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Bell’s theorem, first presented by John Bell in 1964, has been used for many years to prove that no classical theory can ever match verified quantum mechanical predictions for entangled particles. By relaxing the definition of entangled slightly, we have found a mathematical solution for two entangled photons that produces the familiar quantum mechanical counting statistics without requiring a non-local theory such as quantum mechanics. This solution neither is claimed to be unique nor represents an accurate model of photonic interactions. However, it is an existence proof that there are local models of photonic emission that can reproduce quantum statistics.

One of the strongest constraints on a local theory of light is Bell’s theorem [

The logic of Bell’s theorem proceeds as follows: any local theory of light says that for a given E&M pulse from a photonic decay, one can write the probability of detection as a function of polarizer angle

If we assume from rotational symmetry of the source that one of the hidden parameters is an angle

By averaging over

We can now average over the other potential hidden variables

Quantum mechanics says that the result must be Equation (6).

Bell’s theorem says that there is no function

this conclusion is that there is no non-negative real function which can be correlated with itself to give

This principle applies here because

The easy way to prove that there can be no solution

Equation (8) shows the familiar result that when

Now experimental data require that the answer must be Equation (9).

Since the Fourier components of

Equation (10) does not tell us what the phase is for the second Fourier components, only the magnitude, so if we give the phase a name

The minimum of Equation (11) is −0.414, which violates the principle that a probability function must be non-negative. This is why it is said that no classical theory can even come close to the quantum mechanical result—providing a strong argument in favor of quantum mechanics.

Bell’s theorem is a mathematic result which follows from its assumptions. To construct a successful local theory, we need to loosen one of the assumptions which led to the contradiction. One subtle but critical assumption is that the set of hidden variables

In this simplest implementation of this altered model of entangled photons, the pair of photons have different probability functions versus polarization angle,

With this formulation, the next step is to choose two different real, non-negative functions over 0 - 2π which integrate to unity and whose cross-correlation probability distribution is

The only solution we have found where each function is finitely differentiable in the nonzero quadrants is given in Equations (13) and plotted in

Bell’s theorem has long been considered a fundamental barrier to classical theories of light interaction with matter. C.B. Parker [

The original test of Bell’s theorem used entangled electrons, but soon after that entangled photons were the experimental choice. John Clauser and Stuart Freedman [

“We have measured the linear polarization correlation of the photons emitted in an atomic cascade of calcium. It has been shown by a generalization of Bell’s inequality that the existence of local hidden variables imposes restrictions on this correlation in conflict with the predictions of quantum mechanics. Our data, in agreement with quantum mechanics, violate these restrictions to high statistical accuracy, thus providing strong evidence against local hidden-variable theories.”

Quantum mechanics requires the collapse of a probability wave as one of the photons passes through its polarizer, which then determines—faster than the speed of light—whether the other photon would pass through its polarizer. This connection has been probed in many forms over the last half century since Bell’s seminal 1964 paper [

A 2014 paper called Bell Nonlocality by Bruner, N. et al. [

The key advantage of a local theory of light is that this mysterious process that communicates the collapse of a quantum state faster than the speed of light is no longer required. The two photons agree with each other because they are in deterministic interlocked states. What we have found here is that by allowing the entangled photons to have independent but fixed probabilities of detection with the angle of their polarizers, there is indeed a local model of interaction that matches the predictions of quantum mechanics.

The probability functions that result are strange looking to us and not explained or predicted by current theories of light, and yet the experimentally verified zero correlation of counts when the two polarizations are set 90˚ apart forces the two functions to have 90˚ sections with zero probability of detection. If there is a local theory of light interaction, it will have to have probability curves similar to those shown here.

One might conclude that such behavior of detection probability with angle would violate experimental data, but no one has ever measured the probability distribution of a single entangled photon. No one has even proposed a technique to do that. The reason why it is so difficult is that the orientation of a random pair of photons is random and thus always has an equal probability of being detected at any angle of a polarizer. We can relate the probability of one photon detection correlating with the other, but so far we have no way to measure one photon’s probability of detection versus angle by itself.

This paper has shown there are probability functions for individual entangled photons that will match verified quantum mechanical predictions. While many questions come up here, such as what theory of light would predict such probability of detection functions, we suggest that Bell’s theorem as a mathematical result should no longer be considered a definitive proof of non-local interaction.