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 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 Daniel Boyd F.H.Bradley C.D.Broad Michael Burke Lawrence Cahoone C.A.Campbell Joseph Keim Campbell Rudolf Carnap Carneades Nancy Cartwright Gregg Caruso 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 Austin Farrer Herbert Feigl Arthur Fine John Martin Fischer Frederic Fitch Owen Flanagan Luciano Floridi Philippa Foot Alfred Fouilleé Harry Frankfurt Richard L. Franklin Bas van Fraassen 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 Frank Jackson William James Lord Kames Robert Kane Immanuel Kant Tomis Kapitan Walter Kaufmann Jaegwon Kim William King Hilary Kornblith Christine Korsgaard Saul Kripke Thomas Kuhn Andrea Lavazza Christoph Lehner Keith Lehrer Gottfried Leibniz Jules Lequyer Leucippus Michael Levin Joseph Levine George Henry Lewes C.I.Lewis David Lewis Peter Lipton C. Lloyd Morgan John Locke Michael Lockwood Arthur O. Lovejoy E. Jonathan Lowe John R. Lucas Lucretius Alasdair MacIntyre Ruth Barcan Marcus Tim Maudlin James Martineau Nicholas Maxwell 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. Pears Charles Sanders Peirce Derk Pereboom Steven Pinker Plato Karl Popper Porphyry Huw Price H.A.Prichard Protagoras Hilary Putnam Willard van Orman Quine Frank Ramsey Ayn Rand Michael Rea Thomas Reid Charles Renouvier Nicholas Rescher C.W.Rietdijk Richard Rorty Josiah Royce Bertrand Russell Paul Russell Gilbert Ryle JeanPaul Sartre Kenneth Sayre T.M.Scanlon Moritz Schlick Arthur Schopenhauer John Searle Wilfrid Sellars Alan Sidelle Ted Sider Henry Sidgwick Walter SinnottArmstrong J.J.C.Smart Saul Smilansky Michael Smith Baruch Spinoza L. Susan Stebbing Isabelle Stengers George F. Stout Galen Strawson Peter Strawson Eleonore Stump Francisco Suárez Richard Taylor Kevin Timpe Mark Twain Peter Unger Peter van Inwagen Manuel Vargas John Venn Kadri Vihvelin Voltaire G.H. von Wright David Foster Wallace R. Jay Wallace W.G.Ward Ted Warfield Roy Weatherford C.F. von Weizsäcker William Whewell Alfred North Whitehead David Widerker David Wiggins Bernard Williams Timothy Williamson Ludwig Wittgenstein Susan Wolf Scientists David Albert Michael Arbib Walter Baade Bernard Baars Jeffrey Bada Leslie Ballentine Marcello Barbieri Gregory Bateson Horace Barlow John S. Bell Mara Beller Charles Bennett Ludwig von Bertalanffy Susan Blackmore Margaret Boden David Bohm Niels Bohr Ludwig Boltzmann Emile Borel Max Born Satyendra Nath Bose Walther Bothe Jean Bricmont Hans Briegel Leon Brillouin Stephen Brush Henry Thomas Buckle S. H. Burbury Melvin Calvin Donald Campbell Sadi Carnot Anthony Cashmore Eric Chaisson Gregory Chaitin JeanPierre Changeux Rudolf Clausius Arthur Holly Compton John Conway Jerry Coyne John Cramer Francis Crick E. P. Culverwell Antonio Damasio Olivier Darrigol Charles Darwin Richard Dawkins Terrence Deacon Lüder Deecke Richard Dedekind Louis de Broglie Stanislas Dehaene Max Delbrück Abraham de Moivre Bernard d'Espagnat Paul Dirac Hans Driesch John Eccles Arthur Stanley Eddington Gerald Edelman Paul Ehrenfest Manfred Eigen Albert Einstein George F. R. Ellis Hugh Everett, III Franz Exner Richard Feynman R. A. Fisher David Foster Joseph Fourier Philipp Frank Steven Frautschi Edward Fredkin Benjamin GalOr Howard Gardner Lila Gatlin Michael Gazzaniga Nicholas GeorgescuRoegen GianCarlo Ghirardi J. Willard Gibbs James J. Gibson Nicolas Gisin Paul Glimcher Thomas Gold A. O. Gomes Brian Goodwin Joshua Greene Dirk ter Haar Jacques Hadamard Mark Hadley Patrick Haggard J. B. S. Haldane Stuart Hameroff Augustin Hamon Sam Harris Ralph Hartley Hyman Hartman Jeff Hawkins JohnDylan Haynes Donald Hebb Martin Heisenberg Werner Heisenberg John Herschel Basil Hiley Art Hobson Jesper Hoffmeyer Don Howard John H. Jackson William Stanley Jevons Roman Jakobson E. T. Jaynes Pascual Jordan Eric Kandel Ruth E. Kastner Stuart Kauffman Martin J. Klein William R. Klemm Christof Koch Simon Kochen Hans Kornhuber Stephen Kosslyn Daniel Koshland Ladislav Kovàč Leopold Kronecker Rolf Landauer Alfred Landé PierreSimon Laplace Karl Lashley David Layzer Joseph LeDoux Gerald Lettvin Gilbert Lewis Benjamin Libet David Lindley Seth Lloyd Hendrik Lorentz Werner Loewenstein Josef Loschmidt Ernst Mach Donald MacKay Henry Margenau Owen Maroney David Marr Humberto Maturana James Clerk Maxwell Ernst Mayr John McCarthy Warren McCulloch N. David Mermin George Miller Stanley Miller Ulrich Mohrhoff Jacques Monod Vernon Mountcastle Emmy Noether Donald Norman Alexander Oparin Abraham Pais Howard Pattee Wolfgang Pauli Massimo Pauri Wilder Penfield Roger Penrose Steven Pinker Colin Pittendrigh Walter Pitts Max Planck Susan Pockett Henri Poincaré Daniel Pollen Ilya Prigogine Hans Primas Zenon Pylyshyn Henry Quastler Adolphe Quételet Pasco Rakic Nicolas Rashevsky Lord Rayleigh Frederick Reif Jürgen Renn Giacomo Rizzolati Emil Roduner Juan Roederer Jerome Rothstein David Ruelle David Rumelhart Tilman Sauer Ferdinand de Saussure Jürgen Schmidhuber Erwin Schrödinger Aaron Schurger Sebastian Seung Thomas Sebeok Franco Selleri Claude Shannon Charles Sherrington David Shiang Abner Shimony Herbert Simon Dean Keith Simonton Edmund Sinnott B. F. Skinner Lee Smolin Ray Solomonoff Roger Sperry John Stachel Henry Stapp Tom Stonier Antoine Suarez Leo Szilard Max Tegmark Teilhard de Chardin Libb Thims William Thomson (Kelvin) Richard Tolman Giulio Tononi Peter Tse Alan Turing Francisco Varela Vlatko Vedral Mikhail Volkenstein Heinz von Foerster Richard von Mises John von Neumann Jakob von Uexküll C. S. Unnikrishnan C. H. Waddington John B. Watson Daniel Wegner Steven Weinberg Paul A. Weiss Herman Weyl John Wheeler Wilhelm Wien Norbert Wiener Eugene Wigner E. O. Wilson Günther Witzany Stephen Wolfram H. Dieter Zeh Semir Zeki Ernst Zermelo Wojciech Zurek Konrad Zuse Fritz Zwicky Presentations Biosemiotics Free Will Mental Causation James Symposium 
John Stewart Bell
In 1964 John Bell analyzed David Bohm's 1952 suggestion for "hidden variables" added to the 1935 "thought experiments" of Einstein, Podolsky, and Rosen (EPR) which could make them into real experiments. Bell put limits on local "hidden variables" 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 the work of Bohm and Bell has restored the determinism in physics that Einstein had wanted and that Bohm and/or Bell had discovered the "local elements of reality" that Einstein hoped for in EPR. 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. Bohm proposed an improvement on the original EPR experiment (which measured continuous position and momentum variables). 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 new information can come into existence at remote "spacelike separations" at speeds faster then light, if not instantaneously. This is the idea of entanglement. The original EPR paper was based on a question of Einstein's about two particles 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 particle 1's position and would therefore know the position of particle 2 without measuring it explicitly. Einstein implicitly used the conservation of linear momentum to "know" the symmetric position of the other particle. This knowledge implies information about the remote particle that is available instantly. Einstein called this "spooky actionatadistance." It might better be called "knowledgeatadistance." Bohm and his colleague Yakir Aharonov in 1957 proposed a new EPRlike thought experiment using two electrons that are prepared in an initial state of known total spin zero. Instead of measuring continuous variables position and momentum as in EPR, Bohm measures the discrete property of electron 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.
Until the moment that one electron spin is measured, the twoparticle quantum state is spherically symmetric (rotationally invariant). There is no preferred spatial direction. It is described as a superposition a pure quantum state that is a linear combination of particle 1 up, particle 2 down plus particle 1 down with particle 2 up...
ψ_{12} = (1/√2) [ ψ_{+} (1) ψ_{} (2)  ψ_{} (1) ψ_{+} ] (2)
We can simplify the notation
 ψ_{12} > = 1/√2)  +  >  1/√2)   + > (2a)
Note that this combination preserves the total electron spin as zero and it offers no preferred spatial direction. Note also that under exchange of the two indistinguishable fermions, the antisymmetric wave function changes its sign, thus the minus sign in the above equations. Quantum mechanics predicts that a measurement will find the system in either  +  > or   + >, each with 50% probability. So whichever the outcome, the conservation of angular momentum (spin) will still be true. The initial entanglement prepared the system in a state with total spin zero, conserving angular momentum. Quantum mechanics predicts that a measurement will disentangle the particles but leave them randomly in one of two states, either of which have total spin zero, again conserving angular momentum. Many thousands of Bell tests have all confirmed that measurements made at the same angle find opposite spins with total spin zero, experimentally showing that conservation of angular momentum was true at final measurement. As long as there is no environmental interaction between the initial and final states, as long as conditions permit the rotational symmetry to be maintained from start to finish, what mechanism could possibly cause the conservation law for angular momentum to be violated? And how can any physicist be sure that the electrons are not exactly in opposite spin states, so they must be restored to opposite spins by some instantaneous actionatadistance? Finally how could one electron determine precisely the deviant spin of its partner, so it can prepare its remote interaction precisely to realign its deviant partner? The spherical/rotational symmetry is of course broken when one observer (freely) chooses a direction in which to measure a spin component of either particle. Erwin Schrödinger described this measurement as "disentangling" the particles. This is the separation of the two particles that Einstein was hoping for. Replying to Einstein's EPR paper in 1936, Schrödinger wrote This would mean that not only the parts, but the whole system, would be in the situation of a mixture, not of a pure state. It would not preclude the possibility of determining the state of the first system by suitable measurements in the second one or vice versa. But it would utterly eliminate the experimenters influence on the state of that system which he does not touch. Schrödinger then describes the puzzle of entanglement in terms of what one can answer to questions about the two entangled particles. Schrödinger thus set an unfortunate precedent of explaining entanglement in terms of knowledge (epistemology) about the entangled particles rather than what may "really" be going on (ontology). Many modern explanations of entanglement use a logical analysis of yes/no answers to questions or "instruction sets" with those answers thought to be accompanying each particle as "hidden variables."
the result of measuring p_{1} serves to predict the result for p_{1} and vice versa. But of course every one of the four observations in question, when actually performed, disentangles the systems, furnishing each of them with an independent representative of its own. A second observation, whatever it is and on whichever system it is executed, produces no further change in the representative of the other system.
In his 1964 paper "On the EinsteinPodolskyRosen Paradox," Bell made the case for nonlocality. The paradox of Einstein, Podolsky and Rosen was advanced as an 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 describes explicitly how the "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." He also says "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." But Schrödinger, who knew more about twoparticle wave functions than anyone, explains that while the two particles are entangled (with total spin zero), any measurement disentangles them, while it conserves the total spin zero in the agreed upon measurement direction. If Alice measures the electron spin of particle 1 in the xdirection as +ℏ/2, then Bob will measure a perfectly anticorrelated ℏ/2 for particle 2, if (and only if) he measures at the same (preagreed upon) angle as Alice. Note that since it was quantum random whether the two particle state would be projected into  +  > or into  + >, successive measurements by Alice and Bob will generate two perfectly random and anticorrelated strings of + and  (or 0 and 1 bit strings). This is exactly what is needed for the keys needed in quantum cryptography. Each individual string is random, but the two bit stings are perfectly correlated (or anticorrelated). And the strings have been generated in separated locations over a secure communications channel that cannot be eavesdropped, the ideal for quantum key distribution (QKD). A decade later, 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, especially the claim that objects have no properties until they are measured. 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 in his 1916 study of the atomic emission of light quanta. He described their work as "incomplete" because it is based on the statistical results of many experiments so it can only make 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 (as had Bohm) 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 "quantum potentials" traveling faster than light speed, an obvious violation of Einstein's special theory of relativity. In his 1976 review, Bell wrote...
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. Here, 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. Clauser and Shimony wrote... A variant of EPR’s argument was given by Bohm and Aharonov (1957), 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... Clauser and Shimony are wrong to conclude that measuring one spin component would render spin components in all directions definite. 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. The unmeasured components are in a linear combination of + ℏ/2 and  ℏ/2 (with average value zero!). 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. See David Mermin. What is needed is an explanation describing exactly what happens to the quantum particles and their statistics. The most important experiments were likely those done by John Clauser, Michael Horne, Abner Shimony, and Richard Holt (known collectively as CHSH) and later by Alain Aspect, who did even more sophisticated tests.
Now this is true whether σ_{x} or 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. A choice to measure in the x direction gives us a value of the spincomponent in the x direction, σ_{x}. Did the spin in the x direction exist before the measurement? No. Did the spins in the two orthogonal directions exist before the measurement? No. Those orthogonal spins definitely do not exist after the measurement, since the measurement is also a state preparation. σ_{x} now exists, σ_{y} and σ_{z} do not. All three potential spins are latent in the rotationally invariant state with total spin 0, in the sense that whichever direction is chosen for a measurement, if the same direction is chosen for the other particle it will be found to have opposite spin (by conservation of angular momentum). If a different direction is chosen for the other particle, it will no longer be perfectly correlated with the first particle spin. When photons are used, their boson spins are ±1, not ±1/2. But if photons are entangled with opposite spins so the total spin is zero, the results of Bell tests will be the same.
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 any 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 usually add what Bell called "filters," polarization analyzers whose polarization angles can be set, sometimes at high speeds between the socalled "first" and "second" measurements.
On David Bohm's "Impossible" Pilot Wave
John Bell reflected on Bohm's Pilot Wave in 1987... 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 whenever 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, always opposite to one another. 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 always conserved. When one electron is spin up the other is always spin down. The overall twoparticle state remains rotationally symmetric at all times. 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, always either dead or 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. Below is an animation that illustrates the assumption that the two electrons are randomly produced in states that have latent components that conserve spin momentum, and that they remain in those states no matter how far they separate, provided neither interacts with anything else before the measurement. 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.
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. 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 apparent "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 instantaneously 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). We cannot know the specific paths at every instant without measurements. But Einstein has told us that at every instant the particles are conserving momentum, despite our lack of knowledge between 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 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 description, Bob is measuring "the same component" as Alice, meaning that he measures at the same angle as Alice. If Bob should measure in a different spin direction from Alice (a different spin component), his measurements will lose their perfect correlation, slowly at first for a small angle. As the angle between their measurements increases, the correlation falls off as the cosine of the angle. Oddly, Bell's inequality for local hidden variables predicts a linear falloff with angle. We shall try to understand how Bell came up with a linear angle dependence for what he called his ad hoc model and later his "inequality." Supporters of the Copenhagen Interpretation claim that the properties of particles (like angular or linear momentum) do not exist until they are measured. It was Pascual Jordan who claimed the measurement creates the value of a property. This is true when the preparation of the state is in an unknown linear combination (superposition) of quantum states. In our case, the entangled particles have been prepared in a superposition of states, but both of them have total spin zero.
ψ_{12} = (1/√2) [ ψ_{+} (1) ψ_{} (2)  ψ_{} (1) ψ_{+} (2) ]
So whichever of these two states is created by the preparation, it will put the two particles in opposite spin states, randomly +  or  + , but still supporting Bell's view, that they will be perfectly (anti)correlated when measured at exactly the same angle (measuring the same spin component). Wolfgang Pauli called it a "measurement of the first kind" when a system is prepared in a state and if measured again, will be certainly found in the same state. (This is the basis for the quantum zeno effect.) Since our two electrons have been prepared with one spin up and the other down, what could possibly cause them to change, for example, to both spins in the same direction, or as Copenhagen claims, simply to have both spins no longer definite until the next measurement? As long as nothing interferes with either entangled particle as they travel to the distant detectors, they will be found to be still perfectly correlated, if (and only if) they are measured at the same angle. Otherwise, the correlations should fall off as the cosine (or perhaps the square of the cosine?) of the angle difference. We can illustrate the straightline predictions of Bell's inequalities for local hidden variables, the cosine curves predicted by quantum mechanics and conservation of angular momentum, and the odd "kinks" at angles 0°, 90°, 180°, and 270°, with what is called a "PopescuRorhlich box." The "PR Box" shows Bell’s local hidden variables prediction as four straight lines of the inner square. The circular region of quantum mechanics correlations are found outside Bell's straight lines, "violating" his inequalities. Quantum mechanics and Bell's inequalities meet at the corners, where Bell's predictions show a distinctly nonphysical rightangle that Bell called a "kink." All experimental results have been found to lie along the curved quantum predictions called the "Tsirelson bound."
In 1976, Bell gave us this diagram of the "kinks" in his local hidden variables inequality. He says, Unlike the quantum correlation, which is stationary in θ at θ = 0, at the hidden variable correlation must have a kink thereIn his famous 1981 article on "Bertlmann's Socks," Bell explains that the predictions for his "ad hoc" model are linear in the angle difference a  b, and he notes the fact that his inequality only agrees with the quantum predictions at the corners of the square of linear predictions above, and not at intermediate angles. To account then for the EinsteinPodolskyRosenBohm correlations we have only to assume that the two particles emitted by the source have oppositely directed magnetic axes. Then if the magnetic axis of one particle is more nearly along (than against) one SternGerlach field, the magnetic axes of the other particle will be more nearly against (than along) a parallel Stern Gerlach field. So when one particle is deflected up, the other is deflected down, and vice versa. There is nothing whatever problematic or mindboggling about these correlations, with parallel SternGerlach analyzers, from the Einsteinian point of view. What was Bell's "short calculation" that gives "the probabilities of possible results" in his ad hoc model as linearly proportional to the angle ab?? And what exactly was Bell's "quantum mechanical calculation" that gives us probabilities proportional to the cosine of the angle ab squared? Bell does not give us any underlying physical reasons for the linear dependence on angle. He clearly knows that his linear "inequality" is a strong challenge to the curved cosine prediction of quantum mechanics. And Bell's odd prediction of sharp corners or "kinks" where his straight lines turn ninety degrees (it is only at these corners where his linear inequality agrees with the curving quantum mechanics), surely should have prompted Bell to give us a deeper explanation of his theorem? When John Clauser wrote to Bell suggesting an experimental test of his inequality, Bell replied "In view of the general success of quantum mechanics, it is very hard for me to doubt the outcome of such experiments. However, I would prefer these experiments, in which the crucial concepts are very directly tested, to have been done and the results on record."And he added "Moreover, there is always the chance of an unexpected result, which would shake the world."Clauser later recalled to Gilder "Being a young student in this age of revolutionary thinking, I naturally wanted to 'shake the world' ." The dependence on the square of the cosine is the socalled "law of Malus" for crossed polarizers as pointed out by Abner Shimony in his Stanford Encyclopedia article on Bell's Theorem. Paul Dirac taught his "principle of superposition" with crossed polarizers in his 1930 textbook The Principles of Quantum Mechanics.
Can Perfect Correlations Be Explained by Conservation Laws?
We find that David Bohm and John Bell implicitly and Eugene Wigner explicitly, used conservation of angular momentum (or particle spin) to tell us that if one spin1/2 electron is measured up, the other must be down. Just as Albert Einstein implicitly used conservation of linear momentum in his development of the EPR Paradox.
David Bohm and Yakir Aharonov wrote in 1957, We consider a molecule of total spin zero consisting of two atoms, each of spin onehalf. The wave function of the system is therefore Eugene Wigner wrote in 1963 If a measurement of the momentum of one of the particles is carried out — the possibility of this is never questioned — and gives the result p, the state vector of the other particle suddenly becomes a (slightly damped) plane wave with the momentum p. This statement is synonymous with the statement that a measurement of the momentum of the second particle would give the result p, as follows from the conservation law for linear momentum. The same conclusion can be arrived at also by a formal calculation of the possible results of a joint measurement of the momenta of the two particles. John Bell wrote in 1964, With the example advocated by Bohm and Aharonov, the EPR argument is the following. Consider a pair of spin onehalf particles formed somehow in the singlet spin state and moving freely in opposite directions. Measurements can be made, say by SternGerlach magnets, on selected components of the spins σ_{1} and σ_{2}. 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. Now we make the hypothesis, and it seems one at least worth considering, that if the two measurements are made at places remote from one another the orientation of one magnet does not influence the result obtained with the other. Just like Bohm and Wigner, Bell is implicitly using the conservation of total spin. Albert Einstein made the same argument in 1933, shortly before EPR, though with conservation of linear momentum, asking Leon Rosenfeld, Suppose two particles are set in motion towards each other with the same, very large, momentum, and they interact with each other for a very short time when they pass at known positions. Consider now an observer who gets hold of one of the particles, far away from the region of interaction, and measures its momentum: then, from the conditions of the experiment, he will obviously be able to deduce the momentum of the other particle. If, however, he chooses to measure the position of the first particle, he will be able tell where the other particle is. Supporters of the Copenhagen Interpretation claim that the properties of the particles (like angular or linear momentum) do not exist until they are measured. It was Pascual Jordan who claimed the measurement creates the value of a property. This is true when the preparation of the state is in an unknown linear combination (superposition) of quantum states. And in our case, quantum mechanics describes the entangled particles as prepared in a superposition of twoparticle states, but note that both of the states have total spin zero.
ψ_{12} = (1/√2) [ ψ_{+}(1) ψ_{}(2)  ψ_{}(1) ψ_{+}(2)] (1)
Now this initial entangled state is spherically symmetric and rotationally invariant. It has no preferred spin direction that could "predetermine" the directions that will be found by Alice and Bob, as Bell described. The preferred direction is created by Alice's measurement, or by Bob's should he measure first in the "special frame" in which Alice and Bob are "at rest" and equidistant from the location of the initial entanglement. Let's assume that Alice measures first and gets spin +1/2. The prepared state has been projected (randomly) into ψ_{+}(1) ψ_{}(2). But most important, Alice's measurement establishes the angle of her spin measurement  the angle of her SternGerlach magnet in the x,y plane. Werner Heisenberg says it is her free choice to measure the xcomponent. As the Copenhagen Interpretation describes this , Alice brings this xcomponent property into existence. (This was Pascual Jordan's contribution to the interpretation.) There was no x or ycomponent in the rotationally invariant prepared entanglement. Paul Dirac pointed out that the actual value for the property depend's on what he calls "Nature's choice." The initial prepared state (1) might equally have collapsed into ψ_{}(2). This is the source of the quantum randomness which is critically important for quantum encryption. Whichever of the two states is projected by Alice's measurement, it breaks the original symmetry, and puts the two particles in opposite spin states, randomly +  or  +, supporting the views of Bohm, Wigner, and Bell, that particles will be perfectly (anti)correlated when measured. In our example, since Alice measured the xcomponent of spin as +1/2, Bob will necessarily (and because of conservation of angular momentum) measure the xcomponent as 1/2. As we saw above, Wolfgang Pauli called it a "measurement of the first kind" when a system is prepared in a state, so that when measured, it will certainly be found in the same state. As long as nothing interferes with either entangled particle as they travel to the distant detectors (though perhaps decoherence?), they will be found to be perfectly correlated if (and only if) they are measured at the same angle (in our case, the xcomponent). Otherwise. the correlations should fall off as the square of the cosine of the angle difference. It is strange that Bell accepted an inequality that predicts correlations fall off with angle as a nonphysical straightline function with "kinks." In any case, conservation laws tell us that when either particle is measured, we know instantly those properties of the other particle, including its location equidistant from, but on the opposite side of, the entangling interaction, and all other conserved properties such as spin. But this is not "actionatadistance." It's just "knowledgeatadistance." A more recent (2005) study showing that correlations in Bell tests is the result of conservation of angular momentum is "Correlation functions, Bell's inequalities and the fundamental conservation laws," by C. S. Unnikrishnan of the Tata Institute in India. He also discusses the odd "kinks" in Bell's linear predictions of correlations compared to the conservation law curve.
No "Hidden Variables," but Perhaps "Hidden Constants?"
We find no need for "hidden variables," whether local or nonlocal. But we might say that the conservation laws give us a "hidden constant." Conservation of a particular property is often described as a "constant of the motion." Such constants might be viewed as "local," in that they travel along with particles at all times, or as "global," in that they are a property of the twoparticle probability amplitude wave function Ψ_{12} as it spreads out in space. This agrees with Bohm, and especially with Bell, who says that the spin of particle 2 is "predetermined" to be found up if particle 1 is measured to be down. But recall that the Copenhagen Interpretation says we cannot know a spin property until it is measured. So some claim that the spins are in an unknown combination of spin down and spin up until the measurements. It is this that suggests the possibility that both spins might be found in the same direction, violating conservation laws. Although electron spins in this situation are never found to be the same when measured in the same direction, the Copenhagen view gave rise to the idea of a hidden variable as some sort of signal that could travel to particle 2 after the measurement of particle 1, causing it to change its spin to be opposite that of particle 1. What sort of signal might this be? And what mechanism exists in a bare electron that receives the signal and then causes the electron to change its spin without an external force of some kind? Clearly, Wigner's explicit conservation of angular momentum, and the implicit claims of Bohm and Bell that the electron spins were prepared (entangled) in opposite states, give us the simplest and clearest explanations of the entanglement mystery. The intuitive idea that the particles were prepared with spins opposite can be interpreted as the "common cause" of the correlations. Despite accepting that a particular value of some "observables" 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, even other properties, before we measure it? His answer was yes. So Einstein might agree with Wigner, Bohm, and with Bell about conservation, but assume that the two particles have opposite spins from the time of their entangling interaction. Could two "hidden constants" of the motion, one prepared with spin up, the other spin down, explain the fact of perfect correlations of opposing spins? The answer is NO! Electron spin directions can be defined in just one of three dimensions. If spin in the xdirection is known, yspin and zspin are indeterminate. Bohm and Aharonov explained this in 1957... In quantum theory, a difficulty arises, in the interpretation of the above experiment, because only one component of the spin of each particle can have a definite value at a given time. Thus, if the x component is definite, then the y and z components are indeterminate and we may regard them more or less as in a kind of random fluctuation.Given a single direction (or angle) for spins in the initial entanglement preparation, there is zero chance that Alice and Bob would observe in that exact direction or angle. But what if the original entangled state with total spin angular momentum zero is rotationally symmetric, with no preferred direction. The original entangled state with total spin angular momentum zero is rotationally and spherically symmetric, the same in all directions. The preferred direction of measurement or angle for Alice and Bob is introduced by them, agreed to before they measure! Total spin angular momentum is conserved as zero at all times from the initial entanglement to their measurements (assuming that no environmental interaction disturbs either particle). As long as Alice and Bob now measure in the same plane (same direction, same angle), a new planar symmetry replaces the spherical symmetry and defines the one component of electron spin that Bohm and Aharonov say is allowed. The conservation of total angular momentum zero continues. As long as Alice and Bob measure in the chosen plane (preserving the symmetry), they maintain that conservation of total spin angular momentum. This zero total spin is a candidate "hidden constant of the motion." It does not specify which spin is up, which down, only that the total is zero. As we saw above, quantum mechanics describes the entangled particles as prepared in a superposition of twoparticle states, both of which have total spin zero.
ψ_{12} = (1/√2) [ ψ_{+}(1) ψ_{}(2)  ψ_{}(1) ψ_{+}(2)] (1)
That "Nature's" initial choice of updown versus downup is 50/50 quantum random is half the explanation why the bit strings created by Alice and Bob can be used in quantum encryption. The second half is that while each string looks totally random, the two are perfectly anticorrelated but by bit, although this cannot be confirmed unless the strings are shared for comparison over classical communication channels.
These quantum keys have been distributed over a secure quantum communications channel that cannot be "tapped" by an eavesdropper without destroying the perfect correlation of the pair of bit strings.
Principle Theories and Constructivist Theories
In his 1933 essay, "On the Method of Theoretical Physics," Albert Einstein argued that the greatest physical theories would be built on "principles," not on constructions derived from physical experience. His theory of special relativity was based on the principle of relativity, that the laws of physics are the same in all inertial frames, along with the constant velocity of light in all frames. Our explanation of entanglement as the result of "hidden constants" of the motion is based on conservation principles, which, as Emmy Noether showed, are based on still deeper principles of symmetry. Our principle theory explaining entanglement is also supported by the empirical evidence that entangled electron spins are always found in opposite directions, conserving the total spin angular momentum as zero. We think Einstein might have approved. Note that our conservation law explanation of entanglement in no way challenges the amazing generation of the perfectly correlated random bit strings that are used to encrypt secret messages. If anything, the conservation law provides an underlying theory for what is otherwise a mysterious set of "nonlocal" experimental results that has for the last few decades called for simply impossible fasterthanlight interactions at a distance, unacceptable violations of Einstein's principle of relativity. That may remove some of the mystery and sales or marketing value of quantum cryptography, but surely it need not do so. In 1987, Bell contributed an article entitled Are There Quantum Jumps? to a centenary volume for Erwin Schrödinger. Schrödinger strenuously denied quantum jumps or collapses of the wave function. Bell's title was inspired by two articles with the same title written by Schrödinger in 1952 (Part I, Part II). 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 this conference was an article called "Against Measurement." In it he attacked Max Born's statistical interpretation of quantum mechanics (which Born acknowledged was based on an original suggestion of Albert Einstein). And Bell 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. Bell is correct that Schrödinger and before him Einstein thought that light waves and later that wave functions were carrying some kind of "stuff." For Einstein in 1905 it was radiation energy density. For Schrödinger it was electrical energy density. Bell is also correct that the Schrödinger wave function has a kind of "duality" that Schrödinger never liked. The deterministic and unitary evolution of the quantum mechanical wave function in the abstract Hilbert space of eigenvectors in many directions contains nothing that is physically real, no tangible energy, matter, potential, or force that controls the sudden and unpredictable "projection" of the wave function into one of the related eigenstates, thus producing the "expectation values" or eigenvalues of various operators for energy, momentum, or angular momentum. And for Werner Heisenberg, it predicted the radiation intensities of spectral lines given off when atoms "jump" between their discrete energy levels that confirmed his original theory of matrix mechanics. If the wave function does not contain these physical entities, if it is only epistemological knowledge about physical things, if it is only immaterial and nonphysical information with no ontological or "real" physical content, what is it? If quantum mechanics does not include David Bohm's quantum potential traveling at infinite speeds and determining the precise positions and velocities of all particles, if it does not contain Ghirardi, Rimini, and Weber's additional terms to determine the moment of spontaneous collapse, if it does not, as Bell said, describe the electron as being there, but only of the electron being found there, if we have none of Bell's extra variables to determine the exact when and where of the "apparent collapse" to macroscopically definite outcomes, what then is "speakable" about quantum mechanics, as Bell would like to know beyond "for all practical purposes," . The answer may be unacceptable to some who want the deterministic and causal world that our macroscopic experiences and Newton's classical mechanics have led us to expect. It is simply that quantum mechanics is an indeterministic and statistical theory. The abstract mathematics of wave functions and vector spaces provides us only with probabilities. It allows us to calculate those probabilities with extraordinary mathematical precision. It predicts outcomes that experiments confirm to levels of accuracy unheard of in earlier theories. It perfectly predicts the nonlocality and apparent actionatadistance that worried Einstein and Bell. On the 22nd of January 1990, Bell gave a talk explaining his theorem at CERN in Genevaorganized 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 drawing 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 later by an experimenter and therefore be a measurement. References
On the EinsteinPodolskyRosen Paradox, 1964 (PDF)
On the Problem of Hidden Variables in Quantum Mechanics, 1966 (PDF) On the Impossible Pilot Wave (PDF) Are There Quantum Jumps? (PDF, Excerpt) CHSH Inequality, 1969 (PDF) ClauserHorne Inequality, 1973 (PDF) Epistemological Letters, 19731984 Paul_Davies_BBC Interview, 1986 (PDF, Excerpt)
"Quantum Generalizations of Bell's Inequality," Against Measurement, 1987 (PDF) Beables for Quantum Field Theory, 1990 (PDF)
"Correlation functions, Bell's inequalities and the fundamental conservation laws,"
"Why the Tsirelson Bound?,"
"Why the Tsirelson Bound?
Bub's Question and Fuchs' Desideratum," For Teachers
For Scholars
