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The Invalidity of the Tests
of the Bell Theorem for
Charged Particles

A key element of the Copenhagen Interpretation (CI) of Quantum Theory is that particles generally do not have a material existence, but exist only as probability distribution unless they are subject to observation. This unintuitive notion leads to other even more unintuitive notions of the physical world,

There is no contradiction between there being
a probability distribution for a particle and it
maintaining its physical existence

Bell's Theorem

The absence of material existence for particles was initially simply an interpretation of quantum theory; i.e., the Copenhagen Interpretation. Bell's Theorem offered the possibility of empirical verification of that interpretation. The testing using photons seemed to confirm it, but photons are quite different in their nature from electrons, protons and neutrons. Photons do not have a material presence. We know from the studies of solitons and solitary waves that a wave may have particleness as well as being a wave.

Bell derived an inequality based upon a set of assumptions which included the material existence of particles. When that inequality was found not to be satisfied that was taken to mean that the assumption of the material existence of particles wa s found to invalid. But that interpretation of the results is not valid. There are other assumptions of the derivation that may be responsible. Some of the assumptions are explicit but there are others that are implicit.

It must be noted that most of the tests of Bell's Inequality involve photon pairs, but photons do not have a material reality. It is not valid to apply the results concerning nonmaterial particles to particles which do have a material existence. However there have been a few tests of Bell's Inequality us ing pairs of charged psrticles that have found violations. It is contended below that the assumption that charged particles travel in simple straight lines may be at fault. The alternative is that charged particles in the midst of other charged particles are subject to transverse oscillations.

The crucial variable in a Bell's Inequality test is the difference in the setting angles for the devices for measuring the characteristics of paired particles. The angle of the measuring device affects the angle at which the particle impinges upon the measuring device. It is the angle of impingement that is the true crucial variable. The existence of transverse oscillations in the path of the particle can drastically affect its angle of impingement. The existence of transverse oscillations in the paths of charged particles introduces a random element into the angle of impingement that may make it impossible to find violations of Bell's Inequality.

The Source of Transverse Oscillations

Consider three electrons; one located at +a, another at −a and one in the middle at x which is much smaller than a. The net force on the middle electron is

F = ke²/(a + x)² − ke²/(a − x)²
which can be expressed as
F = ke²[1/(a+x) + 1/(a −x)][1/(a+x) − 1/(a −x)]
F = ke²[2a/(a²−x²)][2x//(a²−x²)]
F = ke²ax/(a²−x²)²

Since x<<a

F ≅ ke²ax/a4 = (ke²/a³)x

Thus the electron is esssentially a harmonic oscillator. Its frquency is

ω = ((ke²/(ma³)½

The transverse deviation would then be

x(t) = x(0)cos(ωt)

The angle at which the electron impinges upon the measuring device is also approximately a trigonometric function of time of the same frequency. The difference in the angles of impingment of the two electrons of the pair is also trigonometric function of time.

Thus the experimental results that satisfy Bell's Inequality would look something like the following.

The red shaded area represents the region in which experimental results may lie and still satisfy the Bell Inequality. The width of the interval of uncertainty depends upon the frequency of the transverse oscillations.


So the experimental results concerning the testing of Bell's Theorem must be carefully examined to see if there is an alternate interpretation from the conventional one. If factors important in the physical world has been left out of the derivation then it would not be surprising that a physical experiment does not give results based on a derivation that leaves out those factors.

The predictions of the assumption that the characteristics of a pair of particles are determined at the time of the formation of the pair would have been confirmed innumerable times. These confirmations of this theory were dismissed and its place taken by the implausible apparatus of the Copenhagen Interpretation and the Bell Inequality that involve the communication between paired particles taking place at speeds faster than that of light, even at infinite rates. If such an alternative is considered acceptable then there should not be much difficulty in coming up with an alternate explanation for any experimental violations of Bell's Inequality.

The dilemma is that there are two apparent discrepancies between theories and empirical measurements. One is based upon Bell's Theorem. It is as shown below.

The classical case shown corresponds to material particles having a continual material presence with the characteristic of a pair of particles fixed at the time of the formation of the pair. The quantum case shown is for a pair of particles not to have a material presence until a characteristic of one of the particles is measured. There is a remarkably small difference. On the other hand there is the very great difference between the implied speed of communication between the two paired particles and the evidence supporting the Special Theory of Relativity. The maximum speed of communication classically is the speed of light. And there is communication between the paired particles through their gravitational and electromagnetic fields and this constitutes something in the nature of measurement. The speed of communication implied by the Copenhagen Interpretation has no upper bound.

Which seems more likely to resolve the discrepancy between theory and measurement? On the one hand there could be an expansion of the Bell Theorem analysis to take into account such factors as the transverse oscillation in the path of charged particles or the gradients in the ambient electrical and magnetic field in the vicinity of the measuring apparatus.

On the other hand there could be the search for some mechanism that would account for communication between paired particles at speeds greater than the speed of light and perhaps at infinite rates. There seems to have been no attempt to pursue this later goal. That perhaps is a manifestation of the exasperating true-believer-hood of those who believe in the Copenhagen Interpretation.

Instead what has been of concern among physicists dealing with experimental violation of the Bell Inequality is the existence of loopholes in the interpretation of experimental results. One loophole, called the freedom of choice loophole, has to do with the misalignment of the measuring instruments with the line of travel of the particles. Another loophole is called the superdeterministic Universe loophole. This is the notion that everything in the Universe has been predetermined and the is no freedom in the outcome of experiments.

If the instruments are aligned then the correlation between the readings for the particles and their partners is 100 percent. If the instruments are anti-aligned the correlation is also 100 percent. If the instruments are aligned perpendicular to the line of travel of the particles the correlation has a zero expected value. It is only when the instruments are misaligned that the correlation can give a result at variance with the classical model.


In a recent conference of quantum physicists a survey was done on what interpretation of quantum theory the physicists favored. Only 42 percent of the respondents said the Copenhagen Interpretation. This was more than any other interpretation so the Copenhagen Interpretation is considered the dominant interpretation in physics. Nevertheless the survey indicated that 58 percent found the evidence against the Copenhagen Interpretation more compelling than the evidence for it.


Heisenberg's Uncertainty Principle

The notion of a particle not having a continuous material existence stemmed from the Uncertainty Principle; i.e.,

σx·σp ≥ ½h

where σx is the uncertainty of the particle's location and σp is the uncertainty of the particle's momentum (as determined by the particle's velocity.) h is Planck's constant divided by 2π.

The belief was that the scale of an atom was so limited that the uncertainty of an electron's location had to be small and therefore the uncertainty of its speed so large that an electron could not be said to have a definite trajectory of a material object. This was Heisenberg's contribution to quantum theory.

To counter this notion consider the time-spent distributions for a harmonic oscillator. It is shown that the time-spent probability distributions for a harmonic oscillator satisfy the Uncertainty Principle.

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