Visualization of Bell Experiments
The Mechanism Behind
Author: Gerard van der Ham
Graphics: David Spier
Abstract
This paper shows in a naturel way the particle pairs that give combinations of equal spin outcomes and the particle pairs that give combinations of opposite spin outcomes, resulting in Quantum Mechanical correlations in experiments.
The particles of a pair move in opposite directions and they have opposite spin.
The method is consistent with local-realism because the outcome of a measurement on one particle is not affected by the outcome of a measurement on the second particle and vice versa, and, the outcome of a measurement on one particle is not affected by the setting of the detector used to perform the measurement on the second, no matter how long the choice of setting is delayed.
Apologies for any repeats in the text.
What is the problem?
The problem has to do with the violation of Bell’s inequalities. This violation has never been explained. It seems that the violation is being accepted without further questions. Bell’s inequalities are violated by Quantum Mechanic correlations which are demonstrated in experiments. It seems that the violation is attributed to a weird phenomenon of QM, called entanglement. But this is not really an explanation. A correct explanation of the correlations will also explain why Bell’s inequalities have to be violated.
It will turn out that Bell’s inequalities do not describe the experiments completely correct. The inequalities are all based on examples in which combinations of logical possibilities are being described. However, the experiments only are about directions. They are not about combinations of logical possibilities. Directions differ quite a lot from logical possibilities. To describe the experiments completely a thorough analysis of what happens with the directions in those experiments is needed.
In Bell experiments spin directions of particle pairs are measured. The particles of a pair have opposite spin directions, randomly oriented in space. The particles move in opposite directions. Only particles moving along a certain line (the line of motion) are measured. They are detected by detectors, placed perpendicular on the line of motion, at either side of the source that produces the particles. One of the particles of a pair is detected by detector A and the other by detector B. The detectors give results as +1 (spin up) or - 1 (spin down). The detectors have a measurement orientation (setting) that can be varied in a plane perpendicular on the line of motion of the particles. The particles of a pair are measured at a certain setting of A and a certain setting of B. The settings make an angle (ⱷ) in respect of each other.
In a run of an experiment a large number (N) of pairs is measured at a certain combination of settings (ⱷ is kept constant). When the lists of outcomes from A and B are being compared afterwards a correlation shows up between the number of combinations of equal (or opposite) outcomes and ⱷ, the angle between the settings of A and B (see diagram). The correlation in one experiment is represented by a point of the negative cosine (blue) in the diagram. That point is defined by ⱷ. This correlation is not physically understood. That is the problem.
Source: WikipediaThe solution of the problem
We shall solve the problem in a logical and geometrical, and therefore visible, way. The most simple description of an experiment is addressed. This is the measurement of opposite spin directions of many particle pairs by detectors A and B in fixed settings. This description is simplified even more and made visible. Assumptions are made and numbered in squared brackets: [..]. Movements (translations and rotations) are described and numbered in rounded brackets: (..).
Assumptions:
- [1] Spin of a particle is considered to be a classical rotation around a random axis. The direction of the spin axis is independent from the direction of movement of the particle.
- [2] Spin of a particle is represented by a vector. The direction of the vector is given by the ‘right hand rule’: put the fingers of your right hand around the particle in the rotation direction and then your thumb gives the direction of the vector.
- [3] The spin of a particle (vector) doesn’t change between the moment of production and the moment of measurement. It’s direction is fixed in space.
- [4] (Entangled) particles of a pair have their spin in opposite directions (but random in space). (Entanglement in this paper means: the particles have opposite properties but no interaction).
- [5] Measuring the spin of a particle is considered to be: making a picture of the vector that represents the spin of the particle. Measurement can also be considered to be a projection on the plane in which the setting of the detector can be varied.
- [6] The setting of a detector is considered to be a line with a direction. It could be the direction of an inhomogeneous magnetic field in a Stern Gerlach device. The setting can be varied in a plane perpendicular on the line of motion of the particles.
- [7] Since there is no interaction between the particles of a pair after the moment they have been produced in the source, and since the direction of the spin vectors is fixed, the distances between the detectors and the source make no difference for the outcomes of the measurements.
- [8] The only information that is travelling from the source to the detectors is information transported by the particles. It is information about their real properties. There are no other ‘hidden variables’.
- [9] The outcome of a measurement is defined by the comparison of the vector to the setting of the detector: if the component of the vector along the setting has the same direction as the setting of the detector then the outcome is +1 and if it is opposite to the direction of the setting then the outcome is - 1.
All assumptions are classical and none of them is impossible.
Simplification of the model
For the solution of a problem it might be helpful to simplify it. The already simple model can be simplified even more.
Since the spin direction (vector) of a particle is fixed in space, and the measurements are about spin direction, the distances from the source to the detectors make no difference. Also the time of travelling makes no difference. The experiments are not at all about time. They only are about directions.
In an experiment the position of the detectors is on the line of motion, at either side of the source, ‘looking’ in the direction of the source. It is important to figure out how they got there.
Because the distances between the detectors and the source make no difference for the outcome of the measurements, we can as well make them zero. This means that we translate (1) the detectors in opposite directions along the line of motion to the centre of the source. Now the particles don’t have to travel anymore. The pairs are produced in the centre of the source with their opposite spin directions (vectors) in random directions, and measured at the same time. The randomly distributed opposite vector pairs of the N pairs detected in one run of the experiment, make a vector space in the shape of a sphere around the centre of the source (S). The detectors, being in the middle of the sphere, now measure the vectors from behind: the vectors are ‘seen’, photographed or projected from inside the sphere out instead of from outside the sphere in. This makes no difference for the outcome of a measurement.
We can now show how the line of motion together with the choices of Alice and Bob, and therefore together with the movements of the detectors, define an hourglass shaped vector space from which the QM correlations can be calculated.
Description of the setting up of an experiment
The description is about the setting up of an experiment for the simplified model.
There is a source S which is a fixed point in space and a line of motion which is a fixed line in space through S. And there is a vector space which is a sphere around S, constituted by the opposite vector pairs, in random directions and randomly distributed, representing the spin directions of the N measured particle pairs.
fig. 1In the simplified model both detectors are on the line of motion at S. The settings of the detectors must be perpendicular onto the line of motion. To figure out how they got there we realize that a detector is defined by its setting (a line with one direction) since a spin direction is only compared to the setting of a detector to give an outcome. At this stage of the description we only have one line: the line of motion. So we have to put the settings equivalently (meaning in the same direction in respect of each other) in the position of the line of motion. Then we have to rotate the settings 90°, in a random direction, around S (2). From the line of motion lines can rotate 90° around S in infinitely many ways. The results of these rotated lines make a plane perpendicular onto the line of motion. This is the plane of the detectors. It is the plane in which the settings can vary and it is also the plane on which the vectors can be projected or photographed to compare their direction with the settings.
fig. 2Alice picks one of the lines in this plane as setting for her detector A. Bob picks another line in this plane for his setting of detector B. Note that Alice’s and Bob’s settings are totally independent of each other. But they are linked by the fact that they both have to be perpendicular onto the line of motion at S and end up in the same plane (in this simplified model). In this plane the two picked lines, the settings of the detectors, make an angle ⱷ with each other.
fig. 3Except for the settings the detectors are constructed the same way. Now the plane of the detectors divide the vector space in two equal halves because the centre of the sphere is S and the plane runs through S. Because the vectors of a pair of particles are opposite in respect of S, one of them is in one half of the sphere and the other is in the other half. The detectors can detect a vector from only one direction along the line of motion. Since A and B are constructed the same way, they both detect in the same direction, being able to detect only the vectors in one half of the sphere (the yellow half) and so only one vector of a pair (the red one in this figure).
fig. 4To detect both vectors of a pair one of the detectors must ‘turn around and look in the other direction along the line of motion’. This means that one of the detectors has to rotate 180° (3) in respect of the other. At this point we have to choose a reference detector. We choose A as reference detector. We could equally have chosen B as reference detector. This makes no difference as later will be explained. Detectors are defined by their setting since the setting is the only direction to which the vectors are compared to produce an outcome. Since detectors are defined by their setting, B’s setting has to rotate 180° around the setting of A. During this rotation B’s setting stays at an angle ⱷ in respect of A’s setting. In this way the movement of B’s rotation cuts out a vector space from the sphere, in the shape of two half cones which are situated point-symmetrical in respect of S and have the setting of A as their common axis.
fig. 5View along the line of motion
Note that during the rotation B keeps its setting at ⱷ° in respect of A’s setting. So according to B its setting has not been changed. According to A, however, B’s setting has changed from ⱷ before the rotation to - ⱷ after the rotation. So A and B don’t agree anymore about each other’s setting. This is one reason why the outcome of B’s measurement cannot be known, nor calculated, from the outcome of A’s measurement even though the spin of B’s particle is opposite to the spin of A’s particle and therefore defined.
The rotation can equally be performed in two ways. Performed in both directions, the movement of B’s setting cuts out a vector space in the shape of two opposite cones with a common axis. The cones make the shape of an hourglass, having the setting of A as its axis and its centre in S.
fig. 6View along the line of motion
It will turn out that the vector pairs in this ‘double cone’ yield combinations of equal outcomes. So the condition for pairs of particles to show combinations of equal spin, when they are measured by A and B, is that the angle between their common spin directions and the setting of A as reference detector, is smaller than ⱷ (the angle between the settings of A and B).
The vectors of a pair can be considered to be one object because wherever the particles of a pair are, their vectors (spin directions) are always directed oppositely. It must be recognized and realized that an object can be observed, by one observer, from only one direction. One can turn it around or walk around it but still it is always observed from one direction. Observing an object from another direction is like observing another object.
However, at this point of the description of a simplified experiment the vector pairs are detected from two directions by the detectors A and B. This means that one detector by no means can ‘know’ what the other detector measures. It cannot even be calculated because a coordinate system defined in respect of detector A is totally different from a coordinate system identically defined in respect of detector B.
The fact that one detector cannot know what the outcome of the other detector is, can be demonstrated. In this example the detectors are at either side of the sphere (movement (1), being the translation of the detectors to the centre of the sphere, has not yet taken place). Imagine a pair of particles that have opposite spin and move in opposite directions. Imagine the spin direction of each particle coincides with its direction of movement. (The probability for such a pair is almost zero because their opposite spin direction is one of infinite many directions in space). Then both detectors measure exactly the same. For example: Alice’s detector ‘sees’ a particle approaching spinning left way around and then Bob’s detector also ‘sees’ a particle approaching spinning left way around. Both detectors detect exactly the same, giving equal outcomes. Yet the particles move in opposite directions and have opposite spin. So the detectors do not agree on which particle is measured by which detector and they do not agree on each other’s particle’s spin direction.
We now return to the simplified model in which the detectors have been moved to the centre of the source S in order to explain how QM’s correlation comes about in the experiments.
Entanglement in the sense of instantaneous interaction at a distance is incomprehensible and therefore no explanation.
Superdeterminism too is not an explanation for correlations in Bell experiments. Superdeterminism is applicable to every event in the Universe but it is not an explanation for the physical outcome combinations of measurements and their correlation to the combinations of settings at which the outcomes are obtained. If it were then experiments would not be reproducible. There wouldn’t be any correlation at all. To explain these combinations of outcomes we need to find the mechanism behind the experiments (how the outcomes come about).
First we have to realize that a pair of entangled particles is in fact one object. It consists of two particles that have opposite properties. So when the properties of one particle are known, the properties of the other particle are defined. The two particles are being detected from opposite directions because the particles move in opposite directions and they are detected by two detectors.
Then we have to realize that one observer can observe one object from only one direction. This also goes for measurements by detectors. Since the pairs are being detected by two detectors, the results of the comparison of the outcomes cannot easily be predicted. One has to know the mechanism behind the measurements. The outcomes can only be interpreted correctly when measurements are performed equivalently, meaning that they are being described from one point of view. By studying the outcome of one measurement the outcome of the other measurement can by no means be known because detecting one object from two directions is equal to detect two different objects. Neither can the other outcome be calculated. Yet the two particles of an entangled pair make one object: they have opposite spin directions, which is the only property to be measured. When the spin of one particle is measured or calculated, then the spin of the other particle is defined.
Spin directions are represented by vectors. Vectors occupy a vector space. The randomly distributed opposite vector pairs of all the particle pairs that are being detected in one run of an experiment, make a sphere around S. If we can identify the vector space that contains the vector pairs which give combinations of equal outcomes, we know the outcome of the other vector if one vector of a pair is measured. If the measured vector is in that space, then the outcome of the other vector is the same. If not, then the outcome of the other vector is opposite. We only have to assign the outcome to the measurement of the other vector when we know the outcome of the first vector and the space it is in. There is no other way. This does not mean that the outcome of B depends on the outcome of A. The opposite vectors are being measured completely independent of each other. It is merely the only way to know both outcomes (except for measurement).
How to identify the vector spaces?
To be allowed to compare outcomes, the measurements must be performed equivalently. When the detectors are in the same position (1 viewpoint) their detections are equivalent. But then only one particle of a pair can be detected because the other particle moves in opposite direction. So B has to move (rotate) in respect of A to be able to detect the other particle. Now, if the vector that is to be measured by B moves along with B, then the detections stay being equivalent. The vector that B equivalently detects to be the opposite (the dotted green one from the viewpoint of A) is totally different from the vector that A expects to be the opposite vector (the green one). The vector that A expects to be the opposite vector, is observed by B differently. The opposite vector (the green one) is detected by B as the dotted green one would be detected by A (if that were possible). It is of course the same opposite vector but detected by B it appears to be a different vector from the viewpoint of A. Only described in this way the detections are equivalent and still very real.
fig. 7View along the line of motion
All opposite vector pairs that are measured in a run of an experiment, make a sphere around S. The settings of the detectors in this simplified model are lines, through S, with a direction. When B’s setting differs from A’s setting, they make an angle ⱷ. Now when B rotates 180° around A’s setting to reach a position opposite of A, its setting cuts out, from the vector sphere, two half cones which are point-symmetrical in respect of S and its axis being A’s setting (see fig. 5). It turns out that from the vectors in these half cones that are measured by A (the red in fig. 8), the opposite vectors as detected by B (the dotted green) appear in the half cones that are symmetries of the original half cones in respect of the intersecting plane (which is the plane of the detectors). These four half cones together make an hourglass-shaped vector space: the double cone (the double cone from fig.6).
fig. 8View along the line of motion
The two symmetric half cones are equivalent to the original half cones. The symmetric half cones would have been the original half cones if B’s setting had been rotated 180° around the setting of A in the opposite direction (see fig. 6). So the double cone, consisting of the two half cones plus their symmetries, is a special vector space. When a vector in the double cone is detected by A. the opposite vector also is in the double cone, of course. That vector detected by B, after having been moved along, also is in that double cone, but in the symmetric half cone.
Of course, in reality, a vector doesn’t move along with B. But seen from the viewpoint of A the dotted green vector is the real opposite vector (green) as detected by B. (The green vector that moves along with B becomes the real opposite vector (dotted green), in respect of A’s detected vector (red), from the viewpoint of B. It is detected by B in the same way that the real opposite vector would have been detected by A at - ⱷ if the movement directions of the particles of a pair would have been exchanged.)
It appears to be that the vectors of the pairs that have their opposite spin directions in the double cone, don’t leave the cones and that the vectors of the pairs that have their opposite spin directions outside the cones, don’t enter the cones by moving along with B’s rotation.
It also appears to be that the number of pairs that have their opposite spin directions in the double cone, perfectly correspond to the correlation given by QM, if that pairs show combinations of equal outcomes. Computer programs calculating which of the randomly generated vectors are in the double cone and assigning equal outcomes to the pairs to which the vectors belong, demonstrate this. The correspondence is not very strange because the double cone is a special vector space, constructed only using ⱷ. The double cone is constructed purely based on rotations of the detectors in respect of each other and in respect of the line of motion of the particles.
However, the described double cone is not the one that harbours the real vector pairs that give combinations of equal outcomes. We could equally have constructed a double cone by rotating the setting of A around the setting of B. That would make an identical double cone but in a different position. For a computer program it makes no difference which double cone is used to calculate the numbers. As long as the sizes and the shapes of the double cones are equivalently, they will give about the same numbers.
How to find the double cone that harbours the real vector pairs that give combinations of equal outcomes? It is mentioned that the double cones are constructed based on rotations of the detectors in respect of each other and in respect of the line of motion. Then one more rotation has been taken place: the 90° rotation to get the detectors perpendicular onto the line of motion (rotation (2), see fig. 2). When this rotation is inversely applied to the double cones then they all end up in one double cone, the one from which the axis is the line of motion, irrespective of the shape of the double cone (the width depend on ⱷ). This double cone contains the opposite vector pairs from the pairs of particles that yield combinations of equal outcomes when their spin is measured by oppositely placed detectors.
fig. 9 ATo see why the pairs that have their spin directions in this double cone yield combinations of equal outcomes, we have to see that this double cone, having the line of motion as axis, finds itself in the vector space that is constructed by rotating B 180°, moving its centre perpendicular plane along. Together with the symmetric space (in respect of A’s perpendicular plane), containing the opposite vectors as detected by B, this 3-dimensional vector space between the centre perpendicular planes of A and B looks like a deformed doughnut that is squeezed to the extent that the hole in the middle has just disappeared.
fig. 9 BThe intersection of this space by any plane containing A looks like:
fig. 9 Cfrom any direction perpendicular onto A. This space consists of an infinite number of double cones:
fig. 9 Dand only the one with its axis being the line of motion is the original double cone to which the inversed rotation (2) has been applied.
From the vector pairs that find themselves in the spaces between the centre perpendicular planes of A and B, we can see that they can yield combinations of equal outcomes, even if observed from one direction. The vector pairs outside of these spaces certainly yield combinations of opposite outcomes.
fig. 10 A
fig. 10 B
fig. 10 C
fig. 10 DMeasured from opposite directions, however, not all vector pairs in the vector spaces between the centre perpendicular planes give combinations of equal outcomes. Only those in the double cone do. They show QM’s correlations. Bell’s inequalities, on the other hand, describe vector spaces as were they defined by the centre perpendicular planes of the settings of A and B (fig. 11). The division of the total vector sphere by these planes indeed show Bell’s probabilities but not all pairs in the spaces between the perpendicular planes give combinations of equal outcomes. That is why Bell’s inequalities do not apply and are being violated.
fig. 11 ASphere segments (blue) between the perpendicular planes of A and B. They contain the number of vector pairs that yields Bell’s correlation.
When the halve cones, having the line of motion as axis, are being rotated 90° around A, they fit in those segments but do not fill them up. They contain the number of vector pairs that yields QM’s correlation.
fig. 11 BBlue space: vector pairs that give combinations of equal outcomes. Yellow space: vector pairs that give combinations of opposite outcomes. Green: blue space surrounded by yellow space.
The one double cone, that has the line of motion of the particles as its axis, is the real vector space from which the opposite vector pairs yield QM’s correlation. This double cone provides all symmetries that are needed for the correlations. For the combinations of outcomes, and therefore for the correlation, it makes no difference if:
- - the particles are being exchanged;
- - only their spin is being exchanged;
- - their direction of movement is being exchanged;
- - the position of the detectors is being exchanged;
- - their settings are being exchanged;
- - detector A detects particle 1 of a pair and B detects particle 2 or vice versa;
- - the experiment is being described from one viewpoint or from two viewpoints
Described from two viewpoints the double cones are in a point-symmetrical position in respect of each other and described from one viewpoint in which B had to return to A’s viewpoint, taking its cone with it, the cones coincide with each other at the position of A’s cone.
Assigning outcomes to a pair when one of the vectors of a pair is calculated, suggests imposing non- locality. But non-locality is not imposed. At the moment of measurement the settings of A and B are defined. The settings define the division of the sphere in the two vector spaces from which the vector pairs give combinations of equal or opposite outcomes. Now when a random vector is generated, the opposite vector is defined and the vector pair is defined. So when it is established in which vector space the generated vector is, together with its outcome, then the outcome of the opposite vector measured by the other detector, is defined. There is no non-locality in this whatsoever, except for the fact that detections take place at different places. The settings of the detectors, the outcome of the generated vector and the fact that the spin of a pair is opposite, is the only data we need to establish both outcomes. This is exactly the data we have at our disposal (at the moment of measurement).
The measurements are fully independent of each other. The seeming non-locality is caused by the fact that the detectors have to be placed oppositely in respect of each other (and the object) and because of that they cannot detect the object from one viewpoint (and cannot see it as one object).
The consequences of this have been described here. Comparing outcomes of measurements means describing the measurements from one viewpoint: the outcomes of A and B have to be brought together (one viewpoint) in order to compare them.
To be allowed to compare the outcomes of experiments, we have to describe the measurements equivalently. This means: as if from one viewpoint. For that purpose we analysed the movements of the detectors from the line of motion of the particles to their real positions and settings. The movements of the detectors defined the division of the sphere in two separate vector spaces. The inversed movements applied to the vector spaces, lead to the real positions of these spaces at the moment of measurement. They contain the real vector pairs that give combinations of equal or opposite outcomes, depending on the vector space. When equally distributed in space, the numbers of pairs in these vector spaces yield QM’s correlation as measured in experiments. The number of pairs in the space that gives equal outcomes is: sin²(ⱷ/2)N and the number of pairs in the space that gives opposite outcomes is: cos²(ⱷ/2)N (= (1 - sin²(ⱷ/2))N).
The angle ⱷ, the angle between the settings of the detectors, is not a choice. It is the consequence of the choices for the settings of A and B. Consequently the division of the sphere in the two different vector spaces is also the consequence of these choices. And thus the probabilities for combinations of equal and opposite outcomes are also a consequence of these choices and so is the correlation. The choices for the settings by Alice and Bob are made independently of each other. The moment at which the choices are made make no difference. They can be made at the final moment before the measurement.
In summary: the line of motion of the particles and Alice’s and Bob’s choices for the settings of their detectors define the outcomes of each pair, including the correlation. This can be demonstrated by a computer program.
Note that we didn’t use a coordinate system here. The division of the sphere in the different vector spaces has been achieved by solely analysing the movements of the detectors in respect of the movements of the particles.
This is the mechanism behind Bell experiments. It is the definition of vector spaces by the rotation of both detectors in respect of the line of motion of the particles and in respect of each other, and the definition of the real position of these vector spaces, at the moment of measurement, by the inversed movements of the detectors applied to the vector spaces.
Bell’s probability for combinations of equal outcomes correspond to the pairs that have their spin direction in the vector spaces between the centre perpendicular planes of A and B. Although a vector pair must find itself in these spaces to be able to give a combination of equal outcomes, not all pairs in these spaces give a combination of equal outcomes. Only the pairs in the double cone do. If it were all pairs in these spaces then the number of pairs in these vector spaces would show Bell’s probability for combinations of equal outcomes. That number is: (2ⱷ / 2π)N. (See fig. 11 for example). However, the requirement for a pair of opposite vectors to give a combination of equal outcomes is that the angle between the vectors and the setting of A is smaller than the angle between the settings of A and B. Although the angle between the centre perpendicular planes also is ⱷ degrees, it is not the same angle ⱷ that is between the settings. Rotating ⱷ (that is between the settings) around A gives a (double) cone. Rotating ⱷ (that is between the perpendicular planes) around A gives a deformed doughnut. It are the vector pairs in the cone that yield combinations of equal outcomes, not the vector pairs between the perpendicular planes.
Evidently the sequence of the movements matter. The sequence of the rotations cannot be exchanged. First the 90° rotations have to be performed, establishing the detector planes, before B can rotate 180°. This is only natural. If the setting of B would have been rotated 180° in respect of the setting of A first (when both settings are still in the line of motion), and then rotated 90°, no division of the sphere in different vector spaces at all would have been obtained. In that case the vector spaces between the centre perpendicular planes of A and B would only be the space representing the vector pairs which numbers show Bell’s probabilities. Also the inversed rotations (applied to the obtained vector spaces to find their real position in respect of the line of motion at the moment of measurement) have to be performed in the inversed sequence. If not, then a wrong, non-symmetric vector space is obtained.
To compare the outcomes from A and B, the outcomes have to be brought together. It is like comparing the pictures made by A and B by putting them next to each other. Then they are being described from one viewpoint, as it were. Moving detector B to its original position, and moving B’s detected vectors along, B’s cone end up at the position of A’s cone. To bring B back to its original position the inversed rotation of 180° has to be applied. But now the rotation has to be applied to two axes: A’s setting and the line of motion, being the 90° inversed rotation of A’s setting. Applying these two rotations to B’s cone, this cone ends up, upside down, at the position of A’s cone. In this way the vectors in the half cones measured by B, end up in the half cones of the opposite vectors measured by A. Therefore the outcomes are equal. In this way the outcomes of a pair are being described as if from one viewpoint.
The inversed rotations that have been made to compare the results, have nothing to do with the detections, of course. But it is nice to see that the vectors, that are measured by B and are supposed to give equal outcomes in respect of the vectors measured by A, indeed give equal outcomes when described from one point of view. It is also nice to see that the Principle of Perspective works correctly: inversing the rotations yields the correct outcomes.
Viewed from one viewpoint the cones of the double cone are in the same position (they represent the same vector space). They are visualised, for different angles ⱷ, in fig. 12, viewed from a direction along the line of motion and viewed from a direction perpendicular onto the line of motion. The blue colour represent vector spaces that contain the vector pairs yielding combinations of equal outcomes. The yellow colour represent vector spaces that contain the vector pairs yielding combinations of opposite outcomes. The green colour is a blue space surrounded by yellow space (or vice versa when ⱷ > 90°). The correlations are calculated from the probabilities for the various combinations.
Bell’s probability for a combination of equal outcomes is (ⱷ / π). For a combination of opposite outcomes the probability is then (1 - (ⱷ /π)) = (π ⱷ) / π and Bell’s correlation then is: C Bell = (ⱷ / π) ((π ⱷ) / π) = (2ⱷ - π) / π QM’s probability for a combination of equal outcomes is sin²(ⱷ / 2). For a combination of opposite outcomes the probability is then 1 - sin²(ⱷ / 2) = cos²(ⱷ / 2). QM’s correlation then is: C QM = sin²(ⱷ / 2) -cos²(ⱷ / 2) = -cos ⱷ.
| ⱷ | Bell | C (bell) | QM | C (qm) | ||
|---|---|---|---|---|---|---|
| View in respect of the line of motion | ||||||
| along | perpendicular | along | perpendicular | |||
| 0 |  fig. 12.1.1 |  fig. 12.1.2 | -1 |  fig. 12.1.3 |  fig. 12.1.4 | -1 |
| 30 |  fig. 12.2.1 |  fig. 12.2.2 | -0.67 |  fig. 12.2.3 |  fig. 12.2.4 | -0.87 |
| 60 |  fig. 12.3.1 |  fig. 12.3.2 | -0.33 |  fig. 12.3.3 |  fig. 12.3.4 | -0.50 |
| 90 |  fig. 12.4.1 |  fig. 12.4.2 | 0 |  fig. 12.4.3 |  fig. 12.4.4 | 0 |
| 120 |  fig. 12.5.1 |  fig. 12.5.2 | 0.33 |  fig. 12.5.3 |  fig. 12.5.4 | 0.50 |
| 150 |  fig. 12.6.1 |  fig. 12.6.2 | 0.67 |  fig. 12.6.3 |  fig. 12.6.4 | 0.87 |
| 180 |  fig. 12.7.1 |  fig. 12.7.2 | 1 |  fig. 12.7.3 |  fig. 12.7.4 | 1 |
| 210 |  fig. 12.8.1 |  fig. 12.8.2 | 0.67 |  fig. 12.8.3 |  fig. 12.8.4 | 0.87 |
| 240 |  fig. 12.9.1 |  fig. 12.9.2 | 0.33 |  fig. 12.9.3 |  fig. 12.9.4 | 0.50 |
| 270 |  fig. 12.10.1 |  fig. 12.10.2 | 0 |  fig. 12.10.3 |  fig. 12.10.4 | 0 |
| 300 |  fig. 12.11.1 |  fig. 12.11.2 | -0.33 |  fig. 12.11.3 |  fig. 12.11.4 | -0.50 |
| 330 |  fig. 12.12.1 |  fig. 12.12.2 | -0.67 |  fig. 12.12.3 |  fig. 12.12.4 | -0.87 |
| 360 |  fig. 12.13.1 |  fig. 12.13.2 | -1 |  fig. 12.13.3 |  fig. 12.13.4 | -1 |
At ⱷ = 90° the cone occupies half the sphere, representing the 50 – 50 % probabilities for combinations of equal and combinations of opposite outcomes. When ⱷ > 90° the cone turns ‘inside out’. At ⱷ = 180° the cone fills the whole sphere representing the fact that all measured pairs give combinations of equal outcomes. When ⱷ > 180° the opposite sides of the central perpendicular planes face each other, meaning that when ⱷ > 180° the vector spaces between the perpendicular planes are now the spaces containing the pairs that give combinations of opposite outcomes. At ⱷ = 360° these spaces fill the whole sphere, representing the fact that all measured pairs give combinations of opposite outcomes.
This is the mechanism behind Bell experiments. It is the definition of vector spaces by the rotation of both detectors in respect of the line of motion of the particles and in respect of each other, and the definition of the real position of these vector spaces, at the moment of measurement, by the inversed movements of the detectors.
Concluding we can say that a vector space, consisting of randomly distributed pairs of opposite vectors, can be divided in two different spaces. The division is defined by ⱷ and the line of motion of the particles. One of these spaces contains the pairs that show combinations of equal outcomes and the other contains the pairs that show combinations of opposite outcomes. We showed why opposite vectors of a pair can yield a combination of equal outcomes. We deduced the exact division of the vector space (the shape and position of the resulting spaces) and we demonstrated by means of a computer program that the numbers of pairs in these spaces exactly yield QM’s correlations. This shows that QM’s correlations can be explained without instantaneous interaction at a distance.