Philosophers
Mortimer Adler Rogers Albritton Alexander of Aphrodisias Samuel Alexander William Alston Anaximander G.E.M.Anscombe Anselm Louise Antony Thomas Aquinas Aristotle David Armstrong Harald Atmanspacher Robert Audi Augustine J.L.Austin A.J.Ayer Alexander Bain Mark Balaguer Jeffrey Barrett William Belsham Henri Bergson George Berkeley Isaiah Berlin Richard J. Bernstein Bernard Berofsky Robert Bishop Max Black Susanne Bobzien Emil du BoisReymond Hilary Bok Laurence BonJour George Boole Émile Boutroux F.H.Bradley C.D.Broad Michael Burke C.A.Campbell Joseph Keim Campbell Rudolf Carnap Carneades Ernst Cassirer David Chalmers Roderick Chisholm Chrysippus Cicero Randolph Clarke Samuel Clarke Anthony Collins Antonella Corradini Diodorus Cronus Jonathan Dancy Donald Davidson Mario De Caro Democritus Daniel Dennett Jacques Derrida René Descartes Richard Double Fred Dretske John Dupré John Earman Laura Waddell Ekstrom Epictetus Epicurus Herbert Feigl John Martin Fischer Owen Flanagan Luciano Floridi Philippa Foot Alfred Fouilleé Harry Frankfurt Richard L. Franklin Michael Frede Gottlob Frege Peter Geach Edmund Gettier Carl Ginet Alvin Goldman Gorgias Nicholas St. John Green H.Paul Grice Ian Hacking Ishtiyaque Haji Stuart Hampshire W.F.R.Hardie Sam Harris William Hasker R.M.Hare Georg W.F. Hegel Martin Heidegger Heraclitus R.E.Hobart Thomas Hobbes David Hodgson Shadsworth Hodgson Baron d'Holbach Ted Honderich Pamela Huby David Hume Ferenc Huoranszki William James Lord Kames Robert Kane Immanuel Kant Tomis Kapitan Jaegwon Kim William King Hilary Kornblith Christine Korsgaard Saul Kripke Andrea Lavazza Keith Lehrer Gottfried Leibniz Jules Lequyer Leucippus Michael Levin George Henry Lewes C.I.Lewis David Lewis Peter Lipton C. Lloyd Morgan John Locke Michael Lockwood E. Jonathan Lowe John R. Lucas Lucretius Alasdair MacIntyre Ruth Barcan Marcus James Martineau Storrs McCall Hugh McCann Colin McGinn Michael McKenna Brian McLaughlin John McTaggart Paul E. Meehl Uwe Meixner Alfred Mele Trenton Merricks John Stuart Mill Dickinson Miller G.E.Moore Thomas Nagel Otto Neurath Friedrich Nietzsche John Norton P.H.NowellSmith Robert Nozick William of Ockham Timothy O'Connor Parmenides David F. 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Jay Wallace W.G.Ward Ted Warfield Roy Weatherford William Whewell Alfred North Whitehead David Widerker David Wiggins Bernard Williams Timothy Williamson Ludwig Wittgenstein Susan Wolf Scientists Michael Arbib Walter Baade Bernard Baars Gregory Bateson John S. Bell Charles Bennett Ludwig von Bertalanffy Susan Blackmore Margaret Boden David Bohm Niels Bohr Ludwig Boltzmann Emile Borel Max Born Satyendra Nath Bose Walther Bothe Hans Briegel Leon Brillouin Stephen Brush Henry Thomas Buckle S. H. Burbury Donald Campbell Anthony Cashmore Eric Chaisson JeanPierre Changeux Arthur Holly Compton John Conway John Cramer E. P. Culverwell Charles Darwin Richard Dawkins Terrence Deacon Lüder Deecke Richard Dedekind Louis de Broglie Max Delbrück Abraham de Moivre Paul Dirac Hans Driesch John Eccles Arthur Stanley Eddington Gerald Edelman Paul Ehrenfest Albert Einstein Hugh Everett, III Franz Exner Richard Feynman R. A. Fisher Joseph Fourier Philipp Frank Lila Gatlin Michael Gazzaniga GianCarlo Ghirardi J. Willard Gibbs Nicolas Gisin Paul Glimcher Thomas Gold A.O.Gomes Brian Goodwin Joshua Greene Jacques Hadamard Patrick Haggard Stuart Hameroff Augustin Hamon Sam Harris Hyman Hartman JohnDylan Haynes Donald Hebb Martin Heisenberg Werner Heisenberg John Herschel Art Hobson Jesper Hoffmeyer E. T. Jaynes William Stanley Jevons Roman Jakobson Pascual Jordan Ruth E. Kastner Stuart Kauffman Martin J. 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John Stewart Bell
In 1964 John Bell showed how the 1935 "thought experiments" of Einstein, Podolsky, and Rosen (EPR) could be made into real experiments. He put limits on local "hidden variables" that might restore a deterministic physics in the form of what he called an "inequality," the violation of which would confirm standard quantum mechanics.
Some thinkers, mostly philosophers of science rather than working quantum physicists, think that Bell's work has restored the determinism in physics that Einstein had wanted and that Bell recovered the "local elements of reality" that Einstein hoped for. But Bell himself came to the conclusion that local "hidden variables" will never be found that give the same results as quantum mechanics. This has come to be known as Bell's Theorem. All theories that reproduce the predictions of quantum mechanics will be "nonlocal," Bell concluded. Nonlocality is an element of physical reality and it has produced some remarkable new applications of quantum physics, including quantum cryptography and quantum computing. Bell based his thught of real experiments on the 1952 ideas of David Bohm. Bohm proposed an improvement on the original EPR experiment (which measured position and momentum). Bohm's reformulation of quantum mechanics postulates (undetectable) deterministic positions and trajectories for atomic particles, where the instantaneous collapse happens in a new "quantum potential" field that can move faster than light speed. But it is still a "nonlocal" theory. So Bohm (and Bell) believed that nonlocal "hidden variables" might exist, and that some form of information could come into existence at remote "spacelike separations" at speeds faster then light, if not instantaneously. The original EPR paper was based on a question of Einstein's about two electrons fired in opposite directions from a central source with equal velocities. Einstein imagined them starting from a distance at t_{0} and approaching one another with high velocities, then for a short time interval from t_{1} to t_{1} + Δt in contact with one another, where experimental measurements could be made on the momenta, after which they separate. Now at a later time t_{2} it would be possible to make a measurement of electron 1's position and would therefore know the position of electron 2 without measuring it explicitly. Einstein used the conservation of linear momentum to "know" the symmetric position of the other electron. This knowledge implies information about the remote electron that is available instantly. Einstein called this "spooky actionatadistance." It might better be called "knowledgeatadistance." Bohm's 1952 thought experiment used two electrons that are prepared in an initial state of known total spin. If one electron spin is 1/2 in the up direction and the other is spin down or 1/2, the total spin is zero. The underlying physical law of importance is still a conservation law, in this case the conservation of spin angular momentum.
In his 1964 paper "On the EinsteinPodolskyRosen Paradox," Bell made the case for nonlocality. The paradox of Einstein, Podolsky and Rosen was advanced as a argument that quantum mechanics could not be a complete theory but should be supplemented by additional variables. These additional variables were to restore to the theory causality and locality. In this note that idea will be formulated mathematically and shown to be incompatible with the statistical predictions of quantum mechanics. It is the requirement of locality, or more precisely that the result of a measurement on one system be unaffected by operations on a distant system with which it has interacted in the past, that creates the essential difficulty. There have been attempts to show that even without such a separability or locality requirement no 'hidden variable' interpretation of quantum mechanics is possible. These attempts have been examined [by Bell] elsewhere and found wanting. Moreover, a hidden variable interpretation of elementary quantum theory has been explicitly constructed [by Bohm]. That particular interpretation has indeed a gross nonlocal structure. This is characteristic, according to the result to be proved here, of any such theory which reproduces exactly the quantum mechanical predictions. Bell titled his 1976 review of the first tests of his theorem about his predicted inequalities, "EinsteinPodolskyRosen Experiments." He described his talk as about the "foundations of quantum mechanics," and it was the early days of a movement by a few scientists and many philosophers of science to challenge the "orthodox" quantum mechanics. They particularly attacked the Copenhagen Interpretation, with its notorious speculations about the role of the "conscious observer" and its attacks on physical reality. From the earliest presentations in the late 1920's of the ideas of the supposed "founders" of quantum mechanics, Einstein had deep misgivings of the work going on in Copenhagen, although he never doubted the calculating power of their new mathematical methods, and he came to accept the statistical (indeterministic) nature of quantum physics, which he himself had reluctantly discovered. He described their work as "incomplete" because it is based on the statistical results of many experiments so only makes probabilistic predictions about individual experiments. Nevertheless, Einstein hoped to visualize what is going on in an underlying "objective reality." Bell was deeply sympathetic to Einstein's hopes for a return to the "local reality" of classical physics. He identified the EPR paper's title, "Can QuantumMechanical Description of Physical Reality Be Considered Complete?" as a search for new variables to provide the completeness. Bell thought David Bohm's "hidden variables' were one way to achieve this, though Einstein had called Bohm's approach "too cheap," probably because Bohm included vector potentials traveling faster than light speed, an apparent violation of Einstein's special theory of relativity.
I have been invited to speak on “foundations of quantum mechanics”... Since Bell's original work, many other physicists have defined other "Bell inequalities" and developed increasingly sophisticated experiments to test them. Most recent tests have used oppositely polarized photons coming from a central source. It is the total photon spin of zero that is conserved.
The first experiments that confirmed Bell's Theorem were done by John Clauser and Stuart Freedman in 1971, Clauser and Abner Shimony described the first few experiments in a 1978 review. There they agreed with Bell about measurements on two spin 1/2 particles, as suggested by David Bohm. A variant of EPR’s argument was given by Bohm (1951), formulated in terms of discrete states. He considered a pair of spatially separated spin1/2 particles produced somehow in a singlet state, for example, by dissociation of the spin0 system... If all three x, y, z components of spin had definite values of 1/2, the resultant vector (the diagonal of a cube with side 1/2) would be 3^{½}/2. This is impossible. Spin is always quantized at ℏ/2. Unmeasured components are in a linear combination of + ℏ/2 and  ℏ/2 (average value zero!). The concept of "local" hidden variables is then the simultaneous existence of positive definite values of σ_{y} and σ_{z} (both equal to + /  ℏ/2) at the same time σ_{x} has measured value ℏ/2! Although Bell's Theorem is one of the foundational documents in the "Foundations of Quantum Mechanics," it is cited much more often than the confirming experiments are explained, because they are quite complicated. The most famous explanations are given in terms of analogies, with flashing lights, dice throws, or card games. What is needed is an explanation describing what happens to the quantum particles and their statistics. The most important experiments were likely that done by John Clauser, Michael Horne, Abner Shimony, and Richard Holt (known collectively as CHSH) and later by Alain Aspect, who did more sophisticated tests.
Now this is true whether σ_{x} and σ_{y} is measured (assuming the transmission axis is along the z direction). But keep in mind that if σ_{x} is measured, σ_{y} is then indeterminate. This is why we say that the outcome of a measurement depends on the "free choice" of the experimenter. Measuring in the x direction gives us a value in that direction. Did the spin in the x direction exist before the measurement? Did the spins in the two orthogonal directions exist before the measurement? Those orthogonal spins definitely do not exist after the measurement, since the measurement is also a state preparation. All three potential spins are latent in the sense that whichever direction is chosen, if the same direction is chosen for the other particle it will be found to have opposite spin. If a different direction is chosen for the other particle, it will not be correlated at all with the first particle spin. By latent we mean perfectly opposite correlations are inherent in all three directions. Now the electron spin is limited to This is because under exchange of the two indistinguishable particles, the antisymmetric wave function changes its sign.
Experimental Results
With the exception of some of Holt's early results that were found to be erroneous, no evidence has so far been found of any failure of standard quantum mechanics. And as experimental accuracy has improved by orders of magnitude, quantum physics has correspondingly been confirmed to one part in 10^{18}, and the speed of the probability information transfer between particles has a lower limit of 10^{6} times the speed of light. There has been no evidence for local "hidden variables."
Bell Theorem tests always add what Bell called "filters," polarization analyzers whose polarization angles can be set, sometimes at high speeds between the socalled "first" and "second" measurements. Notice that this represents an interaction that alters the underlying physics, projecting the particles into unpredictable states.
On David Bohm's "Impossible" Pilot Wave
Why is the pilot wave picture ignored in textbooks? Should it not be taught, not as the only way, but as an antidote to the prevailing complacency? To show that vagueness, subjectivity, and indeterminism are not forced on us by experimental facts, but by deliberate theoretical choice? Bohm’s 1952 papers on quantum mechanics were for me a revelation. The elimination of indeterminism was very striking. But more important, it seemed to me, was the elimination of any need for a vague division of the world into “system” on the one hand, and “apparatus” or “observer” on the other. I have always felt since that people who have not grasped the ideas of those papers ... and unfortunately they remain the majority ... are handicapped in any discussion of the meaning of quantum mechanics.
Superdeterminism
During a mid1980's interview by BBC Radio 3 organized by P. C. W. Davies and J. R. Brown, Bell proposed the idea of a "superdeterminism" that could explain the correlation of results in twoparticle experiments without the need for fasterthanlight signaling. The two experiments need only have been predetermined by causes reaching both experiments from an earlier time.
I was going to ask whether it is still possible to maintain, in the light of experimental experience, the idea of a deterministic universe? Bell's superdeterminism would deny the important "free choice" of the experimenter (originally suggested by Niels Bohr and Werner Heisenberg) and later explored by John Conway and Simon Kochen. Conway and Kochen claim that the experimenters' free choice requires that atoms must have free will, something they call their Free Will Theorem. Following John Bell's idea, Nicholas Gisin and Antoine Suarez argue that something might be coming from "outside space and time" to correlate results in their own experimental tests of Bell's Theorem. Roger Penrose and Stuart Hameroff have proposed causes coming "backward in time" to achieve the perfect EPR correlations, as has philosopher Huw Price.
A Preferred Frame?
A little later in the same BBC interview, Bell suggested that a preferred frame of reference might help to explain nonlocality and entanglement. [Davies] Bell's inequality is, as I understand it, rooted in two assumptions: the first is what we might call objective reality  the reality of the external world, independent of our observations; the second is locality, or nonseparability, or no fasterthanlight signalling. Now, Aspect's experiment appears to indicate that one of these two has to go. Which of the two would you like to hang on to? The standard explanation of entangled particles usually begins with an observer A, often called Alice, and a distant observer B, known as Bob. Between them is a source of two entangled particles. The twoparticle wave function describing the indistinguishable particles cannot be separated into a product of two singleparticle wave functions. The problem of fasterthanlight signaling arises when Alice is said to measure particle A and then puzzle over how Bob's (later) measurements of particle B can be perfectly correlated, when there is not enough time for any "influence" to travel from A to B. Now as John Bell knew very well, there are frames of reference moving with respect to the laboratory frame of the two observers in which the time order of the events can be reversed. In some moving frames Alice measures first, but in others Bob measures first. Back in the 1960's, C. W. Rietdijk and Hilary Putnam argued that physical determinism could be proved to be true by considering the experiments and observers A and B in a "spacelike" separation and moving at high speed with respect to one another. Roger Penrose developed a similar argument in his book The Emperor's New Mind. It is called the Andromeda Paradox. If there is a preferred frame of reference, surely it is the one in which the origin of the two entangled particles is at rest. Assuming that Alice and Bob are also at rest in this frame and equidistant from the origin, we arrive at the simple picture in which any measurement that causes the twoparticle wave function to collapse makes both particles appear simultaneously at determinate places (just what is needed to conserve energy, momentum, angular momentum, and spin). Because a "preferred frame" has an important use in special relativity, where all inertial frames are equivalent, we might call this frame a "special frame."
The EPR "paradox" is the result of a naive nonrelativistic description of events. Although the two events (measurements of particles A and B) are simultaneous in our special frame, the spacelike separation of the events means that from Alice's point of view, any knowledge of event B is out in her future. Bob likewise sees Alice's event A out in his future. These both cannot be true. Yet they are both true (and in some sense neither is true). Thus the paradox. Instead of just one particle making an appearance in the collapse of a singleparticle wave function, in the twoparticle case, when either particle is measured, we know instantly those properties of the other particle that satisfy the conservation laws, including its location equidistant from, but on the opposite side of, the source, and its other properties such as spin. Let's look at an animation of the twoparticle wave function expanding from the origin and what happens when, say, Alice makes a measurement.
You can compare the collapse of the twoparticle probability amplitude above to the singleparticle collapse here.
We can enhance our visualization of what might be happening between the time two entangled electrons are emitted with opposite spins and the time one or both electrons are detected. Quantum mechanics describes the state of the two electrons as in a linear combination of  +  > and   + > states. We can visualize the electron moving left to be both spin up  + > and spin down   >. And the electron moving right would be both spin down   > and spin up  + >. We could require that when the left electron is spin up  + >, the right electron must be spin down   >, so that total spin is always conserved. Consider this possible animation of the experiment, which illustrates the assumption that each electron is in a linear combination of up and down spin. It imitates the superposition (or linear combination) with up and down arrows on each electron oscillating quickly. Notice that if you move the animation frame by frame by dragging the dot in the timeline, you will see that total spin = 0 is conserved. When one electron is spin up the other is always spin down. Note that we can challenge the idea that spins are oscillating. Would a force of some kind be needed to change the spins in sync? Perhaps we can see the rapid changes like resonance phenomena in molecular bonds?
Standard quantum mechanics says we cannot know the spin until it is measured, our minimal information estimate is a 50/50 probability between up and down. This is the same as assuming Schrödinger's Cat is 50/50 alive and dead. But what this means of course is simply that if we do a large number of identical experiments, the statistics for live and dead cats will be approximately 50/50%. We never observe/measure a cat that is both dead and alive! As Einstein noted, QM tells us nothing about individual cats. Quantum mechanics is incomplete in this respect. He is correct, although Bohr and Heisenberg insisted QM is complete, because we cannot know more before we measure, and reality is created (they say) when we do measure. Despite accepting that a particular value of an "observable" can only be known by a measurement (knowledge is an epistemological problem, Einstein asked whether the particle actually (really, ontologically) has a path and position before we measure it? His answer was yes.
Since each electron has only one unit of electron spin (a magnetic moment equal to one Bohr magneton), we can only say that if measured in a given direction, the spin will be projected into that direction for the left electron, into the opposite direction for the right electron. The table shows six possible outcomes. This animation measures in the y direction (the second  yellow  row).
Werner Heisenberg and later Paul Dirac and others refer to the "free choice" of the experimenter as to which direction is chosen to measure. But then Dirac adds that nature makes a random choice as to whether to find the electron spin is up or down in that chosen direction. Now entanglement adds the nonlocality and nonseparability that is caused by the (single) twoparticle wave function collapsing symmetrically and simultaneously in our special frame.
How Mysterious Is Entanglement?
Some commentators say that nonlocality and entanglement are a "second revolution" in quantum mechanics, "the greatest mystery in physics," or "science's strangest phenomenon," and that quantum physics has been "reborn." They usually quote Erwin Schrödinger as saying
"I consider [entanglement] not as one, but as the characteristic trait of quantum mechanics, the one that enforces its entire departure from classical lines of thought."Schrödinger knew that his twoparticle wave function Ψ_{12} could not have the same simple interpretation as the single particle, which can be visualized in ordinary 3dimensional configuration space. And he is right that entanglement apparently exhibits a richer form of the "actionatadistance" and nonlocality that Einstein had already identified in the collapse of the single particle wave function. But the main difference is that two particles acquire new properties instead of one, and they appear to do it instantaneously (at faster than light speeds), just as in the case of a singleparticle measurement the probability of finding that particular single particle anywhere else is now zero. Nonlocality and entanglement are thus just another manifestation of Richard Feynman's "only" mystery. In both singleparticle and twoparticle cases paradoxes appear only when we attempt to describe individual particles following specific paths to measurement by observer A (and/or observer B). Wee cannot know the specific paths at every instant without measurements. But Einstein has told us that at every instant the particles are conserving linear momentum and electron spin, despite our lack of knowledge during individual experiments. We can ask what happens if Bob is not at the same distance from the origin as Alice, but farther away. When Alice detects the particle (with say spin up), at that instant the other particle also becomes determinate (with spin down) at the same distance on the other side of the origin. It now continues, in that determinate state, to Bob's measuring apparatus.
Recall Bell's description of the process (quoted above), with its mistaken bias toward assuming first one measurement is made, and the other measurement is made later. If measurement of the component σ_{1} • a, where a is some unit vector, yields the value + 1 then, according to quantum mechanics, measurement of σ_{2} • a must yield the value — 1 and vice versa... Since we can predict in advance the result of measuring any chosen component of σ_{2}, by previously measuring the same component of σ_{1}, it follows that the result of any such measurement must actually be predetermined.Since the collapse of the twoparticle wave function is indeterminate, nothing is predetermined, although σ_{2} is indeed determined to have opposite sign (to conserve spin momentum) once σ_{1} is measured. Here Bell is describing The "following" measurement to be in the same direction as the "previous" measurement. In Bell's termss, Bob is measuring "the same component" as Alice. If Bob should measure even a fraction of a second after Alice, but measure in a different spin direction (a different component), his measurements will be found to be 50/50, up and down, or + and .
To recap our picture of entanglement measurements:
In 1987, Bell contributed an article to a centenary volume for Erwin Schrödinger entitled Just a year before Bell's death in 1990, physicists assembled for a conference on 62 Years of Uncertainty (referring to Werner Heisenberg's 1927 principle of indeterminacy).
John Bell's contribution to the conference was an article called "Against Measurement." In it he attacked Max Born's statistical interpretation of quantum mechanics. And he praised the new ideas of GianCarlo Ghirardi and his colleagues, Alberto Rimini and Tomaso Weber: In the beginning, Schrödinger tried to interpret his wavefunction as giving somehow the density of the stuff of which the world is made. He tried to think of an electron as represented by a wavepacket — a wavefunction appreciably different from zero only over a small region in space. The extension of that region he thought of as the actual size of the electron — his electron was a bit fuzzy. At first he thought that small wavepackets, evolving according to the Schrödinger equation, would remain small. But that was wrong. Wavepackets diffuse, and with the passage of time become indefinitely extended, according to the Schrödinger equation. But however far the wavefunction has extended, the reaction of a detector to an electron remains spotty. So Schrödinger's 'realistic' interpretation of his wavefunction did not survive. On the 22nd of January 1990, Bell gave a talk explaining his theorem at CERN in Geneva organized by Antoine Suarez, director of the Center for Quantum Philosophy.
There are links on the CERN website to the
In this talk, Bell summarizes the situation as follows: It just is a fact that quantum mechanical predictions and experiments, in so far as they have been done, do not agree with [my] inequality. And that's just a brutal fact of nature...that's just the fact of the situation; the Einstein program fails, that's too bad for Einstein, but should we worry about that?Bell gives three reasons for not worrying.
So as a solution of this situation, I think we cannot just say 'Oh oh, nature is not like that.' I think you must find a picture in which perfect correlations are natural, without implying determinism, because that leads you back to nonlocality. And also in this independence as far as our individual experiences goes, our independence of the rest of the world is also natural. So the connections have to be very subtle, and I have told you all that I know about them. Thank you. The work of GianCarlo Ghirardi that Bell endorsed is a scheme that makes the wave function collapse by adding small (order of 10^{24}) nonlinear and stochastic terms to the linear Schrödinger equation. GRW can not predict when and where their collapse occurs (it is simply random), but the contact with macroscopic objects such as a measuring apparatus (with the order of 10^{24} atoms) makes the probability of collapse of order unity. Information physics removes Bell's "shifty split" without "hidden variables" or making ad hoc nonlinear additions like those of GhirardiRiminiWeber to the linear Schrödinger equation. The "moment" at which the boundary between quantum and classical worlds occurs is the moment that irreversible observable information enters the universe. So we can now look at John Bell's diagram of possible locations for his "shifty split" and identify the correct moment  when irreversible information enters the universe.
In the information physics solution to the problem of measurement, the timing and location of Bell's "shifty split" (the "cut" or "Schnitt" of Heisenberg and von Neumann) are identified with the interaction between quantum system and classical apparatus that leaves the apparatus in an irreversible stable state providing information to the observer. As Bell may have seen, it is therefore not a "measurement" by a conscious observer that is needed to "collapse" wave functions. It is the irreversible interaction of the quantum system with another system, whether quantum or approximately classical. The interaction must be one that changes the information about the system. And that means a local entropy decrease and overall entropy increase to make the information stable enough to be observed by an experimenter and therefore be a measurement. References
Against Measurement (PDF)
Beables for Quantum Field Theory (PDF) On the EinsteinPodolskyRosen Paradox (PDF) On the Impossible Pilot Wave (PDF) Are There Quantum Jumps? (PDF, Excerpt) BBC Interview (PDF, Excerpt) For Teachers
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