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The Nature and Significance
of the Bell Theorems


According to the Copenhagen Interpretation of quantum theory, subatomic particles such as electrons generally do not have a physical presence but instead exist only as probability distributions over the set of their allowable states until they are subjected to measurement. The debate over this proposition went on for decades until the Irish physicist, John Stewart Bell, published an article which proposed a method for the empirical testing of the Copenhagen Interpretation. That article was published in 1964. Bell subsequently published a modified version in 1976. Bell died in 1990 at the tragically early age of 62, but other physicists have carried on his line of analysis. The empirical tests using photons as the particles used in the testing seem to vindicate the Copenhagen Interpretation. The nature of photons as particles in the same sense as electrons is in question.

John Stewart Bell

This webpage presents an argument that the Copenhagen Interpretation is a misinterpretation of the nature of the wave function in the Schroedinger's equation for a physical system and hence a misinterpretation of the nature of quantum reality. However the Copenhagen Interpretation is valid for the dynamic appearances of physical systems.

When Erwin Schrödinger revolutionized quantum theory with the brilliant notion that quantization was a matter of discrete eigenvalues of differential equations rather than integral numbers per se he left the wave function variable of his equations unspecified. Max Born of the University of Götingen speculated that the wave function is such that its squared value is equal to a probability density function. When he communicated this speculation to Niels Bohr in Copenhagen the reply was that they had never considered it to be anything else. Erwin Schrödinger himself disagreed with this interpretation, as also did Albert Einstein. Nevertheless Niels Bohr and the Copenhagen Interpretation prevailed.

Niels Bohr

Niels Bohr had a predilection to emphasize the illogic of quantum theory. For example, he said,

If quantum mechanics hasn't profoundly shocked you,
you haven't understood it yet.

Or at another time,

Your theory is crazy, but it's not crazy enough to be true.

It is generally accepted that the squared value of the wave function solution to the Schroedinger equation for a system is a probability density function. The problem is the nature of this probability density function. The Copenhagen Interpretation takes this probability density function to be disembodied probabilities that shows the probability that the system will be in any one of the various allowable states when subjected to measurement. Thus subatomic particles such as electrons, protons and neutrons do not have a physical existence until they subjected to measurement. There are perplexing problems with what happens to the mass and charge when a particle exists only as a probability density function. The noted physicist Murray Gell-Mann remarked that

Gell Mann
Bohr brainwashed a whole generation of physicists into believing the problem had been solved.

There is evidence for and evidence against the Copenhagen Interpretation. There was a recent convention of quantum theorists. They were queried as to which interpretation of quantum mechanics they favored. Only 42 percent said the Copenhagen Interpretation. This was more than any other interpretation so the Copenhagen Interpretation can be considered the dominant philosophy, but still it meant than 58 percent of the respondents thought the evidence against the Copenhagen Interpretation was more significant than the evidence for it. Yet the evidence for the Copenhagen Interpretation is presented by its true believers as settling the issue definitively.

Here it is argued that the probability density function from Schroedinger's time-independent equation when averaged over time is just the proportion of the time-spent in the allowable states given the dynamics of the system. It is thus the probability of finding the system in any of its allowable states at a randomly chosen time. There are no metaphysical perplexities involved with this probability density function. The time-averaged effect of a particle executing a periodic path is the same as if its generic charge (gravitational mass or electric charge) is spread over its path in proportion to its time-spent probability density distribution; i.e., to the proportion of the time it spends at various locations on its path See Static and Dynamic for the details and proof.

The Static versus Dynamic
Appearances of a System

First consider a fan standing still and rapidly rotating.

The still picture is the static appearance of the fan and the blurred disk of the rapidly rotating fan is its dynamic appearance.

The Copenhagen Interpretation (CI) asserts that a system is not in any of its allowable states or alternatively that it is in all of its allowable states simultaneously. Furthermore, according to the CI, a particle does not have a trajectory involving a definite location and velocity as a function of time. Supposedly this is because of the Uncertainty Principle.

Notice how well the dynamic appearance of the fan fits the Copenhagen Interpretation. The fan seems to be smeared over the disk and nowhere and everywhere at once. If one can only observe the dynamic appearance it seems to be unchanged over time. Therefore there is no trajectory for the blurred disk.

The analogy of the blurred disk of the fan may be extended a bit further. Suppose a heavy object like a wrench is plunged into the blurry disk of the rotating fan. The result: CLANG! The fan which had no perceived existence before the intervention is miraculously solidly there. The blurred disk of the rotating fan has, a la the Copenhagen Interpretation, collapsed into a spiked probability density distribution at a particular location.

Thus the Copenhagen Interpretation treats the dynamic appearance of the rotating fan as though it is the static appearance of some object.

The Time-Spent Probability Distribution

The time-spent probability distribution for a particle executing periodic motion is simply 1/(T|v|) where |v| is the absolute value of the velocity of the particle and T is the time period of the motion. In many cases the time-spent probability density distributions for the location and velocity of a particle satisfy the Uncertainty Principle. In the case of the rapidly rotating fan they do not because it is presumed that the fan can be rotating at constant (zero uncertainty) velocity. However in general velocity has a time-spent probability distribution given by

dt = dv/|dv/dt| = dv/|a|

where |a| is the absolute value of acceleration. The time-spent probability density distribution for velocity Pv(x) is then

Pv(x) = 1/(T|a|)

where T is the time period of periodic motion of the particle.

The analysis on this topic is given in Uncertainty. A limited version is given below for the simple case of a harmonic oscillator.


The only reality for a physical system that can be observed is a time averaged one. With electrons revolving about a nucleus at a million billion times per second time averages are all that can be discerned. Rotations at the nuclear level occurring at the much faster rates of billion billions of times per second See Nuclear Rotations.

Quantum Motion

If the probability density function from the time independent Schroedinger equation corresponds to a time-spent probability distribution the Schroedinger probability density can be converted into a an absolute value of the velocity of a particle at the quantum level. What emerges then is a picture of quantum motion as a sequence of slow-slow-fast-slow-slow…. This is in contrast to the Copenhagen picture of a particle resting in an allowable state and then randomly jumping instantaneously to another allowed state. The particle does not rest motionless in an allowed state; an allowed state is just vicinity through which the particle travels relatively slowly. There are not instantaneous jumps but instead vicinities through which the particle travels relatively quickly. When Bohr insisted on the notion of quantum jumps, Schroedinger replied,

If we have to go on with these damned quantum jumps, then I'm sorry that I ever got involved.

Schroedinger later formulated the concept of Zitterbewegung (trembling motion) in the movement of particles at the quantum level.

The Bell Theorem

The debate between the advocates of the Copenhagen Interpretation and its opponents went on through the 1930's, the 1940's, the 1950's and the 1960's without much resolution. In 1964 the Irish physicist, John Stewart Bell, working at CERN in Switzerland publish an article entitled, "On the Einstein Podolsky Rosen paradox", containing what subsequently became known as the Bell Theorem. In 1976 he published a . revised version of his analysis so there are effectively two Bell theorems.These theorems derive limits on the correlation of the readings of two instruments set at an angle to two beams of particles derived from separating paired particles. Bell's analysis presumed that the particles with spin pairing maintained their existence and their spins and he derived an inequality that had to be satisfied. Thus if empirical results did not satisfy the Bell Inequality then the particles did not maintain their physical presence. Thus Bell's Theorem is supposedly of the form If A then B and empirically B is not true hence empirically A is not true. In actuality it is of the form If A, C, D and E then B where A, C, D and E are the assumptions, explicit and implicit, that go into the derivation of B. Then when B is found not to be empirically true it is only A that said to proven not to be true.

For example, in the derivation of the Bell Inequality it is assumed that charged particles travel in straight lines as opposed to having trajectories with transverse oscillations.

Such transverse oscillations occur for charged particles traveling in groups because their repusion from each other. A charged particle is repelled from the nearest charrged until it approaches another charged particle. Such transverse oscillation can drastically affect the angle at which the charged particle impinges upon the measuring device. See Testing if the Bell Theorem.

There is something tenuous about proving something by disproving its supposed opposite. For example, according to the Copenhagen Interpretation a particle does not have a physical presence until observation causes its probability density function to collapse to a spike. But if observation caused a particle to come into physical existence along with its electromagnetic field then there would be a perturbation in the surrounding electromagnetic field; i.e., a photon. Such is not observed so particles cannot exist as nonphysical probability density distributions.

The Metaphysics of
the Bell Theorems

The way the 1964 Bell Theorem is presented is in terms of the concepts of realism and locality. Realism means that particles maintain their physical existence whether or not they are being measured or observed. Locality means there can be no faster than light communication between particles. If the spins of two paired particles are observed at separated locations and the spin of one particle at one location is up and the spin of the other particle is simultaneously observed to be down then this is a violation of locality. Either the spins of the particles were permanent characteristics of the particles, what is called a hidden variable, or the determination of the spin of one particle is instantaneously communicated to the other particle at a separated location. Bell's 1964 theorem assumes realism and locality so if Bell's Inequality is not satisfied then physical reality must be either nonrealistic or nonlocal.

After thinking over the concept of locality for a while quantum physicists decided that it really means local relativistic causality. Realism has come to mean counterfactual definiteness; i.e., measurements or observations which could be made but were not made are just as much a part of the real world as those that were made.

In his 1976 theorem eliminated hidden variables as an option. If his inequality is not satisfied the quantum world of Schroedinger's equation is non-realistic.

Somewhere along the way Bell realized that hidden variables would not be needed if the whole world from the beginning to the end of time was predetermined. The contrary to this superdeterminism is called in the literature freedom.

Bell stated the superdeterminism argument in a 1985 BBC Radio interview:

There is a way to escape the inference of superluminal speeds and spooky action at a distance. But it involves absolute determinism in the universe, the complete absence of free will. Suppose the world is super-deterministic, with not just inanimate nature running on behind-the-scenes clockwork, but with our behavior, including our belief that we are free to choose to do one experiment rather than another, absolutely predetermined, including the ‘decision’ by the experimenter to carry out one set of measurements rather than another, the difficulty disappears. There is no need for a faster-than-light signal to tell particle A what measurement has been carried out on particle B, because the universe, including particle A, already ‘knows’ what that measurement, and its outcome, will be.[5]

An aspect of this superdeterminism is that every particle in the Universe is being observed all of the time through its interaction with every other particle in the Universe. The Moon is always being observed by its gravitational interaction partner, the Earth. This is the Newtonian Universe.

But there is nothinng metaphysical about observation and measurement. Measurement simply stops the motion of the particle. Its dynamic appearance is replaced by its static appearance.

The Nature of the Quantum World of
the Solutions to Schroediner's Equation

Leaving aside the metaphysical imponderables of Bell's theorems, from the previous material it is found that the Bell Theorem concerns measurement and therefore it deals with the dynamic appearance of particles in oscillatory motion and therefore it is dealing with the blurred images of particles in motion and not with the particles themselves. Therefore it is not a surprise that physical systems would appear to only exist as probability distributions. Of course the blurred disks of rapidly rotating fans exist only as probability density distributions and that has no bearing on the unobserved material existence of the fans themselves.

When the physical testing of Bell's theorem appeared to show that separated particles maintained some ties in terms of their spins this was called an entanglement by Erwin Schroedinger. . The alternative to the notion of entanglement is that when particle pairs are created the particles get symmetric characteristics which they maintain throughout the rest of their existence.

Entanglements or Resonances?

A distinctive feature of resonance is that it is not dependent on the magnitude of the stimulation; it only depends on the matching of the frequency of the stimulation with the resonant frequency of the system being stimulated. When two particles are spin paired their oscillations are matched in frequency. If they are separated they remain oscillating in exactly matched frequencies no matter how far they are separated. They continue to be linked through their fields.


Quantum theory in the form of solutions to Schroedinger's equations pertains to the time-spent probability density distributions of physical systems and thus to their dynamic appearance and not to their static nature. The Copenhagen Interpretation is valid for such dynamic appearances but has no relevance concerning the physical realism of the particles making up physical systems. Bell's theorem and its testing may vindicate the Copenhagen Interpretation for the dynamic appearance of quantum systems but that has no relevance to continuous physical reality of particles. The fact that a rapidly rotating fan appears as a blurred disk in which there is no hint of the solid fan that creates the blurred disk has no bearing on the continued existence of that fan. Such a blurred disk does not change over time and thus has no trajectory, just as the Copenhagen Interpretation maintains. .

What is called the entanglement of two particles may be no more than the resonance of two particles oscillating with exactly the same frequency. Such resonance would be maintained at separated distances because of the exact matching of oscillatory frequencies. .

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