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A "Hidden" Constant of the Motion?

In an important article written before the 1964 Bell's Theorem paper, Eugene Wigner in 1963 said 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 1952 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.

One can go even further: instead of measuring the linear momentum of one particle, one can measure its angular momentum about a fixed axis. If this measurement yields the value mℏ, the state vector of the other particle suddenly becomes a cylindrical wave for which the same component of the angular momentum is -mℏ. This statement is again synonymous with the statement that a measurement of the said component of the angular momentum of the second particle certainly would give the value -mℏ. This can be inferred again from the conservation law of the angular momentum (which is zero for the two particles together) or by means of a formal analysis.

We can ask why John Bell, and almost every other physicist, philosopher of science, or science writer, has never explicitly considered the conservation of angular momentum as explaining the perfect correlations between "entangled" particles, and the perfectly opposite electron spins found in Bohm's version of the EPR experiment. It seems likely that many were using the conservation law implicitly, for example, Einstein.

In 2005, C.S.Unnikrishnan of the Tata Institute of Fundamental Research in Mumbai, India was an exception. He proposed 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 would violate conservation of angular momentum.

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 Ψ12 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

Ψ12 = 1/√2) | 1+2- > + 1/√2) | 1-2+ >         (1)

The probability amplitude wave function Ψ12 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 t0 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 Ψ12 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 t0 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 t0, at that moment of new information creation, particle 2 will be found in a spin-down state with probability unity (certainty). And the results of observer B's measurement at t0 or any later time t1 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.

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
| + - > or | - + >.

The two particles cannot already have those spin values before the measurements. That would require them to have spin values in all three x, y, and z directions, which is impossible. 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 superluminal influence or interaction by one particle on the other. It actually requires that there be no actions on either particle, to preserve the symmetry needed for the conservation law. 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 x1 is found less than the expected value by an amount -δ, that x2 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?

We can take the x-axis to be vertical along 0° to 180°, the y-axis horizontal along 270° to 90°, and the z-axis to be into the diagram along the line from observer A through the origin to observer B. Observer A (red line) breaks the rotational symmetry by choosing to measure along angle 0° to 180°, finding her spin in the plane x-z, shown as the line from 0°to 180°. If observer B measures along the same pre-agreed angle 0° to 180°, then both measurements are in the plane x-z, shown superposed (purple line) from 0° to 180°. The observers have perfectly correlated spins.

If both observers measure along a different pre-agreed angle 45° to 225°, then rotational symmetry is again broken, replaced by a new planar symmetry at the 45° angle, but again with perfectly correlated measurements. We see that electron spins cannot exist at 0° and 45° at the same time, but the rotational symmetry of the total spin zero state can be broken and replaced by planar symmetry in any direction.

If observer A measures at 0° to 180° (red line) and observer B measures at a different angle 45°, shown as the blue line, their perfect correlations are lost, reduced by the factor cos45° (wave amplitude), and by cos245° (intensities).

John Bell's theorem maintains that his inequalities for local hidden variables should be a straight-line function of the angle between observer A and observer B, shown as the facets of the square LHV polytope inscribed in the circle of quantum physics predictions. At angle 45°, if Bell's predicted inequality is less than or equal to 1, the distance to the quantum prediction is √2 ≈ 1.414. All the Bell experiments to date have found the the Bell inequalities have been violated by approximately 41%, an amazing confirmation of the accuracy of quantum mechanics.

Summary of the "hidden" constant hypothesis

Standard quantum mechanics plus the principle of angular momentum conservation (true for quantum and classical mechanics) predicts:

  • A measurement of either particle will project both particles into a random product of spin states that did not pre-exist the measurement
  • Outcomes conserve the total spin zero, without influences traveling between the particles
  • Conservation is a joint or shared property of the two particles, not a pre-existing property of the individual particles
  • This conservation of total spin zero has been true at all times from the initial entanglement to the measurements
  • The predicted outcomes (random and perfect spin correlations) have been experimentally confirmed by thousands of Bell test experiments, as long as the two particles are measured at the (pre-agreed upon) same angle.
  • When the measurement angles differ by angle θ, the correlations fall off as cosθ (wave amplitude), and by cos2θ (intensities), as known from measurements of light polarizers crossed at different angles (the "law of Malus")
  • One could not hope for better agreement between hypothesis and experiment
  • This hypothesis is based on deep (conservation) principles and is not what Einstein called a constructed theory
  • It explains the perfectly correlated random bit strings needed for quantum cryptography
  • Do you see a flaw in it? Please write bobdoyle@informationphilosopher.com
  • Of course the constant is "hidden in plain sight" of the conservation law!

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 proven "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 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

Ψ12 = 1/√2) | 1+2- > + 1/√2) | 1-2+ >         (1)
have 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 about the other. he said. But they can no longer interfere with one another and remain correlated in future measurements. They have truly separated, but had not separated earlier, as Einstein hoped with his Trennungsprinzip.

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