The principle of perspective

On Bell's Theorem

Computer Program simulating Bell-test Experiments

Text and program design: Gerard van der Ham
Website design and programming: Olivier Winsemius

Abstract

This site encloses a computer program from which the results simulate exactly the outcomes of EPR- Bohm experiments. The program is based on the description of real vector spaces. The vector spaces have been found by applying a logical principle (the Principle of Perspective). This principle is based on the recognition that one observer can observe an object from only one direction. In the experiments a pair of particles with opposite properties, which can be considered to be one object, is being detected by two detectors. In that case the results of the detections cannot be compared unless they have been made in agreement with each other. To make the results agree with each other it is enough to take into account all movements (rotations) of the detectors in respect of the coordinate system that describes the particles. In that way the real vector spaces can be found. The Principle of Perspective is the deeper reality behind the Quantum Mechanic (QM) concept of ‘entanglement’.

The principle shows exactly how the probabilities (and therefore the correlation) depend on the angle between the settings of the detectors (ⱷ). The vector spaces are defined by the positions and settings of the detectors in respect of the line of motion of the particles that are detected by the detectors. The vector spaces contain the pairs of opposite vectors that yield QM probabilities and correlations. The fact that these real vector spaces exist, shows that ‘instantaneous interaction at a distance’ is not needed to explain the results of QM and the results of the experiments.

Goal of the article

The goal of this article is: to show that the conclusions of Einstein, Podolsky and Rosen, concerning the properties of elementary particles, were correct. The conclusions were that elementary particles must have definite properties also when they are not being measured and that there cannot be immediate interaction between entangled particles when they are far apart. The goal is achieved by describing a model of EPR-Bohm experiments with which a classical physical explanation for QM correlation in these experiments can be given. A classical physical explanation means that no instantaneous interaction at a distance is needed to explain the correlations of entangled particles in the experiments. To ‘prove’ this explanation a computer program has been designed that exactly simulates the outcomes of the experiments. The program is based on the identification of real vector spaces and the assignation of combinations of outcomes to vector pairs in these vector spaces.

Introduction

The program at this website addresses Bell’s theorem. As is well known, Bell’s theorem states that it is not possible to explain correlations in Bell experiments by a local ‘hidden variable’ theory (correlations given by Quantum Mechanics).In other words: Bell [1] stated that, to reach QM correlations, a quantum mechanic effect must exist that produces these correlations. This effect is named entanglement. It is generally believed that this effect cannot be explained classically. Einstein made strong objections against this meaning of entanglement because it means that some kind of instantaneous interaction at a distance exists, which is not possible according to Einstein. Nowadays it is commonly accepted that experiments show that Einstein was wrong. But was he really?

Bell calculated inequalities for EPR Bohm experiments, also known as Bell-test experiments or Bell experiments. Bohm thought of these experiments in connection with the discussion between Einstein and Bohr about the question whether elementary particles have definite properties or not. This discussion culminated in the EPR article of 1935 in which Einstein, Podolsky and Rosen clearly showed that entangled particles must have definite properties. David Bohm thought of a way to prove this experimentally, using the spin of pairs of entangled particles. Then Bell calculated upper limits for the correlation between outcomes in these experiments. Bell’s limits where surpassed by the correlations calculated from QM. Bell stated that QM’s correlations cannot be explained with a local hidden variable theory (read: classically). If a local hidden variable (λ) is considered to be a mathematical factor, then it is rather obvious that this not possible. But what if the local hidden variable is not a factor but a prescription that depends on the measuring system?

When the experiments finally could be performed the results showed QM’s correlation. It was thought that Einstein was wrong and that there must be some unknown quantum mechanic effect, an instantaneous interaction at a distance. Is this conclusion correct?

Reasoning

The reasoning in this article is as follows:

1. Assume opposite spin of entangled particles is represented by opposite vectors (arrows).
2. These spin directions of a pair are being detected by making a picture of the arrows: Alice takes a picture of one arrow from a certain viewpoint and Bob takes a picture of the other arrow from an opposite viewpoint.
3. Comparing the results of the detection afterwards means that Bob goes to Alice, bringing his picture with him.
4. Laying the pictures side by side the arrows don’t point in opposite directions anymore.
5. Describing this way the directions of the arrows in the pictures in a coordinate system and calculated by a computer program, sets of pictures of many pairs produce the exact QM correlations.

The model

The model used in this article is partly borrowed from Bell’s own description in ‘Bertlmann’s Socks and the Nature of Reality’ [1]. My description of the model is as follows: In a source pairs of particles are being produced. Because of conservation laws the particles of a pair have opposite properties. The properties that are important here are the propagation direction and the spin direction: the particles move in opposite directions along a line of motion and they have opposite spin directions along a common axis at the moment of their creation. The spin directions are totally unrelated to the direction of propagation. The spin direction of a particle points in a random direction in space but the spin directions of a pair of particles point in opposite directions of course. It is supposed that spin of a particle can be represented by a vector. (This vector is different from the vector that represents the propagation of the particle. The propagation doesn’t play a role in this model).

It is generally believed that spin of an elementary particle is a quantum mechanic phenomenon that better not can be tried to understand. In this model spin of a particle is just the spinning of the particle around an axis. The vector that represents the spin is an axial vector. Its direction can be found by applying the ‘right hand rule’: put your right hand around the particle, fingers pointing in the spinning direction (you have to let the particle grow a little bit or shrink your hand a little bit) and your thump gives the direction of the vector. You might notice that the particle still can spin ‘right way around’ or ‘left way around’, depending on the direction from which you observe it and yet the vector has unambiguous one direction.

So particles of a pair have opposite spin vectors. It is also supposed that these spin vectors keep their direction in space unchanged until they are being measured by a detector. Each particle of a pair is being measured by a detector. So there are two detectors, each at one side of the source, placed on the line of motion of the particles, both facing towards the source and so ‘looking’ in opposite directions.

Furthermore it is supposed that a pair of particles can be considered to be one object. It is also supposed that the particles stay in the source. Many pairs are being detected in one run of an experiment and it is also supposed that all these pairs are being created at the same time in the source. So if we now forget about all the particles in the source then what is left is a sphere full of pairwise opposite vectors (representing the spin directions of the particle pairs), starting in the centre of the source and pointing in random directions. The sphere is the total vector space that lodges all the vector pairs that are being measured in the run of an experiment.

To be able to calculate on this system of vector pairs and detectors, the whole system is placed in a coordinate system. The coordinate system can be chosen at will. The origin of the coordinate system is placed in the centre of the source. The y-axis is along the line of motion. The line of motion is now only needed to place the detectors on: the particles are gone and the vectors don’t move. The vectors are being measured by the detectors just by looking at them. The detectors in this model represent Stern Gerlach devices. This means that they not only have to face the source, where the vectors are, but also have a ‘setting’. The setting of a detector is the direction in respect of which the vectors are measured. The setting corresponds with the field direction of a Stern Gerlach device. In this model detector A is chosen to be placed at the negative y-axis, facing towards the source at the origin of the coordinate system, and its setting is a fixed setting in the positive vertical direction (parallel to the z-axis). Detector B is then placed at the positive y-axis, facing towards the source, and its setting can be randomly chosen from a direction in the plane that is perpendicular on the line of motion at the position of B. The relative angle between the settings of the detectors is ⱷ.

The measurement of a vector is as follows: the detector looks at the vector and when the vector has a component in the same direction as the setting of the detector, the detector gives as outcome +1. If the vector has a component in a direction opposite to the setting of the detector then the outcome is - 1. The other vector of that pair is then measured by the other detector, following the same procedure. The only restriction is that a detector can see only one vector of a pair.

Remember that a vector represents a spin direction of a particle. The measurement process really is irrelevant. It is about the outcomes and the correlation between numbers of combinations of equal or opposite outcomes and the angle ⱷ between the settings of the detectors. That correlation has to be explained (and therefore the numbers of certain combinations at a certain ⱷ have to be explained).

A random vector is established by a Random Number Generator producing randomly the coordinates of the vector. This vector is one half of a pair of opposite vectors. The other vector of the pair is made by changing the signs of the coordinates of the first produced vector. So now the combinations of outcomes of each vector pair can be calculated in respect of the detector settings, if one knows how to do it. That will be explained hereafter.

The Bell Game Challenge

In ‘Bertlmann’s Socks and the Nature of Reality’ [1], Bell designed a thought experiment, called: The Long Box, in order to explain why the correlations between the outcomes in Bell experiments cannot surpass the limit that he calculated. He argued that the outcomes in The Long Box were being achieved in the same way as in the real experiments and that The Long Box had nothing to do with quantum mechanics or entanglement whatsoever. So if the correlations in the real experiments surpassed the limit, it could not be explained classically according to Bell. One can wonder if The Long Box correctly represents a real experiment if it has nothing to do with entanglement. In my opinion entanglement only means that entangled particles have opposite properties until the moment of detection.

In order to prove Bell’s theorem to be wrong, S. Gull suggested it to be enough to design a computer program, based on a local hidden variable theory, that exactly simulates a Bell experiment and produces the exact results of the experiment. Such a computer program would refute Bell’s theorem and would indicate that a mechanism must exist that could explain the results of the experiments classically. R.D. Gill [2] [3] transformed Gull’s idea into a Challenge: The Bell Game Challenge. The Bell Game Challenge is a challenge to design a computer program that simulates the results of a real experiment. In the experiment the detectors A and B are in four different randomly chosen combinations of settings. So one run of the experiment is in fact a random mixture of four experiments at a fixed combination of settings. The challenge is to calculate outcomes for all these four combinations of settings without knowing the combination of settings. We shall see why this is not possible. However, it is possible to calculate outcomes for each combination of settings. To calculate the outcomes the combination of settings must be known. The Challenge forbids that.

It is true that one particle doesn’t ‘know’ the setting of the other particle’s detector but each particle ‘knows’ the setting of its own detector, so as a pair the particles ‘know’ the combination of settings in a real experiment. In forbidding to use the combination of settings the Challenge goes wrong in respect of real experiments.

The Bell Game Challenge is based on The Long Box. Gill stated that such a program is not possible because it seemed to be statistically proven that it is not possible. However, analyses of the above described model of a Bell experiment based on entangled particles (particles solely having opposite properties without interaction at a distance) shows that it is possible to design a program that produces the exact results of a real experiment. It only doesn’t meet the conditions of Gull and Gill. The Bell Game Challenge is not to win. But that still doesn’t mean that the results of the experiments cannot be explained according to classical physics. The program shows that there is a way.

What does the program do?

The program falsifies Bell’s statement that says that QM correlations cannot be explained classically, meaning: in agreement with local-reality.

The computer program produces the coordinates of random vectors using a Random Number Generator. A random vector represents the spin direction of a particle. As particles of an entangled pair have opposite spin directions, we need pairs of opposite vectors to represent the spin directions of a pair of entangled particles. When a random vector has been produced, the opposite vector is given by changing the signs of the coordinates of the randomly produced vector. Where in a real experiment the spin direction of a particle is not known and we can measure only one component of its direction, in a computer program the direction of a vector is exactly known and we can precisely calculate the outcome of the concerning component of that vector in a correct simulation of the detection. This is what the program has to do.

The computer program produces the coordinates of random vectors using a Random Number Generator. A random vector represents the spin direction of a particle. As particles of an entangled pair have opposite spin directions, we need pairs of opposite vectors to represent the spin directions of a pair of entangled particles. When a random vector has been produced, the opposite vector is given by changing the signs of the coordinates of the randomly produced vector.

Where in a real experiment the spin direction of a particle is not known and we can measure only one component of its direction, in a computer program the direction of a vector is exactly known and we can precisely calculate the outcome of the concerning component of that vector in a correct simulation of the detection. This is what the program has to do.

But the program can’t do this, not for two detectors anyway. The program cannot do this under the conditions of Gull and Gill. The problem is in the combination of ‘the exact calculation of the outcome’ and ‘a correct simulation of the detection’.

To represent a vector and calculate the outcome of its detection we need a coordinate system. A coordinate system defines positions, distances and directions in space. It makes space absolute as it were. Consequently, when vectors are defined in respect of a certain coordinate system, the vectors define the absolute space, not the detectors. We can choose a coordinate system at will and preferably in a convenient way.

In our model of a Bell experiment pairs of entangled particles are being produced in a source. The particles move in opposite directions along the line of motion (horizontally). (We can forget about the movement of the particles but the line of motion is important to place the detectors on). We can choose the source as the origin of the coordinate system, the line of motion as the y-axis and the vertical as z-axis and the x-axis then perpendicular on the y- and the z-axes. We can now define the settings of detector A: setting 1 is along the z-axis (positive vertical) and setting 2 is along the x-axis (positive horizontal). For detector B setting 1 is +45° in respect of the z-axis and setting 2 is - 45° in respect of the z-axis.

However, a detector doesn’t detect according to a coordinate system that is chosen by us. It detects from its own reference frame. The reference frame of a detector is defined by the direction in which it detects the particles (this is the line of motion of the particles moving towards the detector in our model before we forget about them) together with its setting (this is the field direction in case of Stern Gerlach devices). Note that in this case the reference frame of detector A in setting 1 corresponds to the coordinate system that we intentionally chose.

A reference frame is a coordinate system that moves along with the observer or detector. To an observer ‘in front of’ is always ‘in front of’, in whatever direction he looks. Space is relative to an observer or a detector whereas a coordinate system makes space absolute. When detector A is in setting 1 and the coordinate system is chosen corresponding this setting we can calculate the outcomes of the detection of a randomly generated vector pair for detector B in setting 1 as well as in setting 2 and even for detector A in setting 2. But when detector A changes from setting 1 to setting 2, it is no longer possible to calculate the outcomes of that vector for detector B unless the coordinate system moves along with detector A. But this is not possible because then we have two coordinate systems and that cannot be. A coordinate system defines space, it makes space absolute, it is fixed in space, it can’t be changed and there cannot be two of them. The description of a generated vector by coordinates would be meaningless in respect of two coordinate systems.

However, the outcomes of the detection of a vector for detector B can be calculated when detector A is in setting 2 if the coordinate system is chosen corresponding to detector A in setting 2 right from the start. But in that case the outcomes of the detection of the vector for detector B cannot be calculated in respect of detector A in setting 1. Re-calculation (transformation) to the original coordinate system is useless because then it is just calculated from the reference frame of A in setting 1 which was not meant to be. What goes for the settings of A also goes for the settings of B of course.

So we see that in Bell experiments, as described in a Bell Game Challenge way, there are combinations of outcomes that cannot be calculated and there are combinations of outcomes that cannot be detected.

That there are combinations of outcomes that cannot be detected is obvious because a vector can only be detected by a detector in one of the two possible settings.

That there are combinations of outcomes that cannot be calculated for both settings of A (or B) doesn’t mean that the outcomes for the settings of B depend on a setting of A. It only means that there are combinations of outcomes that are not calculable. That is the reason why a computer program that has to meet the conditions of the Bell Game Challenge cannot be designed.

Suppose particle 1 of a random pair is measured by detector A and particle 2 is measured by detector B. At the moment of the (simultaneous) measurement the experimental system ‘knows’ in what settings the detectors are. Particle 1 knows in which setting A is and particle 2 knows in which setting B is. And the detectors know for themselves in which setting they are. If they don’t know it is not possible to make a correct listing of the outcomes or counts of the detections. Of course particle 1 and detector A don’t know in which setting B is at that moment and also particle 2 and detector B don’t know in which setting A is but they don’t need to know. They only need for themselves to know in what setting they are to give an outcome and to make it possible to compare the outcomes afterwards. So at the moment of detection the relative angle between settings is known in real experiments, at least for the detecting system. So it is not reasonable to demand designing a computer program in which those relative settings are not allowed to be used.

It is the spin direction (vector) of a particle relative to the setting of the detector that gives a certain outcome and as spin directions of a pair of particles are linked, a pair gives a combination of outcomes that depends on the relative settings of the detectors (the angle ⱷ). We can also see this in the formulas for correlation, Bell’s formula ((2ⱷ - π) / π) as well as QM’s formula ( -cos ⱷ), both solely depending on ⱷ.

At the moment of the detection of a pair of particles Alice knows the setting of her detector and Bob knows the setting of his detector. At that moment they don’t know the setting of the other detector. However, at the moment the results are being compared, they know both settings and then the relative difference between the settings can be deduced. So there is no reason whatsoever why this difference (ⱷ) should not be allowed to be used in comparing the combination of results. In fact ⱷ is needed to establish the correlation between ⱷ and the combinations of equal or opposite outcomes. We can calculate outcomes for all settings but only if we know the relative difference of the settings, so only if both settings are known. In this way a computer program can be designed that produces the correct outcomes. It seems like the outcomes of one detector depend on the setting of the other but they really do not. It is only the combination of outcomes that depends on the combination of settings.

To design a computer program that correctly simulates an experiment and produces correct outcomes for an experiment, the combination of settings have to be known. One cannot require to calculate outcomes for random settings of detectors. One can for all detectors in every setting calculate results but one has to realize that changing to another coordinate system makes these calculations meaningless. Changing to another coordinate system happens when the detector that corresponds with the original coordinate system (the reference detector), is being changed to another setting. (Remember that as soon as the vectors are defined in respect of a certain coordinate system, it is the vectors that define absolute space, not the detectors).That is why, in a Bell Game Challenge situation, one cannot in advance calculate results for all arbitrary settings of detectors. Some of them are meaningless and wrong. To be able to calculate results, the combination of settings of the detectors must be known as must the detectors be in a combination of definite settings in order to be able to measure outcomes.

Maximum randomness in computer programs can be reached by choosing a coordinate system that corresponds to the reference frame of one detector in a certain setting and then calculate the results for arbitrary settings of the other detector. In this way correct correlations can still be obtained. The randomness required in The Long Box surpasses the maximum randomness, making the Challenge unwinnable. Conditions for randomness in computer programs have to be restricted this way in order to obtain correct correlations. Therefor the conditions in the Bell Game Challenge are too strong. When the condition for arbitrary settings of both detectors is abolished, it is very well possible to design a computer program that simulates Bell experiments and produces correct outcomes.

The Bell Game Challenge describes an experiment in which four combinations of settings are being combined. Two of the combinations are identical: the settings differ 45° in both cases and in the other two combinations the settings differ - 45° and 135°. In the Challenge it is expected to be possible to calculate outcomes for all settings of the detectors irrespective of the setting of the other detector. As explained this is not possible. It is possible to calculate the outcomes for each combination of settings provided that this combination is known and allowed to be used in the calculation. Since this is not allowed the Challenge can not be won. It is of course possible to design a computer program describing such a combined experiment (a combination of four experiments) but it would be meaningless. It would boil down to a combination of four single programs.

The computer program

The question is: can vector pairs be identified from which the detection gives equal spin outcomes in such a way that their numbers result in QM’s correlation? The answer is: yes, that is possible. The vector pairs turn out to be in certain vector spaces that can be identified. The computer program describes randomly generated vector pairs in a coordinate system. It produces outcomes that yield QM correlations, showing that a mechanism exists behind the correlations between the outcomes and the combination of settings in experiments. The computer program calculates which of the generated vector pairs satisfies the prescription to give equal outcomes when being measured by detectors in certain relative settings. The computer program doesn’t meet the conditions of Gull and Gill but it definitely shows that there is a classical explanation for the correlations found in experiments.

At this site there are several programs. Program 1 gives Bell’s correlation as a result at ⱷ = 45°. Bell’s correlation can be considered to be a description of vector pairs observed from one direction. To obtain QM’s correlations a description of the vector pairs from two directions is needed.

Application of a new logical principle leads to Program 2. This program gives QM’s correlation at ⱷ = 45°, using exactly the same vector pairs as in Program 1 when the seed is used and the vectors are copied. This means that a relation exists between Bell’s inequalities and QM’s probabilities in Bell experiments as I explained in my paper: “On the Relation Between Bell’s Inequalities and QM’s Correlations in EPR-Bell Experiments”[4]. In that article I made a mistake: I thought that QM’s probability is a projection density distribution per unit of area. It is not. Well, it is a projection density change per unit of area. QM’s probability for a combination of equal (or opposite) outcomes is a certain vector space, depending on ⱷ, as a part of the total vector sphere. And, as it is a vector sphere, this is equivalent with: a certain part of the surface of the sphere, depending on ⱷ, as part of the total surface of the sphere. The probability is the ratio of that certain vector space to the total vector space. Since the vector pairs are equally distributed in the sphere, the probability also is the ratio of the numbers of pairs in these spaces to the total number of pairs. Note that in this program the numbers of (+) and ( - ) outcomes for both detectors are about 50% of the total number of measurements.

Program 3 shows QM’s correlation in a diagram at angles ⱷ that are multiples of 15°.

The Principle of Perspective

In Program 2 QM’s correlations are obtained. To get QM’s correlations, a newly discovered principle has to be applied. This is the Principle of Perspective. This principle leads to the real vector spaces and to the requirement that opposite vector pairs have to meet, to compose these vector spaces. This requirement depends on ⱷ and that is no problem, as we have seen, because ⱷ is known by the system (not by Alice and Bob) at the moment of detection of a pair. The relative angle ⱷ becomes known to Alice and Bob only at the moment of comparison of the results. The requirement cannot be represented by a mathematical factor λ. It is a prescription.

That there are combinations of outcomes that cannot be calculated is one of the consequences of the Principle of Perspective. Perspective is merely a direction: a direction of observation, a direction of detection or a direction of projection. The Principle of Perspective says: “The observation or detection of one object from different directions is equal to the observation or detection of different objects from one direction”. (The inverse generally is not true). In other words: an object cannot be looked at from two directions by one observer. An observer always looks at an object from one direction.

The Principle of Perspective indicates the difference between an observation (or detection) of an object from one direction and observations (or detections) of that object from two directions by two observers (or two detectors).

This means that the results from the detection of the spin directions of an entangled pair of particles (which can be considered to be one object) from opposite directions, cannot be compared just like that. The detections have to be made in agreement to each other. This is very easy to do: all movements (rotations) of the detectors in respect of the object have to be taken into account, right from the start.

In the way an object can only be looked at from one direction by one observer, also an interaction (or detection) can only be described from one coordinate system. However, it has to be described from two reference frames since there are two detectors needed for detection. So also in case of detections of vector pairs an adaptation is needed to allow for the comparison of the results. To obtain the correct results we have to choose a reference detector in respect of the chosen coordinate system to be able to calculate the results.

As we have seen a reference frame of a detector is defined by the direction from which the particles approach it (line of motion of the particles) and by its setting (its field direction in case of Stern Gerlach devices). Both elements, defining the reference frame of a detector, are subject to the Principle of Perspective. In this case the Principle of Perspective has to be applied because two detectors are needed for the detection of entangled pairs of particles. If the Principle is not applied both elements can be the cause of failure. In case of Bell’s inequalities the detecting directions of the detectors (opposite along the line of motion) are being mixed up resulting in wrong correlations. In case of the Bell Game Challenge the field directions of the reference frames of the detectors are being mixed up resulting in wrong conditions for the Challenge. Both Bell’s inequalities and the Bell Game Challenge do not describe Bell experiments completely and neither does Bell’s Long Box. To describe Bell experiments completely the Principle of Perspective is needed and that is because in Bell experiments two detectors work together to detect one entangled pair of particles (one object).

In Bell experiments the correlation is found by counting the number of combinations of equal spin result. This number divided by the total number of detected combinations is the probability for equal spin result. The correlation is found by subtracting the probability for opposite spin result from the probability for equal spin result. The cases of equal spin result are caused by pairs of entangled particles that have their (opposite) spin directions in certain vector spaces. These vector spaces have been found by trial and error and by applying the Principle of Perspective. The vector spaces are defined by the settings of the detectors and only one prescription for the vectors. The spaces can be drawn on a ball and they can be used in a computer program to calculate the outcome of the measurement of a vector. This computer program shows that the conditions are known for a certain vector(pair) to result in a combination of equal spin outcomes given the relative settings of the detectors and it shows that it is possible, just by generating opposite vector pairs and calculating the outcomes of the combinations of their detections, to obtain the correct QM correlations.

Identification of the vector spaces

The vector pairs, from which the detection results in combinations of equal outcomes, have to meet the requirement that the angle (a) between the vector and the plane that constitutes the reference frame of the detector that measures the vector, is smaller than ⱷ, ⱷ being the angle between the settings of the detectors. The vector pairs that meet this condition compose the wanted vector spaces. These vector spaces differ from the ones described by Bell’s inequalities. The vector spaces that meet Bell’s inequalities are obtained by describing (projections of) vectors from one direction, whereas QM describes the vectors also from one direction but taking into account all rotations. This requirement shows the existence of a simple condition that vector pairs have to meet, to yield QM correlations. This condition is the mechanism in Bell experiments that causes QM’s correlations. The computer program is the result of this condition which could only be found by applying the Principle of Perspective.

When detector A is chosen to be the reference detector the angle a between its reference frame (the y,z-plane, y being the line of motion and z being the setting of A) and the vector to be measured is given by: a = inverse sin(x² / √(x² + y² + z²)). In this equation x, y and z are the coordinates of the randomly generated vector. The vectors, described by angle a in respect of that plane, represent circles on the surface of the sphere that represents the total vector space. The circles are parallel to the y,z-plane. They start at x = 0 with radius 1, going to x = 1 ending in a point at the x-axis. The limit radius in respect of the x-axis of the wanted circle is defined by ⱷ because the angle a between the vector and the y,z-plane must be bigger than (90 - ⱷ) to give a combination of equal outcomes. Vectors that start in the origin of the coordinate system and run to a point on that circle, mark a cone in the sphere. The top of the cone is in the origin of the coordinate system. This cone is the vector space that contains the vector pairs which give combinations of equal outcomes. Because of the Principle of Perspective both vectors of a pair are in that cone, when described from one viewpoint. So when a random generated vector turns out to be in the cone the combination of outcomes will be equal.

The computer program calculates for a random generated vector if it is in the cone and if it is, then it assigns a combination of equal outcomes to the pair. The outcomes are both +1 or -1, depending on the outcome of the randomly generated vector which always is measured by detector A in this model. The opposite vector is always measured by detector B. If the randomly generated vector is not in the cone, then a combination of opposite outcomes is assigned to the pair. Calculated in this way the computer program produces the exact QM correlations.

Those who think that opposite vector pairs don’t fit in a cone and think that opposite vector pairs only fit in an hourglass shaped vector space, forget to apply the Principle of Perspective. When a pair of opposite vectors is detected from opposite directions, they perfectly fit in a cone by the application of the Principle of Perspective (by taking into account all rotations). To describe a pair of opposite vectors from one perspective, the perspective of A, being the reference detector, detector B has to move over to the position of A. Applying the Principle means that the vector that is detected by B, has to move along with B. Then the opposite vectors perfectly fit in a cone shaped vector space.

Described from one position the cone represents two identical halves of an hourglass at the same place. Described from opposite points of view (the real positions of the detectors) these two halves make a complete hourglass with the cones situated point-symmetrical in respect of each other. The hourglass is formed by the settings of the detectors as follows. When the detectors A and B, each in its own setting, are at the same position and ‘looking’ in the same direction, B has to rotate 180° around the setting of A to ‘look’ in the opposite direction. Suppose the position of A and B is in the centre of the vector space (the origin of the coordinate system). By the rotation of B, the line that represents B’s setting cuts out two half cones in the vector space, which are situated point- symmetrical in respect of each other. B can equally rotate 180° in the opposite direction, cutting out two more half cones, which together with the first half cones, form the complete hourglass shaped vector space. In this way the size of the hourglass shaped vector space is defined by ⱷ, the angle between the settings of the detectors.

This procedure is visualized in the next paper at this site.

The fact that the hourglass is determined by ⱷ is consistent with the fact that the correlation only depends on ⱷ (C = -cos ⱷ) because the hourglass determines the probabilities and thus the correlation. Pairs of opposite vectors do fit in this hourglass shaped vector space.

The size of the hourglass shaped vector space determines the probability for combinations of equal outcomes at a certain angle ⱷ and that size depends on ⱷ. Together with the probability for equal outcomes, the probability for opposite outcomes as well as the correlation at that angle ⱷ are determined.

Because of inversed rotation that has to be applied to the vector space (according to the Principle of Perspective), the real position of the vector space is having the line of motion as its axis. This will be shown in the next paper at this site.

Rotating B reversely around A’s setting again, and taking one cone with its content along in that rotation, the cones fuse to the one cone that is used to describe, from one viewpoint, the vector space that contains the vectors that yield combinations of equal outcomes.

Assumptions and their justification

Many assumptions have been made in the description of the model. Most important is the assumption that spin of an elementary particle can be represented by an axial vector that keeps its direction in space unchanged until the particle is being measured. From a classical point of view the assumption is not very strange but that doesn’t mean that it is correct. The fact that entangled particles give opposite outcomes when they are being measured in the same direction and equal outcomes when they are being measured in opposite directions, is a strong indication that the assumption might be correct. But correlations at other angles than 0° or 180° are not so easy to explain at first sight. Although correlations at other angles are difficult to explain, it is clear that there must be a physical mechanism behind them. If the outcomes of both detectors were really purely random then there would be zero correlation. That is clearly not the case. Each repetition of an experiment, performed at the same relative angle of settings, gives about the same non-zero correlation. That is why there must be a physical mechanism behind them.

Some advocate ‘Superdeterminism’ as explanation for the correlations in the experiments. Although Superdeterminsm is true for everything: the Universe is as it is, including the outcomes in Bell experiments, it isn’t really an explanation. Superdeterminism is not a mechanism in itself. It is not a mechanism that accounts for a certain correlation at a certain angle of settings, time and time again.

Essentially a Bell experiment is a probability measurement. The result of such an experiment is a probability estimation. The more experiments are performed, the closer the probability is approached. It is like casting a die. If you want to know the probability for a ‘five’ as result, you can cast a die many times, count the ‘fives’ and calculate the probability estimation. If you do this many times you will find the probability. The physical mechanism behind this is of course that a die has six sides, one of them has a ‘five’ on it and always one side is upside after a cast on a horizontal surface. In Bell experiments one part of this mechanism is that pairs of entangled particles have opposite spin that keeps its direction in space until measurement (or other interactions) so they can give certain combinations of outcomes when they are being measured. But opposite spin cannot be all there is to it. Then Bell’s inequalities could apply and they do not.

The other part of the physical mechanism are the different vector spaces that contain the vector pairs that give the different combinations of outcomes in respect of the detector settings. The vector spaces in which the pairs are that give a combination of equal outcomes, are defined by the requirement that the angle between the pairs and the reference frame of the reference detector, is smaller than the relative angle between the settings of the detectors at the moment of measurement.

The fact that these vector spaces have been exposed is another strong indication that the assumptions were correct.

Goal

In this way the QM correlations have been explained logically and geometrically. The elementary particles of an entangled pair have opposite properties, as EPR told us, and no instantaneous interaction at a distance is needed.

The sheer fact that it is possible to find a prescription that points out exactly the pairs that yield combinations of equal or opposite outcomes in the exact ratio that is found in experiments (QM correlation), shows that Bell’s theorem is false. According to Bell such a prescription is not possible.

Well, Bell’s theorem is not exactly false. Indeed there is no Local Hidden Variable. There is a bi-local relation between opposite spin directions of a pair and the relative settings of the detectors. This relation is exactly described by the prescription for the vector pairs. This prescription is the variable. It is not a mathematical factor (λ). And nothing is hidden: the opposite spin directions are ‘in plain sight’, as EPR showed us. There is no instantaneous action at a distance.

Conclusion

In the end this whole story can be told much easier. We know from the formulas that the correlation in Bell experiments depends on ⱷ only ( C Bell = (2ⱷ - π)/π and C QM = -cos ⱷ ). The angle ⱷ is the angle between the settings of the detectors that measure pairs of particles (opposite vector pairs in the model). To know the relative angle between the settings we need to know both settings. This is against the conditions of the Bell Game Challenge and the reason why the Challenge not can be won. But it is not in conflict with the situation in real experiments.

The vector spaces in which the vectors yield combinations of equal outcomes, are composed of vectors that make an angle ‘a’ with the reference frame of the reference detector and that angle ‘a’ is smaller than ⱷ, the angle between the settings of the detectors.

In the end the results must be described from one perspective. QM describes the real particles with their real spin. Calculating in complex numbers it probably handles the angles correctly to reach the correct results. Experiments measure the real particles with their real spin from two perspectives giving the correct result. Starting from two perspectives and applying the Principle of Perspective the correct results are also obtained when described from one perspective.

Bell described the wrong vector spaces because he didn’t apply the Principle of Perspective, he didn’t take into account the movements of the detectors in respect of the movements of the particles. Therefore he described too big vector spaces resulting in too high probabilities for combinations of equal outcomes and too small correlations. Therefore his inequalities had to be violated.

How to use the program

There are 3 programs. In the upper right corner one can choose to go to the next program. Program 1 gives Bell’s correlations as outcome at ⱷ = 45°, program 2 gives QM’s correlations as outcome at ⱷ = 45° and program 3 gives QM’s correlations for angles ⱷ that are a multiple of 15°, represented in a diagram.

By clicking at “generate” the program generates 1000 random vectors. The number “N = 1000” can be adjusted at will. Then the program calculates an outcome for the detection of each vector by detector A and assigns an outcome for detector B.

By using the ‘seed’ the same set of generated vectors is used in the different programs.

Outcomes are represented in tables. The coordinates of the vector that has been calculated to produce this outcome are shown. In this way it can be checked that different programs use originally the same vector in a certain ‘trial’ if the seed is used.

• A trial is the measurement/calculation plus result of one pair of vectors
• A run is the set of calculated outcomes of a set of N generated vector pairs

At the end of each ‘run’ the number of combinations of equal results (G) and the number of opposite results (T) are counted and represented. The number of combinations of equal results divided by N is represented as probability (P). From the probabilities the correlation (C) is calculated and represented. (C = G/N - T/N).

References:

To my big surprise I found an article by someone who reached the exact same conclusions as I did:
Georgina Woodward; Reality in the Context of Relativity and Quantum Physics
https://vixra.org/abs/2308.0093

As additional support I would like to mention an article by Mark Syrkin:
Quantum Fundamentals and Heuristics.
https://vixra.org/abs/2410.0187