2003). "Selective Impairment of Hippocampal Gamma Oscillations inConnexin-36 Knock-Out Mouse In Vivo". Journal of Neuroscience 23: 1013–1018.Dermietzel, R. (1998). "Gap junction wiring: a `new' principle in cell-to-cell communication in the nervous system?". Brain Research Reviews26: 176–183.Draguhn, A., Traub, R.D., Schmitz, D., and Jefferys, J.G.R. (1998). "Electrical coupling underlies high-frequency oscillations in thehippocampus in vitro". Nature 394: 189–192.Fries, P., Schroder, J.-H., Roelfsema, P.R., Singer, W., and Engel, A.K. (2002). "Oscillatory Neuronal Synchronization in Primary VisualCortex as a Correlate of Stimulus Selection". Journal of Neuroscience 22: 3739–3754.Galarreta, M., and Hestrin, S. (1999). "A network of fast-spiking cells in the neocortex connected by electrical synapses". Nature 402: 72–75.Gibson, J.R., Beierlein, M., and Connors, B.W. (1999). "Two networks of electrically coupled inhibitory neurons in neocortex". Nature 402:75–79.Hormuzdi, S.G., Filippov, M.A., Mitropoulou, G., Monyer, H., and Bruzzone, R. (2004). "Electrical synapses: a dynamic signaling systemthat shapes the activity of neuronal networks". Biochimica et Biophysica Acta 1662: 113–137.LeBeau, F.E.N., Traub, R.D., Monyer, H., Whittington, M.A., and Buhl, E.H. (2003). "The role of electrical signaling via gap junctions in thegeneration of fast network oscillations". Brain Research Bulletin 62: 3–13.Velazquez, J.L.P., and Carlen, P.L. (2000). "Gap junctions, synchrony and seizures". Trends in Neurosciences 23: 68–74.Rozental, R., and de Carvalho, A.C.C. (2000). "Introduction". Brain Research Reviews 32: 1–2.

[11] A 1995 issue of Psyche was devoted to this:Maudlin, T. (1995). "Between The Motion And The Act... A Review of Shadows of the Mind by Roger Penrose" (http:/ / journalpsyche. org/ojs-2. 2/ index. php/ psyche/ article/ view/ 2396/ 2325). Psyche 2. .Klein, S.A. (1995). "Is Quantum Mechanics Relevant To Understanding Consciousness A Review of Shadows of the Mind by Roger Penrose"(http:/ / journalpsyche. org/ ojs-2. 2/ index. php/ psyche/ article/ view/ 2397/ 2326). Psyche 2. .McCullough, D. (1995). "Can Humans Escape Gödel? A Review of Shadows of the Mind by Roger Penrose" (http:/ / journalpsyche. org/ojs-2. 2/ index. php/ psyche/ article/ view/ 2398/ 2327). Psyche 2. .Moravec, H. (1995). "Roger Penrose's Gravitonic Brains A Review of Shadows of the Mind by Roger Penrose" (http:/ / journalpsyche. org/ojs-2. 2/ index. php/ psyche/ article/ view/ 2399/ 2328). Psyche 2. .Baars, B.J. (1995). "Can Physics Provide a Theory of Consciousness? A Review of Shadows of the Mind by Roger Penrose" (http:/ /journalpsyche. org/ ojs-2. 2/ index. php/ psyche/ article/ view/ 2401/ 2330). Psyche 2. .Chalmers, D.J. (1995). "Minds, Machines, And Mathematics A Review of Shadows of the Mind by Roger Penrose" (http:/ / journalpsyche.org/ ojs-2. 2/ index. php/ psyche/ article/ view/ 2402/ 2331). Psyche 2. .McCarthy, J. (1995). "Awareness and Understanding in Computer Programs A Review of Shadows of the Mind by Roger Penrose" (http:/ /journalpsyche. org/ ojs-2. 2/ index. php/ psyche/ article/ view/ 2403/ 2332). Psyche 2. .McDermott, D. (1995). " Penrose is Wrong" (http:/ / journalpsyche. org/ ojs-2. 2/ index. php/ psyche/ article/ view/ 2406/ 2335). Psyche 2..

[12] Grush, R., Churchland, P.S. (1995). "Gaps in Penrose's toilings" (http:/ / mind. ucsd. edu/ papers/ penrose/ penrose. pdf). Journal ofConsciousness Studies 2 (1): 10–29. .

[13] Feferman, S. (1996). "Penrose's Gödelian argument" (http:/ / math. stanford. edu/ ~feferman/ papers/ penrose. pdf). Psyche 2: 21–32. .[14] Tegmark, M.. "Importance of quantum decoherence in brain processes". Physical Review E 61: 4194–4206.

Orch-OR 92

[15] Engel, G.S., Calhoun, T.R., Read, E.L., Ahn, T.-K., Mancal, T., Cheng, Y.-C., Blankenship, R.E., and Fleming, G.R. (2007). "Evidence forwavelike energy transfer through quantum coherence in photosynthetic systems". Nature 446: 782–786.

[16] Hagan, S., Hameroff, S., and Tuszyński, J.. "Quantum Computation in Brain Microtubules? Decoherence and Biological Feasibility" (http:/ /arxiv. org/ abs/ quant-ph/ 0005025). Physical Review E 65: 061901. .

[17] Hameroff, S. (2006), "Consciousness, Neurobiology and Quantum Mechanics", in Tuszynski, Jack, The Emerging Physics of Consciousness,Springer, pp. 193–253

[18] Hameroff, S.R. (1998). "Quantum Computation In Brain Microtubules? The Penrose-Hameroff "Orch OR" model of consciousness" (http:/ /www. quantumconsciousness. org/ penrose-hameroff/ quantumcomputation. html). Philosophical Transactions Royal Society London (A) 356:1869–1896. .

[19] Kikkawa, M., Ishikawa, T., Nakata, T., Wakabayashi, T., Hirokawa, N. (1994). "Direct visualization of the microtubule lattice seam both invitro and in vivo" (http:/ / jcb. rupress. org/ cgi/ content/ abstract/ 127/ 6/ 1965). Journal of Cell Biology 127 (6): 1965–1971.doi:10.1083/jcb.127.6.1965. .

[20] Kikkawa, M., Metlagel, Z. (2006). "A molecular "zipper" for microtubules" (http:/ / dx. doi. org/ 10. 1016/ j. cell. 2006. 12. 009). Cell 127(7): 1302–1304. doi:doi:10.1016/j.cell.2006.12.009. .

[21] Reimers, J.R., McKemmish, L.K., McKenzie, R.H., Mark, A.E., and Hush, N.S. (2009). "Weak, strong, and coherent regimes of Fröhlichcondensation and their applications to terahertz medicine and quantum consciousness". Proceedings of the National Academy of Sciences 106:4219–4224. doi:10.1073/pnas.0806273106.

[22] Georgiev, D.D. (2007). "Falsifications of Hameroff-Penrose Orch OR model of consciousness and novel avenues for development ofquantum mind theory" (http:/ / philsci-archive. pitt. edu/ archive/ 00003049/ ). NeuroQuantology 5 (1): 145–174. .

[23] Georgiev, D.D. (2009). "Remarks on the number of tubulin dimers per neuron and implications for Hameroff-Penrose Orch" (http:/ /precedings. nature. com/ documents/ 3860/ version/ 1). NeuroQuantology 7 (4): 677–679. .

[24] De Zeeuw, C.I., Hertzberg, E.L., Mugnaini, E.. "The dendritic lamellar body: A new neuronal organelle putatively associated withdendrodentritic gap junctions". Journal of Neuroscience 15: 1587–1604.

[25] Georgiev, D.D. (2009). "Tubulin-bound GTP can not pump microtubule coherence in stable microtubules. Towards a revision ofmicrotubule based quantum models of mind" (http:/ / www. neuroquantology. com/ journal/ index. php/ nq/ article/ view/ 358).NeuroQuantology 7 (4): 538–547. .

[26] McKemmish, L.K., Reimers, J.R., McKenzie, R.H., Mark, A.E., and Hush, N.S. (2009). "Penrose-Hameroff orchestrated objective-reductionproposal for human consciousness is not biologically feasible" (http:/ / link. aps. org/ doi/ 10. 1103/ PhysRevE. 80. 021912). Physical ReviewE 80: 021912–021916. doi:10.1103/PhysRevE.80.021912. .

[27] http:/ / online. kitp. ucsb. edu/ plecture/ penrose/[28] http:/ / www. quantum-mind. co. uk

Debye sheath 93

Debye sheathThe Debye sheath (also electrostatic sheath) is a layer in a plasma which has a greater density of positive ions, andhence an overall excess positive charge, that balances an opposite negative charge on the surface of a material withwhich it is in contact. The thickness of such a layer is several Debye lengths thick, a value whose size depends onvarious characteristics of plasma (eg. temperature, density, etc).A Debye sheath arises in a plasma because the electrons usually have a temperature on the order of or greater thanthat of the ions and are much lighter. Consequently they are faster than the ions by at least a factor of .At the interface to a material surface, therefore, the electrons will fly out of the plasma, charging the surface negativerelative to the bulk plasma. Due to Debye shielding, the scale length of the transition region will be the Debye length

. As the potential increases, more and more electrons are reflected by the sheath potential. An equilibrium isfinally reached when the potential difference is a few times the electron temperature.The Debye sheath is the transition from a plasma to a solid surface. Similar physics is involved between two plasmaregions that have different characteristics; the transition between these regions is known as a double layer, andfeatures one positive, and one negative layer.

Description

Positive ion sheaths around grid wires in a thermionic gas tube, where ⊕represents a positive charge (not to scale) (After Langmuir, 1929)

Sheaths were first described by Americanphysicist Irving Langmuir. In 1923 hewrote:

Electrons are repelled from thenegative electrode while positive ionsare drawn towards it. Around eachnegative electrode there is thus asheath of definite thickness containingonly positive ions and neutral atoms.[..] Electrons are reflected from theoutside surface of the sheath while allpositive ions which reach the sheathare attracted to the electrode. [..] itfollows directly that no change occursin the positive ion current reaching theelectrode. The electrode is in factperfectly screened from the dischargeby the positive ion sheath, and itspotential cannot influence thephenomena occurring in the arc, northe current flowing to theelectrode."[1]

Langmuir and co-author Albert W. Hullfurther described a sheath formed in athermionic valve:

"Figure 1 shows graphically the condition that exists in such a tube containing mercury vapor. The spacebetween filament and plate is filled with a mixture of electrons and positive ions, in nearly equal numbers, to

Debye sheath 94

which has been given the name "plasma". A wire immersed in the plasma, at zero potential with respect to it,will absorb every ion and electron that strikes it. Since the electrons move about 600 times as fast as the ions,600 times as many electrons will strike the wire as ions. If the wire is insulated it must assume such a negativepotential that it receives equal numbers of electrons and ions, that is, such a potential that it repels all but 1 in600 of the electrons headed for it."Suppose that this wire, which we may take to be part of a grid, is made still more negative with a view tocontrolling the current through the tube. It will now repel all the electrons headed for it, but will receive all thepositive ions that fly toward it. There will thus be a region around the wire which contains positive ions and noelectrons, as shown diagrammatically in Fig. 1. The ions are accelerated as they approach the negative wire,and there will exist a potential gradient in this sheath, as we may call it, of positive ions, such that the potentialis less and less negative as we recede from the wire, and at a certain distance is equal to the potential of theplasma. This distance we define as the boundary of the sheath. Beyond this distance there is no effect due tothe potential of the wire."[2]

Mathematical treatment

The planar sheath equationThe quantitative physics of the Debye sheath is determined by four phenomena:Energy conservation of the ions: If we assume for simplicity cold ions of mass entering the sheath with avelocity , having charge opposite to the electron, conservation of energy in the sheath potential requires

,

where is the charge of the electron taken positively, i.e. x .Ion continuity: In the steady state, the ions do not build up anywhere, so the flux is everywhere the same:

.Boltzmann relation for the electrons: Since most of the electrons are reflected, their density is given by

.

Poisson's equation: The curvature of the electrostatic potential is related to the net charge density as follows:

.

Combining these equations and writing them in terms of the dimensionless potential, position, and ion speed,

we arrive at the sheath equation:

.

Debye sheath 95

The Bohm sheath criterion

The sheath equation can be integrated once by multiplying by :

At the sheath edge ( ), we can define the potential to be zero ( ) and assume that the electric field isalso zero ( ). With these boundary conditions, the integrations yield

This is easily rewritten as an integral in closed form, although one that can only be solved numerically. Nevertheless,an important piece of information can be derived analytically. Since the left-hand-side is a square, theright-hand-side must also be non-negative for every value of , in particular for small values. Looking at theTaylor expansion around , we see that the first term that does not vanish is the quadratic one, so that we canrequire

,

or

,or

.This inequality is known as the Bohm sheath criterion after its discoverer, David Bohm. If the ions are entering thesheath too slowly, the sheath potential will "eat" its way into the plasma to accelerate them. Ultimately a so-calledpre-sheath will develop with a potential drop on the order of and a scale determined by the physics ofthe ion source (often the same as the dimensions of the plasma). Normally the Bohm criterion will hold withequality, but there are some situations where the ions enter the sheath with supersonic speed.

The Child-Langmuir LawAlthough the sheath equation must generally be integrated numerically, we can find an approximate solutionanalytically by neglecting the term. This amounts to neglecting the electron density in the sheath, or onlyanalyzing that part of the sheath where there are no electrons. For a "floating" surface, i.e. one that draws no netcurrent from the plasma, this is a useful if rough approximation. For a surface biased strongly negative so that itdraws the ion saturation current, the approximation is very good. It is customary, although not strictly necessary, tofurther simplify the equation by assuming that is much larger than unity. Then the sheath equation takes onthe simple form

.

As before, we multiply by and integrate to obtain

,

or

.This is easily integrated over ξ to yield

,

Debye sheath 96

where is the (normalized) potential at the wall (relative to the sheath edge), and d is the thickness of the sheath.Changing back to the variables and and noting that the ion current into the wall is , we have

.

This equation is known as Child's Law, after Clement Dexter Child (1868-1933), who first published it in 1911, oras the Child-Langmuir Law, honoring as well Irving Langmuir, who discovered it independently and published in1913. It was first used to give the space-charge-limited current in a vacuum diode with electrode spacing d. It canalso be inverted to give the thickness of the Debye sheath as a function of the voltage drop by setting :

.

See also• Ambipolar diffusion

References[1] Langmuir, Irving, " Positive Ion Currents from the Positive Column of Mercury Arcs (http:/ / adsabs. harvard. edu/ abs/ 1923Sci. . . . 58. .

290L)" (1923) Science, Volume 58, Issue 1502, pp. 290-291[2] Albert W. Hull and Irving Langmuir, " Control of an Arc Discharge by Means of a Grid (http:/ / www. pubmedcentral. nih. gov/ articlerender.

fcgi?artid=522437)", Proc Natl Acad Sci U S A. 1929 March 15; 15(3): 218–225

John Stewart BellJohn Stewart Bell (28 June 1928 – 1 October 1990) was an Irish physicist, and the originator of Bell's Theorem,one of the most important theorems in quantum physics.

Life and workHe was born in Belfast, Northern Ireland, and graduated in experimental physics at the Queen's University ofBelfast, in 1948. He went on to complete a PhD at the University of Birmingham, specialising in nuclear physics andquantum field theory. His career began with the British Atomic Energy Agency, in Malvern, Britain's, then HarwellLaboratory. After several years he moved to the European Center for Nuclear Research (CERN, Conseil Européenpour la Recherche Nucléaire). Here he worked almost exclusively on theoretical particle physics and on acceleratordesign, but found time to pursue a major avocation, investigating the foundations of quantum theory.In 1964, after a year's leave from CERN that he spent at Stanford University, the University of Wisconsin–Madisonand Brandeis University, he wrote a paper entitled "On the Einstein-Podolsky-Rosen Paradox"[1] . In this work, heshowed that carrying forward EPR's analysis[2] permits one to derive the famous Bell's inequality. This inequality,derived from certain assumptions, conflicts with the predictions of quantum theory.There is some disagreement regarding what Bell's inequality — in conjunction with the EPR analysis — can be saidto imply. Bell held that not only local hidden variables, but any and all local theoretical explanations must conflictwith the predictions of quantum theory: "It is known that with Bohm's example of EPR correlations, involvingparticles with spin, there is an irreducible nonlocality."[3] According to an alternative interpretation, not all localtheories in general, but only local hidden variables theories (or "local realist" theories) have shown to beincompatible with the predictions of quantum theory.Bell's interest in hidden variables was motivated by the existence in the formalism of Quantum Mechanics of a "movable boundary" between the quantum system and the classical apparatus[4] : "A possibility is that we find

John Stewart Bell 97

exactly where the boundary lies. More plausible to me is that we will find that there is no boundary. ... The wavefunctions would prove to be a provisional or incomplete description of the quantum-mechanical part, of which anobjective account would become possible. It is this possibility, of a homogeneous account of the world, which is forme the chief motivation of the study of the so-called 'hidden variable' possibility". Bell was impressed that in theformulation of Bohm’s nonlocal hidden variable theory, no such boundary is needed, and it was this which sparkedhis interest in the field of research. Bell also criticized the standard formalism of Quantum Mechanics on the groundsof lack of physical precision[5] : "For the good books known to me are not much concerned with physical precision.This is clear already from their vocabulary. Here are some words which, however legitimate and necessary inapplication, have no place in a formulation with any pretension to physical precision: system, apparatus,environment, microscopic, macroscopic, reversible, irreversible, observable, information, measurement. .... On thislist of bad words from good books, the worst of all is 'measurement'."But if he were to thoroughly explore the viability of Bohm's theory, Bell needed to answer the challenge of theso-called impossibility proofs against hidden variables. Bell addressed these in a paper entitled "On the Problem ofHidden Variables in Quantum Mechanics".[6] Here he showed that von Neumann’s argument[7] does not proveimpossibility, as it claims. The argument fails in this regard due to its reliance on a physically unreasonableassumption. In this same work, Bell showed that a stronger effort at such a proof (based upon Gleason's theorem)also fails to eliminate the hidden variables program. (The flaw in von Neumann's proof was previously discovered byGrete Hermann in 1935, but did not become common knowledge until rediscovered by Bell.)If these attempts to disprove hidden variables failed, can Bell's resolution of the EPR paradox be considered asuccess? According to Bell's interpretation, quantum mechanics itself has been demonstrated to be irreduciblynonlocal. Therefore, one cannot fault a hidden variables scheme if, as in the pilot wave theory of de Broglie andBohm, it includes a violation of local causality.In 1972 the first of many experiments that have shown (under the extrapolation to ideal detector efficiencies) aviolation of Bell's Inequality was conducted. Bell himself concludes from these experiments that "It now seems thatthe non-locality is deeply rooted in quantum mechanics itself and will persist in any completion."[8] This, accordingto Bell, also implied that quantum theory is not locally causal and cannot be embedded into any locally causaltheory.Bell remained interested in objective 'observer-free' quantum mechanics. He stressed that at the most fundamentallevel, physical theories ought not to be concerned with observables, but with 'be-ables': "The beables of the theoryare those elements which might correspond to elements of reality, to things which exist. Their existence does notdepend on 'observation'."[9] He remained impressed with Bohm's hidden variables as an example of such a schemeand he attacked the more subjective alternatives such as the Copenhagen interpretation. [10]

John Stewart Bell 98

Blue plaque honouring John Bell at the Queen'sUniversity of Belfast

Bell seemed to be quite comfortable with the notion that futureexperiments would continue to agree with quantum mechanics andviolate his inequalities. Referring to the Bell test experiments, heremarked:

"It is difficult for me to believe that quantummechanics, working very well for currently practicalset-ups, will nevertheless fail badly withimprovements in counter efficiency ..."[11]

Some people continue to believe that agreement with Bell'sinequalities might yet be saved. They argue that in the future muchmore precise experiments could reveal that one of the knownloopholes, for example the so-called "fair sampling loophole", hadbeen biasing the interpretations. This latter loophole, firstpublicized by Philip Pearle in 1970[12] , is such that increases incounter efficiency decrease the measured quantum correlation,eventually destroying the empirical match with quantummechanics. Most mainstream physicists are highly skeptical aboutall these "loopholes", admitting their existence but continuing tobelieve that Bell's inequalities must fail.

Bell died unexpectedly of a cerebral hemorrhage in Belfast in 1990. His contribution to the issues raised by EPR wassignificant. Some regard him as having demonstrated the failure of local realism (local hidden variables). Bell's owninterpretation is that locality itself met its demise.

See also• Bell's theorem, published in the mid-1960s• Bell's spaceship paradox• EPR paradox, a thought experiment by Einstein, Podolsky, and Rosen published in 1935 as an attack on quantum

theory• CHSH Bell test, an application of Bell's theorem• Quantum mechanical Bell test prediction• Quantum entanglement• Local hidden variable theory• Bell state• Superdeterminism• Afshar experiment

References• Aczel, Amir D. (2001) Entanglement: The Greatest Mystery in Physics. New York: Four Walls Eight Windows• Bell, John S. (1987) Speakable and Unspeakable in Quantum Mechanics. Cambridge Univ. Press, ISBN

0-521-36869-3, 2004 edition with introduction by Alain Aspect and two additional papers: ISBN 0-521-52338-9.• Albert Einstein, Podolsky, Rosen, (1935) "Can Quantum Mechanical Description of Physical Reality Be

Considered Complete?" Phys. Rev. 47: 777.• Gilder, Louisa (2008) The Age of Entanglement: When Quantum Physics Was Reborn. New York: Alfred A.

Knopf.• Pearle, Philip (1970) "Hidden-Variable Example Based upon Data Rejection," Physical Review D 2: 1418-25.

John Stewart Bell 99

• John von Neumann (1932) Mathematical Foundations of Quantum Mechanics. Princeton Univ. Press. 1996 ed.:ISBN 0-691-02893-1.

External links• MacTutor profile (University of St. Andrews) [13]

• John Bell and the most profound discovery of science (December 1998) [14]

• The Most Profound Discovery of Science (September 2006) [15]

References[1] John Bell, Speakable and Unspeakable in Quantum Mechanics, p. 14[2] Einstein, et al., "Can Quantum Mechanical Description of Physical Reality Be Considered Complete?"[3] Bell, p. 196[4] Introduction to the hidden-variable question, pg. 30, in Speakable and Unspeakable in Quantum Mechanics.[5] Against 'measurement' , pg. 215, in Speakable and Unspeakable in Quantum Mechanics.[6] Bell, p.1[7] John von Neumann, Mathematical Foundations of Quantum Mechanics[8] Bell, p. 132[9] Bell, p. 174[10] Bell, p. 92, 133, 181[11] Bell, p. 109[12] Philip Pearle, Hidden-Variable Example Based upon Data Rejection[13] http:/ / www-groups. dcs. st-and. ac. uk/ ~history/ Biographies/ Bell_John. html[14] http:/ / physicsweb. org/ articles/ world/ 11/ 12/ 8[15] http:/ / www. rds. ie/ home/ index. aspx?id=1755

Karl H. Pribram

Karl Pribram in 2008

Karl H. Pribram (born February 25, 1919 in Vienna, Austria) is aprofessor at Georgetown University , and an emeritus professor ofpsychology and psychiatry at Stanford University and Radford University.Board-certified as a neurosurgeon, Pribram did pioneering work on thedefinition of the limbic system, the relationship of the frontal cortex to thelimbic system, the sensory-specific "association" cortex of the parietal andtemporal lobes, and the classical motor cortex of the human brain. To thegeneral public, Pribram is best known for his development of theholonomic brain model of cognitive function and his contribution toongoing neurological research into memory, emotion, motivation andconsciousness. American best selling author Katherine Neville is hissignificant other.

Karl H. Pribram 100

Holonomic modelPribram's holonomic model of brain processing states that, in addition to the circuitry accomplished by the largefiber tracts in the brain, processing also occurs in webs of fine fiber branches (for instance, dendrites) that formwebs. This type of processing is properly described by Gabor quanta of information, wavelets that are used inquantum holography, the basis of fMRI, PET scans and other image processing procedures.Gabor wavelets are windowed Fourier transforms that convert complex spatial (and temporal) patterns intocomponent waves whose amplitudes at their intersections become reinforced or diminished. Fourier processes are thebasis of holography. Holograms can correlate and store a huge amount of information - and have the advantage thatthe inverse transform returns the results of correlation into the spatial and temporal patterns that guide us innavigating our universe.David Bohm had suggested that were we to view the cosmos without the lenses that outfit our telescopes, theuniverse would appear to us as a hologram. Pribram extended this insight by noting that were we deprived of thelenses of our eyes and the lens like processes of our other sensory receptors, we would be immersed in holographicexperiences.

Other contributionsIn the late 1940s and early 1950s, Pribram's neurobehavioral experiments established the composition of the limbicsystem and the executive functions of the prefrontal cortex. Pribram also discovered the sensory specific systems ofthe association cortex, and showed that these systems operate to organize the choices we make among sensorystimuli, not the sensing of the stimuli themselves.

Bibliography• Miller, George; Galanter, Eugene, & Pribram, Karl (1960). Plans and the structure of behavior. New York: Holt,

Rinehart and Winston. ISBN 0030100755.• Pribram, Karl H. (1969). Brain and behaviour. Hammondsworth: Penguin Books. ISBN 0140805214.• Pribram, Karl (1971). Languages of the brain; experimental paradoxes and principles in neuropsychology.

Englewood Cliffs, N. J.: Prentice-Hall. ISBN 0135227305.• Pribram, Karl; Gill, Morton M. (1976). Freud's "Project" re-assessed: preface to contemporary cognitive theory

and neuropsychology. New York: Basic Books. ISBN 0465025692.• Pribram, Karl (1991). Brain and perception: holonomy and structure in figural processing. Hillsdale, N. J.:

Lawrence Erlbaum Associates. ISBN 0898599954.• Globus, Gordon G.; Pribram, Karl H., & Vitiello, Giuseppe (2004-09-30). Brain And Being: At The Boundary

Between Science, Philosophy, Language, And Arts (Advances in Consciousness Research, 58). John BenjaminsPublishing Co.. ISBN 158811550X.

• Pribram, Karl (ed.) (1969). On the biology of learning. New York: Harcourt Brace & World. ISBN 0155675206.• Pribram, Karl, & Broadbent, Donald (eds.) (1970). Biology of memory. New York: Academic Press.

ISBN 0125643500.• Pribram, K. H., & Luria, A. R. (eds.) (1973). Psychophysiology of the frontal lobes. New York: Academic Press.

ISBN 0125643403.• Pribram, Karl, & Isaacson, Robert L. (eds.) (1975). The Hippocampus. New York: Plenum Press.

ISBN 0306375354.• Pribram, Karl (ed.) (1993). Rethinking neural networks: quantum fields and biological data. Hillsdale, N. J.:

Erlbaum. ISBN 0805814663.• Pribram, Karl (ed.) (1994). Origins: brain and self organization. Hillsdale, N. J.: Lawrence Erlbaum.

ISBN 0805817867.

Karl H. Pribram 101

• King, Joseph, & Pribram, Karl (eds.) (1995). Scale in conscious experience: Is the brain too important to be leftto the specialists to study?. Mahwah, N. J.: Lawrence Erlbaum Associates. ISBN 0805821783.

• Pribram, Karl, & King, Joseph (eds.) (1996). Learning as self-organization. Mahwah, N. J.: L. ErlbaumAssociates. ISBN 080582586X.

• Pribram, Karl (ed.) (1998). Brain and values: is a biological science of values possible. Mahwah, N. J.: LawrenceErlbaum Associates. ISBN 0805831541.

• Pribram, Karl (2004). "Brain and Mathematics" [1]. Pari Center for New Learning. Retrieved 2007-10-25.• "Like Bohm, Karl Pribram sees the holographic nature of reality" [2]. The Ground of Faith. October 2003.

Retrieved 2007-10-25.• Mishlove, Jeffrey (1998). "The Holographic Brain with Karl Pribram, MA; Ph.D." [11]. TWM.co.nz. Retrieved

2007-10-25.

External links• "The Holographic Brain" [3] - Dr. Jeffrey Mishlove interviews Karl Pribham• "Comparison between Holographic Brain Theory and conventional models of neuronal computation" [15] –

academic paper on Pribham's work• "Pribram Receives Havel Prize For His Work in Neuroscience" [4] – news article• "Winner 1998 Noetic Medal for Consciousness & Brain Research - For Lifetime Achievement" [5]

• Global Lens Interview [6] (Video)• [28] quantum mind

References[1] http:/ / www. paricenter. com/ library/ papers/ pribram01. php[2] http:/ / homepages. ihug. co. nz/ ~thegroundoffaith/ issues/ 2003-10/ pribram. html[3] http:/ / homepages. ihug. co. nz/ ~sai/ pribram. htm[4] http:/ / www. katherineneville. com/ karl_havel_prize. htm[5] http:/ / www. mindspring. com/ ~quantum. computing/[6] http:/ / www. immaginehdv. com/ detail. php?c=2& i=b90c95bf29b3909ced9b95a10d865cd329684d33

Implicate and explicate order according to David Bohm 102

Implicate and explicate order according to DavidBohmDavid Bohm proposed a cosmological order radically different from generally accepted conventions, which heexpressed as a distinction between the implicate and explicate order, described in the book Wholeness and theImplicate Order:

In the enfolded [or implicate] order, space and time are no longer the dominant factors determining therelationships of dependence or independence of different elements. Rather, an entirely different sort of basicconnection of elements is possible, from which our ordinary notions of space and time, along with those ofseparately existent material particles, are abstracted as forms derived from the deeper order. These ordinarynotions in fact appear in what is called the "explicate" or "unfolded" order, which is a special and distinguishedform contained within the general totality of all the implicate orders (Bohm, 1980, p. xv).

David Bohm's challenges to some generally prevailing viewsIn proposing this new notion of order, Bohm explicitly challenged a number of tenets that are fundamental to muchscientific work. The tenets challenged by Bohm include:1. That phenomena are reducible to fundamental particles and laws describing the behaviour of particles, or more

generally to any static (i.e. unchanging) entities, whether separate events in space-time, quantum states, or staticentities of some other nature.

2. Related to (1), that human knowledge is most fundamentally concerned with mathematical prediction of statisticalaggregates of particles.

3. That an analysis or description of any aspect of reality (e.g. quantum theory, the speed of light) can be unlimitedin its domain of relevance.

4. That the Cartesian coordinate system, or its extension to a curvilinear system, is the deepest conception ofunderlying order as a basis for analysis and description of the world.

5. That there is ultimately a sustainable distinction between reality and thought, and that there is a correspondingdistinction between the observer and observed in an experiment or any other situation (other than a distinctionbetween relatively separate entities valid in the sense of explicate order).

6. That it is, in principle, possible to formulate a final notion concerning the nature of reality; e.g. a Theory ofEverything.

Implicate and explicate order according to David Bohm 103

A hydrogen atom and its constituent particles: an example ofa small collection of posited building blocks of the universe

Bohm’s proposals have at times been dismissed largely on thebasis of such tenets, without due consideration necessarilygiven to the fact that they had been challenged by Bohm.

Bohm’s paradigm is inherently antithetical to reductionism, inmost forms, and accordingly can be regarded as a form ofontological holism. On this, Bohm noted of prevailing viewsamong physicists: "the world is assumed to be constituted of aset of separately existent, indivisible and unchangeable'elementary particles', which are the fundamental 'buildingblocks' of the entire universe … there seems to be anunshakable faith among physicists that either such particles, orsome other kind yet to be discovered, will eventually makepossible a complete and coherent explanation of everything"(Bohm, 1980, p. 173).

In Bohm’s conception of order, then, primacy is given to theundivided whole, and the implicate order inherent within thewhole, rather than to parts of the whole, such as particles, quantum states, and continua. For Bohm, the wholeencompasses all things, structures, abstractions and processes, including processes that result in (relatively) stablestructures as well as those that involve metamorphosis of structures or things. In this view, parts may be entitiesnormally regarded as physical, such as atoms or subatomic particles, but they may also be abstract entities, such asquantum states. Whatever their nature and character, according to Bohm, these parts are considered in terms of thewhole, and in such terms, they constitute relatively autonomous and independent "sub-totalities". The implication ofthe view is, therefore, that nothing is entirely separate or autonomous.

Bohm (1980, p. 11) said: "The new form of insight can perhaps best be called Undivided Wholeness in FlowingMovement. This view implies that flow is, in some sense, prior to that of the ‘things’ that can be seen to form anddissolve in this flow". According to Bohm, a vivid image of this sense of analysis of the whole is afforded by vortexstructures in a flowing stream. Such vortices can be relatively stable patterns within a continuous flow, but such ananalysis does not imply that the flow patterns have any sharp division, or that they are literally separate andindependently existent entities; rather, they are most fundamentally undivided. Thus, according to Bohm’s view, thewhole is in continuous flux, and hence is referred to as the holomovement (movement of the whole).

Quantum theory and relativity theoryA key motivation for Bohm in proposing a new notion of order was what he saw as the incompatibility of quantumtheory with relativity theory, with respect to certain features of the theories as observed in relevant experimentalcontexts. Bohm (1980, p. xv) summarised the state of affairs he perceived to exist:

…in relativity, movement is continuous, causally determinate and well defined, while in quantummechanics it is discontinuous, not causally determinate and not well-defined. Each theory is committedto its own notions of essentially static and fragmentary modes of existence (relativity to that of separateevents connectible by signals, and quantum mechanics to a well-defined quantum state). One thus seesthat a new kind of theory is needed which drops these basic commitments and at most recovers someessential features of the older theories as abstract forms derived from a deeper reality in which whatprevails is unbroken wholeness.

Bohm maintained that relativity and quantum theory are in basic contradiction in these essential respects, and that a new concept of order should begin with that towards which both theories point: undivided wholeness. This should not be taken to mean that he advocated such powerful theories be discarded. He argued that each was relevant in a

Implicate and explicate order according to David Bohm 104

certain context—i.e. a set of interrelated conditions within the explicate order—rather than having unlimited scope,and that apparent contradictions stem from attempts to overgeneralize by superposing the theories on one another,implying greater generality or broader relevance than is ultimately warranted. Thus, Bohm (1980, pp. 156-167)argued: "... in sufficiently broad contexts such analytic descriptions cease to be adequate ... 'the law of the whole' willgenerally include the possibility of describing the 'loosening' of aspects from each other, so that they will berelatively autonomous in limited contexts ... however, any form of relative autonomy (and heteronomy) is ultimatelylimited by holonomy, so that in a broad enough context such forms are seen to be merely aspects, relevated in theholomovement, rather than disjoint and separately existent things in interaction".

Hidden variable theoryBohm proposed a hidden variable theory of quantum physics (see Bohm interpretation). According to Bohm, a keymotivation for doing so was purely to show the possibility of such theories. On this, Bohm (1980, p. 81) said "... itshould be kept in mind that before this proposal was made there had existed the widespread impression that noconceptions of hidden variables at all, not even if they were abstract, and hypothetical, could possibly be consistentwith the quantum theory". Bohm (1980, p. 110) also claimed that "the demonstration of the possibility of theories ofhidden variables may serve in a more general philosophical sense to remind us of the unreliability of conclusionsbased on the assumption of the complete universality of certain features of a given theory, however general theirdomain of validity seems to be". Another aspect of Bohm's motivation was to point out a confusion he perceived toexist in quantum theory. On the dominant approaches in quantum theory, he said: "...we wish merely to point out thatthis whole line of approach re-establishes at the abstract level of statistical potentialities the same kind of analysisinto separate and autonomous components in interaction that is denied at the more concrete level of individualobjects" Bohm (1980, p. 174).

Quantum entanglementCentral to Bohm's schema are correlations between observables of entities which seem separated by great distancesin the explicate order (such as a particular electron here on earth and an alpha particle in one of the stars in the Abell1835 galaxy, the farthest galaxy from Earth known to humans), manifestations of the implicate order. Withinquantum theory there is entanglement of such objects.This view of order necessarily departs from any notion which entails signalling, and therefore causality. Thecorrelation of observables does not imply a causal influence, and in Bohm's schema the latter represents 'relatively'independent events in space-time; and therefore explicate order.He also used the term unfoldment to characterise processes in which the explicate order becomes relevant (or"relevated"). Bohm likens unfoldment also to the decoding of a television signal to produce a sensible image on ascreen. The signal, screen, and television electronics in this analogy represent the implicate order whilst the imageproduced represents the explicate order. He also uses an interesting example in which an ink droplet can beintroduced into a highly viscous substance (such as glycerine), and the substance rotated very slowly such that thereis negligible diffusion of the substance. In this example, the droplet becomes a thread which, in turn, eventuallybecomes invisible. However, by rotating the substance in the reverse direction, the droplet can essentially reform.When it is invisible, according to Bohm, the order of the ink droplet as a pattern can be said to be implicate withinthe substance.Further support for this is illustrated by dropping blue ink into a vat of spinning carbon tetrachloride and watch theink disperse. Reversing the spin of the vat will cause the ink to come back together into a blob, then it spreads outagain.In another analogy, Bohm asks us to consider a pattern produced by making small cuts in a folded piece of paper andthen, literally, unfolding it. Widely separated elements of the pattern are, in actuality, produced by the same originalcut in the folded piece of paper. Here the cuts in the folded paper represent the implicate order and the unfolded

Implicate and explicate order according to David Bohm 105

pattern represents the explicate order.

The hologram as analogy for the implicate order

In a holographic reconstruction, each region of a photographic platecontains the whole image

Bohm employed the hologram as a means ofcharacterising implicate order, noting that each regionof a photographic plate in which a hologram isobservable contains within it the wholethree-dimensional image, which can be viewed from arange of perspectives. That is, each region contains awhole and undivided image. In Bohm’s words:

There is the germ of a new notion of orderhere. This order is not to be understoodsolely in terms of a regular arrangement ofobjects (eg., in rows) or as a regulararrangement of events (e.g. in a series).Rather, a total order is contained, in someimplicit sense, in each region of space andtime. Now, the word 'implicit' is based onthe verb 'to implicate'. This means 'to foldinward' ... so we may be led to explore thenotion that in some sense each regioncontains a total structure 'enfolded' withinit".[1]

Bohm noted that although the hologram conveys undivided wholeness, it is nevertheless static.In this view of order, laws represent invariant relationships between explicate entities and structures, and thus Bohmmaintained that in physics, the explicate order generally reveals itself within well-constructed experimental contextsas, for example, in the sensibly observable results of instruments. With respect to implicate order, however, Bohmasked us to consider the possibility instead "that physical law should refer primarily to an order of undividedwholeness of the content of description similar to that indicated by the hologram rather than to an order of analysisof such content into separate parts …".[2]

A common grounding for consciousness and matter

Implicate and explicate order according to David Bohm 106

Karl Pribram and colleagues have presented evidencethat indicates that memories do not in general appear to

be localized in specific regions of brains

The implicate order represents the proposal of a generalmetaphysical concept in terms of which it is claimed that matterand consciousness might both be understood, in the sense that it isproposed that both matter and consciousness: (i) enfold thestructure of the whole within each region, and (ii) involvecontinuous processes of enfoldment and unfoldment. For example,in the case of matter, entities such as atoms may representcontinuous enfoldment and unfoldment which manifests as arelatively stable and autonomous entity that can be observed tofollow a relatively well-defined path in space-time. In the case ofconsciousness, Bohm pointed toward evidence presented by KarlPribram that memories may be enfolded within every region of thebrain rather than being localized (for example in particular regions

of the brain, cells, or atoms).

Bohm went on to say:As in our discussion of matter in general, it is now necessary to go into the question of how inconsciousness the explicate order is what is manifest ... the manifest content of consciousness is basedessentially on memory, which is what allows such content to be held in a fairly constant form. Ofcourse, to make possible such constancy it is also necessary that this content be organized, not onlythrough relatively fixed association but also with the aid of the rules of logic, and of our basic categoriesof space, time causality, universality, etc. ... there will be a strong background of recurrent stable, andseparable features, against which the transitory and changing aspects of the unbroken flow of experiencewill be seen as fleeting impressions that tend to be arranged and ordered mainly in terms of the vasttotality of the relatively static and fragmented content of [memories].[3]

Bohm also claimed that "as with consciousness, each moment has a certain explicate order, and in addition it enfoldsall the others, though in its own way. So the relationship of each moment in the whole to all the others is implied byits total content: the way in which it 'holds' all the others enfolded within it". Bohm characterises consciousness as aprocess in which at each moment, content that was previously implicate is presently explicate, and content whichwas previously explicate has become implicate.

One may indeed say that our memory is a special case of the process described above, for all that isrecorded is held enfolded within the brain cells and these are part of matter in general. The recurrenceand stability of our own memory as a relatively independent sub-totality is thus brought about as part ofthe very same process that sustains the recurrence and stability in the manifest order of matter ingeneral. It follows, then, that the explicate and manifest order of consciousness is not ultimately distinctfrom that of matter in general.[4]

Connections with other worksMany have seen strong connections between his ideas and ideas traditionally associated with Eastern Philosophy andreligion. Bohm himself seemed also to see such connections, as evidenced by his close relationship with JidduKrishnamurti. There are particularly strong connections to Buddhism. Some proponents of new age beliefs (such asshamanism) claim a connection with their belief systems as well. Bohm's ideas are also used in certain meditationpractices.Bohm may have known that his idea is a striking analogy to "intentional and extensional aboutness" to which R. A. Fairthorne (1969) insightfully referred information scientists. Searle's concept of aboutness is in sharp contrast to, and is as odd as Bohm's idea of wholeness. As the former is to the content, so the latter is to the context as the

Implicate and explicate order according to David Bohm 107

ultimate determiner of meaning. The holistic view of context, hence another striking analogy of wholeness, was firstput forward in The Meaning of Meaning by C. K. Ogden & I. A. Richards (1923), including the literary,psychological, and external. These are respectively analogous to Karl Popper's world 3, 2, and 1 appearing in hisObjective Knowledge (1972 and later ed.). Bohm's worldview of "undivided wholeness" is contrasted with Popper'sthree divided worlds.

Cells stained for keratin and DNA: suchparts of life exist because of the whole,

but also to sustain it

Bohm's views bear some similarities to those of Immanuel Kant, according toWouter Hanegraaff. For example, Kant held that the parts of an organism,such as cells, simultaneously exist to sustain the whole, and depend upon thewhole for their own existence and functioning. Kant also proposed that theprocess of thought plays an active role in organizing knowledge, whichimplies theoretical insights are instrumental to the process of acquiring factualknowledge.

Kant restricted knowledge to appearances only and denied the existence ofknowledge of any "thing in itself," but Bohm believed that theories in scienceare "forms of insight that arise in our attempts to obtain a perception of adeeper nature of reality as a whole" (Bohm & Hiley, 1993, p. 323). Thus forBohm the thing in itself is the whole of existence, conceived of not as acollection of parts but as an undivided movement. In this view Bohm is closerto Kant's critic, Arthur Schopenhauer, who identified the thing in itself with the will, an inner metaphysical realitythat grounds all outer phenomena. Schopenhauer's will plays a role analogous to that of the implicate order; forexample, it is objectified (Bohm might say it is "made explicate") to form physical matter. And Bohm's concept thatconsciousness and matter share a common ground resembles Schopenhauer's claim that even inanimate objectspossess an inward noumenal nature. In The World as Will and Representation, Schopenhauer (1819/1995) describedthis ground thus:

When I consider the vastness of the world, the most important factor is that this existence-in-itself, of whichthe world is the manifestation, cannot, whatever it may be, have its true self spread out and dispersed in thisfashion in boundless space, but that this endless extension belongs only to its manifestation, whileexistence-in-itself, on the contrary, is present entire and undivided in everything in nature and in everythingthat lives. (p. 60)

Implicate and explicate order according to David Bohm 108

For Bohm, life is a continuous flowing process of enfoldment and unfoldment involvingrelatively autonomous entities. DNA 'directs' the environment to form a living thing. Life

can be said to be implicate in ensembles of atoms that ultimately form life.

See also

• Implicature• Holographic principle• The Holographic Universe• Holomovement• Arthur Schopenhauer• Brahman• Buddhism• Immanuel Kant• Kabbalism• Laminar flow• Meditation for Spiritual Unfoldment• Mind's eye• Noumenon• Parable of the cave• Plato• Samsara• Taoism

• Unobservables

References• Bohm, D. (1980). Wholeness and the Implicate Order. London: Routledge. ISBN 0-7100-0971-2• Bohm, D., & Hiley, B. J. (1993). The Undivided Universe. London: Routledge. ISBN 0-415-06588-7• Kauffman, S. (1995). At Home in the Universe. New York: Oxford University Press. hardcover: ISBN

0-19-509599-5, paperback ISBN 0-19-511130-3• Kauffman, S. (2000). Investigations. New York: Oxford University Press.• Kuhn, T.S. (1961). The function of measurement in modern physical science. ISIS, 52, 161-193.• Schopenhauer, A. (1819/1995). The World as Will and Idea. (D. Derman, Ed.; J. Berman, Trans.). London:

Everyman. ISBN 0-460-87505-1

Further reading• Michael Talbot. The Holographic Universe, Harpercollins (1991)

External links• Interview with David Bohm [5] – An interview with Bohm concerning this particular subject matter conducted by

F. David Peat.• Excerpt from The Holographic Universe [6]– Parallels some of the experiences of 18th century Swedish mystic,

Emanuel Swedenborg, with David Bohm's ideas.

Implicate and explicate order according to David Bohm 109

References[1] (Bohm, 1980, p. 149)[2] (1980, p. 147)[3] (1980, p. 205)[4] (Bohm, 1980, p. 208)

Hidden variable theoryHistorically, in physics, hidden variable theories were espoused by a minority of physicists who argued that thestatistical nature of quantum mechanics indicated that quantum mechanics is "incomplete". Albert Einstein, the mostfamous proponent of hidden variables, insisted that, "I am convinced God does not play dice"[1] — meaning that hebelieved that physical theories must be deterministic to be complete.[2] Later, Bell's theorem would suggest (in theopinion of most physicists and contrary to Einstein's assertion) that local hidden variables are impossible. It wasthought that if hidden variables exist, new physical phenomena beyond quantum mechanics are needed to explain theuniverse as we know it.The most famous such theory (because it gives the same answers as quantum mechanics, thus invalidating thefamous theorem by von Neumann that no hidden variable theory reproducing the statistical predictions of QM ispossible) is that of David Bohm. It is most commonly known as the Bohm interpretation or the Causal Interpretationof quantum mechanics. Bohm's (nonlocal) hidden variable is called the quantum potential. Nowadays Bohm's theoryis considered to be one of many interpretations of quantum mechanics which give a realist interpretation, and notmerely a positivistic one, to quantum-mechanical calculations. It is in fact just a reformulation of conventionalquantum mechanics obtained by rearranging the equations and renaming the variables. Nevertheless it is a hiddenvariable theory.The major reference for Bohm's theory today is his posthumous book with Basil Hiley[3] .

MotivationUnder the Copenhagen interpretation, quantum mechanics is nondeterministic, meaning that it generally does notpredict the outcome of any measurement with certainty. Instead, it tells us what the probabilities of the outcomes are.This leads to the situation where measurements of a certain property done on two apparently identical systems cangive different answers. The question arises whether there might be some deeper reality hidden beneath quantummechanics, to be described by a more fundamental theory that can always predict the outcome of each measurementwith certainty. In other words if the exact properties of every subatomic particle and smaller were known the entiresystem could be modeled exactly using deterministic physics similar to classical physics.In other words, the Copenhagen interpretation of quantum mechanics might be an incomplete description of reality.Physicists supporting the Bohmian interpretation of quantum mechanics maintain that underlying the probabilisticnature of the universe is an objective foundation/property — the hidden variable. Others, however, believe that thereis no deeper reality in quantum mechanics — experiments have shown a vast class of hidden variable theories to beincompatible with observations. Kirchmair and colleagues show that, in a system of trapped ions, quantummechanics conflicts with hidden variable theories regardless of the quantum state of the system.[4]

Although determinism was initially a major motivation for physicists looking for hidden variable theories,nondeterministic theories trying to explain what the supposed reality underlying the quantum mechanics formalismlooks like are also considered hidden variable theories; for example Edward Nelson's stochastic mechanics.

Hidden variable theory 110

EPR Paradox & Bell's TheoremIn 1935, Einstein, Podolsky and Rosen wrote a four-page paper titled "Can quantum-mechanical description ofphysical reality be considered complete?" that argued that such a theory was in fact necessary, proposing the EPRParadox as proof. In 1964, John Bell showed through his famous theorem that if local hidden variables exist, certainexperiments could be performed where the result would satisfy a Bell inequality. If, on the other hand, Quantumentanglement is correct the Bell inequality would be violated. Another no-go theorem concerning hidden variabletheories is the Kochen-Specker theorem.Physicists such as Alain Aspect and Paul Kwiat have performed experiments that have found violations of theseinequalities up to 242 standard deviations[5] (excellent scientific certainty). This rules out local hidden variabletheories, but does not rule out non-local ones (which would refute quantum entanglement). Theoretically, there couldbe experimental problems that affect the validity of the experimental findings.

Some hidden-variable theoriesAssuming the validity of Bell's theorem, any hidden-variable theory which is consistent with quantum mechanicswould have to be non-local, maintaining the existence of instantaneous or faster than light acausal relations(correlations) between physically separated entities. The first hidden-variable theory was the pilot wave theory ofLouis de Broglie, dating from the late 1920s. The currently best-known hidden-variable theory, the CausalInterpretation, of the physicist and philosopher David Bohm, created in 1952, is a non-local hidden variable theory.Those who believe the Bohm interpretation to be actually true (rather than a mere model or interpretation), and thequantum potential to be real, refer to Bohmian mechanics.What Bohm did, on the basis of an idea of Louis de Broglie, was to posit both the quantum particle, e.g. an electron,and a hidden 'guiding wave' that governs its motion. Thus, in this theory electrons are quite clearly particles. Whenyou perform a double-slit experiment (see wave-particle duality), they go through one slit rather than the other.However, their choice of slit is not random but is governed by the guiding wave, resulting in the wave pattern that isobserved.Such a view does not contradict the idea of local events that is used in both classical atomism and relativity theory asBohm's theory (and indeed quantum mechanics, with which it is exactly equivalent) are still locally causal but allownonlocal correlations (that is information travel is still restricted to the speed of light). It points to a view of a moreholistic, mutually interpenetrating and interacting world. Indeed Bohm himself stressed the holistic aspect ofquantum theory in his later years, when he became interested in the ideas of Jiddu Krishnamurti. The Bohminterpretation (as well as others) has also been the basis of some books which attempt to connect physics withEastern mysticism and consciousness. Nevertheless this nonlocality is seen as a weakness of Bohm's theory by somephysicists.Another possible weakness of Bohm's theory is that some feel that it looks contrived. It was deliberately designed togive predictions which are in all details identical to conventional quantum mechanics. Bohm's aim was not to make aserious counterproposal but simply to demonstrate that hidden-variable theories are indeed possible. His hope wasthat this could lead to new insights and experiments that would lead beyond the current quantum theories.Gerard 't Hooft has disputed the validity of Bell's theorem on the basis of the superdeterminism loophole andproposed some ideas to construct local deterministic models.[6]

Hidden variable theory 111

See also• Local hidden variable theory• Bell's theorem• Bell test experiments• Quantum mechanics• Bohm interpretation

References[1] private letter to Max Born, 4 December 1926, Albert Einstein Archives (http:/ / www. alberteinstein. info/ db/ ViewDetails.

do?DocumentID=38009) reel 8, item 180[2] Einstein, A., Podolsky, B. and Rosen, N. (1935) Can Quantum-Mechanical Description of Physical Reality Be Considered Complete? (http:/ /

prola. aps. org/ abstract/ PR/ v47/ i10/ p777_1), Phys. Rev. 47, 777-780[3] D.Bohm and B.J.Hiley, The Undivided Universe, Routledge, 1993, ISBN 0-415-06588-7.[4] Kirchmair, G., et al. (2009) State-independent experimental test of quantum contextuality, Nature 460, 494-497[5] Kwiat, P. G., et al. (1999) Ultrabright source of polarization-entangled photons, Physical Review A 60, R773-R776[6] G 't Hooft, The Free-Will Postulate in Quantum Mechanics (http:/ / arxiv. org/ abs/ quant-ph/ 0701097); Entangled quantum states in a local

deterministic theory (http:/ / arxiv. org/ abs/ 0908. 3408)

Local hidden variable theoryIn quantum mechanics, a local hidden variable theory is one in which distant events are assumed to have noinstantaneous (or at least faster-than-light) effect on local ones.According to the quantum entanglement theory of quantum mechanics, on the other hand, distant events may undersome circumstances have instantaneous correlations with local ones. As a result of this it is now generally acceptedthat there can be no interpretations of quantum mechanics which use local hidden variables. (There are those whodispute this. Their arguments are called loophole theories, referring to loopholes in the presuppositions of Bell's localhidden variable theory, implying Bell's theorem to be not sufficiently general to draw general conclusions from itwith respect to locality or nonlocality of the quantum world.) The term is most often used in discussions of the EPRparadox and Bell's inequalities. It is effectively synonymous with the concept of local realism, which can onlycorrectly be applied to classical physics and not to quantum mechanics.

Local hidden variables and the Bell testsThe principle of "locality" enables the assumption to be made in Bell test experiments that the probability of acoincidence can be written in factorised form:

(1)

where is the probability of detection of particle with hidden variable by detector , set indirection , and similarly is the probability at detector , set in direction , for particle , sharingthe same value of . The source is assumed to produce particles in the state with probability .Using (1), various Bell inequalities can be derived, giving restrictions on the possible behaviour of local hiddenvariable models.When John Bell originally derived his inequality, it was in relation to pairs of indivisible "spin-1/2" particles, every one of those emitted being detected. In these circumstances it is found that local realist assumptions lead to a straight line prediction for the relationship between quantum correlation and the angle between the settings of the two detectors. It was soon realised, however, that real experiments were not feasible with spin-1/2 particles. They were conducted instead using photons. The local hidden variable prediction for these is not a straight line but a sine curve,

Local hidden variable theory 112

similar to the quantum mechanical prediction but of only half the "visibility".

The difference between the two predictions is due to the different functions and involved. Byassuming different functions, a great variety of other realist predictions can be derived, some very close to thequantum-mechanical one. The choice of function, however, is not arbitrary. In optical experiments usingpolarisation, for instance, the natural assumption is that it is a cosine-squared function, corresponding to adherence toMalus's Law.

Bell tests with no "non-detections"Consider, for example, David Bohm's thought-experiment (Bohm, 1951), in which a molecule breaks into two atomswith opposite spins. Assume this spin can be represented by a real vector, pointing in any direction. It will be the"hidden variable" in our model. Taking it to be a unit vector, all possible values of the hidden variable arerepresented by all points on the surface of a unit sphere.Suppose the spin is to be measured in the direction a. Then the natural assumption, given that all atoms are detected,is that all atoms the projection of whose spin in the direction a is positive will be detected as spin up (coded as +1)while all whose projection is negative will be detected as spin down (coded as −1). The surface of the sphere will bedivided into two regions, one for +1, one for −1, separated by a great circle in the plane perpendicular to a.Assuming for convenience that a is horizontal, corresponding to the angle a with respect to some suitable referencedirection, the dividing circle will be in a vertical plane. So far we have modelled side A of our experiment.Now to model side B. Assume that b too is horizontal, corresponding to the angle b. There will be second greatcircle drawn on the same sphere, to one side of which we have +1, the other −1 for particle B. The circle will beagain be in a vertical plane.The two circles divide the surface of the sphere into four regions. The type of "coincidence" (++, −−, +− or −+)observed for any given pair of particles is determined by the region within which their hidden variable falls.Assuming the source to be "rotationally invariant" (to produce all possible states λ with equal probability), theprobability of a given type of coincidence will clearly be proportional to the corresponding area, and these areas willvary linearly with the angle between a and b. (To see this, think of an orange and its segments. The area of peelcorresponding to a number n of segments is roughly proportional to n. More accurately, it is proportional to the anglesubtended at the centre.)The formula (1) above has not been used explicitly — it is hardly relevant when, as here, the situation is fullydeterministic. The problem could be reformulated in terms of the functions in the formula, with ρ constant and theprobability functions step functions. The principle behind (1) has in fact been used, but purely intuitively.

Local hidden variable theory 113

Fig. 1: The realist prediction (solid lines) for quantum correlation when there areno non-detections. The quantum-mechanical prediction is the dotted curve.

Thus the local hidden variable prediction forthe probability of coincidence isproportional to the angle (b − a) between thedetector settings. The quantum correlation isdefined to be the expectation value of theproduct of the individual outcomes, and thisis

(2)    E = P++ + P−− − P+− −P−+

where P++ is the probability of a '+' outcomeon both sides, P+− that of a + on side A, a '−'on side B, etc..

Since each individual term varies linearlywith the difference (b − a), so does theirsum.

The result is shown in fig. 1.

Optical Bell testsIn almost all real applications of Bell's inequalities, the particles used have been photons. It is not necessarilyassumed that the photons are particle-like. They may be just short pulses of classical light (Clauser, 1978). It is notassumed that every single one is detected. Instead the hidden variable set at the source is taken to determine only theprobability of a given outcome, the actual individual outcomes being partly determined by other hidden variableslocal to the analyser and detector. It is assumed that these other hidden variables are independent on the two sides ofthe experiment (Clauser, 1974; Bell, 1971).In this stochastic model, in contrast to the above deterministic case, we do need equation (1) to find the local realistprediction for coincidences. It is necessary first to make some assumption regarding the functions and

, the usual one being that these are both cosine-squares, in line with Malus' Law. Assuming the hiddenvariable to be polarisation direction (parallel on the two sides in real applications, not orthogonal), equation (1)becomes:

(3) , where .

The predicted quantum correlation can be derived from this and is shown in fig. 2.

Local hidden variable theory 114

Fig. 2: The realist prediction (solid curve) for quantum correlation in an opticalBell test. The quantum-mechanical prediction is the dotted curve.

In optical tests, incidentally, it is not certainthat the quantum correlation is well-defined.Under a classical model of light, a singlephoton can go partly into the + channel,partly into the − one, resulting in thepossibility of simultaneous detections inboth. Though experiments such as Grangieret al.'s (Grangier, 1986) have shown that thisprobability is very low, it is not logical toassume that it is actually zero. Thedefinition of quantum correlation is adaptedto the idea that outcomes will always be +1,−1 or 0. There is no obvious way ofincluding any other possibility, which is oneof the reasons why Clauser and Horne's1974 Bell test, using single-channelpolarisers, should be used instead of the CHSH Bell test. The CH74 inequality concerns just probabilities ofdetection, not quantum correlations.

Generalizations of the modelsBy varying the assumed probability and density functions in equation (1) we can arrive at a considerable variety oflocal realist predictions.

Time effectsPreviously some new hypotheses were conjectured concerning the role of time in constructing hidden variablestheory. One approach is suggested by K. Hess and W. Philipp (Hess, 2002) and discusses possible consequences oftime dependences of hidden variables, previously not taken into account by Bell's theorem. This hypothesis has beencriticized by R.D. Gill, G. Weihs, A. Zeilinger and M. Żukowski (Gill, 2002).Another hypothesis suggests to review the notion of physical time (Kurakin, 2004). Hidden variables in this conceptevolve in so called 'hidden time', not equivalent to physical time. Physical time relates to 'hidden time' by some'sewing procedure'. This model stays physically non-local, though the locality is achieved in mathematical sense.

Optical models deviating from Malus' LawIf we make realistic (wave-based) assumptions regarding the behaviour of light on encountering polarisers andphotodetectors, we find that we are not compelled to accept that the probability of detection will reflect Malus' Lawexactly.We might perhaps suppose the polarisers to be perfect, with output intensity of polariser A proportional to cos2(a −λ), but reject the quantum-mechanical assumption that the function relating this intensity to the probability ofdetection is a straight line through the origin. Real detectors, after all, have "dark counts" that are there even whenthe input intensity is zero, and become saturated when the intensity is very high. It is not possible for them toproduce outputs in exact proportion to input intensity for all intensities.By varying our assumptions, it seems possible that the realist prediction could approach the quantum-mechanical onewithin the limits of experimental error (Marshall, 1983), though clearly a compromise must be reached. We have tomatch both the behaviour of the individual light beam on passage through a polariser and the observed coincidencecurves. The former would be expected to follow Malus' Law fairly closely, though experimental evidence here is not

Local hidden variable theory 115

so easy to obtain. We are interested in the behaviour of very weak light and the law may be slightly different fromthat of stronger light.

References• Bell, 1971: J. S. Bell, in Foundations of Quantum Mechanics, Proceedings of the International School of Physics

“Enrico Fermi”, Course XLIX, B. d’Espagnat (Ed.) (Academic, New York, 1971), p. 171 and Appendix B. Pages171-81 are reproduced as Ch. 4, pp 29–39, of J. S. Bell, Speakable and Unspeakable in Quantum Mechanics(Cambridge University Press 1987)

• Bohm, 1951: D. Bohm, Quantum Theory, Prentice-Hall 1951• Clauser, 1974: J. F. Clauser and M. A. Horne, Experimental consequences of objective local theories, Physical

Review D, 10, 526-35 (1974)• Clauser, 1978: J. F. Clauser and A. Shimony, Bell’s theorem: experimental tests and implications, Reports on

Progress in Physics 41, 1881 (1978)• Gill, 2002: R.D. Gill, G. Weihs, A. Zeilinger and M. Żukowski, No time loophole in Bell's theorem; the

Hess-Philipp model is non-local (http:/ / arxiv. org/ abs/ quant-ph/ 0208187), quant-ph/0208187 (2002)• Grangier, 1986: P. Grangier, G. Roger and A. Aspect, Experimental evidence for a photon anticorrelation effect

on a beam splitter: a new light on single-photon interferences, Europhysics Letters 1, 173–179 (1986)• Hess, 2002: K. Hess and W. Philipp, Europhys. Lett., 57:775 (2002)• Kurakin, 2004: Pavel V. Kurakin, Hidden variables and hidden time in quantum theory (http:/ / www. geocities.

com/ bellstheorem/ ), a preprint #33 by Keldysh Inst. of Appl. Math., Russian Academy of Sciences (2004)• Marshall, 1983: T. W. Marshall, E. Santos and F. Selleri, Local Realism has not been Refuted by

Atomic-Cascade Experiments, Physics Letters A, 98, 5–9 (1983)

See also• Hidden variable theory

Bell's theorem 116

Bell's theoremIn theoretical physics, Bell's theorem is a no-go theorem, loosely stating that:

No physical theory of local hidden variables can ever reproduce all of the predictions of quantum mechanics.It is the most famous legacy of the late physicist John S. Bell.Bell's theorem has important implications for physics and the philosophy of science as it proves that every quantumtheory must violate either locality or counterfactual definiteness.A series of experiments has demonstrated the quantum predictions that form the basis of Bell's Theorem and somewould therefore claim that not only the predictions of quantum theory but also experimental results now prove, usingBell's Theorem, that the universe must violate either locality or counterfactual definiteness. However, interpretationof these experiments is still the subject of some debate.

OverviewBell’s theorem shows that there are limits that apply to local hidden-variable models of quantum systems, and thatquantum mechanics (QM) predicts that these limits will be exceeded by measurements performed on entangled pairsof particles. This article discusses Bell’s theorem in the context of experiments that show that the predictions ofquantum mechanics are consistent with the results of experiments, and inconsistent with local hidden variablemodels of quantum mechanics.

Illustration of Bell test for spin 1/2 particles. Source produces spin singlet pair, oneparticle sent to Alice another to Bob. Each performs one of the two spin

measurements.

As in the situation explored in theEinstein–Podolsky–Rosen (EPR) paradox,Bell considered an experiment in which asource produces pairs of correlated particles.For example, a pair of particles withcorrelated spins is created; one particle issent to Alice and the other to Bob. Theexperimental arrangement differs from theEPR arrangement in that the same type ofmeasurement is performed on both particlesof a pair. On each trial, each of theobservers independently chooses between various detector settings and then performs an independent measurementon the particle arriving at their position. Hence, Bell’s theorem can be tested by coincidence measurements on pairsof entangled particles in which the correlation between two independently chosen measurements is determined.

When Alice and Bob measure the spin† of entangled particles along the same axis (but in opposite directions), theyget identical results 100% of the time. When Bob measures at orthogonal (right) angles to Alice’s measurements, hismeasurement matches hers 50% of the time. In terms of mathematics, the two measurements have a correlation of 1,or perfect correlation when read along the same axis (in opposite direction); when read at right angles, they have nocorrelation.

Bell's theorem 117

Same axis: pair1

pair2

pair3

pair4

...n

Alice, 0°: + − − + ...

Bob, 180°: + − − + ...

Correlation: ( +1 +1 +1 +1 ...)/n = +1

(100%identical)

Orthogonalaxes:

pair1

pair2

pair3

pair4

...n

Alice, 0°: + − + − ...

Bob, 45°: − − + + ...

Correlation: ( −1 +1 +1 −1 ...)/n = 0.0

(50% identical)

So far, (when the analyzers are aligned, orthogonal, or 45 degrees) the measurement results can be modeled byproposing physical attributes, referred to as local hidden variables, within each particle that determine the outcomeof a measurement. If the analyzers are aligned, and the source only emits pairs of particles with identical propertiesthen Alice and Bob’s measurements will match (+,+) and (-,-). Similarly, the results will be anti-correlated (+,-) (-,+)if their analyzers are aligned on orthogonal axes.Now, consider Alice or Bob set their analyzers so that their axes are at some arbitrary angle between 0 and 90degrees. A quantum mechanical calculation of the degree to which Alice and Bob’s measurements will correlateresults in a curve that varies as the cosine of twice the angle between the analyzers. That is, using entangled particles,there is a cosine curve from correlated (at zero degrees) and anti-correlated at 90 degrees.In contrast, Bell’s theorem places a straight-line limit on the curve that any local hidden variable model (involvingidentical particles) can follow from correlated to anti-correlated. The QM prediction for entangled particles breaksthis limit. For example, when the relative analyzer alignment is 22.5 degrees QM gives 0.71 correlation whereas thestraight-line limit (implied by Bell’s theorem) is 0.5. From this, one may conclude that the outcome of quantummeasurements on entangled particles cannot be replicated by a model that employs identical particles that havehidden attributes/properties which locally determine the outcome of measurements.One possible way for a hidden variable system to break the limit imposed by Bell’s theorem, is to suppose that somenon-local process or communication acts to increase the degree of correlation above the limits imposed by Bell’stheorem. To test this possibility, the analyzer angles are set at arbitrary angles before measuring the particles, evenafter the particles leave the source. In this case, this supposed non-local interaction or communication would have tooccur instantaneously (i.e. travelling faster than light) in order to reproduce the behavior observed in quantumsystems. Note that this does not necessarily mean that QM itself involves non-local or instantaneous communication,it just means that hidden variable accounts of QM would require these, or similar, drastic elements to be viable.Multiple researchers have performed equivalent experiments using different methods. It appears [1] that most of theseexperiments refute local-hidden-variable theories and support the notion that QM involves some degree ofnonlocality. Not everyone agrees with these findings.[2] There have been two loopholes found in the earlier of theseexperiments, the detection loophole[1] and the communication loophole[1] with associated experiments to close theseloopholes. After all current experimentation it seems these experiments support quantum mechanical non-locality [1]

or disprove the ‘no enhancement’ assumption (below).  Although the correlated property used here is the effect a particle’s spin has on its direction through an analyzer, it could alternatively be

any correlated "quantum state" that encodes exactly one quantum bit.

Bell's theorem 118

Importance of the theoremBell's theorem, derived in his seminal 1964 paper titled On the Einstein Podolsky Rosen paradox,[3] has been called"the most profound in science".[4] The title of the article refers to the famous paper by Einstein, Podolsky andRosen[5] purporting to prove the incompleteness of quantum mechanics. In his paper, Bell started from essentiallythe same assumptions as did EPR, viz. i) reality (microscopic objects have real properties determining the outcomesof quantum mechanical measurements) and ii) locality (reality is not influenced by measurements simultaneouslyperformed at a large distance). Bell was able to derive from these assumptions an important result, viz. Bell'sinequality, violation of which by quantum mechanics implying that at least one of the assumptions must beabandoned if experiment would turn out to satisfy quantum mechanics.In two respects Bell's 1964 paper was a big step forward compared to the EPR paper: i) it considered more generalhidden variables than the elements of physical reality of the EPR paper, ii) more importantly, Bell's inequality wasliable to be experimentally tested, thus yielding the opportunity to lift the discussion on the completeness of quantummechanics from metaphysics to physics. Whereas Bell's 1964 paper deals only with deterministic hidden variablestheories, Bell's theorem was later generalized to stochastic theories[6] as well, and it was realized[7] that the theoremcan even be proven without introducing hidden variables.After EPR (Einstein–Podolsky–Rosen), quantum mechanics was left in an unsatisfactory position: either it wasincomplete, in the sense that it failed to account for some elements of physical reality, or it violated the principle offinite propagation speed of physical effects. In a modified version of the EPR thought experiment, two observers,now commonly referred to as Alice and Bob, perform independent measurements of spin on a pair of electrons,prepared at a source in a special state called a spin singlet state. It was equivalent to the conclusion of EPR that onceAlice measured spin in one direction (e.g., on the x axis), Bob's measurement in that direction was determined withcertainty, with opposite outcome to that of Alice, whereas immediately before Alice's measurement, Bob's outcomewas only statistically determined. Thus, either the spin in each direction is an element of physical reality, or theeffects travel from Alice to Bob instantly.In QM, predictions were formulated in terms of probabilities — for example, the probability that an electron mightbe detected in a particular region of space, or the probability that it would have spin up or down. The idea persisted,however, that the electron in fact has a definite position and spin, and that QM's weakness was its inability to predictthose values precisely. The possibility remained that some yet unknown, but more powerful theory, such as a hiddenvariables theory, might be able to predict those quantities exactly, while at the same time also being in completeagreement with the probabilistic answers given by QM. If a hidden variables theory were correct, the hiddenvariables were not described by QM, and thus QM would be an incomplete theory.The desire for a local realist theory was based on two assumptions:1. Objects have a definite state that determines the values of all other measurable properties, such as position and

momentum.2. Effects of local actions, such as measurements, cannot travel faster than the speed of light (as a result of special

relativity). If the observers are sufficiently far apart, a measurement taken by one has no effect on themeasurement taken by the other.

In the formalization of local realism used by Bell, the predictions of theory result from the application of classicalprobability theory to an underlying parameter space. By a simple argument based on classical probability, he thenshowed that correlations between measurements are bounded in a way that is violated by QM.Bell's theorem seemed to put an end to local realist hopes for QM. Per Bell's theorem, either quantum mechanics orlocal realism is wrong. Experiments were needed to determine which is correct, but it took many years andimprovements in technology to perform them.Bell test experiments have been interpreted as showing that Bell inequalities are violated, in favor of QM. The no-communication theorem proves that the observers cannot use the inequality violations to communicate

Bell's theorem 119

information to each other faster than the speed of light. But the ‘fair sampling’ and ‘no enhancement’ assumptionsrequire more careful consideration (below).John Bell's paper examines both John von Neumann's 1932 proof of the incompatibility of hidden variables with QMand the seminal paper on the subject by Albert Einstein and his colleagues.

Bell inequalitiesBell inequalities concern measurements made by observers on pairs of particles that have interacted and thenseparated. According to quantum mechanics they are entangled while local realism limits the correlation ofsubsequent measurements of the particles. Different authors subsequently derived inequalities similar to Bell´soriginal inequality, collectively termed Bell inequalities. All Bell inequalities describe experiments in which thepredicted result assuming entanglement differs from that following from local realism. The inequalities assume thateach quantum-level object has a well defined state that accounts for all its measurable properties and that distantobjects do not exchange information faster than the speed of light. These well defined states are often called hiddenvariables, the properties that Einstein posited when he stated his famous objection to quantum mechanics: "God doesnot play dice."Bell showed that under quantum mechanics, which lacks local hidden variables, the inequalities may be violated.Instead, properties of a particle are not clear to verify in quantum mechanics but may be correlated with those ofanother particle due to quantum entanglement, allowing their state to be well defined only after a measurement ismade on either particle. That restriction agrees with the Heisenberg uncertainty principle, a fundamental andinescapable concept in quantum mechanics.In Bell's work:

Theoretical physicists live in a classical world, looking out into a quantum-mechanical world. The latter wedescribe only subjectively, in terms of procedures and results in our classical domain. (...) Now nobody knowsjust where the boundary between the classical and the quantum domain is situated. (...) More plausible to me isthat we will find that there is no boundary. The wave functions would prove to be a provisional or incompletedescription of the quantum-mechanical part. It is this possibility, of a homogeneous account of the world,which is for me the chief motivation of the study of the so-called "hidden variable" possibility.(...) A second motivation is connected with the statistical character of quantum-mechanical predictions. Oncethe incompleteness of the wave function description is suspected, it can be conjectured that random statisticalfluctuations are determined by the extra "hidden" variables — "hidden" because at this stage we can onlyconjecture their existence and certainly cannot control them.(...) A third motivation is in the peculiar character of some quantum-mechanical predictions, which seemalmost to cry out for a hidden variable interpretation. This is the famous argument of Einstein, Podolsky andRosen. (...) We will find, in fact, that no local deterministic hidden-variable theory can reproduce all theexperimental predictions of quantum mechanics. This opens the possibility of bringing the question into theexperimental domain, by trying to approximate as well as possible the idealized situations in which localhidden variables and quantum mechanics cannot agree

In probability theory, repeated measurements of system properties can be regarded as repeated sampling of randomvariables. In Bell's experiment, Alice can choose a detector setting to measure either or and Bob canchoose a detector setting to measure either or . Measurements of Alice and Bob may be somehowcorrelated with each other, but the Bell inequalities say that if the correlation stems from local random variables,there is a limit to the amount of correlation one might expect to see.

Bell's theorem 120

Original Bell's inequalityThe original inequality that Bell derived was:[3]

where C is the "correlation" of the particle pairs and a, b and c settings of the apparatus. This inequality is not usedin practice. For one thing, it is true only for genuinely "two-outcome" systems, not for the "three-outcome" ones(with possible outcomes of zero as well as +1 and −1) encountered in real experiments. For another, it applies only toa very restricted set of hidden variable theories, namely those for which the outcomes on both sides of theexperiment are always exactly anticorrelated when the analysers are parallel, in agreement with the quantummechanical prediction.There is a simple limit of Bell's inequality which has the virtue of being completely intuitive. If the result of threedifferent statistical coin-flips A, B, and C have the property that:1. A and B are the same (both heads or both tails) 99% of the time2. B and C are the same 99% of the timethen A and C are the same at least 98% of the time. The number of mismatches between A and B (1/100) plus thenumber of mismatches between B and C (1/100) are together the maximum possible number of mismatches betweenA and C.In quantum mechanics, by letting A, B, and C be the values of the spin of two entangled particles measured relativeto some axis at 0 degrees, θ degrees, and 2θ degrees respectively, the overlap of the wavefunction between thedifferent angles is proportional to . The probability that A and B give the same answer is , where is proportional to θ. This is also the probability that B and C give the same answer. But A and C are thesame 1 − (2ε)2 of the time. Choosing the angle so that , A and B are 99% correlated, B and C are 99%correlated and A and C are only 96% correlated.Imagine that two entangled particles in a spin singlet are shot out to two distant locations, and the spins of both aremeasured in the direction A. The spins are 100% correlated (actually, anti-correlated but for this argument that isequivalent). The same is true if both spins are measured in directions B or C. It is safe to conclude that any hiddenvariables which determine the A,B, and C measurements in the two particles are 100% correlated and can be usedinterchangeably.If A is measured on one particle and B on the other, the correlation between them is 99%. If B is measured on oneand C on the other, the correlation is 99%. This allows us to conclude that the hidden variables determining A and Bare 99% correlated and B and C are 99% correlated. But if A is measured in one particle and C in the other, theresults are only 96% correlated, which is a contradiction. The intuitive formulation is due to David Mermin, whilethe small-angle limit is emphasized in Bell's original article.

CHSH inequalityIn addition to Bell's original inequality,[3] the form given by John Clauser, Michael Horne, Abner Shimony and R. A.Holt,[8] (the CHSH form) is especially important[8] , as it gives classical limits to the expected correlation for theabove experiment conducted by Alice and Bob:

where C denotes correlation.Correlation of observables X, Y is defined as

This is a non-normalized form of the correlation coefficient considered in statistics (see Quantum correlation).In order to formulate Bell's theorem, we formalize local realism as follows:

Bell's theorem 121

1. There is a probability space and the observed outcomes by both Alice and Bob result by random sampling ofthe parameter .

2. The values observed by Alice or Bob are functions of the local detector settings and the hidden parameter only.Thus

• Value observed by Alice with detector setting is • Value observed by Bob with detector setting is

Implicit in assumption 1) above, the hidden parameter space has a probability measure and the expectation of arandom variable X on with respect to is written

where for accessibility of notation we assume that the probability measure has a density.Bell's inequality. The CHSH inequality (1) holds under the hidden variables assumptions above.For simplicity, let us first assume the observed values are +1 or −1; we remove this assumption in Remark 1 below.Let . Then at least one of

is 0. Thus

and therefore

Remark 1. The correlation inequality (1) still holds if the variables , are allowed to take on anyreal values between −1 and +1. Indeed, the relevant idea is that each summand in the above average is boundedabove by 2. This is easily seen to be true in the more general case:

To justify the upper bound 2 asserted in the last inequality, without loss of generality, we can assume that

In that case

Bell's theorem 122

Remark 2. Though the important component of the hidden parameter in Bell's original proof is associated withthe source and is shared by Alice and Bob, there may be others that are associated with the separate detectors, theseothers being independent. This argument was used by Bell in 1971, and again by Clauser and Horne in 1974,[9] tojustify a generalisation of the theorem forced on them by the real experiments, in which detectors were never 100%efficient. The derivations were given in terms of the averages of the outcomes over the local detector variables. Theformalisation of local realism was thus effectively changed, replacing A and B by averages and retaining the symbol

but with a slightly different meaning. It was henceforth restricted (in most theoretical work) to mean only thosecomponents that were associated with the source.However, with the extension proved in Remark 1, CHSH inequality still holds even if the instruments themselvescontain hidden variables. In that case, averaging over the instrument hidden variables gives new variables:

on which still have values in the range [−1, +1] to which we can apply the previous result.

Bell inequalities are violated by quantum mechanical predictionsIn the usual quantum mechanical formalism, the observables X and Y are represented as self-adjoint operators on aHilbert space. To compute the correlation, assume that X and Y are represented by matrices in a finite dimensionalspace and that X and Y commute; this special case suffices for our purposes below. The von Neumann measurementpostulate states: a series of measurements of an observable X on a series of identical systems in state produces adistribution of real values. By the assumption that observables are finite matrices, this distribution is discrete. Theprobability of observing λ is non-zero if and only if λ is an eigenvalue of the matrix X and moreover the probabilityis

where EX (λ) is the projector corresponding to the eigenvalue λ. The system state immediately after the measurementis

From this, we can show that the correlation of commuting observables X and Y in a pure state is

We apply this fact in the context of the EPR paradox. The measurements performed by Alice and Bob are spinmeasurements on electrons. Alice can choose between two detector settings labelled a and a′; these settingscorrespond to measurement of spin along the z or the x axis. Bob can choose between two detector settings labelled band b′; these correspond to measurement of spin along the z′ or x′ axis, where the x′ – z′ coordinate system is rotated135° relative to the x – z coordinate system. The spin observables are represented by the 2 × 2 self-adjoint matrices:

These are the Pauli spin matrices normalized so that the corresponding eigenvalues are +1, −1. As is customary, wedenote the eigenvectors of Sx by

Let be the spin singlet state for a pair of electrons discussed in the EPR paradox. This is a specially constructedstate described by the following vector in the tensor product

Now let us apply the CHSH formalism to the measurements that can be performed by Alice and Bob.

Bell's theorem 123

Illustration of Bell test for spin 1/2 particles. Source produces spin singlet pairs, oneparticle of each pair is sent to Alice and the other to Bob. Each performs one of the two

spin measurements.

The operators , correspond to Bob's spin measurements along x′ and z′. Note that the A operatorscommute with the B operators, so we can apply our calculation for the correlation. In this case, we can show that theCHSH inequality fails. In fact, a straightforward calculation shows that

and

so that

Bell's Theorem: If the quantum mechanical formalism is correct, then the system consisting of a pair of entangledelectrons cannot satisfy the principle of local realism. Note that is indeed the upper bound for quantummechanics called Tsirelson's bound. The operators giving this maximal value are always isomorphic to the Paulimatrices.

Bell's theorem 124

Practical experiments testing Bell's theorem

Scheme of a "two-channel" Bell testThe source S produces pairs of "photons", sent in opposite directions. Each photon

encounters a two-channel polariser whose orientation (a or b) can be set by theexperimenter. Emerging signals from each channel are detected and coincidences of four

types (++, −−, +− and −+) counted by the coincidence monitor.

Experimental tests can determinewhether the Bell inequalities requiredby local realism hold up to theempirical evidence.Bell's inequalities are tested by"coincidence counts" from a Bell testexperiment such as the optical oneshown in the diagram. Pairs ofparticles are emitted as a result of aquantum process, analysed withrespect to some key property such aspolarisation direction, then detected.The setting (orientations) of theanalysers are selected by the

experimenter.Bell test experiments to date overwhelmingly violate Bell's inequality. Indeed, a table of Bell test experimentsperformed prior to 1986 is given in 4.5 of Redhead, 1987.[10] Of the thirteen experiments listed, only two reachedresults contradictory to quantum mechanics; moreover, according to the same source, when the experiments wererepeated, "the discrepancies with QM could not be reproduced".

Nevertheless, the issue is not conclusively settled. According to Shimony's 2004 Stanford Encyclopedia overviewarticle:[1]

Most of the dozens of experiments performed so far have favored Quantum Mechanics, but not decisivelybecause of the 'detection loopholes' or the 'communication loophole.' The latter has been nearly decisivelyblocked by a recent experiment and there is a good prospect for blocking the former.

To explore the 'detection loophole', one must distinguish the classes of homogeneous and inhomogeneous Bellinequality.The standard assumption in Quantum Optics is that "all photons of given frequency, direction and polarization areidentical" so that photodetectors treat all incident photons on an equal basis. Such a fair sampling assumptiongenerally goes unacknowledged, yet it effectively limits the range of local theories to those which conceive of thelight field as corpuscular. The assumption excludes a large family of local realist theories, in particular, Max Planck'sdescription. We must remember the cautionary words of Albert Einstein[11] shortly before he died: "Nowadays everyTom, Dick and Harry ('jeder Kerl' in German original) thinks he knows what a photon is, but he is mistaken".Objective physical properties for Bell’s analysis (local realist theories) include the wave amplitude of a light signal.Those who maintain the concept of duality, or simply of light being a wave, recognize the possibility or actuality thatthe emitted atomic light signals have a range of amplitudes and, furthermore, that the amplitudes are modified whenthe signal passes through analyzing devices such as polarizers and beam splitters. It follows that not all signals havethe same detection probability[12] .

Two classes of Bell inequalitiesThe fair sampling problem was faced openly in the 1970s. In early designs of their 1973 experiment, Freedman and Clauser[13] used fair sampling in the form of the Clauser-Horne-Shimony-Holt (CHSH[8] ) hypothesis. However, shortly afterwards Clauser and Horne[9] made the important distinction between inhomogeneous (IBI) and homogeneous (HBI) Bell inequalities. Testing an IBI requires that we compare certain coincidence rates in two separated detectors with the singles rates of the two detectors. Nobody needed to perform the experiment, because

Bell's theorem 125

singles rates with all detectors in the 1970s were at least ten times all the coincidence rates. So, taking into accountthis low detector efficiency, the QM prediction actually satisfied the IBI. To arrive at an experimental design inwhich the QM prediction violates IBI we require detectors whose efficiency exceeds 82% for singlet states, but havevery low dark rate and short dead and resolving times. This is well above the 30% achievable[14] so Shimony’soptimism in the Stanford Encyclopedia, quoted in the preceding section, appears over-stated.

Practical challengesBecause detectors don't detect a large fraction of all photons, Clauser and Horne[9] recognized that testing Bell'sinequality requires some extra assumptions. They introduced the No Enhancement Hypothesis (NEH):

a light signal, originating in an atomic cascade for example, has a certain probability of activating adetector. Then, if a polarizer is interposed between the cascade and the detector, the detectionprobability cannot increase.

Given this assumption, there is a Bell inequality between the coincidence rates with polarizers and coincidence rateswithout polarizers.The experiment was performed by Freedman and Clauser[13] , who found that the Bell's inequality was violated. Sothe no-enhancement hypothesis cannot be true in a local hidden variables model. The Freedman-Clauser experimentreveals that local hidden variables imply the new phenomenon of signal enhancement:

In the total set of signals from an atomic cascade there is a subset whose detection probability increasesas a result of passing through a linear polarizer.

This is perhaps not surprising, as it is known that adding noise to data can, in the presence of a threshold, help revealhidden signals (this property is known as stochastic resonance[15] ). One cannot conclude that this is the onlylocal-realist alternative to Quantum Optics, but it does show that the word loophole is biased. Moreover, the analysisleads us to recognize that the Bell-inequality experiments, rather than showing a breakdown of realism or locality,are capable of revealing important new phenomena.

Theoretical challengesMost advocates of the hidden variables idea believe that experiments have ruled out local hidden variables. They areready to give up locality, explaining the violation of Bell's inequality by means of a "non-local" hidden variabletheory, in which the particles exchange information about their states. This is the basis of the Bohm interpretation ofquantum mechanics, which requires that all particles in the universe be able to instantaneously exchange informationwith all others. A recent experiment ruled out a large class of non-Bohmian "non-local" hidden variable theories.[16]

If the hidden variables can communicate with each other faster than light, Bell's inequality can easily be violated.Once one particle is measured, it can communicate the necessary correlations to the other particle. Since in relativitythe notion of simultaneity is not absolute, this is unattractive. One idea is to replace instantaneous communicationwith a process which travels backwards in time along the past Light cone. This is the idea behind a transactionalinterpretation of quantum mechanics, which interprets the statistical emergence of a quantum history as a gradualcoming to agreement between histories that go both forward and backward in time[17] .A few advocates of deterministic models have not given up on local hidden variables. E.g., Gerard 't Hooft hasargued that the superdeterminism loophole cannot be dismissed[18] .The quantum mechanical wavefunction can also provide a local realistic description, if the wavefunction values areinterpreted as the fundamental quantities that describe reality. Such an approach is called a many-worldsinterpretation of quantum mechanics. In this view, two distant observers both split into superpositions whenmeasuring a spin. The Bell inequality violations are no longer counterintuitive, because it is not clear which copy ofthe observer B observer A will see when going to compare notes. If reality includes all the different outcomes,locality in physical space (not outcome space) places no restrictions on how the split observers can meet up.

Bell's theorem 126

This implies that there is a subtle assumption in the argument that realism is incompatible with quantum mechanicsand locality. The assumption, in its weakest form, is called counterfactual definiteness. This states that if the resultsof an experiment are always observed to be definite, there is a quantity which determines what the outcome wouldhave been even if you don't do the experiment.Many worlds interpretations are not only counterfactually indefinite, they are factually indefinite. The results of allexperiments, even ones that have been performed, are not uniquely determined.

Final remarksThe phenomenon of quantum entanglement that is behind violation of Bell's inequality is just one element ofquantum physics which cannot be represented by any classical picture of physics; other non-classical elements arecomplementarity and wavefunction collapse. The problem of interpretation of quantum mechanics is intended toprovide a satisfactory picture of these non-classical elements of quantum physics.The EPR paper "pinpointed" the unusual properties of the entangled states, e.g. the above-mentioned singlet state,which is the foundation for present-day applications of quantum physics, such as quantum cryptography. Thisstrange non-locality was originally supposed to be a Reductio ad absurdum, because the standard interpretation couldeasily do away with action-at-a-distance by simply assigning to each particle definite spin-states. Bell's theoremshowed that the "entangledness" prediction of quantum mechanics have a degree of non-locality that cannot beexplained away by any local theory.In well-defined Bell experiments (see the paragraph on "test experiments") one can now falsify either quantummechanics or Einstein's quasi-classical assumptions: currently many experiments of this kind have been performed,and the experimental results support quantum mechanics, though some believe that detectors give a biased sample ofphotons, so that until nearly every photon pair generated is observed there will be loopholes.What is powerful about Bell's theorem is that it doesn't come from any particular physical theory. What makes Bell'stheorem unique and powerful is that it relies only on the general properties of quantum mechanics. No physicaltheory which assumes a deterministic variable inside the particle that determines the outcome, can account for theexperimental results, only assuming that this variable cannot acausally change other variables far away.

See also• Bell test experiments• CHSH Bell test• Clauser and Horne's 1974 Bell test• Counterfactual definiteness• Leggett–Garg inequality• Local hidden variable theory• Mott problem• Quantum entanglement• Quantum mechanical Bell test prediction• Measurement in quantum mechanics• Renninger negative-result experiment• GHZ State

Bell's theorem 127

References• A. Aspect et al., Experimental Tests of Realistic Local Theories via Bell's Theorem, Phys. Rev. Lett. 47, 460

(1981)• A. Aspect et al., Experimental Realization of Einstein-Podolsky-Rosen-Bohm Gedankenexperiment: A New

Violation of Bell's Inequalities, Phys. Rev. Lett. 49, 91 (1982).• A. Aspect et al., Experimental Test of Bell's Inequalities Using Time-Varying Analyzers, Phys. Rev. Lett. 49, 1804

(1982).• A. Aspect and P. Grangier, About resonant scattering and other hypothetical effects in the Orsay atomic-cascade

experiment tests of Bell inequalities: a discussion and some new experimental data, Lettere al Nuovo Cimento 43,345 (1985)

• B. D'Espagnat, The Quantum Theory and Reality [19], Scientific American, 241, 158 (1979)• J. S. Bell, On the problem of hidden variables in quantum mechanics, Rev. Mod. Phys. 38, 447 (1966)• J. S. Bell, Introduction to the hidden variable question, Proceedings of the International School of Physics 'Enrico

Fermi', Course IL, Foundations of Quantum Mechanics (1971) 171–81• J. S. Bell, Bertlmann’s socks and the nature of reality, Journal de Physique, Colloque C2, suppl. au numero 3,

Tome 42 (1981) pp C2 41–61• J. S. Bell, Speakable and Unspeakable in Quantum Mechanics (Cambridge University Press 1987) [A collection

of Bell's papers, including all of the above.]• J. F. Clauser and A. Shimony, Bell's theorem: experimental tests and implications, Reports on Progress in Physics

41, 1881 (1978)• J. F. Clauser and M. A. Horne, Phys. Rev D 10, 526–535 (1974)• E. S. Fry, T. Walther and S. Li, Proposal for a loophole-free test of the Bell inequalities, Phys. Rev. A 52, 4381

(1995)• E. S. Fry, and T. Walther, Atom based tests of the Bell Inequalities — the legacy of John Bell continues, pp

103–117 of Quantum [Un]speakables, R.A. Bertlmann and A. Zeilinger (eds.) (Springer, Berlin-Heidelberg-NewYork, 2002)

• R. B. Griffiths, Consistent Quantum Theory', Cambridge University Press (2002).• L. Hardy, Nonlocality for 2 particles without inequalities for almost all entangled states. Physical Review Letters

71 (11) 1665–1668 (1993)• M. A. Nielsen and I. L. Chuang, Quantum Computation and Quantum Information, Cambridge University Press

(2000)• P. Pearle, Hidden-Variable Example Based upon Data Rejection, Physical Review D 2, 1418–25 (1970)• A. Peres, Quantum Theory: Concepts and Methods, Kluwer, Dordrecht, 1993.• P. Pluch, Theory of Quantum Probability, PhD Thesis, University of Klagenfurt, 2006.• B. C. van Frassen, Quantum Mechanics, Clarendon Press, 1991.• M.A. Rowe, D. Kielpinski, V. Meyer, C.A. Sackett, W.M. Itano, C. Monroe, and D.J. Wineland, Experimental

violation of Bell's inequalities with efficient detection,(Nature, 409, 791–794, 2001).• S. Sulcs, The Nature of Light and Twentieth Century Experimental Physics, Foundations of Science 8, 365–391

(2003)• S. Gröblacher et al., An experimental test of non-local realism,(Nature, 446, 871–875, 2007).• D. N. Matsukevich, P. Maunz, D. L. Moehring, S. Olmschenk, and C. Monroe, Bell Inequality Violation with Two

Remote Atomic Qubits, Phys. Rev. Lett. 100, 150404 (2008).• The comic Dilbert, by Scott Adams, refers to Bell's Theorem in the 1992-09-21 [20] and 1992-09-22 [21] strips.

Bell's theorem 128

Further readingThe following are intended for general audiences.• Amir D. Aczel, Entanglement: The greatest mystery in physics (Four Walls Eight Windows, New York, 2001).• A. Afriat and F. Selleri, The Einstein, Podolsky and Rosen Paradox (Plenum Press, New York and London, 1999)• J. Baggott, The Meaning of Quantum Theory (Oxford University Press, 1992)• N. David Mermin, "Is the moon there when nobody looks? Reality and the quantum theory", in Physics Today,

April 1985, pp. 38–47.• Louisa Gilder, The Age of Entanglement: When Quantum Physics Was Reborn (New York: Alfred A. Knopf,

2008)• Brian Greene, The Fabric of the Cosmos (Vintage, 2004, ISBN 0-375-72720-5)• Nick Herbert, Quantum Reality: Beyond the New Physics (Anchor, 1987, ISBN 0-385-23569-0)• D. Wick, The infamous boundary: seven decades of controversy in quantum physics (Birkhauser, Boston 1995)• R. Anton Wilson, Prometheus Rising (New Falcon Publications, 1997, ISBN 1-56184-056-4)• Gary Zukav "The Dancing Wu Li Masters" (Perennial Classics, 2001, ISBN 0-06-095968-1)

External links• An explanation of Bell's Theorem [22], based on N. D. Mermin's article, "Bringing Home the Atomic World:

Quantum Mysteries for Anybody [23]," Am. J. of Phys. 49 (10), 940 (October 1981)• Quantum Entanglement [24] Includes a simple explanation of Bell's Inequality.• Bell's theorem on arXiv.org [25]

• Disproofs of Bell, GHZ, and Hardy Type Theorems and the Illusion of Entanglement [26] Disproof of Bell'sTheorem

• Interactive experiments with single photons: entanglement and Bell´s theorem [27]

References[1] Article on Bell's Theorem (http:/ / plato. stanford. edu/ entries/ bell-theorem) by Abner Shimony in the Stanford Encyclopedia of Philosophy,

(2004).[2] Caroline H. Thompson The Chaotic Ball: An Intuitive Analogy for EPR Experiments Found.Phys.Lett. 9 (1996) 357-382

arXiv:quant-ph/9611037[3] J. S. Bell, On the Einstein Podolsky Rosen Paradox (http:/ / www. drchinese. com/ David/ Bell_Compact. pdf), Physics 1, 195-200 (1964)[4] Stapp, 1975[5] A. Einstein, B. Podolsky and N. Rosen, Can quantum-mechanical description of physical reality be considered complete? Phys. Rev. 47,

777--780 (1935).[6] J.F. Clauser and M.A. Horne, Experimental consequences of objective local theories, Phys. Rev. D 10, 526-535 (1974).[7] P.H. Eberhard, Bell's theorem without hidden variables, Nuovo Cimento 38B, 75-80 (1977).[8] J. F. Clauser, M. A. Horne, A. Shimony and R. A. Holt, Proposed experiment to test local hidden-variable theories, Physical Review Letters

23, 880–884 (1969)[9] J. F. Clauser and M. A. Horne, Experimental consequences of objective local theories, Physical Review D, 10, 526–35 (1974)[10] M. Redhead, Incompleteness, Nonlocality and Realism, Clarendon Press (1987)[11] A. Einstein in Correspondance Einstein–Besso, p.265 (Herman, Paris, 1979)[12] Marshall and Santos, Semiclassical optics as an alternative to nonlocality (http:/ / www. crisisinphysics. co. uk/ optrev. pdf) Recent Research

Developments in Optics 2:683-717 (2002)[13] S. J. Freedman and J. F. Clauser, Experimental test of local hidden-variable theories, Phys. Rev. Lett. 28, 938 (1972)[14] Brida et al. Experimental tests of hidden variable theories from dBB to Stochastic Electrodynamics ournal of Physics: Conference Series 67

(2007) 012047, arXiv:quant-ph/0612075[15] Gammaitoni et al., Stochastic resonance (http:/ / prola. aps. org/ abstract/ RMP/ v70/ i1/ p223_1) Rev. Mod. Phys. 70, 223 - 287 (1998)[16] S. Gröblacher et al., An experimental test of non-local realism (http:/ / www. nature. com/ nature/ journal/ v446/ n7138/ abs/ nature05677.

html) Nature 446, 871–875, 2007[17] Cramer, John G. "The Transactional Interpretation of Quantum Mechanics", Reviews of Modern Physics 58, 647–688, July 1986[18] G 't Hooft, Entangled quantum states in a local deterministic theory (http:/ / arxiv. org/ abs/ 0908. 3408); The Free-Will Postulate in

Quantum Mechanics (http:/ / arxiv. org/ abs/ quant-ph/ 0701097)

Bell's theorem 129

[19] http:/ / www. sciam. com/ media/ pdf/ 197911_0158. pdf[20] http:/ / www. dilbert. com/ strips/ comic/ 1992-09-21/[21] http:/ / www. dilbert. com/ strips/ comic/ 1992-09-22/[22] http:/ / www. ncsu. edu/ felder-public/ kenny/ papers/ bell. html[23] http:/ / dx. doi. org/ 10. 1119/ 1. 12594[24] http:/ / www. ipod. org. uk/ reality/ reality_entangled. asp[25] http:/ / xstructure. inr. ac. ru/ x-bin/ theme3. py?level=2& index1=369244[26] http:/ / arxiv. org/ abs/ 0904. 4259[27] http:/ / www. didaktik. physik. uni-erlangen. de/ quantumlab/ english/ index. html

Bell test experimentsThe Bell test experiments serve to investigate the validity of the entanglement effect in quantum mechanics byusing some kind of Bell inequality. John Bell published the first inequality of this kind in his paper "On theEinstein-Podolsky-Rosen Paradox". Bell's Theorem states that a Bell inequality must be obeyed under any localhidden variable theory but can in certain circumstances be violated under quantum mechanics. The term "Bellinequality" can mean any one of a number of inequalities — in practice, in real experiments, the CHSH or CH74inequality, not the original one derived by John Bell. It places restrictions on the statistical results of experiments onsets of particles that have taken part in an interaction and then separated. A Bell test experiment is one designed totest whether or not the real world obeys a Bell inequality. Such experiments fall into two classes, depending onwhether the analysers used have one or two output channels.

Conduct of optical Bell test experimentsIn practice most actual experiments have used light, assumed to be emitted in the form of particle-like photons(produced by atomic cascade or spontaneous parametric down conversion), rather than the atoms that Bell originallyhad in mind. The property of interest is, in the best known experiments, the polarisation direction, though otherproperties can be used.

A typical CHSH (two-channel) experiment

Scheme of a "two-channel" Bell testThe source S produces pairs of "photons", sent in opposite directions. Each photon

encounters a two-channel polariser whose orientation can be set by theexperimenter. Emerging signals from each channel are detected and coincidences

counted by the coincidence monitor CM.

The diagram shows a typical opticalexperiment of the two-channel kind forwhich Alain Aspect set a precedent in 1982(Aspect, 1982a). Coincidences(simultaneous detections) are recorded, theresults being categorised as '++', '+−', '−+' or'−−' and corresponding counts accumulated.

Four separate subexperiments areconducted, corresponding to the four termsE(a, b) in the test statistic S ((2) below). Thesettings a, a′, b and b′ are generally inpractice chosen to be 0, 45°, 22.5° and 67.5°respectively — the "Bell test angles" — these being the ones for which the quantum mechanical formula gives thegreatest violation of the inequality.

For each selected value of a and b, the numbers of coincidences in each category (N++, N--, N+- and N-+) arerecorded. The experimental estimate for E(a, b) is then calculated as:(1)        E = (N++ + N-- − N+- − N-+)/(N++ + N-- + N+- + N-+).

Bell test experiments 130

Once all four E’s have been estimated, an experimental estimate of the test statistic(2)       S = E(a, b) − E(a, b′) + E(a′, b) + E(a′ b′)can be found. If S is numerically greater than 2 it has infringed the CHSH inequality. The experiment is declared tohave supported the QM prediction and ruled out all local hidden variable theories.A strong assumption has had to be made, however, to justify use of expression (2). It has been assumed that thesample of detected pairs is representative of the pairs emitted by the source. That this assumption may not be truecomprises the fair sampling loophole.The derivation of the inequality is given in the CHSH Bell test page.

A typical CH74 (single-channel) experiment

Setup for a "single-channel" Bell testThe source S produces pairs of "photons", sent in opposite directions. Each photonencounters a single channel (e.g. "pile of plates") polariser whose orientation can

be set by the experimenter. Emerging signals are detected and coincidencescounted by the coincidence monitor CM.

Prior to 1982 all actual Bell tests used"single-channel" polarisers and variations onan inequality designed for this setup. Thelatter is described in Clauser, Horne,Shimony and Holt's much-cited 1969 article(Clauser, 1969) as being the one suitable forpractical use. As with the CHSH test, thereare four subexperiments in which eachpolariser takes one of two possible settings,but in addition there are othersubexperiments in which one or otherpolariser or both are absent. Counts aretaken as before and used to estimate the test statistic.(3)       S = (N(a, b) − N(a, b′) + N(a′, b) + N(a′, b′) − N(a′, ∞) − N(∞, b)) / N(∞, ∞),where the symbol ∞ indicates absence of a polariser.If S exceeds 0 then the experiment is declared to have infringed Bell's inequality and hence to have "refuted localrealism".The only theoretical assumption (other than Bell's basic ones of the existence of local hidden variables) that has beenmade in deriving (3) is that when a polariser is inserted the probability of detection of any given photon is neverincreased: there is "no enhancement". The derivation of this inequality is given in the page on Clauser and Horne's1974 Bell test.

Experimental assumptionsIn addition to the theoretical assumptions made, there are practical ones. There may, for example, be a number of"accidental coincidences" in addition to those of interest. It is assumed that no bias is introduced by subtracting theirestimated number before calculating S, but that this is true is not considered by some to be obvious. There may besynchronisation problems — ambiguity in recognising pairs due to the fact that in practice they will not be detectedat exactly the same time.Nevertheless, despite all these deficiencies of the actual experiments, one striking fact emerges: the results are, to a very good approximation, what quantum mechanics predicts. If imperfect experiments give us such excellent overlap with quantum predictions, most working quantum physicists would agree with John Bell in expecting that, when a perfect Bell test is done, the Bell inequalities will still be violated. This attitude has led to the emergence of a new sub-field of physics which is now known as quantum information theory. One of the main achievements of this new branch of physics is showing that violation of Bell's inequalities leads to the possibility of a secure information

Bell test experiments 131

transfer, which utilizes the so-called quantum cryptography (involving entangled states of pairs of particles).

Notable experimentsOver the past thirty or so years, a great number of Bell test experiments have now been conducted. Theseexperiments are subject to assumptions, in particular the ‘no enhancement’ hypothesis of Clauser and Horne (above).The experiments are commonly interpreted to rule out local hidden variable theories, but they could also be said todemonstrate ‘signal enhancement’, which relates to the stochastic resonance phenomenon(Two_classes_of_Bell_inequalities). Advancements in technology have led to significant improvement inefficiencies, as well as a greater variety of methods to test the Bell Theorem.Some of the best known:

Freedman and Clauser, 1972This was the first actual Bell test, using Freedman's inequality, a variant on the CH74 inequality.

Aspect, 1981-2Aspect and his team at Orsay, Paris, conducted three Bell tests using calcium cascade sources. The first andlast used the CH74 inequality. The second was the first application of the CHSH inequality, the third thefamous one (originally suggested by John Bell) in which the choice between the two settings on each side wasmade during the flight of the photons.

Tittel and the Geneva group, 1998The Geneva 1998 Bell test experiments showed that distance did not destroy the "entanglement". Light wassent in fibre optic cables over distances of several kilometers before it was analysed. As with almost all Belltests since about 1985, a "parametric down-conversion" (PDC) source was used.

Weihs' experiment under "strict Einstein locality" conditionsIn 1998 Gregor Weihs and a team at Innsbruck, led by Anton Zeilinger, conducted an ingenious experiment thatclosed the "locality" loophole, improving on Aspect's of 1982. The choice of detector was made using a quantumprocess to ensure that it was random. This test violated the CHSH inequality by over 30 standard deviations, thecoincidence curves agreeing with those predicted by quantum theory.

Pan et al.'s experiment on the GHZ stateThis is the first of new Bell-type experiments on more than two particles; this one uses the so-called GHZ state ofthree particles; it is reported in Nature (2000)

Gröblacher et al. (2007) test of Leggett-type non-local realist theoriesThe authors interpret their results as disfavouring "realism" and hence allow QM to be local but "non-real". Howeverthey have actually only ruled out a specific class of non-local theories suggested by Anthony Leggett.[1] [2]

Salart et al. (2008) Separation in a Bell TestThis experiment filled a loophole by providing an 18 km separation between detectors, which is sufficient to allowthe completion of the quantum state measurements before any information could have traveled between the twodetectors.[3] [4]

Bell test experiments 132

LoopholesThough the series of increasingly sophisticated Bell test experiments has convinced the physics community ingeneral that local realism is untenable, there are still critics who point out that the outcome of every singleexperiment done so far that violates a Bell inequality can, at least theoretically, be explained by faults in theexperimental setup, experimental procedure or that the equipment used does not behave as well as it is supposed to.These possibilities are known as "loopholes". The most serious loophole is the detection loophole, which means thatparticles are not always detected in both wings of the experiment. It is possible to "engineer" quantum correlations(the experimental result) by letting detection be dependent on a combination of local hidden variables and detectorsetting. Experimenters have repeatedly stated that loophole-free tests can be expected in the near future(García-Patrón, 2004). On the other hand, some researchers point out that it is a logical possibility that quantumphysics itself prevents a loophole-free test from ever being implemented (Gill, 2003; Santos, 2006).

References• Aspect, 1981: A. Aspect et al., Experimental Tests of Realistic Local Theories via Bell's Theorem, Phys. Rev.

Lett. 47, 460 (1981)• Aspect, 1982a: A. Aspect et al., Experimental Realization of Einstein-Podolsky-Rosen-Bohm

Gedankenexperiment: A New Violation of Bell's Inequalities, Phys. Rev. Lett. 49, 91 (1982),• Aspect, 1982b: A. Aspect et al., Experimental Test of Bell's Inequalities Using Time-Varying Analyzers, Phys.

Rev. Lett. 49, 1804 (1982),• Barrett, 2002 Quantum Nonlocality, Bell Inequalities and the Memory Loophole: quant-ph/0205016 [5] (2002).• Bell, 1987: J. S. Bell, Speakable and Unspeakable in Quantum Mechanics, (Cambridge University Press 1987)• Clauser, 1969: J. F. Clauser, M.A. Horne, A. Shimony and R. A. Holt, Proposed experiment to test local

hidden-variable theories, Phys. Rev. Lett. 23, 880-884 (1969),• Clauser, 1974: J. F. Clauser and M. A. Horne, Experimental consequences of objective local theories, Phys. Rev.

D 10, 526-35 (1974)• Freedman, 1972: S. J. Freedman and J. F. Clauser, Experimental test of local hidden-variable theories, Phys.

Rev. Lett. 28, 938 (1972)• García-Patrón, 2004: R. García-Patrón, J. Fiurácek, N. J. Cerf, J. Wenger, R. Tualle-Brouri, and Ph. Grangier,

Proposal for a Loophole-Free Bell Test Using Homodyne Detection [6], Phys. Rev. Lett. 93, 130409 (2004)• Gill, 2003: R.D. Gill, Time, Finite Statistics, and Bell's Fifth Position: quant-ph/0301059 [7], Foundations of

Probability and Physics - 2, Vaxjo Univ. Press, 2003, 179-206 (2003)• Kielpinski: D. Kielpinski et al., Recent Results in Trapped-Ion Quantum Computing [8] (2001)• Kwiat, 1999: P.G. Kwiat, et al., Ultrabright source of polarization-entangled photons [9], Physical Review A 60

(2), R773-R776 (1999)• Rowe, 2001: M. Rowe et al., Experimental violation of a Bell’s inequality with efficient detection [10], Nature 409,

791 (2001)• Santos, 2005: E. Santos, Bell's theorem and the experiments: Increasing empirical support to local realism:

quant-ph/0410193 [11], Studies In History and Philosophy of Modern Physics, 36, 544-565 (2005)• Tittel, 1997: W. Tittel et al., Experimental demonstration of quantum-correlations over more than 10 kilometers

[12], Phys. Rev. A, 57, 3229 (1997)• Tittel, 1998: W. Tittel et al., Experimental demonstration of quantum-correlations over more than 10 kilometers

[12], Physical Review A 57, 3229 (1998); Violation of Bell inequalities by photons more than 10 km apart [13],Physical Review Letters 81, 3563 (1998)

• Weihs, 1998: G. Weihs, et al., Violation of Bell’s inequality under strict Einstein locality conditions [14], Phys.Rev. Lett. 81, 5039 (1998)

Bell test experiments 133

References[1] Quantum physics says goodbye to reality (http:/ / physicsworld. com/ cws/ article/ news/ 27640)[2] An experimental test of non-local realism (http:/ / www. nature. com/ nature/ journal/ v446/ n7138/ abs/ nature05677. html)[3] Salart, D.; Baas, A.; van Houwelingen, J. A. W.; Gisin, N.; and Zbinden, H. “Spacelike Separation in a Bell Test Assuming Gravitationally

Induced Collapses.” Physical Review Letters 100, 220404 (2008). (http:/ / arxiv. org/ abs/ 0803. 2425)[4] http:/ / www. physorg. com/ news132830327. html[5] http:/ / arxiv. org/ abs/ quant-ph/ 0205016[6] http:/ / arxiv. org/ abs/ quant-ph/ 0403191[7] http:/ / arxiv. org/ abs/ quant-ph/ 0301059[8] http:/ / arxiv. org/ abs/ quant-ph/ 0102086[9] http:/ / arXiv. org/ abs/ quant-ph/ 9810003[10] http:/ / www. nature. com/ nature/ journal/ v409/ n6822/ abs/ 409791a0. html[11] http:/ / arxiv. org/ abs/ quant-ph/ 0410193[12] http:/ / arxiv. org/ abs/ quant-ph/ 9707042[13] http:/ / arxiv. org/ abs/ quant-ph/ 9806043[14] http:/ / arXiv. org/ abs/ quant-ph/ 9810080

Hidden variablesHidden variables may refer to:• In physics, Hidden variable theories are a class of theories trying to explain away the statistical nature of quantum

mechanics.• In statistics, Latent variables are variables that are inferred from other observed variables.• In Computer Science, Hidden transformation is a way to transform a generic constraint satisfaction problem into a

binary one by introducing new hidden variables.

Article Sources and Contributors 134

Article Sources and ContributorsDavid Bohm  Source: http://en.wikipedia.org/w/index.php?oldid=358755906  Contributors: A Kit, Abeltje, Acroterion, Ael 2, Ahalani, Alan XAX Freeman, Amanda.nelson12, And4e, AndriuZ,Art Carlson, Auréola, Belinrahs, BhangraGirl, BillBell, Billinghurst, Birchmore, Blainster, Bobblehead, Carlo.Ierna, Cgingold, Charles Matthews, Csberger, D6, Demiurge, Duendeverde,ELApro, EPadmirateur, Ealconchel, Fastfission, Floorsheim, Franis, Franis Engel, GangofOne, Gary D, Georgewilliamherbert, Gilisa, Goatasaur, Goethean, Good Olfactory, Grazia11, Gregbard,Headbomb, Heah, Hmains, Ig0774, InquireConsciously, Io, J Di, JEN9841, Jansci Tilleman, Jb849, Jiang, Joelwest, John Z, Jpbowen, KYPark, Kalki, Karol Langner, Kbdank71, Krash,L7HOMAS, Lambiam, Leptons, Liontooth, Liquidhuman, Looxix, Lotte Monz, MER-C, MarcAurel, Marokwitz, Masterpiece2000, Maurice Carbonaro, Mbahrami, Medlat, MessinaRagazza,Midgley, Mmarci, Modify, Muchness, Mwanner, NickBush24, Nietzsche 2, Northwesterner1, Npepperell, Olessi, Ombudsman, Outriggr, Paul A, Pedant17, Persephone19, Piano non troppo,Postdlf, PremRasal, QuantumOne, RKiddzz, Renata3, Rich Farmbrough, Richard Taytor, Rjwilmsi, Robofish, Rsabbatini, SGGH, Sadi Carnot, Sc147, Smithfarm, Splash, Studentofisless,Tarotcards, The wub, TheMadBaron, Thinkg, Threepounds, Timrollpickering, Togo, TonyClarke, Torrazzo, Treybien, Twas Now, Udzu, Versageek, Victor Gijsbers, Victor Lopes, W guice,Waelder, Wasell, Wayward, Wereon, Wik, Woohookitty, Wuhwuzdat, Zereshk, 165 ,רופביני anonymous edits

Aharonov- Bohm effect  Source: http://en.wikipedia.org/w/index.php?oldid=341663792  Contributors: Andejons, Andyspring, Arnero, BenRG, Bender235, Bobblehead, Brews ohare, CYD,Casey boy, Charles Matthews, Dratman, Dysprosia, E.pajer, Fapae, Gaius Cornelius, GregAsche, Gulmammad, Headbomb, Helixblue, Henry Delforn, Karol Langner, Ladypine, Linas, Liontooth,Longone, MFNickster, Martin TB, Michael C Price, Omegatron, Pixelface, Reddi, Richsid, SPat, Sbyrnes321, Siddhant, Stevenj, The wub, Tim Starling, Timb66, Tlabshier, Vssun, Wolfkeeper,50 anonymous edits

Bohm diffusion  Source: http://en.wikipedia.org/w/index.php?oldid=357971977  Contributors: ABCD, Agricola44, Art Carlson, Cypa, DabMachine, Deglr6328, GangofOne, Karol Langner,Koffieyahoo, Matticus78, PKnight, RoundemUpJeff, Simeon H, Venny85, 11 anonymous edits

Bohm interpretation  Source: http://en.wikipedia.org/w/index.php?oldid=326542520  Contributors: 1ForTheMoney, A.C. Norman, Agger, Alessandro70, Alphatronic, Andersæøå, Andrewpmk,AoS1014, Arjen Dijksman, AshtonBenson, Barbara Shack, Benja, Borat fan, CSTAR, Charles Matthews, Cmdulya, Cojoco, DV8 2XL, Dan Gluck, Dataweaver, Deadly Nut, Dhemm, Dmr2,DomenicDenicola, Dragon's Blood, Duduong, Duendeverde, Dvtausk, ESkog, Ebitnet, Editorius, Edward, Emurphy42, Evand, Extremophile, Eyv, Falcorian, Floorsheim, Freakofnurture,GangofOne, Giftlite, Goethean, Gregbard, GregorB, Holon, Hypnosifl, IRevLinas, Ilja Schmelzer, Iridescent, Itangalo, Jambaugh, Jason Davies, Jefffire, Jfire, Jmundo, John Reaves, Jostylr,KSchutte, Kalonymos, Kevin aylward, Kkchang, Kripkenstein, Kuratowski's Ghost, L0rents, L33tminion, LC, Leafyplant, Lexivore, Likebox, Linas, Loren Rosen, Lumidek, M0rph, MarSch,Michael C Price, Michael Devore, Michael Hardy, Michael Rogers, Mike Peel, Mir Harven, MistySpock, Murf42, Noah Salzman, Nsomnia03, Olleicua, Pfalstad, Phys-demystifier, Plumbago,Pretzelpaws, Quantumobserver, RDBury, RJN, RandomHumanoid, Razimantv, Rednblu, Rich Farmbrough, Rjwilmsi, Roadrunner, RogueNinja, SQL, Scentoni, Scottie 000, Sfwild, SiegeLord,Skewyou, Smithfarm, Stephen B Streater, SuezanneC Baskerville, Sverdrup, Swiftly, The Anome, The Anonymous One, Tim Starling, Timwi, Tjic, Tnorsen, Togo, Tomixdf, Tonymec, Tumulka,Ummonk, Vegaswikian, Venny85, Vgy7ujm, Voorlandt, Vuen, W1k13rh3nry, Waleswatcher, Weierstrass, Were-Bunny, Wireader, WolfmanSF, Yafujifide, ZRPerry, Zicovich, Zoicon5, 189anonymous edits

Correspondence principle  Source: http://en.wikipedia.org/w/index.php?oldid=320470658  Contributors: Ahoerstemeier, AllenUNC86, Bdesham, Bobathon71, CYD, Charles Matthews,Claudiodib, Colonies Chris, Dan Gluck, Fastfission, Giftlite, Headbomb, Jakob.scholbach, Jctaylr, Johnstone, Jrdioko, Keenan Pepper, Knucmo2, Likebox, Lixy, Michael C Price, Michael Hardy,Mpatel, Naturalnumber, Oleg Alexandrov, Radagast83, Rayhaith, Rbj, RepublicanJacobite, Rinconsoleao, Rorro, Semako, Shalom Yechiel, Smmurphy, Snoyes, Venny85, WMdeMuynck,Yevgeny Kats, Youandme, 49 anonymous edits

De Broglie–Bohm theory  Source: http://en.wikipedia.org/w/index.php?oldid=361431514  Contributors: 1ForTheMoney, A.C. Norman, Agger, Alessandro70, Alphatronic, Andersæøå,Andrewpmk, AoS1014, Arjen Dijksman, AshtonBenson, Barbara Shack, Benja, Borat fan, CSTAR, Charles Matthews, Cmdulya, Cojoco, DV8 2XL, Dan Gluck, Dataweaver, Deadly Nut,Dhemm, Dmr2, DomenicDenicola, Dragon's Blood, Duduong, Duendeverde, Dvtausk, ESkog, Ebitnet, Editorius, Edward, Emurphy42, Evand, Extremophile, Eyv, Falcorian, Floorsheim,Freakofnurture, GangofOne, Giftlite, Goethean, Gregbard, GregorB, Holon, Hypnosifl, IRevLinas, Ilja Schmelzer, Iridescent, Itangalo, Jambaugh, Jason Davies, Jefffire, Jfire, Jmundo, JohnReaves, Jostylr, KSchutte, Kalonymos, Kevin aylward, Kkchang, Kripkenstein, Kuratowski's Ghost, L0rents, L33tminion, LC, Leafyplant, Lexivore, Likebox, Linas, Loren Rosen, Lumidek,M0rph, MarSch, Michael C Price, Michael Devore, Michael Hardy, Michael Rogers, Mike Peel, Mir Harven, MistySpock, Murf42, Noah Salzman, Nsomnia03, Olleicua, Pfalstad,Phys-demystifier, Plumbago, Pretzelpaws, Quantumobserver, RDBury, RJN, RandomHumanoid, Razimantv, Rednblu, Rich Farmbrough, Rjwilmsi, Roadrunner, RogueNinja, SQL, Scentoni,Scottie 000, Sfwild, SiegeLord, Skewyou, Smithfarm, Stephen B Streater, SuezanneC Baskerville, Sverdrup, Swiftly, The Anome, The Anonymous One, Tim Starling, Timwi, Tjic, Tnorsen,Togo, Tomixdf, Tonymec, Tumulka, Ummonk, Vegaswikian, Venny85, Vgy7ujm, Voorlandt, Vuen, W1k13rh3nry, Waleswatcher, Weierstrass, Were-Bunny, Wireader, WolfmanSF, Yafujifide,ZRPerry, Zicovich, Zoicon5, 189 anonymous edits

EPR paradox  Source: http://en.wikipedia.org/w/index.php?oldid=358449842  Contributors: -- April, .:Ajvol:., 207.171.93.xxx, ASCWiki, Agge1000, Agger, Alan McBeth, Alkivar, Amakuha,AmarChandra, Andejons, Apoc2400, Ark, Arpingstone, AxelBoldt, B4hand, Ballhausflip, Bevo, Bigbluefish, Blauhart, Brews ohare, Brianga, Brrk.3001, Bryan Derksen, CSTAR, CYD,Canadian-Bacon, Carbuncle, Cardmagic, Caroldermoid, Caroline Thompson, Chas zzz brown, Chris 73, Clarityfiend, CobbSalad, Complexica, Cortonin, Craig Pemberton, Crowsnest, DBooth,Darktaco, David R. Ingham, Declare, Dogcow, Dpbsmith, Dr Smith, DrBob, DrChinese, DragonflySixtyseven, Dryke, Dzhim, Długosz, EdH, Edward, Eequor, Egmontaz, Eiffel, Ejrh, Falcorian,Finlay McWalter, Fredkinfollower, Frish, Fulldecent, Fwappler, GeorgeMoney, Goldfritter, Graham87, GregorB, Gretyl, Hackwrench, Hephaestos, Hirak 99, Houftermann, Hugo Dufort, Hydnjo,IAdem, J-Star, JaGa, JamesMLane, Jan-Åke Larsson, Jcajacob, Jinxman1, Jnc, JocK, JohnBlackburne, Jpittelo, Jwrosenzweig, KHamsun, Kapalama, Karada, Karol Langner, KasugaHuang,Keenan Pepper, Kuratowski's Ghost, Larsobrien, Lethe, Lf89, LiDaobing, Linas, Linus M., Looxix, Lumidek, Maher27777, Marek.zukowski, Marie Poise, Mark J, Masudr, Maurice Carbonaro,Mav, Metron4, Michael C Price, Michael Hardy, Moink, MrJones, Msridhar, Naddy, Natevw, NathanHurst, Nickyus, Noca2plus, ObsidianOrder, Owen, Pace212, Paranoid, Pateblen, Peashy,Pekka.virta, Peter Erwin, PeterBFZ, PierreAbbat, Prezbo, Publicly Visible, Pwjb, Pérez, RTreacy, Radiofriendlyunitshifta, Rama, Razimantv, Rich Farmbrough, Roadrunner, Robert K S, Robertd,Robertefields, Ronjoseph, Rracecarr, Ryft, SGBailey, Sanders muc, Schneelocke, ScienceApologist, Seb, Shadanan, Shakyshake, Shalom Yechiel, Sidasta, Skierpage, Snoyes, Srleffler, Stain,Steve Quinn, Suisui, Sundar, Syko, Tarotcards, Tempshill, Tercer, Texture, ThomasK, Thrain2, Timwi, Tlabshier, Tsop, Ty8inf, Vasiľ, Victor Gijsbers, Vodex, Voyajer, WMdeMuynck,Weekwhom, Wik, Wile E. Heresiarch, William M. Connolley, XJamRastafire, Xgrrr, YUL89YYZ, Yill577, Zeycus, Zootm, Александър, 264 anonymous edits

Holographic paradigm  Source: http://en.wikipedia.org/w/index.php?oldid=344204183  Contributors: Artaxiad, Cgingold, D-rew, Horkana, Innv, JohnCD, Kurtan, Len Raymond, Maxrempel,Pigsonthewing, Rich Farmbrough, Tarotcards, Tassedethe, Wndl42, 42 anonymous edits

Holographic principle  Source: http://en.wikipedia.org/w/index.php?oldid=357123504  Contributors: 2over0, Addps4cat, Al Lemos, Alexnye, Andrw, Armeria, AugPi, Bcrowell, Bender235,Bmdavll, Borat fan, Brianhe, Bryan Derksen, Charvest, Christopher Thomas, Ckatz, Cmcelwain, Conversion script, Crus4d3, Derek Ross, Dougweller, Dr. Salvia, Fcady2007, Flash.starwalker,Foresee, Gpvos, Graeme Bartlett, Gzabers, Headbomb, HorsePunchKid, Hypnosifl, Jfraatz, JocK, Johnfn, Joke137, Jynus, Kermit2, Knotwork, Kungfoofairy, Lambiam, LeYaYa, Leibniz, LenRaymond, Likebox, Ln2069, LoveEncounterFlow, M4gnum0n, Mark Germine, M.D., Maurice Carbonaro, McSly, Mcarling, Mhs5392, Michael C Price, Mindmatrix, Mmortal03, Mpatel, NatKrause, NerdyNSK, PEiP, Peterdjones, Phil Boswell, PhysPhD, Pink18, Pjacobi, Pohta ce-am pohtit, Potatoswatter, Prari, Reddi, Richard1968, Rjwilmsi, Robma, Scentoni, ScienceApologist,Sheliak, Simonm223, Sligocki, Stevertigo, StradivariusTV, Sverdrup, Systemizer, Tassedethe, Tdent, The Anome, TimBentley, Timo Honkasalo, Togo, Tromer, UncleDouggie, Utopos, Verbal,Vicki Rosenzweig, Vyznev Xnebara, Wndl42, Xaven, 105 anonymous edits

Holomovement  Source: http://en.wikipedia.org/w/index.php?oldid=357670681  Contributors: Ahalani, Colonies Chris, Kitten b, Leshughb, MBisanz, Misarxist, RHaworth, Sfwild, Skysmith,Tassedethe, Timrollpickering, Wingedsubmariner, Wod observer, 10 anonymous edits

Holonomic brain theory  Source: http://en.wikipedia.org/w/index.php?oldid=358711142  Contributors: 1ForTheMoney, A Ramachandran, Aaron Brenneman, Algebraist, Antaeus Feldspar,Arkansas001, Ask123, Badguy21, Blainster, Bookandcoffee, Catalyst2007, Cgingold, Crzrussian, DannyDaWriter, FT2, GangofOne, Gary D, Goethean, GraemeL, Grafen, Helvetius, Holon,Jammus, Johnbod, KYPark, Karada, Looie496, Lordvolton, Loxley, Mattisse, Mccready, Midgley, Mramz88, Natalya, Ombudsman, Phoenix-forgotten, Pnrj, RichardF, Scott5834, Srleffler, TheAnome, Three887, TimBentley, Travis.Thurston, USAjp22, Valarians, Whatever404, Woohookitty, Zazaban, 25 ,کشرز anonymous edits

Implicate and Explicate Order  Source: http://en.wikipedia.org/w/index.php?oldid=326124478  Contributors: Alloranleon, ArnoldReinhold, Banno, Belovedfreak, Blainster, Bwhack, Cgingold,Charles Matthews, Ciprianman, Commander Keane, DO'Neil, DerEikopf, Dispenser, Draicone, Dreadstar, Eaefremov, Eep², Esowteric, Falcorian, Floorsheim, Goethean, Holon, Howcheng,Igiffin, J. Finkelstein, J04n, JLaTondre, Jeff3000, Josh Parris, KYPark, Kh7, Lisatwo, Macmelvino, MarekTT, Matthew Yeager, Mayagaia, Mbell, Michael Hardy, Midgley, Mitar, Mysdaao,Nealparr, Nonphixion, Philippe, Popsracer, R Lee E, RichardF, Robofish, Rosenkreuz, Sfwild, Stevertigo, Sundar, Tassedethe, The Anome, The wub, ThePeg, Togo, Unfree, 29 anonymous edits

Implicate order  Source: http://en.wikipedia.org/w/index.php?oldid=326124435  Contributors: Alloranleon, ArnoldReinhold, Banno, Belovedfreak, Blainster, Bwhack, Cgingold, CharlesMatthews, Ciprianman, Commander Keane, DO'Neil, DerEikopf, Dispenser, Draicone, Dreadstar, Eaefremov, Eep², Esowteric, Falcorian, Floorsheim, Goethean, Holon, Howcheng, Igiffin, J.Finkelstein, J04n, JLaTondre, Jeff3000, Josh Parris, KYPark, Kh7, Lisatwo, Macmelvino, MarekTT, Matthew Yeager, Mayagaia, Mbell, Michael Hardy, Midgley, Mitar, Mysdaao, Nealparr,Nonphixion, Philippe, Popsracer, R Lee E, RichardF, Robofish, Rosenkreuz, Sfwild, Stevertigo, Sundar, Tassedethe, The Anome, The wub, ThePeg, Togo, Unfree, 29 anonymous edits

Article Sources and Contributors 135

Membrane paradigm  Source: http://en.wikipedia.org/w/index.php?oldid=321366194  Contributors: Alai, Bm gub, Damacdonald, Drbreznjev, ErkDemon, Gaius Cornelius, Kasparov, KenArromdee, Len Raymond, Loren Rosen, Michael C Price, Paul August, Phys, Roadrunner, SDC, SimonP, Thehotelambush, 6 anonymous edits

Orch- OR  Source: http://en.wikipedia.org/w/index.php?oldid=360158729  Contributors: AugustinMa, Bardon Dornal, Bboyneko, BenRG, Bigmantonyd, Brookie, C S, Ciphergoth,Clicketyclack, Cmallett, DV8 2XL, Danko Georgiev MD, Deathphoenix, Deglr6328, Delta G, El C, Euphrosyne, Full Shunyata, Gary D, Gene Nygaard, Gil987, Giraffedata, HEL, Hameroff,Hhmaung, Iridescent, Jacobolus, Jefffire, Jkasd, JonathanD, Kablamo2007, Karch, MCB, Maury Markowitz, Mbell, Mgiganteus1, Michael Hardy, Nagualdesign, Night Gyr, OlgaMats,Patrickwilken, Pennarin, Persephone19, Peterdjones, Pollinosisss, QueenAdelaide, ReluctantPhilosopher, Rich Farmbrough, Rickybuchanan, Robma, Sperxios, Tarcieri, The Anome, Three887,Tijmz, Tim Shuba, Ufoolme, Vald, Weirdy, WereSpielChequers, Wingedsubmariner, 57 anonymous edits

Debye sheath  Source: http://en.wikipedia.org/w/index.php?oldid=331805750  Contributors: Alai, Art Carlson, Frank101, Iantresman, Jaganath, Linas, Stannered, 16 anonymous edits

John Stewart Bell  Source: http://en.wikipedia.org/w/index.php?oldid=357224211  Contributors: A-Ge0, A5b, AGK, Alansohn, Angela, Ardfern, Bkalafut, Blainster, Blutfink, Brownshoes,CRKingston, Caroline Thompson, ChKa, Charles Matthews, Cmdrjameson, D6, Dammit, Deathphoenix, Dhemm, Diberri, Drewarrowood, Dvtausk, EugeneZelenko, Evidentialist, Fasach Nua,Georgia guy, Gershwinrb, Harald88, Interintel, Ipatrol, Jaraalbe, John, Jpgordon, Kalki, Keithbowden, Kevin Forsyth, KingTT, Lethe, Looxix, Lumidek, Marasmusine, Marek.zukowski,MartinRobinson, Masterpiece2000, Mayumashu, Mbahrami, Miaers, MisterSheik, Mukkakukaku, Nfr-Maat, Notjim, Paul Silverman, Rl, SJLPP, Sanders muc, Sannse, Stepa, The Fashion Icon,TheParanoidOne, Thejerm, Timrollpickering, Tuesdaily, Waltpohl, Wikiliki, XJamRastafire, 59 anonymous edits

Karl H. Pribram  Source: http://en.wikipedia.org/w/index.php?oldid=361425443  Contributors: 6birc, A314268, Allen3, Antaeus Feldspar, Bomac, Cgingold, Cuchullain, D6, DMG413, Denny,FT2, Gamahucheur, Gareth Jones, Gary D, Goethean, Hu, Iaai 99, J'raxis, Johnpacklambert, Jonegan, KrakatoaKatie, LMBM2012, Lotte Monz, M.Ajnhorn, MarcAurel, Midgley, Mike Dillon,Mpulier, Nanouk, Nv8200p, Olessi, Ombudsman, Pedrovitorh2, Persephone19, PeterStJohn, RogDel, Rsabbatini, SCEhardt, Sadi Carnot, Siddhi.powers, Suidafrikaan, Tassedethe, VerbumVeritas, Viriditas, WhatamIdoing, Xmarquez, Zahd, 29 anonymous edits

Implicate and explicate order according to David Bohm  Source: http://en.wikipedia.org/w/index.php?oldid=349663213  Contributors: Alloranleon, ArnoldReinhold, Banno, Belovedfreak,Blainster, Bwhack, Cgingold, Charles Matthews, Ciprianman, Commander Keane, DO'Neil, DerEikopf, Dispenser, Draicone, Dreadstar, Eaefremov, Eep², Esowteric, Falcorian, Floorsheim,Goethean, Holon, Howcheng, Igiffin, J. Finkelstein, J04n, JLaTondre, Jeff3000, Josh Parris, KYPark, Kh7, Lisatwo, Macmelvino, MarekTT, Matthew Yeager, Mayagaia, Mbell, Michael Hardy,Midgley, Mitar, Mysdaao, Nealparr, Nonphixion, Philippe, Popsracer, R Lee E, RichardF, Robofish, Rosenkreuz, Sfwild, Stevertigo, Sundar, Tassedethe, The Anome, The wub, ThePeg, Togo,Unfree, 29 anonymous edits

Hidden variable theory  Source: http://en.wikipedia.org/w/index.php?oldid=358050867  Contributors: Agge1000, Antonielly, Arensb, BeatePaland, Benhocking, Blainster, Bubbleboys,CannibalSmith, Cardmagic, Caroline Thompson, Charles Matthews, Chetvorno, Chippy87, Christopher Cooper, Ciphers, ConradPino, Count Iblis, DJIndica, Deadly Nut, Deathphoenix, DiegoMoya, Dragon's Blood, Dvtausk, Elsweyn, Falcorian, Fastily, Felixaldonso, Foober, Freakofnurture, Fulldecent, GaeusOctavius, Goethean, Guettarda, Hornplease, Hwasungmars, Inquam,Iridescent, Jim E. Black, Keithbowden, Lethe, Linas, Lumidek, Mckaysalisbury, Michael C Price, Michael Hardy, Orangedolphin, PChalmer, Peterdjones, Plumbago, Roadrunner, Ryan Cable,Sheliak, Shoecream, Smithfarm, StuRat, Tarotcards, The wub, Thecurran91, Trevor mendham, Vnvnfls, Vyznev Xnebara, Whitepaw, Wildspell, Zeeber78, Zoz, 51 ,התאו ינא anonymous edits

Local hidden variable theory  Source: http://en.wikipedia.org/w/index.php?oldid=361286376  Contributors: Agge1000, Antonielly, Asterion, Brighterorange, Caroline Thompson, CharlesMatthews, DrChinese, Filemon, Fuhghettaboutit, Gazpacho, GregorB, Headbomb, Keithbowden, Linas, Mlessard, Orangedolphin, Pavel Vozenilek, Pfalstad, Pilgrims tale, Plumbago, Reinderien,Richard75, Roadrunner, Shminux, Sophroniscus, Tothebarricades.tk, WMdeMuynck, Whitepaw, Zahd, 35 anonymous edits

Bell's theorem  Source: http://en.wikipedia.org/w/index.php?oldid=360942204  Contributors: A. di M., AC+79 3888, AdamSiska, Agge1000, Aldux, AmarChandra, Anakin101,Andrewthomas10, Android Mouse, Antonielly, Aranel, ArnoldReinhold, Ashmoo, Avb, B9 hummingbird hovering, Ballhausflip, Barticus88, Baxxterr, BeatePaland, Bender235, BeteNoir, BoJacoby, BobKawanaka, Bobo192, Bongwarrior, Bryan Derksen, Byrgenwulf, CSTAR, CYD, Cacycle, Calvero JP, Caroline Thompson, Catchanil, Cgingold, Chalst, Charles Matthews,Christopher Cooper, Complexica, Count Iblis, Cremepuff222, Crocodealer, Curious1i, Dauto, Dave Runger, Deathphoenix, Deniz195, Dfrg.msc, Dirac66, Dmr2, DomenicDenicola, DrChinese,Drilnoth, Drmies, EcoMan, EdC, Eequor, Egg, Endlessmike 888, Fastily, Franl, Frish, Fulldecent, Fwappler, GangofOne, Gene Nygaard, Geremia, Giftlite, Gmusser, Gregbard, GregorB, Grok42,Gsjaeger, Guy Harris, Gzornenplatz, HaeB, Hairy Dude, Harald88, Headbomb, Hmonroe, IWillNeverLearn, Informatorium, Interintel, Isaacdealey, Jcobb, Jim E. Black, Jim.belk, Jpittelo, Jwpitts,KHamsun, Keenan Pepper, Keithbowden, Kingmundi, Kmmertes, KnowledgeOfSelf, Konstable, Kyrisch, LC, Laudaka, Leobold1, Lethe, Likebox, Linas, Lumidek, Lycurgus, Malcohol,Maniatis, Marek.zukowski, MathKnight, Maxw41, Michael C Price, Michael Hardy, Mike2vil, Mister fu-ck you, MisterSheik, Mmyotis, Monslucis, Nihil, Oakwood, Oleg Alexandrov, OlekBijok, Orangedolphin, Ott2, Paddu, Patrick0Moran, Paul August, Paulginz, Petri Krohn, Physicskev, Physis, Pjacobi, Plrk, Pokipsy76, RJFJR, RK, Rich Farmbrough, Richard75, Richwil, RickardVogelberg, Rl, Roadrunner, Robert1947, Rodasmith, Roke, Rror, Ryanmcdaniel, SJLPP, Salsb, Seth Bresnett, Simetrical, StevenJohnston, Stevertigo, Stirling Newberry, Suffusion of Yellow,Susvolans, Tethys sea, Tothebarricades.tk, Tox, TriTertButoxy, Ueit, Ulrich Utiger, Vessels42, WMdeMuynck, Waleswatcher, Wfku, Wwoods, Ybband, Yecril, Zeycus, Zvis, 214 anonymousedits

Bell test experiments  Source: http://en.wikipedia.org/w/index.php?oldid=360743504  Contributors: 1pezguy, A.Ou, Agge1000, CSTAR, Cactus.man, Caroline Thompson, Christopherjfoster,Cmdrjameson, DFrankstein, DJIndica, DaveMudstain, Deathphoenix, Deviant Paleoart, Download, DrBob, DrChinese, Dreish, Giftlite, Gill110951, Glenn, Harald88, Headbomb, Hmonroe, J SLundeen, Jan-Åke Larsson, Jaraalbe, Joel7687, Julesd, KDesk, Linas, Matthew Mattic, Maxw41, Mhims, Nwbeeson, PaddyLeahy, Pjacobi, RayTomes, Rich Farmbrough, Richardbrucebaxter,Sheliak, Snowdog, Venny85, Wikiliki, Ybband, 37 anonymous edits

Hidden variables  Source: http://en.wikipedia.org/w/index.php?oldid=251546317  Contributors: Antonielly, Diego Moya, Discospinster, Skomorokh, 1 anonymous edits

Image Sources, Licenses and Contributors 136

Image Sources, Licenses and ContributorsImage:David Bohm.jpg  Source: http://en.wikipedia.org/w/index.php?title=File:David_Bohm.jpg  License: Attribution  Contributors: Original uploader was Karol Langner at en.wikipediaImage:Flag of the United Kingdom.svg  Source: http://en.wikipedia.org/w/index.php?title=File:Flag_of_the_United_Kingdom.svg  License: Public Domain  Contributors: User:Zscout370Image:aharonov-bohm.png  Source: http://en.wikipedia.org/w/index.php?title=File:Aharonov-bohm.png  License: GNU Free Documentation License  Contributors: Original uploader wasStevenj at en.wikipediaImage:doppelspalt.jpg  Source: http://en.wikipedia.org/w/index.php?title=File:Doppelspalt.jpg  License: Public Domain  Contributors: AnonMoos, Glenn, OpassonImage:EPR-paradox-illus.png  Source: http://en.wikipedia.org/w/index.php?title=File:EPR-paradox-illus.png  License: GNU Free Documentation License  Contributors: Original uploader wasCSTAR at en.wikipediaImage:Hydrogen.svg  Source: http://en.wikipedia.org/w/index.php?title=File:Hydrogen.svg  License: GNU Free Documentation License  Contributors: Mets501, Mion, Soeb, Treisijs, Xxxx00, 5anonymous editsImage:Holography-reconstruct.png  Source: http://en.wikipedia.org/w/index.php?title=File:Holography-reconstruct.png  License: GNU Free Documentation License  Contributors: Originaluploader was DrBob at en.wikipediaImage:Human brain NIH.jpg  Source: http://en.wikipedia.org/w/index.php?title=File:Human_brain_NIH.jpg  License: unknown  Contributors: OldakQuill, StuRat, Talgraf777, Ysangkok, 1anonymous editsImage:Epithelial-cells.jpg  Source: http://en.wikipedia.org/w/index.php?title=File:Epithelial-cells.jpg  License: GNU Free Documentation License  Contributors: Dbc334, Duesentrieb,GeorgHH, Helix84, JWSchmidt, Martin H., ViperSnake151, 2 anonymous editsImage:Common clownfish.jpg  Source: http://en.wikipedia.org/w/index.php?title=File:Common_clownfish.jpg  License: unknown  Contributors: User:JanderkImage:plasma-sheath.svg  Source: http://en.wikipedia.org/w/index.php?title=File:Plasma-sheath.svg  License: Creative Commons Attribution 2.5  Contributors: User:StanneredImage:John Stewart Bell's Blue plaque.JPG  Source: http://en.wikipedia.org/w/index.php?title=File:John_Stewart_Bell's_Blue_plaque.JPG  License: Public Domain  Contributors: Originaluploader was Fasach Nua at en.wikipediaImage:Pribram Tucson2.jpg  Source: http://en.wikipedia.org/w/index.php?title=File:Pribram_Tucson2.jpg  License: Public Domain  Contributors: User:ZereshkImage:StraightLines.png  Source: http://en.wikipedia.org/w/index.php?title=File:StraightLines.png  License: Public Domain  Contributors: Brighterorange, Caroline Thompson, Mike Rosoft,Nv8200p, RedWolf, StanneredImage:MalusQC.png  Source: http://en.wikipedia.org/w/index.php?title=File:MalusQC.png  License: unknown  Contributors: Original uploader was Caroline Thompson at en.wikipediaImage:Bells-thm.png  Source: http://en.wikipedia.org/w/index.php?title=File:Bells-thm.png  License: GNU Free Documentation License  Contributors: Bdesham, It Is Me Here,Joshbaumgartner, Karelj, Maksim, Mdd, Pieter Kuiper, Tano4595Image:Bell-test-photon-analyer.png  Source: http://en.wikipedia.org/w/index.php?title=File:Bell-test-photon-analyer.png  License: GNU Free Documentation License  Contributors: Chetvorno,Glenn, Joshbaumgartner, Karelj, Maksim, MddFile:Two channel.png  Source: http://en.wikipedia.org/w/index.php?title=File:Two_channel.png  License: GNU Free Documentation License  Contributors: MaksimFile:Single-channel Bell test.png  Source: http://en.wikipedia.org/w/index.php?title=File:Single-channel_Bell_test.png  License: GNU Free Documentation License  Contributors: CarolineThompson, RedWolf

License 137

LicenseCreative Commons Attribution-Share Alike 3.0 Unportedhttp:/ / creativecommons. org/ licenses/ by-sa/ 3. 0/