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How a Common Cause, a Constant of the Motion, and Spherical Symmetry Can Explain Einstein's "Spooky Action at a Distance"
Abstract
Whether a calcium cascade or a spontaneous parametric down-conversion, at initial entanglement the spins of two particles are projected into a singlet state whose wave function is spherically symmetric with total spin zero. It has no preferred spatial direction. As the particles travel to the distant measurement devices the spherical symmetry and total spin zero are conserved in the absence of environmental interactions and is a constant of the motion. We can picture the two particle spins as at all times perfectly opposite, because otherwise they would be violating the fundamental law conserving angular momentum. As long as the two measurement devices are set at the same pre-agreed upon arbitrary angle, their planar symmetry and the wave function spherical symmetry ensure that measurements will produce perfectly correlated results.
The Arguments Against a Common Cause
The basic argument against a common cause that creates a definite spin direction in the initial entanglement is that the spin direction chosen for final measurement is arbitrary, so it's nearly impossible to be that initial spin direction. Initial spins in any fixed direction φ could not produce the perfect correlations seen in the arbitrary direction θ agreed upon for the experiments. Measurements would be degraded by the "law of Malus" cos²(θ-φ).
David Bohm gave the earliest and best known (but mistaken) criticism of a common cause explanation. He said that the electron spins would need to have pre-determined values in all three x, y, z directions at initial entanglement, but that this is impossible.
Only one spin component can have a definite value, say sz. When sz has the value ℏ, the other two components sy and sx are indeterminate.
Bohm and his colleague Yakir Aharonov wrote in 1957 that in classical mechanics, the molecule could have all three components of the spin well-defined, but this is impossible for quantum mechanics, since at most one component of the spin can be well-defined...
If this were a classical system, there would be no difficulty in interpreting the above results, because all components of the spin of each particle are well defined at each instant of time. Thus, in the molecule, each component of the spin of particle A has, from the very beginning, a value opposite to that of the same component of B; and this relationship does not change when the atom disintegrates. In other words, the two spin vectors are correlated. Hence, the measurement of any component of the spin of A permits us to conclude also that the same component of B is opposite in value. The possibility of obtaining knowledge of the spin of particle B in this way evidently does not imply any interaction of the apparatus with particle B or any interaction between A and B. 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.John Clauser and Abner Shimony published a variation of Bohm's criticisms of a common cause in 1978. Suppose that one measures the spin of particle 1 along the x axis. The outcome is not predetermined by the description [wave function] Ψ12. But from it, one can predict that if particle 1 is found to have its spin parallel to the x axis, then particle 2 will be found to have its spin antiparallel to the x axis if the x component of its spin is also measured. Thus, an experimenter can arrange the apparatus in such a way that he can predict the value of the x component of spin of particle 2 presumably without interacting with it (if there is no action-at-a-distance). Likewise, he can arrange the apparatus so that he can predict any other component of the spin of particle 2. The conclusion of the argument is that all components of spin of each particle are definite, which of course is not so in the quantum-mechanical description. Hence, a hidden-variables theory seems to be required.N David Mermin made a similar argument for photons in 1988, claiming that in the absence of spooky actions, it appears that both photons must have definite polarizations along every conceivable direction... Both photons must have had definite polarizations along α. Furthermore, since the conclusion that one photon has a definite polarization along the direction α does not require an actual measurement of the polarization of the other along that direction (again, in the absence of spooky connections), and since not measuring polarization along a direction α is the same as not measuring it along any other direction, we are led to conclude that both photons must have definite polarizations along every conceivable direction.
Arguments For Momentum Conservation and a Constant of the Motion
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.In 2004, C.S.Unnikrishnan of the Tata Institute of Fundamental Research in Mumbai, India wrote Bell’s inequalities can be obeyed only by violating a conservation law.Although Wigner and Unnikrishnan are explicit that the correlations are the result of conservation of spin angular momentum, the other great physicists who discovered nonlocality and entanglement, Bohm, Bell, and even Einstein himself, were using conservation of momentum implicitly with statements like these: "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,"
The Critical Importance of Spherical Symmetry
The spherical symmetry of the two-particle wave function ensures that if measurements are made symmetrically (viz, at the same angle in the same central plane) the two spins will be exactly opposite and the total spin zero of the particles conserved. If the two measurement angles (or planes) differ by θ, the correlations will be reduced by cos2θ, but this is not because overall angular momentum is not strictly conserved! The lost angular momentum of one particle is gained by the massive measurement device at the divergent angle.
This author discovered the spherical symmetry of the two-particle wave function while working on his Ph.D. thesis The Continuous Spectrum of the Hydrogen Quasi-Molecule at Harvard in 1968. The hydrogen "quasi-molecule" is two hydrogen atoms absorbing and emitting light while they are colliding and separating. Their lowest energy molecular state is a spherically symmetric 1Σg singlet state.
These molecular orbitals are spheres with the electrons equally likely to appear at any point, though if both could be found they would most likely be located opposite one another because of their mutual repulsion. The total spin angular momentum is zero.
As the hydrogen atoms "separate," the quasi-molecular wave function remains rotationally symmetric around the molecular axis. The total spin angular momentum remains zero at all times, unless the atoms are disturbed by the environment or a measurement is made. This is exactly like David Bohm's 1952 proposal for a hidden variables experiment with a hydrogen molecule that dissociates into two hydrogen atoms.
The Discovery of Nonlocality (But No Action at a Distance!)
Between 1905 and 1927 Albert Einstein had concerns about instantaneous interactions or "influences" between his light quanta and light waves. By 1933 his concerns were about two particles that he thought had been long "separated." In 1935 his colleagues Boris Podolsky and Nathan Rosen formulated the EPR paradox. In 1947 Einstein called it "spooky action at a distance." Such events are today called "nonlocal" or "entangled." There is no superluminal and physically causal action at a distance.
Today hundreds of experiments with two entangled particles confirm that measurements made at arbitrarily large separations show the two particles' properties are perfectly correlated, even though the individual particle properties are found to be random, an unusual combination of determined and indeterministic.
Widely separated events, happening simultaneously in a frame of reference that includes the entanglement preparation at the center between the particles, can give the mistaken impression of one event influencing the other at speeds much greater than the velocity of light.
Despite Einstein's great physical insight, and despite his deep understanding of conservation principles and mathematical and geometrical symmetries, he may have introduced a false asymmetry into a symmetric situation.
In the 1935 Einstein-Podolsky-Rosen paper, the properties measured were momentum and position, which are continuous variables. Since David Bohm in 1952, the entangled property studied is spin angular momentum, which has discrete values much easier to measure precisely.
Bohm proposed "hidden variables" might be found traveling along "locally" with the particles to explain their perfect correlations. No working physical model for such hidden variables has been found.
Entanglement experiments are usually described with two widely separated experiments at points A and B with experimenters, Alice and Bob, typically located symmetrically about the center C where the two particles are initially entangled.
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