San José State University
Thayer Watkins
Silicon Valley,
Tornado Alley
& the Gateway
to the Rockies

The Testing of the
Bell Theorem as
Support for the

In 1926 Ernest Schrödinger published four articles establishing Wave Mechanics. Shortly later he published two articles showing that Wave Mechanics is exactly equivalent to Werner Heisenberg's Matrix Mechanics. Wave Mechanics soon became the foundation for Quantum Physics.

The only problem was that Schrödinger's included an undefined variable subsequently called the wave function. Niels Bohr of Copenhagen and his German associates chose to interpret the wave function as such that its squared value is the probability density function for the particle. This became known as the Copenhagen Interpretation. The Copenhagen School however went well beyond this. They asserted and maintained that because a particle had a probability density function it could not have a material existence; i.e., unless subjected to observation a particle exists only as a probability distribution. This was entirely unjustified. Any particle in motion has a probability density distribution based upon the proportion of time it spends in the various intervals of its trajectory. This time-spent probability distribution reflects the dynamic appearance of the particle in motion. This is what the solutions to Schrödinger's equations represent.

In 1927 Heisenberg formulated the concept of the Uncertainty Principle which said that the location of a particle and its momentum cannot simultaneously be known to arbitrary degrees of precision. He used this justify the Copenhagen Interpretation's position that a particle generally does not have material existence but exists only as a probability distribution. This too is unjustified. The time-spent probability distributions for a particle can satisfy the Uncertainty Principle even though they represent particles with a continued physical existence.

That leaves only the Bell Theorem and its empirical testing as supposedly providing any support for the Copenhagen Interpretation. John Stuart Bell, an Irish physicist, derived an inequality that the correlation between two measuring instruments aligned at different angles to the stream of particles was supposed to satisfy. Bell's derivation assumed the particles had a continuous material existence.

When experiments found Bell's Inequality did not hold this was taken as proof that particles generally do not have continuous material existence. What a profound result?! Here the value of an obscure correlation reveals an amazing aspect of the nature of physical reality.

However consider how difficult it would be to prove the truth of a proposition by showing the nonconfirmation of a contrary proposition. Suppose Proposition A can be derived from Assumptions B&C. If A is not found to be true by empirical testing it means that B and C cannot both be true. B or C or both might be false, but one cannot say that the nonconfirmation of A definitely establishes the truth of not-B or not-C. But this is what has happened with the testing of Bell's Theorem. Proposition A is Bell's Inequality and Assumption B is the material existence of particles. When it was found that Bell's Inequality was not satisfied in experiments it was taken to be proof that particles do not have a continuous material existence. Consider how weak this sort of experimental confirmation is.

Suppose a simplistic model implies some definite result and one of the assumptions is that space is flat as opposed to being curved. Simplistic means there are some strong assumptions involved that make it easy to derive the definite result. If the definite result is found not to be true empirically the proposition that space is curved does not necessarily follow. Yet that is the way the testing of the Bell Inequality has been handled. And the only remaining empirical support for the Copenhagen Interpretation was the testing of Bell's Inequality.

Most of the tests of Bell's Inequality involved photons and photons in fact do not have a material existence. It is only the tests involving charged particles that have the profound implication that particles with mass have no material existence until they are subject to observation and their material characteristics measured.

The apparatus for the testing of Bell's Inequality is of the nature of the following.

But this presumes among other things that charged particles travel in strictly straight lines. Charged particles in a group are subject to repulsion away from their nearest neighbor, but that may take them closer to some other neighbor and reverse their direction of travel. The net result is that the charged particles may have paths that involve transverse oscillations.

The crucial variable involved in the testing of Bell's Inequality is the difference in the angles of the detectors with respect to the general path of the particles.

The heavy line is the limit of Bell's Inequality and the light line is the supposedly quantum theoretic value. The difference is so small that experimental error may make it impossible to distinguish between the two.

The angle settings of the detectors are important because they determine the angles at which the particles impinge upon the detectors.

Transverse oscillations can drastically affect the angles of impingement of the particles on the detectors and thus their difference, as shown below.

Thus the nonsatisfaction of Bell's Inequality may be solely due to the transverse oscillations in the charged particles paths. The implication of the nonsatisfaction of Bell's Inequality may just be that charged particles do not move in strictly straight lines. Thus there is no basis for the Copenhagen Interpretation that even massive, charged particles have no material existence until they are subjected to observation and measurement.

The Correspondence Principle

The Correspondence Principle wa articulated by Niels Bohr in the 1920's. He observed that the validity of classical analysis is well documented. Therefore for quantum analysis to be valid it must be such that when extended by increased energy its results must approach those of classical analysis.

Since the results of quantum analysis based upon Schrödiger's equation are probability density functions the classical results of classical analysis they must approach are the time-spent probability density functions for the physical systems under analysis.

This is aptly illustrated by the case of the harmonic oscillator.

The light line is for the quantum anaysis and the heavy line for the classical analysis. As can be seen spatial averages of the quantum figures are very close to the classical figures.

It is shown elsewhere that generally the solution to Schrödinger's equation for a particle in a potential field asymptotically approaches a probability density which is the product of the time-spent probability density distribution and a purely oscillatory function. Thus when the purely oscillatory function is eliminated by spatial averaging the result is the classical time-spent probability density function.


There was no basis for the original Copenhagen Interpretation. Material particles iin motion have perfectly legitimate probability density functions in the form of time-spent probability functions based upon the proportion of the time they spend in the various intervals of their trajectories. These time-spent probability density functions can satisfy the Uncertainty Principle even though they are for particles with a continuous material existence. Also these time-spent probability density functions are compatible with the Correspondence Principle of quantum physics.

Supposedly the testing of Bell's Theorem provided support for the notion that even charged, massive particles do not have a material existence util they are subjected to observation and measurement. But the derivation of Bell's Inequality is based upon a number of assumptions besides the materiality of particles. The experimental nonsatisfaction of Bell's Inequality only means that one or more of the assumptions upon which its derivation is based are physically invalid but not necessarily the materiality of particles.

Thus the Copenhagen Interpretation of quantum physics has no empirical foundation. It is in fact flat wrong and has been based upon unjustified assertions.

(To be continued.)

HOME PAGE OF applet-magic
HOME PAGE OF Thayer Watkins,