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How a "Hidden" Constant of the Motion Can Explain Einstein's "Spooky Action at a Distance"
Abstract Albert Einstein had concerns for decades about apparently instantaneous interactions or "influences" between quantum particles or fields that he thought had been long "separated."
Introduction
As Louisa Gilder has written,^{11} we live in "The Age of Entanglement." Entanglement is the basis of powerful new technologies like quantum cryptography, quantum teleportation, and quantum computing, which Lars Jaeger^{20} and others call "The Second Quantum Revolution."
Although these technologies are new, the "weirdness" of entanglement is not something new. We will show that it is partly based on what Richard Feynman called the "central mystery of quantum mechanics," the "general mystery," and the "only mystery" in quantum mechanics, which led Feynman to famously say "nobody understands quantum mechanics." Feynman's central mystery is the two-slit experiment, the problem of a particle (or its "wave") appearing to go through two slits and "interfering" with itself to produce characteristic diffraction patterns on a distant screen. Feynman told us we cannot hope to "understand" what is going on in quantum mechanics in terms of processes we have learned from our everyday experiences of classical mechanics. But he said he could describe what is happening for us in the two-slit experiment. In a similar vein, Paul Dirac argued that the non-intuitive concepts of quantum mechanics, though impossible to understand in terms of classical concepts, could be mastered through long familiarity with them. The new theories, if one looks apart from their mathematical setting, are built up from physical concepts which cannot be explained in terms of things previously known to the student, which cannot even be explained adequately in words at all. Like the fundamental concepts (e.g. proximity, identity) which every one must learn on his arrival into the world, the newer concepts of physics can be mastered only by long familiarity with their properties and uses. By contrast with Feynman and Dirac, Niels Bohr and Werner Heisenberg, in their Copenhagen interpretation of quantum mechanics, said that the results of quantum measurements must be expressible in classical concepts because it is the language that humans can understand. Sadly, the Copenhagen Interpretation has led to a great deal of confusion. Following Feynman's and Dirac's suggestion of simply making you familiar with quantum phenomena, we try to describe, with words, diagrams, and a minimum of equations, several essential quantum phenomena... 1) Feynman's "only mystery," the two-slit experiment. 3) Erwin Schrödinger's "wave function" ψ that describes the wave as well as 4) | ψ |^{2}, the absolute square of the wave function, which first Einstein and later Max Born interpreted as giving us the probabilities of finding the particle at one of its possible locations (which is now the "standard" interpretation of quantum mechanics). Einstein said this about the wave function...
|ψ|^{2} expresses the probability that there exists at the point considered a particular particle of the cloud, for example at a given point on the screen. 5) the so-called "collapse" of the wave function which describes the random appearance of a particle at a certain location and with random properties when measured. This indeterminism and probabilistic or statistical aspect of quantum mechanics greatly bothered Einstein, who hoped such properties existed before their measurement. But we are not there yet. It takes two particles to be entangled. The two-slit experiment involves a single particle and its wave (function) going through two slits. Entanglement involves two particles measured when their two-particle wave function ψ_{12} has spread out in all directions at the speed of light, so that when measured they are most likely to be found a great distance apart, but not yet physically "separated!" Schrödinger said that a wave function describing two entangled particles must be in what Paul Dirac called a 6) Superposition of two-particle states. Consider two particles we can number 1 and 2 which can each be in one of two states, + or -.
| ψ_{12} > = 1/√2) | 1_{+} 2_{-} > - 1/√2) | 1_{-} 2_{+} >
We can read the first two-particle state | 1_{+} 2_{-} > as particle 1 being plus (or spin-up) and particle 2 being minus (or spin-down). The particles are in "opposite" states.
The second two-particle state | 1_{-} 2_{+} > reverses the signs of the two particles. We can also simplify Schrödinger's notation for later discussions as
| ψ_{12} > = (1/√2) | + - > - (1/√2) | - + >
We must also explain the deep meaning of superposition of states, which is frequently and mistakenly described as "being in two states (or places) at the same time." Schrödinger poked fun at Dirac's superposition and its probabilities (which Schrödinger and Einstein did not like) with his famous cat that was dead and alive at the same time. Although he was skeptical about superpositions of states in the late 1920's, Schrödinger wrote Einstein in 1935 that the two-particle wave function would keep the particles entangled and "not separated" until a measurement would produce either | + - > or | - + > which he said could be rewritten as the product of two single-particle states, either | + > | - > or | - > | + >, both of which will have decohered and disentangled and become independent of one another. Note that the random appearance of either | + - > or | - + > means that the bit string of values for each particle is a random sequence, but the two particles are always in opposite states, never violating the conservation of total spin zero. Beyond superposition, we must also say something about 7) Werner Heisenberg's 1927 uncertainty principle. Heisenberg said his principle introduced indeterminism and acausality into quantum mechanics, but Einstein had shown in 1916 that the emission of light quanta involves ontological chance. And the collapse of the two-particle superposition of states into a product of single-particle states is ontologically random. Heisenberg said the product of uncertainty in the position ∆x multiplied by the uncertainty in momentum ∆p is equal to Planck's constant ℏ. The more certain we are of position, the less certain we are of momentum. He wrote
∆p∆x = ℏ
We may think about Heisenberg's uncertain position as the dimension in position space where Schrödinger's wave function is non-zero, and the uncertain momentum as the dimension of momentum space where Schrödinger's wave function is non-zero, so the dimension of so-called phase space (position x momentum) as ℏ, and the minimum phase-space volume that can be occupied by two spin 1/2 particles is ℏ^{3}, the volume of the first electron shell in atoms. We should note that Heisenberg's uncertainty in 1927 was already implicit in Schrödinger's wave function, which Einstein had long interpreted as the probability of finding particles, though he gave full credit to Max Born for the idea, which became known as Born's Rule. And Einstein's interpretation of quantum theory as probabilistic and quantum experiments as statistical goes back to his 1916 discovery of ontological chance in the emission and absorption of light quanta by atoms. Einstein called this fundamental randomness a "weakness in the theory" and spent the rest of his life exploring the possibility of a deterministic theory underlying indeterministic quantum mechanics, making him the foremost critic of quantum mechanics, though all his life he was more concerned about nonlocal effects between particles and between particles and their waves. Finally, we need to describe principles that are fundamental to both quantum mechanics and classical mechanics. 8) The Conservation Laws The principle is the conservation of some fundamental properties of physics, like mass, energy, and momentum. According to Emmy Noether these conserved properties emerge from certain symmetries in space. Our proposed explanation of entanglement as resulting from the conservation of spin angular momentum as a "constant of the motion" will therefore not be simple, but we hope to make it as clear as possible. In part 1 we show that Einstein's concerns about nonlocality can be glimpsed as early as his "miracle year" of 1905 in his photoelectric paper. In 1909 he described how a high-energy electron could eject a high energy light wave from a metal plate and radiate in all directions. A small part of the wave could hit a second plate and eject an electron with energy almost equal to the original electron, but in a random direction. Einstein drew an illustration of the nonlocality problem on a blackboard in 1927. He challenged Leon Rosenfeld to explain it in 1933, just two years before his colleagues Boris Podolsky and Nathan Rosen published the famous "EPR Paradox," the most cited scientific paper in history. In a letter to Max Born in 1947 Einstein described it as "spooky action at a distance." And in his 1949 "Autobiography" he said the measurement of one particle "telepathically changes the real situation" of the second particle. In part 2, we discuss the famous EPR paradox of 1935 and Einstein's fears about possible violations of special relativity. In part 3, we see that Erwin Schrödinger replied to the EPR paper by saying that his two-particle wave function ψ_{12} would keep the particles "entangled" until a measurement projects them (i.e., collapses the two-particle wave function) into a product of two single-particle wave functions ψ_{1} and ψ_{2}, only after which the particles would be "disentangled" (decohered) and could then actually "separate," as Einstein insisted. In part 4, we see how David Bohm in 1952 replaced Einstein's continuous variables of position and momentum with the discrete variables of electron spins, making future experimental measurements practically possible. Bohm also revived Louis de Broglie's "pilot-wave" theory which offered a way of imagining hidden variables that are "guiding" the particles and suggested a new (Bohmian) mechanics that could restore determinism, eliminating quantum indeterminism. And we can now show that Dirac's principle of superposition of states <1/√2) | 1_{+}2_{-} > + 1/√2) | 1_{-}2_{+} >, which for Schrödinger led to his famous "cat both dead and alive," can explain both the 50/50 randomness of outcomes and the perfect correlations (or anti-correlations) of spin measurements. In part 5, we explain why the perfectly correlated electron spins could not have been determined at the time of the initial entanglement preparation, with one spin up and the other down in a particular direction. Some physicists said that the electron spins would have to have been defined in all three dimensions - x-spin, y-spin, and z-spin. But this is impossible. In 1957 Bohm and Aharonov showed that that electron spins could be defined in at most a single direction, and this would not be the random freely chosen direction of the experimenters... 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. But having opposite spins in even a single preferred direction would not explain the experimental data! Had there been a single preferred direction (or angle) for electron spins in the initial entanglement preparation, there is essentially zero chance that Alice and Bob could observe in precisely that (unknown) direction or angle. And observations at angle θ to that initial preferred direction will produce decorrelated results + + and - - proportional to sin^{2}θ. In part 6, we will show that no preferred direction exists in the initial entanglement preparation. 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. The conserved total spin zero is a candidate for a "hidden constant of the motion." In part 7, we examine John Bell's theorem predicting how correlations between measurements made at differing angles could distinguish between local hidden variables and quantum mechanics. We show how the correlations would be reduced if Alice and Bob's measurement angles differ by θ degrees. And we will examine Bell's hypothesis about how the falloff of correlations with increasing angle θ is different for local hidden variables (straight line function of θ) and for quantum mechanics (curved cosθ function). In part 8, we comment on the origin of the flawed idea that something must be traveling from Alice to Bob. Assuming, for the moment, that the entangled particles do not have perfectly correlated opposing spins, what kind of interaction could Alice's measurement produce to bring Bob's spin angular momentum into agreement. Would it not first have to detect the amount of angular deviation? And then what kind of torque could it exert from far away to rotate the remote particle precisely into agreement? In part 9, we criticize the argument that perfectly opposed spins at the time of the Alice and Bob measurements (and in the arbitrary direction chosen by them) would require that the spins be defined in all three axial directions, which is impossible. When spins are defined in one direction, they are necessarily undefined in the other two directions. In part 10, we review the work of some physicists, starting with Einstein, who used conservation principles to explain some aspects of entanglement, but without them seeing exactly how conservation of momentum could produce the appearance of Einstein's "spooky action at a distance." In part 11, we discuss the relation between conservation principles and geometric symmetries. Einstein was normally quite aware of symmetries and he greatly favored theories based on "principles" over those that are "constructed" to fit experimental data. In part 12, we comment on the idea of "instruction sets" accompanying the particles as "hidden variables." We trace the origin of this idea to Schrödinger's suggestion that the particles seemed to know the answers to all possible questions that would be asked of them when measured. What process could prepare such information and what mechanism could read the information in the "instructions" and then change the electron spin of the distant particle as required? In part 13, we conclude with a summary of the "hidden constant" hypothesis.
Part 1 Einstein and Nonlocality, from 1905 to 1927
We show that Einstein's concerns about nonlocality can be glimpsed as early as his "miracle year" of 1905 in his light quantum/photoelectric effect paper. Einstein wrote...
How can energy spread continuously over a large volume and later be absorbed in its entirety, without contradicting
his principle of relativity? Einstein sees this here, but does not say so explicitly until 1927. The energy of a ponderable body cannot be subdivided into arbitrarily many or arbitrarily small parts, while the energy of a beam of light from a point source (according to the Maxwellian theory of light or, more generally, according to any wave theory) is continuously spread over an ever increasing volume. In 1909 Einstein described how a high-energy electron could eject a high energy light wave radiating in all directions, a small part of which could hit a plate and eject an electron with energy almost equal to the original electron. In this work, Einstein gave us the first clear picture of wave-particle duality. In a presentation at the Salzburg conference in September, 1909, Einstein argued that the interaction of radiation and matter involves elementary processes that are not "invertible," a deep insight into the irreversibility of natural processes. While incoming spherical waves of radiation are mathematically possible, they are not practically achievable. Nature appears to be asymmetric in time. He speculated that the continuous electromagnetic field might be made up of large numbers of light quanta - singular points in a field that superimpose collectively to create the wavelike behavior. Although he could not formulate a mathematical theory that does justice to both the continuous oscillatory waves and the discrete particle pictures, Einstein argued that they could be "fused" and made compatible. This was over a decade before Erwin Schrödinger's wave mechanics and Werner Heisenberg's quantum mechanics. And because gases behave statistically, he knows that the connection between the wave and particles may involve probabilistic behavior.
When light was shown to exhibit interference and diffraction, it seemed almost certain that light should be considered a wave. In 1927, Einstein drew an illustration of the nonlocality problem on a blackboard at the fifth Solvay Conference in Brussels, Belgium. Sadly, despite Einstein's two decades of pioneering work on the interaction of photons and electrons, his ideas and concerns were given little attention at this Solvay, even though the conference was dedicated to electrons and photons! The conference was dominated by papers on the new quantum theory delivered by Louis de Broglie, Niels Bohr, Max Born, and Werner Heisenberg. It is best known for Einstein's after-hours suggestions to Bohr and Heisenberg probing for faults in the uncertainty principle. Accounts of these events have been told largely by the victors (there are no holes in uncertainty) but Einstein has said they often missed or ignored his important point. That point was the nonlocal behavior of a spherical light wave as it collapses to get absorbed by a single electron. This was Einstein's only contribution mentioned in the published proceedings. Here are the notes on Einstein's original remarks at the conference and Bohr's brief and confused response. They contain much of Einstein's future 1935 EPR paper, except in 1927 only one particle is involved. Entanglement in EPR requires two identical particles. Notice how Einstein's diagram clearly shows his concerns of over two decades about reconciling a spherical wave (his example is now an electron) and its collapse to being measured at just one point where it appears as a particle. At this point in the history of quantum mechanics, wave-particle duality is seen as the debate between Schrödinger's wave mechanics and Heisenberg's particle mechanics.
Bohr's reaction to Einstein's presentation has been preserved. He didn't understand a word! He disingenuously claims he does not know what quantum mechanics is. His response is vague and ends with his ideas on complementarity and the inability to describe a causal spacetime reality.
Twenty-two years later, in his contribution to the Schilpp memorial volume on Einstein, Bohr had no better response to Einstein's 1927 concerns. But he does remember vividly and provides a picture of what Einstein drew on the blackboard. Here is Bohr's 1949 recollection of Einstein's 1927 presentation:
At the general discussion in Como, we all missed the presence of Einstein, but soon after, in October 1927, I had the opportunity to meet him in Brussels at the Fifth Physical Conference of the Solvay Institute, which was devoted to the theme "Electrons and Photons." Although Bohr seems to have missed Einstein's point completely, Werner Heisenberg at least came to explain the 1927 presentation well. In his 1930 lectures at the University of Chicago, Heisenberg presented a critique of both particle and wave pictures, including a new example of nonlocality that Einstein had apparently developed since 1927. It includes Einstein's concern about "action-at-a-distance" that might violate his principle of relativity, and Heisenberg anticipates the Einstein-Podolsky-Rosen paradox. Heisenberg wrote: In relation to these considerations, one other idealized experiment (due to Einstein) may be considered. We imagine a photon which is represented by a wave packet built up out of Maxwell waves. It will thus have a certain spatial extension and also a certain range of frequency. By reflection at a semi-transparent mirror, it is possible to decompose it into two parts, a reflected and a transmitted packet. There is then a definite probability for finding the photon either in one part or in the other part of the divided wave packet. After a sufficient time the two parts will be separated by any distance desired; now if an experiment yields the result that the photon is, say, in the reflected part of the packet, then the probability of finding the photon in the other part of the packet immediately becomes zero. The experiment at the position of the reflected packet thus exerts a kind of action (reduction of the wave packet) at the distant point occupied by the transmitted packet, and one sees that this action is propagated with a velocity greater than that of light. However, it is also obvious that this kind of action can never be utilized for the transmission of signals so that it is not in conflict with the postulates of the theory of relativity.
Part 2 EPR, Entanglement, and Suspected Violations of Special Relativity
Just two years before his colleagues Boris Podolsky and Nathan Rosen published the famous "EPR Paradox," the most cited scientific paper in history, in 1933 Einstein challenged Leon Rosenfeld to explain two-particle nonlocality. You will see that EPR is a version of this challenge to Rosenfeld. Shortly before Einstein left Germany to emigrate to America, Einstein attended a lecture on quantum electrodynamics by Leon Rosenfeld. Keep in mind that Rosenfeld was perhaps the most dogged defender of the Copenhagen Interpretation. After the talk, Einstein asked Rosenfeld, “What do you think of this situation?” 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. We can diagram a simple case of Einstein’s question as follows after the particles have interacted and separate from the center. We use electrons instead of Einstein's generic particles, an anachronism introduced by David Bohm in 1952.
Recall that it was Einstein who discovered in 1924 the identical nature, indistinguishability, and interchangeability of some quantum particles. He found that identical particles are not independent, altering their quantum statistics.
After the particles interact at t_{1}, quantum mechanics describes them with a single two-particle wave function that is not the product of independent single-particle wave functions. In the case of electrons, which are indistinguishable and interchangeable particles, it is not proper to say electron 1 goes this way and electron 2 that way. (Nevertheless, it is convenient to label the particles, as we do in the illustration.)
Note the anachronism of electrons as Einstein's generic particles. It was David Bohm in 1952 who proposed that Einstein's EPR problem use electrons. Today many if not most accounts of the EPR paradox describe it with electrons. Many mistakenly say that Einstein was describing electrons. Einstein then asked Rosenfeld, “How can the final state of the second particle be influenced by a measurement performed on the first after all interaction has ceased between them?” This was the germ of the EPR paradox, and ultimately the paradoxical problem of two-particle entanglement. Why does Einstein question Rosenfeld and describe this as an “influence,” suggesting an “action-at-a-distance?” One Einstein concern could see a paradox in the context of Rosenfeld’s Copenhagen Interpretation, since the second particle is not itself measured and yet we know something about its properties, whereas the Copenhagen Interpretation says we cannot know any properties without an explicit measurement... Einstein was clearly correct to tell Rosenfeld that at a later time t_{2}, a measurement of one particle's position would instantly establish the position of the other particle - without measuring it. Einstein was implicitly using the conservation of linear momentum to calculate (and to know) the position of the second particle. Another of Einstein's concerns may have been Werner Heisenberg's uncertainty principle. It suggests that knowing the exact position of the second particle is not possible. In any case, if the second particle were to be measured, it should be found in a range of positions Δx ≥ h/2πΔp. And this might contradict the conservation of linear momentum, as Nathan Rosen appears to have considered in the later EPR paper. Two years later, reacting to EPR, Schrödinger described two such particles as becoming "entangled" (verschränkt) at their first interaction, so "nonlocal" phenomena are also known as "quantum entanglement." Although conservation laws are rarely cited as the explanation, we argue that they are the physical reason that entangled particles always produce correlated results for all properties, linear momentum, angular momentum, and energy. If the results were not always correlated, the implied violation of a fundamental conservation law would cause a much bigger controversy than entanglement itself, as puzzling as that phenomenon is. The hypothesis that the conservation principle gives us a "hidden constant" of the motion like the above linear momentum (perhaps three hidden constants to include angular momentum and energy?), can explain the above results. In his 1935 EPR paper, Einstein cleverly introduced two particles instead of one, and in the same year Erwin Schrödinger would create a two-particle wave function that describes both particles. EPR described the particles as widely separated, one "here" and measurable "now" and the other distant and to be measured "later." Here we must explain the asymmetry that Einstein mistakenly introduced into a perfectly symmetric situation, making entanglement such a mystery. The two entangled particles must normally be measured at the same time (synchronously) to ensure that just one entangled pair is involved. This means the measuring devices are placed equidistant (symmetrically) from the central location of initial entanglement. Most accounts of entanglement say that Alice measures "first" and then must interact instantaneously with Bob's particle to align its spin with hers. This is a false asymmetry and an impossible interaction, as it is a violation of special relativity. At creation, the wave function of the entangled pair is rotationally and spherically symmetric. The measurement devices must be arranged symmetrically. in the same plane with the origin. If the measurements differ by an angle θ, the perfect correlations would be reduced by cos^{2}θ. The normal interactions between the separating particles (electromagnetic forces) do not exceed the speed of light and have the translational, rotational, and temporal invariance symmetries needed to conserve the total spin zero (as well as total linear momentum zero and total energy) created at initial entanglement. All these symmetries are essential to creating perfectly correlated outcomes. What could have led Einstein to see an asymmetry as an instantaneous action a distance. In his 1905 photoelectric study it was likely the view that the light wave was radiation energy spread out in space that had to collect itself into the photoelectron. In his 1909 wave-particle duality electron article, it was likely the view shared by Schrödinger that the wave was spread out electric charge. It lies behind the current concern with the collapse of the wave function. The EPR paradox has been reframed since David Bohm in 1952 as two entangled electrons instead of generic particles conserving their total linear momentum zero. In this case, the particles are identical, indistinguishable, and interchangeable, with indeterminate positions according to the uncertainty principle. We can diagram a simple case of indistinguishable electrons separating from their point of entanglement to be measured at a later time when they are far apart.
As we also saw, Einstein had considered nonlocal phenomena as early as 1905 and objected publicly to nonlocal phenomena at the Solvay Conference of 1927, when he criticized the collapse of the wave function as "instantaneous-action-at-a-distance." He said that the probability wave | ψ |^{2} could not act simultaneously at different places on the screen, which are in a "space-like" separation, without violating his theory of relativity. Einstein's criticism resembles the criticisms made of Newton's theory of gravitation. Newton's opponents charged that his theory was instantaneous "action at a distance." Einstein's new theory of general relativity shows that gravitational "influences" travel at the speed of light and are mediated by a gravitational field that shows up as curved space-time. An allowable "action-at-a-distance" is one that is caused by "local" events, those in its past light-cone. Both gravitation and electromagnetism are field theories, in which physical variables are functions of the four "local" space-time coordinates of Einstein's theory of relativity. Any disturbance at one point in the field can only "influence" another distant point by propagating to that point at the speed of light or less. But the quantum-mechanical wave-function is different. It is neither matter nor energy, nothing physical, only information and a "probability amplitude" whose square gives us probabilities and intensities of quantum phenomena. Can we say probabilities also can only travel at the speed of light? Note that when a probability function collapses to unity in one place and zero elsewhere, nothing physical, no matter or energy, is moving from one place to another. When the nose of one horse crosses the finish line, its probability of winning goes to certainty, and the finite probabilities of the other horses, including the one in the rear, instantaneously drop to zero. This happens faster than the speed of light, since the last horse is in a "space-like" separation. Quantum theory, especially Paul Dirac's "transformation theory," describes the two electrons as in a superposition of states, either state equally likely to be randomly found by an experiment. The first state | + - > is spin up ( + ) and spin down ( - ), the second | - + > is spin down ( - ) and spin up ( + ),
| ψ_{12} > = (1/√2) | + - > - (1/√2) | - + >
The coefficients 1/√2, when squared, tell us the probabilities of either state being found is 1/2. Dirac's superposition) can provide only the probabilities of finding the entangled system in either the | + - > state or the | - + > state. Quantum mechanics does not and cannot describe the paths or the spins of the individual particles as the wave function expands at the speed of light. Note that should measurements result in | + + > or | - - > state, that would violate the conservation of angular momentum. We call the conserved total spin zero a "hidden constant of the motion". It is the implicit reason why David Bohm, John Bell, and many others say that when one particle is measured spin-up, we instantly know the other must be spin-down.
Part 3 Erwin Schrödinger, Entanglement, and his Two-Particle Wave Function
Erwin Schrödinger replied to the EPR paper by telling Einstein that his two-particle wave function ψ_{12} would keep the particles coherently "entangled" until a measurement projects them into a product of single-particle wave functions ψ_{1} and ψ_{2}. We show that the Schrödinger/Dirac superposition of states <1/√2) | 1_{+}2_{-} > - 1/√2) | 1_{-}2_{+} > can explain both the 50/50 randomness of outcomes and the perfect correlations (or anti-correlations) of measurements, provided the spin angular momentum is conserved.
Schrödinger said that the entangled particles have jointly shared properties that cannot be changed independently, until after they have become disentangled, decoherent, and independent of one another. Though Schrödinger did not mention it, one of those properties is total linear momentum, key to the EPR paper. But like Einstein's other reactions to quantum mechanics, he could not accept that particles he regarded as separated could change their properties simultaneously. Einstein did not explicitly recognize that the joint properties of the two particles would need to obey conservation laws, though he had implicitly used the conservation of linear momentum in 1933 and 1935, when he said the second particle position must be symmetrically placed. Conservation laws are the consequence of extremely deep properties of nature that arise from simple considerations of symmetry. We regard these laws as "cosmological principles." Physical laws do not depend on the absolute place and time of experiments, nor their particular direction in space. Conservation of linear momentum depends on the translation invariance of physical systems, conservation of energy the independence of time, and conservation of angular momentum the invariance under rotations. Conservation laws are the consequence of symmetries, as explained by Emmy Noether. Recall that the EPR experiment (Bohm version) starts with two electrons (or photons) prepared in an entangled state that is a mixture of pure two-particle states, each of which conserves the total angular momentum and, of course, conserves the linear momentum as in Einstein's original EPR example. This information about the linear and angular momenta is established by the initial state preparation (a measurement). Quantum mechanics describes the probability amplitude wave function Ψ_{12} of the two-particle system as in a superposition of two-particle states. It is not a product of single-particle states, and there is no information about the identical indistinguishable electrons traveling along distinguishable paths. With slightly different notation, we can write equation (1) as
Ψ_{12} = 1/√2) | 1_{+}2_{-} > - 1/√2) | 1_{-}2_{+} > (2)
The probability amplitude wave function Ψ_{12} travels away from the source (at the speed of light or less). Let's assume that at t_{0} observer A finds an electron (e_{1}) with spin up.
At the time of this "first" measurement, by observer A or B, new information comes into existence telling us that the wave function Ψ_{12} has "collapsed" into the state | 1_{+}2_{-} > And conservation of linear momentum tells us that at t_{0} the second electron is equidistant from the source in the opposite direction.
Unlike the two-slit experiment, where the "collapse" goes to a specific point in 3-dimensional configuration space, the "collapse" here is a "jump" (Bohr) or a "projection" (Dirac) or a "reduction" (Heisenberg) into one of the two possible 6-dimensional two-particle quantum states | + - > or | - + >. This makes "visualization" (Schrödinger's Anschaulichkeit) difficult or impossible, but the parallel with the collapse in the two-slit case provides an intuitive insight of sorts. In both cases, the local values of the probability wave function change instantly over a wide range of space. This is what Einstein saw in 1905 with the light wave spread out everywhere collapsing into a photon, in 1927 with the probability of a particle being found everywhere on a spherical surface collapses to the spot where it is found, and again in 1933 where observing a particle in one place tells us exactly where the other one is far away. If the measurement finds an electron (call it electron 1) as spin-up, then at that moment of new information creation, the two-particle wave function collapses to the state | + - > and electron 2 "jumps" into a spin-down state with probability unity (certainty). The results of observer B's measurement at a later time t_{1} is therefore determined to be spin down. Notice that Einstein's intuition that the result seems already "determined" or "fixed" before the second measurement is in fact correct. The result is determined by the law of conservation of momentum.
But as with the distinction between determinism and pre-determinism in the free-will debates, the measurement by observer B was not pre-determined before observer A's measurement.
Part 4 David Bohm, Atoms, Electron Spins, and Hidden Variables
As we saw in part 2, the first practical and workable experiments to test the EPR paradox were suggested in 1952 by David Bohm (though experiments were not realized for almost two decades). Instead of using continuous linear momenta in opposite directions, Bohm proposed using two atoms (or electrons) in discrete electronic spin states. The atoms are prepared in an initial molecular state of known total spin zero.
Bohm also revived Louis de Broglie's "pilot-wave" theory which offered a way of imagining hidden variables "guiding" the particles and suggested a new (Bohmian) mechanics that could restore determinism and eliminate quantum indeterminism. The Bohm version of EPR starts with a hydrogen molecule in a spherically symmetric "singlet" state, with total spin angular momentum zero, which disassociates into two hydrogen atoms. Bohm-EPR is describable by a two-particle wave function we can call ψ_{12} which is the linear combination of two terms, each the product of two single-particle wave functions, ψ_{1} and ψ_{2}.
ψ_{12} = 1/√2(ψ_{1}ψ_{2}) - 1/√2(ψ_{2}ψ_{1})
The coefficients 1/√2, when squared, tell us there is a 50-50 chance of finding the separated atoms in ψ_{1}ψ_{2} or in ψ_{2}ψ_{1}. In either case the atomic spins are always found in opposite directions, when measurements are made in the same direction, preserving the symmetry, and conserving the total spin angular momentum as zero, the same as the original molecule. It is this physical fact that let Bohm in 1951 write Suppose that we have a molecule containing two atoms in a state in which the total spin is zero and that the spin of each atom is ℏ/2. Roughly speaking, this means that the spin of each atom points in a direction exactly opposite to that of the other, insofar as the spin may be said to have any definite direction at all. [Indeed, we can only say that the total spin zero of the two atoms means that their two-particle wave function ψ_{12} is spherically symmetric with no preferred direction - the singlet state.] Now suppose that the molecule is disintegrated by some process that does not change the total angular momentum. The two atoms will begin to separate and will soon cease to interact appreciably. [But Erwin Schrödinger tells us that they will not become independent and "cease to interact" until some measurement disentangles them, decohering their phases, and allowing their two-particle wave function to be replaced with the product of two single-particle wave functions ψ_{1}ψ_{2} or ψ_{2}ψ_{1}.] Their combined spin angular momentum, however, remains equal to zero, because no torques have acted on the system... Bohm and Yakir Aharonov said the equivalent in 1957 Then, because the total spin is still zero, it can immediately be concluded that the same component of the spin of the other particle (B) is opposite to that of A. Finally, in John Bell's first paper, he wrote in 1964 With the example advocated by Bohm and Aharonov, the EPR argument is the following. Consider a pair of spin one-half particles formed somehow in the singlet spin state and moving freely in opposite directions. Measurements can be made, say by Stern-Gerlach 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 why, we may ask, did not Bohm, or Bell, or modern commentators on Bell's theorem, consider the conservation of total spin angular momentum zero as a "common cause" explanation for the perfect measurement correlations? The conservation is true at all times for the "separating" particles (now known as a "quasi-molecule" because it is described with a molecular wave function and not the product of two atomic wave functions). Conservation will remain true as long as the forces involved in Stern-Gerlach spin measurements are symmetric, the two SG devices oriented in the same plane to preserve the symmetry. When the devices measure in planes at an angle to one another, the correlations fall off as the cosine squared of that angle (Malus's law). This "common cause" is just a shared/joint property of the two particles, true at all times, not just for measurements. It is not one particle's measurement acting (instantaneously at a distance) on the other particle. The constant total spin zero is thus a "constant of the motion." Since it achieves the goal of hypothetical "local hidden variables" traveling with the particles, we might call it a "hidden constant of the motion."
Part 5 Were the Spin Variables Created at the Initial Entanglement?
We comment on the flawed idea that something (hidden variables?) must be traveling between Alice and Bob. Assuming, for the moment, that the entangled electrons do not already have perfectly correlated opposing spins (which they will have if the electrons are conserving total spin angular momentum zero), 1) what kind of sensor would Alice need to measure the deviation between their electron spins and 2) what kind of interaction could Alice's measurement generate and send to Bob to bring his spin angular momentum into agreement? Alice must first detect the amount of Bob's angular deviation from her measurement angle. (Keep in mind that an electron spin measuring device, for example, a Stern-Gerlach device, would change the electron spin to be in the direction of the SG device. But Alice could not send an SG device to Bob, especially instantaneously!) Otherwise, what kind of torque could Alice exert on Bob's remote electron from far away to rotate it precisely into agreement? Any such two-part interaction, first signals about his orientation (pure information) from Bob to Alice and then mechanical forces back from Alice to Bob, applied to his electron in a distant space-like separation. Both violate the no-signaling theorem and the principle of relativity.
We explain why the perfectly correlated electron spins in the direction of measurements could not have been determined at the time of the initial entanglement preparation, with one spin up and the other down in the particular direction of measurements. Some physicists say that the electron spins would have to have been defined in all three dimensions - x-spin, y-spin, and z-spin. But this is impossible. Bohm in 1951 and Bohm and Aharonov in 1957 argued that electron spins could be defined in at most a single direction, and this direction would very likely not be the random freely chosen direction of the experimenters. This is a serious misunderstanding...
Here, the investigator can measure either the x, y, or z component of the spin of article No.1, but not more than one of these components, in any particular experiment. Nevertheless, it still turns out as we shall see that whichever component is measured, the results are correlated, so that if the same component of the spin of No.2 is measured, it will always turn out to have the opposite value. This means that a measurement of any component of the spin of atom No.1 provides, as in classical theory, an indirect measurement of atom No.2. Since, by hypothesis, the two particles no longer interact, we have obtained a way of measuring an arbitrary component of the spin of particle No.2 without in any way disturbing that particle. Since this can be accomplished without in any way disturbing the second atom, we conclude that...precisely defined elements of reality must exist in the second atom, corresponding to the simultaneous definition of all three components of its spin. [This is a gravely flawed conclusion. It is not that components of spin are defined in all (3) direction, but that total spin zero means the spins will be found with opposite values in whatever agreed-upon direction they are measured!] ...Because the wave function can specify, at most, only one of these components at a time with complete precision, we are then led to the conclusion that the wave function does not provide a complete description of all elements of reality existing in the second atom. [The spherically symmetric (singlet state) wave function actually has no preferred direction, so as long as measurements are made symmetrically by the Stern-Gerlach devices, the atoms will be found to have perfectly correlated opposite spin values in that direction (which the observers must freely choose by previous agreement to be in the same direction, otherwise correlations will be reduced by the square of the cosine of the angle between them.] In 1957, Bohm and Aharonov expanded this description...
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. But having opposite spins in even a single preferred direction would not explain the experimental data! Had there been a single preferred direction (or angle) for electron spins in the initial entanglement preparation, there is essentially zero chance that Alice and Bob could observe in precisely that (unknown) direction or angle. And observations at angle θ to that initial preferred direction will produce decorrelated results + + and - - proportional to sin^{2}θ.
Part 6 No Preferred Direction in the Spherically Symmetric Wave Function
No preferred direction exists in the initial entanglement preparation. The original entangled state with total spin angular momentum zero is rotationally and spherically symmetric, the same in all directions. The arbitrary 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 spin angular momentum zero continues. As long as Alice and Bob measure in their "freely chosen" plane (preserving the symmetry), conservation of total spin angular momentum continues. There are no "local hidden variables" traveling with the particles to maintain perfect correlations, but conserved total spin zero might be called a "hidden constant of the motion" that provides the needed "common cause" for the correlations.
Part 7 John Bell and his Inequalities Theorem
We examine John Bell's theorem predicting how correlations between measurements made at differing angles could distinguish between local hidden variables and quantum mechanics. Bell predicted how the correlations would be reduced if Alice and Bob did not measure in the same direction, but chose measurement angles different by θ degrees. And we will examine Bell's hypothesis about how the falloff of correlations with increasing angle θ is different for local hidden variables (a straight line function of θ) and for quantum mechanics (a curved cosθ function).
We can illustrate the straight-line 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 "Popescu-Rorhlich 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 non-physical right-angle 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 Einstein-Podolsky-Rosen-Bohm 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 Stern-Gerlach 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 mind-boggling about these correlations, with parallel Stern-Gerlach 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 |a-b|? 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 physical explanation of his theorem? And what exactly was Bell's "quantum mechanical calculation" that gives us probabilities proportional to the cosine of the angle |a-b| squared? This more physical picture coincides with falloff of light polarization when polarizers are crossed at an angle (Malus's Law). Conservation of total spin angular momentum zero (our "hidden constant of the motion") perfectly matches Bell's quantum mechanics predictions.
Part 8 Were the Correlations Predetermined?
Could the measurement outcomes have been predetermined at the initial entanglement?
John Bell considered this a possibility. In his very first paper on EPR, he wrote... 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. Over a decade later, Bell still considered predeterminism... During a mid-1980'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 two-particle experiments without the need for faster-than-light signaling. The two experiments need only have been pre-determined 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? The freely chosen direction of measurement angle previously agreed upon by Alice and Bob can only be implemented at the time of their measurements and not before. As shown in parts 5 and 6, the only dependence on the original entanglement is that its spherically symmetric singlet state has total spin zero and no preferred direction. The specific measurement direction/angle, freely chosen by Alice and Bob, and the specific correlations at that measurement angle (specifically either + - or - +), results from the "collapse" of the Dirac/Schrödinger two-particle wave function (a "superposition" of + - and - + states) one of those states which is randomly realized. These random results are "created" long after the particles were together at entanglement, when the particles have apparently become "separated" by arbitrarily large distances. The phenomenon therefore appears distinctly "nonlocal," but it does not involve any physical interaction or information transfer between the particles. It is "caused" by a condition (the conserved total spin angular momentum zero) that has been true from the moment of initial entanglement, as long as neither particle is disturbed by any asymmetric environmental interaction. In his 1951 textbook Quantum Theory, Bohm wrote...Suppose that we have a molecule containing two atoms in a state in which the total spin is zero and that the spin of each atom is ℏ/2. Roughly speaking, this means that the spin of each atom points in a direction exactly opposite to that of the other, insofar as the spin may be said to have any definite direction at all. [Indeed, we can only say that the total spin zero of the two atoms means that their two-particle wave function ψ_{12} is spherically symmetric with no preferred direction - the singlet state.] Now suppose that the molecule is disintegrated by some process that does not change the total angular momentum. The two atoms will begin to separate and will soon cease to interact appreciably. [But Erwin Schrödinger tells us that they will not become independent and "cease to interact" until some measurement disentangles them, decohering their phases, and allowing their two-particle wave function ψ_{12} to be replaced with the product of two single-particle wave functions ψ_{12}ψ_{2} or ψ_{2}ψ_{1}.] Their combined spin angular momentum, however, remains equal to zero, because no torques have acted on the system...The Bohm and Aharonov argument that perfectly opposed spins at the time of the Alice and Bob measurements (and in the arbitrary direction chosen by them) would require that the spins would have to be defined in all three axial directions (which is impossible). This is a flawed argument. When spins are measured in one direction, they are necessarily undefined in the other two directions, as Bohm and Aharonov later said. The original entangled state with total spin zero is rotationally symmetric, with no preferred direction, so that the preferred plane or direction of measurement for Alice and Bob is introduced by them. This symmetry ensures the conservation of total angular momentum zero until the time of measurement. As long as Alice and Bob measure in the same plane (preserving the symmetry), they maintain the conservation of angular momentum, our proposed "hidden constant of the motion."
Part 10 Physicists Who May Have Seen Conservation Principles
We review the work of some physicists to show they were clearly using conservation principles to explain some aspects of entanglement, but without their necessarily seeing how conservation of linear or angular momentum could produce the appearance of Einstein's "spooky action at a distance" without any interaction or "influence" between the particles.
David Bohm
Bohm and his Israeli student Yakir Aharonov reformulated the original EPR argument in terms of electron spin. They said experimental tests with continuous variables would be much more difficult than tests with discrete quantities, such as the spin of electrons or polarization of photons.
Bohm and Aharonov described the preparation of two particles, such that a measurement of one at a later time determines a measurement in the same direction of the other particle at any distance away.
We consider a molecule of total spin zero consisting of two atoms, each of spin one-half. The wave function of the system is therefore Their statement "because the total spin is still zero, it can immediately be concluded that the same component of the spin of the other particle (B) is opposite to that of A." surely is true because of the conservation of spin angular momentum.
Eugene Wigner
In an important article written after Bohm and before Bell's 1964 "theorem" paper, Eugene Wigner in 1963 said very clearly that the symmetrically placed positions (for the EPR paper) are caused by conservation of linear momentum and the perfectly opposite electron spins (for David Bohm's version of nonlocality with electrons) are the result of conservation of angular momentum. Wigner wrote
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
We can ask why John Bell, with the prior writings of Bohm and Wigner, never explicitly considered the conservation of linear momentum in the EPR experiment, and conservation of spin angular momentum as explaining the perfectly opposite electron spins found in Bohm's version of EPR. But we can see in Bell's description that, like Bohm, he is implicitly using conservation of angular momentum.
In his 1964 paper "On the Einstein-Podolsky-Rosen Paradox," Bell made the case for nonlocality. With the example advocated by Bohm and Aharonov, the EPR argument is the following. Consider a pair of spin one-half particles formed somehow in the singlet spin state and moving freely in opposite directions. Measurements can be made, say by Stern-Gerlach 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.During a mid-1980'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 two-particle experiments without the need for faster-than-light signaling. The two experiments need only have been pre-determined 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?But the obvious fact that the total spin angular momentum at zero is conserved for all times since the initial entanglement means that the two particles are indeed "determined by causes reaching both experiments from an earlier time," though not "superdetermined from the origin of the universe.
C.S.Unnikrishnan
In 2005, C.S.Unnikrishnan of the Tata Institute of Fundamental Research in Mumbai, India was an exception. He proposed explicitly that the conservation law of angular momentum can correlate measurements of entangled electrons, explaining the perfect correlations of entangled particles, without the faster-than-light interactions-at-a-distance or "hidden variables" often invoked to explain nonlocaiity.
Unnikrishnan wrote Bell’s inequalities can be obeyed only by violating a conservation law. Unnikrishnan argues that conservation of angular momentum (electron spin) produces the same perfect correlations (or anti-correlations) found in all Bell test experiments when both experimenters measure at the same (pre-agreed upon) measurement angle. However, Unnikrishnan is concerned that "for individual measurements of the two-point correlation, the conservation law cannot be invoked, since only the conditional probabilities are predicted by quantum mechanics." He uses instead averages of measurements. Unnikrishnan's apparent concern is that individual measurements will have random outcomes of up-down, down-up, and possibly even some up-up and down-down. He mistakenly thinks quantum mechanics predicts separate probabilities for each electron. The latter two product states (up-up and down-down) would violate conservation of angular momentum.
Part 10 Conservation Laws and Constants of the Motion
Conservation laws are the consequence of extremely deep properties of nature that arise from simple considerations of symmetry. We regard these laws as "cosmological principles." Physical laws do not depend on the absolute place and time of experiments, nor their particular direction in space. Conservation of linear momentum depends on the translation invariance of physical systems, conservation of energy the independence of time, and conservation of angular momentum the invariance under rotations. Conservation laws are the consequence of these spatial symmetries, as explained by Emmy Noether.
David Bohm's version of the EPR experiment starts with two electrons (or photons) prepared in an entangled state that is a superposition of two-particle states, each of which conserves the total angular momentum and, of course, conserves the linear momentum as in Einstein's original EPR example. Quantum mechanics describes the probability amplitude wave function Ψ_{12} of the two-particle system as in a superposition of two-particle states. It is not a product of single-particle states, as Erwin Schrödinger told Einstein in his reaction to the EPR paper. We can write this as
Ψ_{12} = 1/√2) | 1_{+}2_{-} > + 1/√2) | 1_{-}2_{+} > (1)
The probability amplitude wave function Ψ_{12} travels away from the source (at the speed of light or less). The total spin zero wave function is rotationally symmetric and isotropic, the same in all directions. Let's assume that at t_{0} an observer finds a particle with spin up in the x direction. This measurement breaks the rotational symmetry. The new symmetry is planar, including the chosen x direction and the z direction back to the origin of the entangled particles. Before the measurement, the spin has a possibility of being found in any direction. Rotational symmetry says the probability is the same in all directions. This does not mean a particle has spins in all directions at all times, which is impossible. At the time of this "first" measurement, say by observer A, new information comes into existence telling us that the wave function Ψ_{12} has "collapsed" into the state | 1_{+}2_{-} >. Probabilities have now become certainties, one possibility is now an actuality. If the first measurement finds particle 1's x-component spin is up, so the same spin component of entangled particle 2 must be down to conserve total angular momentum. And conservation of linear momentum tells us that at t_{0} the second electron is equidistant from the source in the opposite direction. As with any wave-function "collapse", the probability amplitude information changes. Nothing really "collapses." Nothing physical, no matter or energy, is moving. Only information is changing. The wave function is updated to reflect the new information that comes into existence as the result of the measurement. When the "first" measurement finds particle 1 as spin-up at t_{0}, at that moment of new information creation, particle 2 will be found (if measured simultaneously or later) in a spin-down state with probability unity (certainty). And the results of observer B's measurement at t_{0} or any later time t_{1} is therefore determined to be spin down (if and only if, B measures in a pre-agreed upon same direction). Notice that Einstein's intuition that observer B's result seems already "determined" or "fixed" before the second measurement is in fact correct. Observer B's outcome is determined by the law of conservation of momentum. But the measurement by observer B was not pre-determined before observer A's measurement. It was simply determined by her measurement. The measured values of particle 1 spin-up and particle 2 spin-down did not exist before the "free choice" of observer A brought them into existence, as Werner Heisenberg insisted. Which of the two-particle quantum states | + - > or | - + > occurs is completely random. It is the result of "Nature's choice," as Paul Dirac described it. Note also that before the measurement the two-particle wave function was rotationally symmetric. No preferred angular direction existed. The preferred angle also comes into existence as a result of what Heisenberg called the "free choice" of the ("first") experimenter. Of course neither Alice nor Bob measures first, as their measurements of a single entangled pair reaching their experiments are normally made simultaneously.
This "free choice" of a measurement angle breaks the rotational symmetry of the original two-particle wave function. As Erwin Schrödinger described it to Einstein in his 1935 response to the EPR paper, the measurement disentangles the particles and projects the pure-state superposition into a mixed-state product of single-particle wave functions, either Many critics have pointed out that the two particles cannot already have opposite spin values in a specific direction long before the measurements. That would require them to have spin values in all three x, y, and z directions, say some physicists (e.g., David Bohm) which is impossible, they say. They only need to acquire opposite spins when measured along an agreed upon direction that breaks the rotational symmetry of the two-particle wave function. Finally note that conservation of total spin zero requires no instantaeous superluminal influence or interaction by one particle on the other. It only requires that interactions between the particles preserve the symmetry needed for the conservation law. Many other critics have insisted that some mechanism will be found to explain the mysterious entanglement That symmetry has become linear and planar, as the rotational symmetry disappears, leaving symmetry only along the plane between the electrons that includes the chosen measurement direction. If the two particles did not conserve total spin zero (and every Bell test shows that they do conserve total spin), the violation of the conservation law would likely be met with more criticism than hypothetical superluminal interactions, which are of course impossible. What about the uncertainty principle? In the case of EPR measuring the position x (or the momentum p) of the two particles, won't their values be "fuzzy" (ΔxΔp ~ħ) and therefore not conserve momentum exactly, but only statistically for large numbers of examples? No, the conservation laws require that if x_{1} is found less than the expected value by an amount -δ, that x_{2} would be greater by the opposite amount +δ, so that the two particles are equidistant from the origin. This ensures the conservation of linear momentum, just as Einstein in 1924 proved that the Bohr-Kramers-Slater theory that energy is only conserved statistically was wrong. Momentum is conserved exactly in every measurement, although the uncertainty principle may prevent this from being shown experimentally. Furthermore, in the Bohm version of nonlocality the quantities are discrete spins, not continuous positions or momenta. Electron spins are always measured to be either up or down, with nothing fuzzy about these values. And all the experimental results from all the Bell tests have always found the two spins opposite as long as both measurements are made in exactly the same direction, thus conserving total spin zero. There is still indeterminism (uncertainty) in the spin measurement results. We don't know which electron will be up and which down. It is this property that "does not exist" before the measurement. Physics has not found any hidden variables, local or nonlocal, as the cause of the perfect opposite spins. Is the conserved total spin zero acceptable as what we call a "hidden constant of the motion" that completely accounts for the perfectly opposite spins?
Part 11 David Mermin's Instruction Sets as Hidden Variables
We trace the modern idea of "instruction sets" accompanying the particles as "hidden variables" to Erwin Schrödinger's suggestion that the particles seem to know the answers to all possible questions that would be asked of them when measured. What process could prepare such information for the particles and what mechanism could read the information in the "instructions" and then change the particle properties as required?
In his 1936 paper "Property Relations Between Separated Systems," Schrödinger wrote Years ago I pointed out that when two systems separate far enough to make it possible to experiment on one of them without interfering with the other, they are bound to pass, during the process of separation, through stages which were beyond the range of quantum mechanics as it stood then... Schrödinger refers to the central claim of the EPR paradox, in terms of the information available to the particles...
the result of measuring p_{1} serves to predict the result for p_{2} 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.
Part 12. Summary of the "Hidden Constant" hypothesis
Standard quantum mechanics plus the principle of angular momentum conservation (true for quantum and classical mechanics) predicts:
The "hidden constant of the motion" was introduced in chapter 45 of My God, He Plays Dice, How Albert Einstein Invented Most of Quantum Mechanics, 2019, p.376. If the conservation of angular momentum (spin) is not the "common cause" of the perfect correlations, the vast experimental evidence for those correlations, so critical to the twin random bit strings needed for quantum cryptography, tells us that conservation of spin is an experimentally proven fact of the matter! We can also note that a subsequent measurement by either observer at a different angle will destroy the planar symmetry. The original linear combination or superposition of states
Ψ_{12} = 1/√2) | 1_{+}2_{-} > + 1/√2) | 1_{-}2_{+} > (1)
will become a mixture of product states that have decohered(their pure-state coherent phase relations lost). They are disentangled, as Erwin Schrödinger argued in 1936. Measuring one can still tell us something about the other. he said. But they have decohered and can no longer interfere with one another and cannot remain correlated after future measurements. They have then truly separated and "independent," but they had not become independent before the disentanglement, as Einstein had hoped with his separability principle (Trennungsprinzip).
References
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