Bell's Theorem

[Duder]

Greetings, Science Guys -

For years I have encountered the subject of Bells' Theorem in my reading, but I was never quite smart enough to follow along with it. Last night, however, I stumbled upon a "Bell's Theorem for Dummies" kind explaination, and I want to discuss it here from memory, first to see if I properly grasp it, and second, to make it available to others who have wrestled unsuccessfuly with Bell's Theorem.

Bell's Theorem has to do with the way in which things are connected to each other. If I kick a soccer ball, then it and I have influenced each other. We achieved a connection when it and my foot interacted.

There are two kinds of connection of interest to us in Bell's Theorem - local connections and non-local connections. A local connection is the everday sort of connection we are all familiar with, such as my kicking a soccer ball. Another example would be a telephone conversation. My voice makes a connection with your ear by means of particles and fields that influence each other in a normal cause-and effect sort of way.

A non-local connection, however, would be quite different. In a non-local connection, there is no discernable medium that carries the connection. We don't see the cause-and-effect chain of events that connects one thing with the other. Voodoo would be an example of a non-local connection. My sticking a pin in this doll instantaneously causes my victim many miles away to feel pain.

In physics, no influence may travel faster than the speed of light. Thus if an event happens here and now, there is no possibility that this event could influence anything a light-year away for at least one year. If you and I are a light-year apart, we are each oblivious to what happens to the other for at least one year. However, if something violated this, and we were able to influence each other or communicate with each other sooner than one year, that would be a non-local connection - since there is no discernable medium which could connect us so quickly.

Bell's Theorem proves that things are connected non-locally - even though we don't directly perceive the non-local connection. In other words, a physical model of reality that says things are connected locally must be wrong.

Now I want to explain how Bell's Theorem gets there, as best I can from my memory of the "dummy" explaination I found. This is bound to be lengthy, so I will post this introduction and work on my explaination.

Be back soon!

[technomage]

In physics, no influence may travel faster than the speed of light.

Not considered to be true in one particular instance--the collapse of a quantum waveform is seen as not being affected by the speed of light, so if the N possibilities in the waveform are lightyears apart, the verification of one leads to the immediate nullification of all others.

[NeilUnreal]

I'm also only familiar with the broad outlines of Bell's theorem. I guess I have two questions:

1) For entanglement occur, isn't it necessary for the two particles to have been together at some point?

2) Isn't Bell's theorem often used not so much as a theorem of non-local interaction, as it is an example of how macro-world concepts like "local" and "non-local" are themselves only loose analogies for what is actually going on at the quantum level? Like Bohr's view of the Copenhagen Interpretation. In other words, we get confused when we leave the equations behind and take our macro-scale analogies for truth. Local and non-local "mean" something to us, and the particles' behavior seems to violate that meaning. It seems spooky to us only because we take that meaning for something that must hold true at all scales (the classical view).

-Neil

[Duder]

All good points and questions - but first, if I may, I'd like to try and explain the Theorem so that science dummies like me can understand what it says.
___________________

Bell's Theorem is a simple mathamatical prediction about what we should observe in a certain experiment, if the real world is connected locally and not non-locally. The experimental result does not match the prediction, and so therefore the world is connected non-locally - says Bell's Theorem.

Here is the idiot's version of the experiment. Imagine first that we have a detector that gives a yes or no answer to a question about an electron (a binary detector). If the answer is yes, a green light flashes, if the answer is no, a red light flashes.

What attribute of the electron are we measuring with this yes-or-no detector? The answer is spin - but that is unimportant to this dummy's explaination. It is only important that we know it is measuing some actual attribute of the electron. Since we are dummies (I am, anyway), let's call the attribute being measured "happiness". If the electron is happy about its interaction with the detector, it says "yay" and the green light flashes. If it is unhappy about its interaction with the detector, it says "boo", and the red light flashes.

Now the experiment gets a little bit more complicated. Our "happiness detector" can be oriented in one of three ways. It can be set to the "up" orientation, the "down-left" orientation or the "down-right" orientation. We can call these orientations 1,2 and 3, respectively (see fig. 1, below).

The electron may like some detector orientations, and dislike others. For instance, it may be happy when it encounters the detector in orientation 1, causing the green light to flash. But if the detector is set to orientation 2, maybe it will be unhappy, and the red light will flash. But maybe it likes the detector at orientation 3, which would cause the green light to flash.

Picture each electron as carrying a little sign. The sign carried by the electron indicates whether it will say "yay" or "boo" about each of the three detector settings. For instance, if the electron's sign says Y,Y,B, the green light will flash if it hits a detector in orientation 1, the green light will flash if it hits a detector in orientation 2, and the red light will flash if it hits a detector in orientation 3 (see fig. 2, below).

There is one final compliction to this experiement. We are going to fire pairs of electrons in opposite directions. The two electrons in each pair are twins because they are exactly alike. In other words, if one particle in the pair has a sign that says YBY, the other electron in the pair will also carry a sign that says YBY. Not all pairs will be alike. Some pairs will carry signs different than those of other pairs. But the crucial thing is that each electron in a twin pair carries the same sign.

For the sake of simplicty, disregard electrons that either like or dislike all three detector settings. In other words, forget about electrons with YYY and BBB signs - and consider only those with one Y and two Bs, or two Ys and one B.

We place two happiness detectors some distance apart, and fire electron pairs at the two detectors from a spot midway between the two. In addition, each happiness detector shall be randomly oriented. In other words, you are operating detector A. Before we fire an electron pair, you roll a die to determine whether your detector will be oriented 1, 2 or 3. At detector B, I roll a die to determine my detector's orientation. Our two detectors may or may not be set to the same orientation. We fire an electron pair, randomize the settings again, and repeat.

Here is where Bell's Theorem comes in. Statistically, as we fire pairs of electrons, how often should the two detectors flash the same color? Bell's Theorem is nothing more than the math that predicts the result. I have worked this out myself, and I find that all of the different signs the electrons may be carrying will result in one of the following nine results in the two detectors.

green-green
green-green
green-red
green-green
green-green
green-red
red-green
red green
red-red

Five out of those nine possibilities are matches. So this means that 5/9 of the time, the detectors will flash the same color. If we fire nine million electron pairs, the lights will flash the same on about five million occasions.

However, if we actually do the experiment, this does not happen. The lights match only half the time. How can that be?

When there is only one detector, the ratio of red to green flashes is exactly what we would expect. However, when we add a second detector, the result gets skewed for no reason that we can conceive. Somehow, the sudden introduction of a second detector skews the result. Somehow, the interaction of one electron with detector A can change the interaction of the other electron at detector B, and this would happen even if the detectors are light years apart - out of each other's normal sphere of local influence.

Bell concludes that the electrons are connected non-locally - by a kind of voodoo, as it were.

Those of you who are intimately familiar with Bell's Theorem - have I correctly understood the essence of it (disregarding the silly aspects of how the experiment is described)?


.

[geochron]

All good points and questions - but first, if I may, I'd like to try and explain the Theorem so that science dummies like me can understand what it says.
___________________

Bell's Theorem is a simple mathamatical prediction about what we should observe in a certain experiment, if the real world is connected locally and not non-locally. The experimental result does not match the prediction, and so therefore the world is connected non-locally - says Bell's Theorem.

Here is the idiot's version of the experiment. Imagine first that we have a detector that gives a yes or no answer to a question about an electron (a binary detector). If the answer is yes, a green light flashes, if the answer is no, a red light flashes.

What attribute of the electron are we measuring with this yes-or-no detector? The answer is spin - but that is unimportant to this dummy's explaination. It is only important that we know it is measuing some actual attribute of the electron. Since we are dummies (I am, anyway), let's call the attribute being measured "happiness". If the electron is happy about its interaction with the detector, it says "yay" and the green light flashes. If it is unhappy about its interaction with the detector, it says "boo", and the red light flashes.

Now the experiment gets a little bit more complicated. Our "happiness detector" can be oriented in one of three ways. It can be set to the "up" orientation, the "down-left" orientation or the "down-right" orientation. We can call these orientations 1,2 and 3, respectively (see fig. 1, below).

The electron may like some detector orientations, and dislike others. For instance, it may be happy when it encounters the detector in orientation 1, causing the green light to flash. But if the detector is set to orientation 2, maybe it will be unhappy, and the red light will flash. But maybe it likes the detector at orientation 3, which would cause the green light to flash.

Picture each electron as carrying a little sign. The sign carried by the electron indicates whether it will say "yay" or "boo" about each of the three detector settings. For instance, if the electron's sign says Y,Y,B, the green light will flash if it hits a detector in orientation 1, the green light will flash if it hits a detector in orientation 2, and the red light will flash if it hits a detector in orientation 3 (see fig. 2, below).

There is one final compliction to this experiement. We are going to fire pairs of electrons in opposite directions. The two electrons in each pair are twins because they are exactly alike. In other words, if one particle in the pair has a sign that says YBY, the other electron in the pair will also carry a sign that says YBY. Not all pairs will be alike. Some pairs will carry signs different than those of other pairs. But the crucial thing is that each electron in a twin pair carries the same sign.

For the sake of simplicty, disregard electrons that either like or dislike all three detector settings. In other words, forget about electrons with YYY and BBB signs - and consider only those with one Y and two Bs, or two Ys and one B.

We place two happiness detectors some distance apart, and fire electron pairs at the two detectors from a spot midway between the two. In addition, each happiness detector shall be randomly oriented. In other words, you are operating detector A. Before we fire an electron pair, you roll a die to determine whether your detector will be oriented 1, 2 or 3. At detector B, I roll a die to determine my detector's orientation. Our two detectors may or may not be set to the same orientation. We fire an electron pair, randomize the settings again, and repeat.

Here is where Bell's Theorem comes in. Statistically, as we fire pairs of electrons, how often should the two detectors flash the same color? Bell's Theorem is nothing more than the math that predicts the result. I have worked this out myself, and I find that all of the different signs the electrons may be carrying will result in one of the following nine results in the two detectors.

green-green
green-green
green-red
green-green
green-green
green-red
red-green
red green
red-red

Five out of those nine possibilities are matches. So this means that 5/9 of the time, the detectors will flash the same color. If we fire nine million electron pairs, the lights will flash the same on about five million occasions.

However, if we actually do the experiment, this does not happen. The lights match only half the time. How can that be?

When there is only one detector, the ratio of red to green flashes is exactly what we would expect. However, when we add a second detector, the result gets skewed for no reason that we can conceive. Somehow, the sudden introduction of a second detector skews the result. Somehow, the interaction of one electron with detector A can change the interaction of the other electron at detector B, and this would happen even if the detectors are light years apart - out of each other's normal sphere of local influence.

Bell concludes that the electrons are connected non-locally - by a kind of voodoo, as it were.

Those of you who are intimately familiar with Bell's Theorem - have I correctly understood the essence of it (disregarding the silly aspects of how the experiment is described)?


.

Each electron carrying a sign would constitute a local hidden variable. Thinking about the experiment in terms of each electron carrying a sign from the get-go leads to the wrong answer. Hence each electron doesn't carry a sign from the get-go. Hence no local hidden variables.

Is that what you're saying? Itseems like a pretty good explnation of Bell's theorem to me.

[Duder]

Each electron carrying a sign would constitute a local hidden variable. Thinking about the experiment in terms of each electron carrying a sign from the get-go leads to the wrong answer. Hence each electron doesn't carry a sign from the get-go. Hence no local hidden variables.

Is that what you're saying? Itseems like a pretty good explnation of Bell's theorem to me.

I think get what you are saying. The electron, after it has left the electron gun and before it hits the detector, is in an indeterminate state. In terms of how I described things above, it carries a "blank sign".

The other electron in the twin pair also carries a blank sign. When the electrons hit their detectors, then symbols appear on the signs.

The natural question that follows is, when the electrons arrive at their detectors and symbols appear on their respective signs, how is it that the same symbols appear on both paired particles, unless there was something intrinsic to the particles that would determine the attributes that would appear at the detectors?

[technomage]

I think get what you are saying. The electron, after it has left the electron gun and before it hits the detector, is in an indeterminate state. In terms of how I described things above, it carries a "blank sign".

The other electron in the twin pair also carries a blank sign. When the electrons hit their detectors, then symbols appear on the signs.

The natural question that follows is, when the electrons arrive at their detectors and symbols appear on their respective signs, how is it that the same symbols appear on both paired particles, unless there was something intrinsic to the particles that would determine the attributes that would appear at the detectors?

The theory that Bell was trying to establish is that the electrons are, somehow, linked--but not because of any intrinsic attributes of the individual electrons themselves (in other words, there are no "local hidden variables").

Beyond that ... heck, Duder, I barely understood the "for Dummys" version you gave, and I'm making absolutely no headway on the Wikipedia article. I'm lost.

[Duder]

The theory that Bell was trying to establish is that the electrons are, somehow, linked--but not because of any intrinsic attributes of the individual electrons themselves (in other words, there are no "local hidden variables").

Beyond that ... heck, Duder, I barely understood the "for Dummys" version you gave, and I'm making absolutely no headway on the Wikipedia article. I'm lost.

Don't blame ya. I've tried to get BT for years in several different books, and it always eluded me. Here - try this:

http://www.ncsu.edu/felder-public/ke...pers/bell.html

My explaination was a paraphrase of it, from memory.

[geochron]

I think get what you are saying. The electron, after it has left the electron gun and before it hits the detector, is in an indeterminate state. In terms of how I described things above, it carries a "blank sign".

The other electron in the twin pair also carries a blank sign. When the electrons hit their detectors, then symbols appear on the signs.

When one electon hits its detector, the symbols appear on the other electron's sign.

The natural question that follows is, when the electrons arrive at their detectors and symbols appear on their respective signs, how is it that the same symbols appear on both paired particles, unless there was something intrinsic to the particles that would determine the attributes that would appear at the detectors?

I guess this is where the classical world turns out to be a poor analogy to the quantum world.

[Duder]

I'm also only familiar with the broad outlines of Bell's theorem. I guess I have two questions:

1) For entanglement occur, isn't it necessary for the two particles to have been together at some point?

Right - and unless I miss my guess, every particle that exists within our light cone "has been together" with every other, in that it has had some effect on every other, however slight. Hence, there would be a kind of global, non-local interconnectedness between all of these particles forever and ever after.

Moreover, wasn't every particle together-with or at least very close to every other particle in the first moments of the big bang, before inflation pulled them all apart into seperate realms?

2) Isn't Bell's theorem often used not so much as a theorem of non-local interaction, as it is an example of how macro-world concepts like "local" and "non-local" are themselves only loose analogies for what is actually going on at the quantum level? Like Bohr's view of the Copenhagen Interpretation. In other words, we get confused when we leave the equations behind and take our macro-scale analogies for truth. Local and non-local "mean" something to us, and the particles' behavior seems to violate that meaning. It seems spooky to us only because we take that meaning for something that must hold true at all scales (the classical view).

-Neil

I think that's a fair assesment, as far as my limited understanding. Rather than thinking of spooky, voodoo connections at a distance, we could think in terms of Bohm's undivided whole model of quantum reality, or Wheeler's interconnectedness model of quantum reality.

[Bald Ape]

The results of Bell's experiments cannot be explained with neo-Bohrian physics, and only make sense if we posit an intelligent light-turner-onner that exists at all possible detector locations simultaneously.

[Duder]

The results of Bell's experiments cannot be explained with neo-Bohrian physics, and only make sense if we posit an intelligent light-turner-onner that exists at all possible detector locations simultaneously.

Now that was FUNNY!

[NeilUnreal]

Moreover, wasn't every particle together-with or at least very close to every other particle in the first moments of the big bang, before inflation pulled them all apart into seperate realms?

Good point.

we could think in terms of Bohm's undivided whole

For some time now, I've been wanting to read Wholeness and the Implicate Order. I think I'll add it to my holiday reading list.

-Neil

[Duder]

Good point.



For some time now, I've been wanting to read Wholeness and the Implicate Order. I think I'll add it to my holiday reading list.

-Neil

I found it a little impenatrable in places, but very interesting in other places. Let me know what you think of it.