Contents
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| Note To The Reader
This paper is a revision of one I circulated to some friends in the fall of 2004. At this stage, it should still be considered as a draft. I have posted it online in an attempt to facilitate the gathering of comments and suggestions from a network of friends who are scattered around the globe. It is written for a general audience so that everyone can judge its merits. |
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| Summary
This paper demonstrates that Bell's theorem misses the mark in its attempt to establish a rift between the behaviors of quanta and so-called 'everyday objects.' It also includes a description of the EPR/Bohm/Bell experiments, derives a generalized equation for the correspon dences between 'spin' detectors in those experiments, and uncovers some 'hidden variables.' |
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| 1 According to one interpretation of Quantum physics, there is an ever-increasing number of universes where I have gotten this wrong. Let us hope that you, as an observer, have timed your encounter with this paper well, so that everything turns out right for us in this universe. If things do go well for us, they will not be going well for Bell's theorem and certain quantum speculations. This is not to say that Quantum Physics will be rendered untenable by the things said here, but that it will be significantly demystified, and, I hope, rendered more complete, more understandable, and more intuitively plausible. | Introduction1
In a 1935 paper, Einstein, Rosen, and Podolsky jointly issued a challenge to Quantum Mechanics in a thought experiment which has come to be known as the EPR experiment. In their opinion, this experiment revealed that nature still had something to hide which was eluding Quantum Theory. In the years that followed the publication of their paper, David Bohm proposed a modification of the experiment that involved using pairs of electrons both of which were evaluated for what is called 'spin.' In Bohm's experiment (as Figure 1 shows), the detecting apparati2 (involving a deflector and detectors) are parallel along their yz planes and aligned perpendicularly along the x axis, with the x axis passing through their midpoints. The detectors may be separated as far as you like along the x axis. |
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| 2 Throughout this paper, I will use the term 'detector' to mean the entire apparatus, except on the rare occasion when context makes it clear that I mean the detecting portion of the apparatus. | John Bell then devised a modification of this experiment by thinking
of rotating the detecting apparati in the yz plane, and comparing two sets
of readings. One set with only one detector rotated; another set with both
detectors rotated. He then argued that the correlation predicted by traditional
physics would have a maximum value of ±2. Quantum Mechanics, on
the other hand, predicts a range that reaches to ±2√2.
With a difference between the two predictions amounting to √2, there was a clear possibility that an experiment could determine which approach provides a better description of reality. The theories were now headed for a showdown. |
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| Soon enough it became technologically possible to conduct the experiment. The results are within striking range of ±2√2. Quantum physics apparently triumphed. What this appears to mean in lay terms is that we have experimental proof that quanta do not act like everyday objects (pens, stones, pine trees, fish…). | |||||||||||||||||||||||||||||||||
| 3Lurçat has recently proposed something called a 'biphoton' and separate realms of reality. You may see my response to his approach here. | The responses to this victory have varied from an unconcerned shrug of the shoulders from those who have placed their trust in whatever the equations dictate, to hypotheses of logic-defying superpositions, magical action at a distance, superluminal interactions, and even the constant creation of new universes.3 | ||||||||||||||||||||||||||||||||
| The occult qualities of some of these responses can be enticing, but science is—or ought to be—the antithesis of occultism. The scientist, then, has a responsibility to make some sense of these findings. In science, 'making sense' has traditionally been bound by certain parameters. One of these is the adherence to certain rules of thought, such as those outlined by logicians. Another is the avoidance, wherever and whenever possible, of appeals to things that lie outside of normal, repeatable observation. When dealing with things that are hidden or too small to see, the general course has been to appeal, by analogy, to things that are perceptible. And a third parameter is to avoid any direct or implied appeal to lazy explanations such as 'it was magic' or 'God did it.' | |||||||||||||||||||||||||||||||||
| I do not think that I would be saying anything unusual if I suggested that superpositions, action at a distance, and the creation of new universes violate these traditional parameters. No one has ever observed a superposition, and, in fact, like the toothfairy, superpositions are conveniently defined as something that cannot be observed. Action at a distance may have been embraced popularly as a way of thinking about Newton's gravity, but real scientists (including Newton himself) were wise enough to avoid endorsing such an hypothesis. And the hypothesis of many worlds involves action at a distance, prolific creation ex nihil, and an imaginary appeal to not merely individual items but to entire universes, all of which are conveniently unobservable. These paths do not seem to me to lead to what I would term a scientific explanation. | |||||||||||||||||||||||||||||||||
| 4 For those who are unfamiliar, I reject Relativity Theory's light postulate as unnecessary. See, for example, my letter to the Post-Gazette. | As for superluminal interactions, those who know my work know that I fully endorse the idea of the possibility of superluminal velocities,4 but appealing to them in this context is not only unnecessary, but also simply wrong. | ||||||||||||||||||||||||||||||||
| 5 Equations can be grounded or ungrounded. Grounded equations take their meaning from 'common language' theories that make some kind of sense. In such cases, the equations can be used as shorthand for the explanation. Ungrounded equations, on the other hand, lack this foundation in theory and so, ultimately, lack meaning. This lack of meaning persists no matter how successful the equations may be at predicting or affirming results. Ungrounded equations cannot serve as a substitute for an explanation because there is no theory for which they serve as the shorthand. | Nor does the act of simply referring to the equations qualify
as 'making sense of the findings,' because the equations themselves are
simply a codification (and perhaps a mystical one at that) of the outcomes.
Equations are not explanations.5 They are
tools. In doing physics, these tools are designed to help us understand
the presumably hidden inner workings of nature. When we have equations
that wondrously map onto real events, but no further connection, we do
not have an explanation; we have a coincidence. We do not
have something that assists us to understand nature; we have, instead,
a
second mystery. When the tools you are using only serve to complicate
the task you are attempting to accomplish, it is time to reassess their
usefulness; time to reach for a new tool.
All these approaches—superpositions, magical action at a distance, creation of new universes, superluminal interactions, and raw appeals to 'the equations'—are, then, flawed. What these hypotheses and questionable tactics suggest to me is that theorists have hit an impasse. I, however, do not regard encountering an apparent impasse as a license to violate the parameters of 'making sense' that have guided us thus far. Abandoning reason is not the route to a better explanation, even if the alternative fits all the facts. If it were, then we could have saved ourselves a lot of effort and stuck with pagan superstitions, or taken the logician's shortcut, which starts with a deliberately false premise that renders every deduction true. |
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| The task at hand for someone with my point of view, therefore, is to come up with a way to explain the experimental results in a way that fits the parameters outlined above. I have, to be sure, been assured on a number of fronts that this task is simply impossible. To my mind, that means it is worth a try, because I am convinced that nature is really not all that complicated. What follows is one such attempt. If it fails, it is likely that I will try again. | |||||||||||||||||||||||||||||||||
| 6 A note to Classicists: as
my play, The Fog, indicates, I do not know what the physicists are
doing with the word quantum, so pardon me if I follow their lead and use
the singular as an adjective and the plural as a noun and otherwise fail
to inflect it.
7We might wish to reconsider the wisdom of this, for if we are all striving to come up with an analogy that does not make sense, how are we ever going to hit upon one that does? |
Overview
The EPR/Bohm/Bell experiment involves two major problems. We can term one 'the sorting issue' and the other 'non-locality.' These issues have been described in a number of ways. In order to emphasize the divide between everyday objects and quantum phenomena,6 it is common—not only in popular treatments, but even in advanced graduate courses—to try to present the behavior of quanta in terms of everyday objects. This invariably leads to what those with 'common sense' would call total nonsense. And even the physicists agree that what they are describing violates common sense. But things become odd when the physicists go on to call for a redefinition of common sense in light of their findings—or rather, in light of their interpretation of their findings. In order to get acquainted with this problem, we can take a moment to review one of these stories. The point of these stories—whether told here, or in other treatments of quantum phenomena—is to lead the reader to see that things are not making sense in any common (e.g., non-quantum) understanding of what 'making sense' means.7 |
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| 8 Albert, passim.
9 Peat, 77, 99-100. 10 Sadly, I cannot recall the author or the title of the book. As I elaborate on this analogy below, I clearly owe him or her a great debt for providing fodder for inspiration. Other analogies include playing cards, throwing dice, flipping coins, etc. 11 A note to cattlemen: I am not well acquainted with raising cattle, so pardon me if my ignorance shows. |
We can examine one such story here, but the reader is welcome to explore
such treatments elsewhere. David Albert describes it in terms of balls
that are either black or white, and hard or soft.8 F.
David Peat uses images of twins, princes, and wizards.9
Yet another author gives an analogy to cows and horses.10
I find the barnyard example particularly ludicrous, so I will elaborate on a modification of it here. Let's say that Old MacDonald had a farm, and on his farm he had some cows and horses. One day, he had to sort them out, so he got his men to herd everything up and drive them to the corral. At the corral he had his oldest son guide the horses into one pen, and the cows into another. Now it turned out that some of the cattle were dairy cows and the others were raised for beef.11 So his second son was appointed to separated the dairy cows from the cattle raised for beef. Old MacDonald, however, had a third son, who was a perfect rascal. Just to make extra work for his brothers, this third son keeps sneaking dairy cows back out of the barn and keeps brings them back around to the corral where the first son is sorting cattle and horses. The marvelous thing that happens, on this analogy, is that, on their second pass through the sorting process, half of the dairy cows turn out to be horses! |
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| This is a marvelous story. The mystics must love it. Somewhere in heaven (or on the head of a pin) the angels are taking wicked delight in bedeviling us humans with a puzzle of this magnitude. But this is only part of the devilry we are dealing with. We still have to consider the non-locality problem. | |||||||||||||||||||||||||||||||||
| Back at Old MacDonald's farm, the devilish son decides to make extra work for everyone. He decides that they now have to separate the males from the females. He tells the men to get the animals and bring them up to him two by two, like Noah's Ark. He then sends one to his right, where his oldest brother has the nasty job of checking for gender, and one to his left, where his other brother has the same task. It turns out, even when he blindfolds the men who are sending the pairs of animals to him, that every time one brother has a male, the other has a female, and vice versa. It gets to the point that the third brother tells one of his brothers to keep quiet. When the oldest yells, "Male," the third brother turns to the other and says, "Female, right?" and gets a confirmation. The oldest catches on, and starts to call, upon completing his inspection, what his brother has, "Male here, female there!". They get so confident with the system, that, when it comes time to sell some of the male dairy cattle, they just run through this routine and every time the oldest son gets a female on his end, he yells, "Load him up!" to the workers on the other end, who just put the animal that comes their way on the truck without paying any attention to its gender. These animals are transported by truck and train to various points in the country—in some cases the journey involves thousands of miles—and Old MacDonald's farm never gets a complaint that they shipped a female instead of a male. The system works like magic, and anytime they start running short of male cattle, they just sort through the female horses and instantly find more cattle, half of which are males! | |||||||||||||||||||||||||||||||||
| Old MacDonald thinks something is strange in all this and so takes up drinking. One day while he is in the bar, Madame Q. M. Belle, the local psychic, comes in and says, "You are puzzled about what your sons are doing at the farm." He agrees. She says, "What do you think would happen if you had them check for gender with a double test that was a bit oblique; one boy checking for gender by seeing if the animal is pregnant, or has a calf, the other by looking at muscle mass and horns?" Old MacDonald said, "The best you could hope for from any double test would be twice the correspondence." Madame Q laughed and said, "There is more on heaven and earth than are dreamt of in your philosophy! I tell you, the correspondence will be nearly thrice!" | |||||||||||||||||||||||||||||||||
| Now Old MacDonald had a test that he could run that would determine if his expectations matched, or fell short of, real outcomes. When he ran the tests, Madame Q turned out to be right. Old MacDonald spent the rest of his days at the bar. | |||||||||||||||||||||||||||||||||
| Digression: Lousy Analogies
The preceding story makes it abundantly clear that what is happening is totally incomprehensible, and that Old MacDonald's solution is probably the best one. Yet, it might be wise to consider what this means. Does it, for example, mean that the problem is unsolvable, or that this particular approach is the wrong way? There are many steep cliffs in the world which look like forbidding barriers, but when one approaches them from another angle, the slope is easily managed even without any climbing gear. Maybe we need to approach the problem in some other way. |
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| 12 Lucretius 2.441ff.
13 This is not to say that Lucretius was right, but just to point out that his approach was helpful. 14 Fish, for example, do not act like trees, but this does not mean that fish do not act like macroscopic objects. If quanta do not act like cows and horses, that does not mean that there are no macroscopic objects that might serve to model their behavior. We may have to search a bit to find the right analogy, but it would be better to search a bit than to throw in the towel. |
One of the major tools that we are using to help us understand the problems involved in the EPR/Bohm/Bell experiment is that of analogy. Analogies help to make the unfamiliar understandable by pointing to parallels in familiar objects. When Lucretius talks about how atoms link up, for example, he speaks of them as if they are balls or rocks equipped with hooks, rings, jagged edges, holes, and so forth,12 and so by appealing to what everyone can readily imagine, he enables them to understand and visualize the invisible process of atoms accumulating to form larger bodies.13 It is important to realize, however, that analogies can be misleading. Not everything is like everything else, and sometimes what looks like a helpful analogy is actually an impediment to further understanding.14 The examples given above are lousy analogies. They do not promote understanding; instead, they create confusion and promote a sense of mysticism, magic, counter-intuitiveness, and irrationality.15 | ||||||||||||||||||||||||||||||||
| 15 This is not to say that I am unappreciative of the efforts of those who have provided such analogies. I would never have learned of this problem, nor been inspired to try to solve it without the accounts I have cited. As, among other things, a classicist, I recognize that we all stand on the shoulders of giants (or on the shoulders of those who are on the shoulders of giants, or...). When, however, we see farther or clearer than they do, it can be helpful to point out why, so that those who will eventually step all over us will have some idea of the kinds of places that they may put their feet. | Presumably, these analogies are offered because no one has
found any that are better, but the suggestion that accompanies them is
that it is impossible to find an analogy that fits. This is an error which
is impeding the progress of science. In fact, there are many analogies
that fit the facts. I will offer one here that I think is particularly
well suited to the experiment.
The phenomena we are dealing with in the EPR/Bohm/Bell experiment are things that can be done with electrons and electromagnetic radiation. We do not, therefore, have to search as far as the barnyard to find items on a macroscopic scale that can help illustrate what is going on. It is well established that a moving charge creates a magnetic field, so if we think of the experiment with electrons, we can regard them as small magnets, since they are in motion throughout the experiment. As the experiment has many distinct stages, we can follow this magnet analogy very carefully through each stage of the experiment.16 |
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| 16The following experiment should be possible on the macroscopic level with magnets that are light enough in weight to be deflected by a sufficiently strong magnet. In setting up the experiment, one will discover that there are a range of variables that are not being addressed here (the weight of the magnets, their velocity, the strength of the magnetic field of the gateway, the degree of deflection, and so forth). These additional quantities show how much further we can push our investigation of quanta using the same experimental setup, but all this is, for what we are concerned with here, beside the point. | Stage 1: Random Distribution of
Property 1
(Old MacDonald's Cows and Horses are All Mixed Together) In one version of the EPR/Bohm/Bell experiment, the sorting problem is done by taking pairs of electrons, separating them and passing them through a magnetic field that is created by having two magnets stand as the pillars of a gateway. One magnet has its north pole by the opening; the other has its south pole there (Figure 2). If we think of the electrons as small magnets, then as they pass, they are deflected either towards the north pole or towards the south pole, depending on how they are oriented when they first reach the area near the gateway where the gateway's magnetic field begins to affect them. To avoid confusions, we speak of the deflections being up or down, or left or right depending on the orientation of the gateway. In order to mathematize this, 'up', 'down', 'left', and 'right' are given numerical values like this:
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Figure 2 |
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| To see this more clearly, let's take a small bar magnet and fly it like an airplane towards the gateway, with its north pole serving as the nose. As the magnet gets close enough to the gateway, its magnetic field will begin to interact with that of the gateway. The result will be that its nose will be deflected away from the north pole of the gateway and attracted to the south pole. Like a comet passing the sun, which keeps its head toward the sun and its tail away as it swings around, the magnet keeps its nose aligned with the south pole of the gateway as it passes. The attraction to the south pole causes a deflection in the magnet's trajectory, which we can call 'down' if we have the north pole above the magnets trajectory and the south pole below it. Once the magnet is free of the field around the gateway, the rotation induced by the gateway does not stop. It is like a javelin which strikes a branch or wire just above its leading edge and so cartwheels along for the rest of its trajectory, even though the contact was only momentary. Let's say that the spin is in a clockwise direction (See animation 1: curve.gif). | |||||||||||||||||||||||||||||||||
| What happens if the magnet begins by leading with its south pole? In this case, the magnet is attracted to the north pole of the gateway and so is deflected 'up' and it begins to spin as well. The spinning, however, is in the opposite direction to that of our original magnet. It is counterclockwise (See animation 2: up.gif). | |||||||||||||||||||||||||||||||||
| 17 In the case we first considered, the magnet's axis was perpendicular to the axis of the gateway, and in line with its direction of travel. In other words, the magnet approached the gateway like the body of an airplane; in this case, the magnet is positioned like the wing. | When the magnets come in at a tilt (that is, their north-south axis is not perpendicular to the north-south axis of the gateway), their deflection is determined by analyzing how the vectors of magnetic force from the gateway interact with the magnetic field of the magnet. The most difficult cases are when the north-south axis of the magnet is parallel to the north-south axis of the gateway or when the magnet's axis is perpendicular to both the axis of the gateway and to its path of travel.17 Figures 3-6 and animation 3 show how such cases are resolved. | ||||||||||||||||||||||||||||||||
Figure 3
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Figure 3 depicts a magnet that is oriented
so as to be attracted to both poles. The magnetic field of the gateway
is perfectly balanced on either side of the path it is traveling. The magnet
is not deflected, and no spin is induced. In Figure 4, the magnet would
travel along the same path, but its orientation causes it to be repelled
instead of attracted through the gateway. In Figures 5 and 6, we see that
proximity to one gateway leads to both deflection and rotation as the magnets
pass through the gateway.
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| Animation 3 shows the interaction of forces when the magnet's axis is perpendicular to both the axis of the gateway and to its path of travel (like the wing of an airplane). In these cases, the magnet is realigned so as to become like the magnet shown in Figures 3-6. If the alignment of the magnet's axis is not perfectly perpendicular to its direction of travel, the case becomes a variation of the first set of cases we examined (where the magnet's axis was in line with its direction of travel), though it may be reasonable to treat small deviations as being equivalent to true perpendicularity (experimentation or mathematical considerations can determine if 1o or 5o or 15o or more qualify as a 'small deviation'—more on this later). | |||||||||||||||||||||||||||||||||
| 18 Well, almost, magnets such as those depicted in Figures 3, and 4, and animation 3 will either pass right through without deflection, or be rejected entirely. Consequently, the percentages will be under 50% if we count the 'duds'. | Since we are doing nothing to control the orientation of the magnets as they approach the gateway, it is reasonable to expect that they will be oriented randomly as they enter the gateway. We should, therefore, expect the deflections to be distributed randomly. So, for example, if our gateway tests for up/down deflection, then we should get, over many trials, 50% up and 50% down readings,18 with the ups and downs occurring randomly. | ||||||||||||||||||||||||||||||||
| Stage 2: Repeatable Verification of Property
1
(Double Checking the Cows and Horses Shows They've Been Sorted Correctly) In the EPR/Bohm/Bell experiment, if an electron deflects, say, upwards, it will consistently deflect upwards on repeated tests. If we run our spinning magnets through additional gateways which are oriented in the same way, we find the same result: all of those we labeled 'up' come through as 'ups' and all of those we labeled 'down' come through as 'downs'. How does this happen? It is simple: as we noted in the previous section, the gateways induce clockwise (down) or counterclockwise (up) rotation, since, on their second thime through, the magnets already have such a rotation going in, the gateways let them roll right through. Even if the magnet is oriented poorly when it first interacts with the gateway's field (for example, say it is a 'downer' that should curl around the south pole of the gateway, but it enters near the north pole and is leading with its south pole—which should call for attraction to the north pole and conversion to an 'up'), the momentum of its spin may still overcome the initial attraction (or repulsion) that it experiences at the fringes of the gateways field (where the force is relatively weak) and so quickly align its trajectory in such a way that it is repelled (or attracted) into a path that suits its direction of spin. |
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| Stage 3: Random Distribution of a Second Property
(Sorting the Dairy Cows from the Beef Cattle) In the EPR/Bohm/Bell experiment, if a detector is set up behind the detector that tests for up/down deflection such that, say, an 'up' electron passing through the first detector will also pass through the second, and this second detector is rotated 90° around the axis that lies on the path that the 'up' electron travels, so that it now selects for 'right' or 'left' deflections, it turns out that, if many tests are run, about 50% of the 'up' electrons go left and 50% go right. Again, the sequence is completely random. |
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Figure 7 19 Roughly, the same caveat about rejected magnets and those that pass through undeflected (figures 3, and 4, and animation 3) applies here as well. |
By setting up a second magnetic gateway behind the first in just this same way (Figure 7), we can achieve the same result with our rotating magnets. Half of those that pass through the second detector will go left, and half right.19 How does it happen? In this case, it is a matter of chance. If the spinning magnet enters the field of the gateway (that is, where the field becomes strong enough to 'catch' it) with its north pole leading, then it will be attracted to the south pole of the gateway, deflected in that direction (we'll call it 'left'), and have a corresponding spin induced in it (we'll call it 'sundialwise' since it is at right angles to the previous spin). If the magnet enters the field with its south pole leading, it is deflected 'right' and has a 'counter-sundialwise' spin induced. Because the spin induced by the first detector is at right angles to the spin which the second detector requires, there is no vector interaction which leads to a predisposition to go either right or left. | ||||||||||||||||||||||||||||||||
| Stage 4: Property 2 Now Repeatably Verifiable
(Double Checking the Dairy Cows Shows They've Been Sorted Correctly) Just as was the case with the test for up/down deflection (where once an electron [or magnet] was deflected, say, 'up', it would continue to deflect upwards on repeated passes through an 'up/down' detector), if an electron (or magnet) is deflected, say, 'left', by the second detector, it will register as a 'lefty' in repeated tests with a left/right detector. In the case of our magnets, there is again the inducement of a spin as they pass through the gateway. In this case, since the rotation is at right angles to the clockwise (or counterclockwise) spin of the original detector, we can refer to the spin as 'sundialwise' (or 'counter-sundialwise') |
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Figure 8 |
Aside 1: Quantum Magic Trick Number 1 Revealed
(How to Turn Cows into Horses) We can now refer to these magnets with compound names ('up-left', 'down-left', 'up-right', and 'down-right'). We have checked their 'upness,' 'downness,' 'rightness,' or 'leftness' as much as we liked. Or have we? What happens if we take our 'up-left' electrons (or magnets) and now double-check their 'upness' as in Figure 8? In the EPR/Bohm/Bell experiment, half of the 'up-lefts' turn out to be 'downs'! |
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| 20A similar error lies at the heart of the
indeterminacy principle, which looks for simultaneous readings on position
and momentum. Momentum is mass in motion (mv). How can a moving
object have a fixed location? This is nothing but a modern encounter
with Zeno's paradoxes about motion. The confusion stems from the fact that
modern physicists have never come to grips with the reality that motion
is incompatible with stillness. Historically, it was possible to ignore
the problem by pushing it into ever smaller distances, but on the subatomic
scale, there is nowhere left to hide it.
21 The fuller meaning of this qualification will become clearer below. Table 2 provides some insight into the range. |
Is this surprising? Is this counter-intuitive? Is this freaky?
Have half of our cows turned into horses? Or is this a magician's trick?
Magicians trick us by leading us to make assumptions that appear to be
plausible, natural, even reasonable, but these appearances are misleading,
and the assumptions we are led to adopt are false. In this case, assigning
names such as 'up-left,' and making statements such as "half of the 'ups'
turn out to be 'lefties' as well" are not justified.20
Such locutions involve the assumption that the inducement of the new spin
(sundialwise or counter-sundialwise) does not cancel or otherwise disturb
the initial spin (clockwise or counterclockwise). We have no evidence
that this is the case, and experimentation reveals that such an assumption
is an error.
With our magnets, we can get the same result if we take seriously the idea that the magnets are 'caught' by the magnetic field as they pass through a gateway (see animations 1-3 for examples of such 'catching'). This 'catching' suppresses any prior spin that is unsuited to the orientation of the gateway and induces a new spin that is suited to it. The spin that is induced is an end over-end-spin that keeps one pole oriented towards the pole of the gateway to which it is attracted. Any secondary end-over-end spin would involve moving these attracting poles apart, which will not happen outside of a given range21 unless sufficient forces are involved. In this case, the secondary spin that is present (that is, the spin induced by the prior detector) is there due only to inertia, whereas the new primary spin is being imposed by the forces of the interacting magnetic fields. Since the detectors are of equal strength, and the moving magnet's field remains constant, the secondary spin cannot persist through the imposition of the new primary spin.22 |
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| 22Because the
first gateway establishes a specific spin, while the electrons (magnets)
all have the same velocity going in, their north poles, for example, will
all have the same periodic or loopy trajectory as they move toward the
second detector. Consequently, because 'conversions' will be, in part,
a function of the orientation of the electron (magnet) in relation to the
second gateway, the distance between the two detectors will play a periodic
role in the number of 'conversions' that occur (because this will determine
at which stage of its loops the electron [magnet] will encounter the gateway).
Other factors involved in 'conversions' include the relative strength of
the magnets on the detectors vs those of the electrons (or moving magnets),
and the proximity of the incoming electron (magnet) to one of the poles
of the second gateway.
23 'Suppressed' is a precise term. 'Stopped', 'Canceled' or 'eliminated' would be too strong. What is likely to happen in these cases is that the new spin (in this case, let's say 'left') dominates the 'spin characteristics' of the electron (magnet) but, since things can rotate on two axes, the 'original' spin (let's say, 'up' in this case) continues to have a residual or latent presence. This residual 'up' spin is not strong enough to express itself in a new test by an up/down detector. However, if we take a 'left' with residual 'up' and combine it with a 'right' with residual 'up', the two form a pair (like interlocking gears), and the spin characteristic of this pair functioning as a unit is their combined residual 'up' spin. We should be able to tweak this residual spin so that it causes the pair to have 'down' spin just by sufficiently modifying the residual spin of one of the pair such that, when we combine them, the vector resolution of their residual spins is 'down'. Albert obliquely describes such a measurement sequence (p 7-12), but does not see through the mystery. |
Cows have not turned into horses. Black has not magically
become white. Instead, the poles of magnets have been realigned by up to
90°, and, because we did not expected this to have happened, we were
surprised when they no longer exhibited behaviors consistent with their
former alignment. We can easily see, however, that the error lies in
our expectations. The magnets (or the quanta) themselves have not done
anything out of the ordinary.
Stage 5: Verified Property 1 Now Found to be
Randomly Distributed
Since, as we have seen, if we take a group of our misnamed 'up-right' magnets, for example, and test them for 'upness' by passing them through a gateway which is aligned like those we originally used (as In Figure 8), we find that half of these 'up-right' magnets turn out to be 'downs'. Since the clockwise (or counterclockwise) spin of these magnets has been suppressed23 and replaced by a sundialwise (or counter-sundialwise) spin, they enter the active portion field of the up/down gateway with the axis of their poles randomly oriented in the 'sundial plane.' It is, then, a matter of random chance whether the magnet is 'caught' by the north or south pole of the gateway. As this is a 50-50 proposition, there is no surprise that half of the magnets deflect one way, and the other half go the other way,24 and that the distribution is random. Stage 6, Part I: Generalizing Stage 5
To this point, we have been using special cases. The up/down or left/right
deflection tests represent a 90° rotation of the second detector. What
happens if, for example, we take 'up' electrons (or magnets), pass them
(as shown in Figure 9) through a detector that has been rotated through
any angle, θ, and then retest for upness? This is an important
test because, in geometry, right angles often serve as special cases which
do not fully reveal what is going on. To examine this in detail, let
us consider Figure 10.
|
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| 24 Again, roughly speaking. There
will be those magnets that are rejected, and others that will pass through
unde flected.
25 The values of +1 and -1 are the ones we arbitrarily set (following the conventions of Bell's Theorem) for each electron (or magnet) in Table 1 above. |
This figure is a helpful composite of several events. If we imagine that we take two rides on an electron (or on one of our flying magnets) and head right down the x axis (represented by B) towards the y axis of the detector (represented by AD for our first flight, and CF for our second, when the detector is rotated to some angle θ), and take a snapshot just before reaching the y axis of the detector each time, and then super-impose them, we would get something like Figure 10. In the figure, the segment BA does double duty as a representation of the top half of the central axis of the detector when it is in its upright position (in which case AD is the whole axis of the detector), and as a an indication of the number of 'ups' we record when we run the experiment with the detector set at θ=0°. The line CF (where CF = AD with B as midpoint for each) indicates only the size and orientation of the rotated detector. The line BE indicates the number of electrons (or magnets) we 'catch' with the detector rotated to this angle. We can also regard BE as a vector with BA and AE as its components, and notice that BE > BC. This is as it should be: BC = BA, and BA represents the number of electrons (or magnets) we catch when θ=0° and AE = 0; as θ increases toward 90°, AE increases to a maximal value25 of 1 or -1, and it is only after it reaches that value (at &theta=45°), and the 'catch rate' (BE) reaches √2 that the BA vector component begins to diminish towards 0. | ||||||||||||||||||||||||||||||||
| 26 I can only hazard a guess, but this seems
like a reasonable kind of mistake that one could easily make.
27 Hess and Philipp raise a similar issue,
though they focus on time as the important component instead of vector
values.
|
Aside #2: Revealing a Fault in Bell's Theorem
We are now in a position to make some observations that are crucial to parting ways with Bell's theorem. Bell's hypothesis treats BE as equal to BC and fails to recognize that we are testing for the vector components, BA and AE, of BE. How has this happened? This is probably26 the result of relativistic considerations which led Bell and others to regard rotating the device to any angle θ as being the same as leaving it where it is and tilting their heads: surely the angle of observation should not change the readings in any way. The error that is occurring in this approach is that they did not start with a generalized case (i.e., θ set to an angle that is not a whole number multiple of 90). This has led them to treating both BE and BA as having assigned values of +1 or -1. As we can see, the only way that this can happen is if the units used to measure BE do not equal those that measure BA,27 but mixing units like that would simply be bad practice. Presented visually (as in Figure 11), Bell conceives the situation as a circle (rotate the apparatus to any angle, get the same result). In actuality, it is a square (when you rotate the apparatus away from a right angle, you can detect more of what is happening). |
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| 28 This is where talking in terms of 'ups'
and 'downs' becomes ambiguous and tricky because, when the detector is
rotated, the readings are, strictly speaking, neither 'up' nor 'down'.
29 Or, which for our purposes amounts to the same thing, -100% correspondence. |
Stage 6, Part II: Generalizing Stage 5 Continued
We are seeking to find the percentage of magnets (or electrons) that will 'still'—or, more precisely, again register as 'ups'28 after being deflected 'leftishly' by a detector that was rotated in the yz plane to any angle, θ. We already know that if we 'rotate' the second detector through 0°, we get 100% correspondence. If we rotate it through 90° (or 270°), we get 50% correspondence. At 180°, we get 0% correspondence.29 |
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| 30 Right angles are notorious special
cases in geometry: you cannot safely use induction to generalize what happens
at right angles to cover what happens at any given angle.
|
These, however, are easy, special cases.30
We can get more exact figures for any θ. In Figure 10, BA represents
the vertical component of BE. It is likely, then, that there is a relationship
between the extent of this vertical component and the likelihood that a
electrons (or magnets) will 'retain' their 'upness.' That is, since an
up/down detector compels an up or down result, it is reasonable to expect
that the percentage (P) of electrons (or magnets) that have already been
oriented by a θ-rotated detector and which next register as up
or down—'ups' (u) where the vertical component is positive, 'downs' (d)
where it is negative—is directly related to the vertical component of θ.
Expressed as an equation:
P(uθ+dθ) = k(BAθ). Since we are concerned only with 'ups' here, and they will be half of the total: P(uθ) = (½)k(BAθ). As BA is proportional to cosθ, this suggests: P(uθ) = (½)K(cosθ). The values for special cases (0°, 90°, 180° and 270°, see Table 2) suggest that K=(1+[1/|cosθ|]), yielding P(uθ)=(½)(cosθ+1). The whys and whats of this equation, as well as a fuller exposition of the makeup of K, will become clear in the end. Since cosine values are known, we can predict the outcomes for various rotations of the apparatus. |
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| 31 Where θ > 60° (see Table 2), we can expect small deviations from these percentages due to 'conversions' such as those discussed in Stage 2. The equation given does not factor in 'conversions.' |
Table 2:
Experimental values should approximate these calculated values.31 |
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|
32 Plato, Phaedo 99d |
Stage 7: Distant Correlations with Two Detectors
(Old MacDonald's Sons Can Predict Gender without Checking) From Stage 1, we understand that the 'up/down' readings we get from the up/down detector are random. We have, however, only been dealing with half of the issue. So now we have to make, as Socrates says, "a second sailing."32 |
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Figure 1 |
In the EPR/Bohm/Bell experiment, pairs of electrons are released in opposite directions, each to its own 'up/down' detector (as in Figure 1). We can consider each detector as operating as we discussed in Stage 1, randomly producing 50% 'up' readings and 50% 'down' readings over many trials. The detectors operate independently, and, aside from being aligned to the path of the separated electrons, and oriented to detect the same deflection (that is, both set to 'up/down,' not one set to 'up/down' and the other to some degree of 'leftish/rightish'), there is no correlation between the detectors. One can be set a few feet from the point of separation, the other miles and miles away. The detectors are not connected by wires or any mechanism. In fact, we can position them such that the laws of relativity forbid any interaction between them. Now comes the spooky part: every time detector 1 randomly registers an 'up,' detector 2 randomly registers a 'down;' when detector 1 randomly registers a 'down', detector 2 randomly registers an 'up'; and vice versa for both cases. It looks as though either the electrons or the detectors are somehow in cahoots, or there is some invisible hand of the devil that we have not accounted for. This is the position of EPR. Quantum adherents, however, argue that that is just the way it is, quanta do not behave like familiar objects of experience. There is, however, a less magical explanation for all this. | ||||||||||||||||||||||||||||||||
| Aside #3: Quantum Magic Trick Number 2 Revealed
Why is an 'up' here always correlated with a 'down' over there? How does that electron (or magnet) 'know' what this one has done? How are we to explain it? Superluminal interactions? Hidden variables? Black magic? Experiments prove and confirm that this correlation occurs no matter how fast or how far apart we put our detectors. Is there a hidden variable, or is Quantum non-locality still something to be reckoned with? |
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| As the heading of this section suggests, there is a hidden variable. This variable has remained hidden because of a technique that magicians employ: misdirection. I do not mean to suggest that we have been deliberately tricked (unless, perhaps, by some divine agent), but to point out that we humans are often all too ready to look for answers in all the wrong places. In this case, we have been looking for an answer at the detectors, after a reading has been made on one of the electrons (or magnets). The solution to the mystery is not there. The missing variable we seek comes into play at the source, before the electrons (or magnets) are parted and sent their separate ways. | |||||||||||||||||||||||||||||||||
| 33Although, as I show in my play, The Fog, there are plenty of examples that one can find (packaged shoes, light bulbs, styrofoam cups, etc.) once one knows what one is looking for. | When setting up the experiment, care is taken not to prejudice the results. The electrons (or magnets) are put into a neutral environment and allowed to normalize. Here is where talking about magnets instead of balls, horses and twins can really pay off.33 If we put magnets into a magnetically neutral environment, and let them come into a state of mutual equilibrium in terms of velocity, energy, temperature, and whatever else we can think of, and then emit them, two by two, toward a wedge that separates them, we will get the same anti-correspondence that we have been getting with electrons and photons and whatnot. | ||||||||||||||||||||||||||||||||
Figure 12 |
Why will this happen? Because, as they are paired up, the magnets naturally align in such a way that the north pole of one magnet will be adjacent to the south pole of the other. Figure 12 shows such correlations. When the magnets are separated, one heads to the gateway with one orientation, the other with the opposite orientation. In Figure 12, magnets oriented as in A, will have the magnet on the left enter the one gateway north pole first, the other will enter south pole first. Since the poles of the gateways are aligned the same way, the deflections will be opposite. If B depicts two vertical magnets, they will pass as described above in Stage 1 (flying towards the gateway like the wing of an airplane). If B is a picture of two magnets stacked horizontally, they will fail to separate at the wedge and so be disqualified from the experiment. The same thing will happen if the A pair approaches the wedge rotated 90° (that is, vertically, straight up into it). | ||||||||||||||||||||||||||||||||
| Since this will happen for every magnet (or electron) that we can get a reading for (i.e., excluding 'duds'), we can say we get a 100% correspondence, C=1. Since, with a pair of magnets, the NS axis of one magnet will line up with the SN axis of the other (that is, the NS orientation, F of one magnet is matched, by the other magnet, with an orientation of -F), the correspondence is negative, C=-1. | |||||||||||||||||||||||||||||||||
Figure 13
|
Stage 8: Distant Correlations When One Detector
is Rotated
(Madame Q. M. Belle's Oblique Test Part II) While Stage 7 expands the experiment to include two detectors giving reports on separated pairs of magnets (electrons), it still only describes a special case: when the two detectors are yz parallel, and in-line along x. We can take a step towards generalizing this by considering what happens when the detectors are in-line along x, and yz skew, that is, one is rotated to any θ while the other remains upright, such as is shown in Figure 13. Figure 10 can also be understood as representing this situation as one looks down the x axis (B): DA standing for one detector, and CF for the other. In this case, the correspondence—no longer surprisingly, given our discussion in Stage 6—turns out to be a function of cosθ. Because the electrons (or magnets) are anti-correlated (that is, an 'up' on one side matches, if anything, a 'down' on the other side), the correspondence is a function of -cosθ. The special case values (for 0°, 90°, 180° and 270°) suggest: |
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Table 3
|
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Figure 14 |
Stage 9, Part I: Distant Correlations When
Both Detectors Are Rotated
(Madame Q. M. Belle's Oblique Test Part III) The next step is to take these anti-correlated electrons (or magnets) and see what happens when we rotate both detectors, D1 to angle θ, and D2 to angle φ (Figure 14). It would seem plausible to regard this situation as being exactly the same as leaving the apparatus as it is in Figure 15 and observing it from an oblique vantage point, or the same as leaving D2 where it is, and rotating D1 to angle θ+φ. That is, for example, leaving D2 in line with AD in Figure 15 and rotating D1 to BF; rather than rotating it to BC, and D2 to BE. Using what we have found so far, this would lead us to expect that the correspondence at θ+φ to equal -cos(&theta+φ), and for the values to fall within the range bounded by -1 and +1. |
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| Yet this turns out not to be the case. It is easy to get confused at this stage. Some principle of simple relativity seems to be being violated: why should rotating two detectors be different than rotating only one? Yet the mystery is not so deep; the conundrum not so convoluted. In order to understand why, we must consider what we have, and have not, been measuring. | |||||||||||||||||||||||||||||||||
Figure 16 |
Aside #4: Quantum Magic Trick Number 3 Revealed
To get a firmer grip on what is going on, let us review what is happening with some helpful graphics. Figure 16 shows the 'hot zones' for an electron (or magnet) traveling down the x axis and entering the gateway of a detector that has its axis set in line with the y axis. The 'hot zone' is the range of axis-tilt that the moving electron (or magnet) can have as it enters the gateway and still get 'caught' by the detector. Table 2 suggests that the hot zone can extend up to 60° on either side of the axis. |
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| The situation will be just about identical for the hot zone of a electron (or magnet) approaching a detector that has its axis set in line with the z axis. To see this, all one has to do is turn Figure 16 on its side. In either case, the hot zone only overlaps one axis, so we can only detect y or z spin, not both. | |||||||||||||||||||||||||||||||||
Figure 17 |
When, however, the detector is set with its axis set at 45° to the z (or y) axis, the hot zone overlaps both the z and y axis, and the detector can now 'catch' both the electrons (or magnets) with a predominantly vertical orientation and those with a predominantly horizontal orientation, and all those in-between (Figure 17). The result is that, with this orientation, we can easily exceed the '100%' catch rate that we get when the detector is set at a right angle, which means that we can find more correspondences. | ||||||||||||||||||||||||||||||||
Figure 18 34 See Herbert p. 182-185. |
To recognize the importance of this, it will be helpful to consider a slightly different experiment for a minute. When an experiment is conducted with electromagnetic radiation (photons) passing though polarizing filters, we find that if we take two filters and position them so that their axes of polarization are skew at right angles to each other (Figure 18), then the radiation (photon stream) does not pass through them. If, however, we place a third filter between them that is skew to 45° (Figure 19), the light passes through. If we opt to think of this as strange (two barriers can stop it, but three let it pass),34 then we set ourselves up for a failure to understand what is happening. If however, we regard each filter as redirecting a vector orientation (first to vertical, then to 45°, and then to horizontal), and realize that the first filter induces, say, a y-only orientation, then the second, 'catches' most of these and induces an zy orientation, and then the third catches most of these zy oriented 'photons' and induces in them a z-only orientation, the process is no stranger than watching a car turn a corner: sure, cars racing east crash if they suddenly try to drive north, but cars that have gone east, and then northeast, can go north, even if they are traveling quite fast (think of a banked turn on a race track). | ||||||||||||||||||||||||||||||||
| 35 One way to think of this 'disappearance' (which might, at first glance sound like magic) is to think about what we'd call an 'up' reading if we brought the apparatus out into space, set it to 'vertical' but then laid on our sides while observing it. Would our cows become horses if, in that position, we found it more convenient to start calling the readings 'rights' and 'lefts' instead of 'ups' and 'downs'? | It seems that, because of the perceived incompatibility with, say, driving east and driving north (or registering as an 'up' or a 'left'), physicists have leapt to the wrong conclusion and have tried to regard these as separate, permanent properties instead of transient properties that can commingle (e.g., 'up' and 'left' commingle to 'up-left' when the detector is set at 45°, and 'up' disappears35 when the detector only allows for 'left or right' results). This small change in our approach to the problem makes all the difference between regarding it as an unfathomable mystery and seeing it as something quite mundane. | ||||||||||||||||||||||||||||||||
| To this point, the difference that changing the orientation of the detector makes, though important, will only show up in the number of 'catches' or readings that the detector makes. This is a factor that is easily overlooked or dismissed. When, however, we use two detectors in combination, and look to the rate of correspondence between them, the difference introduced by changing the orientation of the detectors causes a set of remarkable findings. | |||||||||||||||||||||||||||||||||
|
Figure 16 |
We can now regard Figure 16 in a new way: rather than being a look at one detector, let us consider it as illustrating the 'hot zones' of two detectors when both are aligned with, say, the y axis. When the two detectors are stationary and parallel, our results are restricted to correspondences with the y (or z) component of the electron's (or magnet's) spin. Only two readings are possible, 'up' or 'down' (or 'left' and 'right'). According to convention, we have given these values of +1 and -1. | ||||||||||||||||||||||||||||||||
Figure 20 36 Except for occasional, accidental correspondences. |
When one detector is rotated, we are still comparing, say, only the y component of spin. The process is, however, more sophisticated because we can now detect a range of values for BCθ, from -1 to +1. As we have seen, these values vary with the cosine of the angle of rotation, θ. Rotating one detector restricts the 'hot zones' of the pair to those areas where the 'hot zones' of the individual detectors overlap (see the reddest portions in Figure 20), because, while the one detector can now pick up both y and z components, the other can still only detect y components, consequently, correspondences can only36 occur when both detect a y component. With the second detector rotated to 90°, only the weakest portions of the 'hot zones' overlap. | ||||||||||||||||||||||||||||||||
| These readings lull us into a false sense of what is going on. When we rotate one detector, the correspondences diminish. This leads us to conclude that the highest reading we can get occurs when the detectors are aligned as in Figure 13. If we set the numerical value for such readings as +1 and -1 and develop the expectation that all other readings will fall within the range bounded by +1 and -1, we lock ourselves into an error, and set ourselves up to be confounded by the phenomena. Yet this error goes undetected as long as we rotate only one detector. | |||||||||||||||||||||||||||||||||
Figure 21 37 Figure 21 is a little deceiving. It should be, as is hinted at by figure 20, as much blue as it is red. What makes the difference is that everything that registers as a 'catch' on one detector, registers as a 'catch' on the other (and everything that misses on the one misses on the other). This configuration (setting the detectors at +45° and =-45°) is the one that works best when the electrons (magnets) are anti-correlated ('up' here and 'down' there), when the correlation is direct, the detectors should be rotated to the same side. |
When, however, we rotate both detectors, D1
to angle θ, and D2 to angle φ,
we open up a whole new horizon. Now we can detect both the y and
the z components with both detectors. That is to say, we can now
detect y and z correspondences. In Figure 20, all the reddish zones
now become detectable. The detection zone expands, and correspondences
increase.
Figure
21 shows the best case (θ=45°, φ=-45°), and illustrates
how the situation has changed dramatically.37
Since we arbitrarily set the values for correspondences as +1 and -1 for a detector set as shown in Figure 16, we should not be surprised if the correspondences found when the detector is set as shown in Figure 18 exceed this range, which is what, if fact happens. We can now return to the final stage of the experiment. Stage 9, Part II: Distant Correlations When Both
Detectors Are Rotated Continued
As we have seen, when we rotate both detectors, D1 to angle θ, and D2 to angle φ, we can detect both the y and the z components with both detectors, and so detect y and z correspondences. To this point, our equations for these correspondences have featured cosine values, because the cosine corresponds to the y component. We now have to include the z component, which is reflected by sine values. |
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Figure 15 |
If we return to Figure 15, we can now look at it as representing a couple of different cases: one in which one detector is rotated through θ to position BC, and the other is rotated in the opposite direction through φ to position BE; the other case shows one detector rotated to position BC and the other detector rotated through θ+φ to position BF. These cases should help us to develop a fully generalized equation for all phases of the experiment (i.e., the equations given above, featuring cosines, have been special cases where sinφ has had the value 0 or 1). Let us consider the case of BC and BE first. | ||||||||||||||||||||||||||||||||
| 38 m can also serve to account for the factors discussed in sidenote 22 above. | As vectors, lines BC and BE have both a y and a z component. For BC,
z is proportional to sinθ or, rather, in order to prevent negative values of the sine or cosine from diminishing the length of BC, |BC|=m(|sinθ|+|cosθ|), Similarly, for BE: |BE|=m(|sinφ|+|cosφ|). In these equations, m serves to scale the quantities and provide the proper units of measure. The correspondences we are recording, C, will also be a function of the y and z components that we have detected. The sign will be negative because an 'up here corresponds to a 'down' there. That is, C=-(correspondencey + correspondencez), but correspondencey = N(|cosθ|+|cosφ|)
C=-N[(|cosθ|+|cosφ|)+(|sinθ|+|sinφ|)], or, by commutation, C=-N[(|cosθ|+|sinθ|)+(|cosφ|+|sinφ|)], C, therefore, is proportional to BC+BE: C=-N[(1/m)(BC+BE)]. |
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| We know that when φ=0° and θ=0°, BC=BA=1, and we have called the correspondence rate at this setting '1'; likewise, when φ=0° and θ=90°, BC=1. This suggests N/m = 1. With N/m=1, we get the following values for correspondences when both detectors are rotated to the same, but various, angles (Table 4): | |||||||||||||||||||||||||||||||||
Table 4:
|
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| We can make this table more general by considering cases where θ and φ have different values, but, as such a table can quickly become immense, we will only supply a handful of cases here (Table 5): | |||||||||||||||||||||||||||||||||
|
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| As it would make sense to set the highest degree of correspondence to 100%, we should adjust the values of table 4 by a factor of 1/1.41, or, reset N/m from 1 to 1/1.414. This shows that, if we wish to make any angle Θ the standard unit for comparison, N/m must contain the component n=1/(|sinΘ|+|cosΘ|). An additional component of N/m is p, which can adjust the percentage of correspondences we get depending on conditions set prior to running the experiment (see sidenote 21 above). In the cases we are considering, p=1. The reason for this will be given below. | |||||||||||||||||||||||||||||||||
Figure 22
39 Here and in the equations that follow, the
sign '±' indicates adjustments that are to be made depending on
as the correlation is direct or opposite.
|
We can now give a generalized equation for the correspondences that
relies entirely on traditional, local physics. The equation governing these
correspondences (C) is:
C=θcorre spondence+ φcorrespondence
=-np[(|cosθ|+|sinθ|)+(|cosφ|+|sinφ|)] =[-p/(|sinΘ|+|cosΘ|)][(|cosθ|+
With p=1: C=[-1/(|sinΘ|+|cosΘ|)][(|cosθ|+|sinθ|) +(|cosφ|+|sinφ|)] When the detectors are not anti-correlated, as when both detectors are on the same side, as in Stages 1-6, the negative is removed, leaving: C=1/(|sinΘ|+|cosΘ|)[(|cosθ|+|sinθ|)+
When both detectors are in their upright position as in Figure 22, θ = φ = 0°, then sinφ=0=cosφ, and sinθ=1=cosφ so39 C=±(1 + 1) As we rotate the one detector, as in Figure 7, the equation becomes C=±(|cosθ| +1+|sinθ|), but the apparatus is not able to detect correspondences to the |sinθ| component in this configuration, so our measurements correspond to C=±(cosθ +1), If we want only, for example, the 'ups' of an up/down detector, we get C=±(½)(cosθ +1), as in Stage 6. As we compare the correspondence rates at different θs, while leaving the φ detector upright, the cosφ component (+1) remains constant, and so we get: C1-C2=(cosθ1
+1)-(cosθ2 +1)
which leaves the misleading impression that, as we found in Stage 8, C=±cosθ. |
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| These results, which are reached without any recourse to Quantum arguments (or anything else magical or mystical), agree with both Quantum Mechanics and experimental results. The Bell Inequality, therefore, is shown to house an error either arising from regarding BA as a unit length for all orientations of the apparatus or, which may amount to the same thing, from failing to regard BCθ as a vector. | |||||||||||||||||||||||||||||||||
| 40 Montaigne, Essays
Bk 1 Ch1.
41 As we've done throughout this paper (due to the experimental setup we've inherited), we take coincidence with the +y axis as 0°. |
Alternates to Stage 9:
As Montaigne observed,40 there is more than one way to reach a given goal. Another way to look at this entire discussion is this. When we have an 'up/down' detector, and set41 Θ = θ = 0°, we get readings of +1 and -1, that is z=0 and y=±1, but this does not mean that we are registering 100% of the emitted electrons (or magnets). It only means that we have arbitrarily named an 'up' result +1, and a 'down' -1. Indeed, it is reasonable to expect that we are completely missing a significant percentage of the emitted electrons: namely, those electrons (or magnets) that would register for 'left/right' spin if we were measuring for it, but, at this setting, pass undetected. As we discussed above in Stage 1, the apparatus's ability to 'catch' electrons depends on their orientation to the gateway. Table 2 suggests that the apparatus will have difficulty registering electrons (or magnets) oriented with a yz tilt (that is, the angle formed by their north-south axis with the north-south axis of the gateway when viewed along the x axis) in the range defined by ±60° (See Figure 16). |
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|
Figure 21 42 In this case, the angles are in the opposite direction, with other phenomena (e.g. polarized light) θ and φ have to be rotated in the same direction. 43 We use the Pythagorean theorem here (not vector addition) because we are working with percentages, not vectors. |
If we want to detect closer to 100% of the emitted electrons (or magnets),
we have to orient the detectors so that the electrons (or magnets) with
'left/right' spin are caught as well as those with 'up/down' spin. At θ=45°
and φ=-45°42, this becomes
possible because the detectors are now in position to 'catch' and register
both types of spin equally (Figure 21). Since
we can now 'catch' just about 100% of those with 'up/down' spin that we
can 'catch' when θ = 0° (y=±1), as well as just about
100% of those with 'left/right' spin that we can 'catch' when θ
= 90° (z=±1), we can figure the value for BC45°:43
BC45° =√(z²+y²)=±√2. We have, however, two such BC45°s, one corresponding to BCθ the other to BCφ. The situation, then, gives us a value of ±2√2. This is the value that Quantum equations have been predicting, and which experiments bear out. Again, we see that the same value can be achieved the old-fashioned way, and without recourse to anything weird or unimaginable. |
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| 44 Some other quantum 'mysteries' can be dealt with quite simply. The double slit experiment which 'reveals' the wave-particle duality of an electron, for example, is nothing but experimental evidence of the existence of an aether (a ship and its wake can reproduce the phenomena exactly). The relation of energy to frequency becomes obvious when one realizes that increases in frequency are our only direct way of measuring changes in velocities greater than c (sound waves produced by supersonic jets do the same thing). | Conclusion
As I noted earlier, the analogy between magnets and moving electrons is a fairly thorough one, and having a good analogy can clearly make the difference between understanding and wallowing in confusion.44 If I have gotten this right, for the past forty-two years Bell's theorem has misled us by a small, devilishly misleading flaw wrapped up in three layers of magic. The keys to solving this mystery were the firm belief that there was a solution, the identification of the right kind of analogy with which to analyze it, and the realization that Bell's hypothesis does not cover every possible classical model. Like a magician's trick, the theorem's flaw, and the secrets of the hidden variables have baffled and amazed, but now, exposed, they may seem embarrassingly obvious. Let's hope we can all bear this news with some good humor. |
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| On a sad note, if I have gotten this right, we may have, in the course of this realization, completely annihilated a couple bazillion universes. Perhaps we should observe a moment of silence. | |||||||||||||||||||||||||||||||||
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Works Cited: Albert, D. Quantum Mechanics and Experience Harvard University Press (Cambridge, MA) 1992 Bell, J. "On the Einstein-Podolsky-Rosen Paradox" Physics 1 (1964) 195-200. Einstein, A. Podolsky, B. and Rosen, N., "Can Quantum Mechanical Description of Reality Be Considered Complete?" Physical Review 47 (1935) 777-780. Herbert, N. Quantum Reality Beyond the New Physics Anchor Books (New York) 1985. Karl Hess, K., and Philipp, W., "Breakdown of Bell's theorem for certain objective local parameter spaces" Proc Natl Acad Sci U S A. 2004 February 17; 101(7): 1799-1805. Published online 2004 January 22. doi: 10.1073/pnas.0307479100. http://www.pubmedcentral.gov/articlerender.fcgi? tool=pmcentrez&artid=357007 Lucretius, De Rerum Natura http://etext.library.adelaide.edu.au/ mirror/classics.mit.edu/Carus/nature_things.html. Montaigne, Essays. http://www.gutenberg.org/files/3600/3600-h/3600-h.htm Newell, J. The Fog Mywitz Books, 2005 Peat, D. Einstein's Moon Bell's Theorem and the Curious Quest for Quantum Reality Contemporary Books (Chicago) 1990. Plato, Phaedo http://www.gutenberg.org/files/13726/13726-h/13726-h.htm http://en.wikipedia.org/wiki/Bell's_Theorem http://en.wikipedia.org/wiki/Bell_test_experiments Clausen, B. L., "Quantum Mechanics: The Strange World at Small Dimensions" Origins 18(1):39-47 (1991). http://www.grisda.org/origins/18039.htm Rowe, M. A., Kielpinski, D., Meyer, V., Sackett, C. A., Itano, W. M., Monroe, C., and Wineland, D. J., "Experimental violation of a Bell's inequality with efficient detection" Nature 409, 791-794 (15 February 2001) | doi: 10.1038/35057215 http://www.nature.com/nature/journal/v409/n6822/ full/409791a0.html |
