May 24, 2005

part 10 - bell's inequality

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John Bell became interested in the EPR paradox. He went in thinking that Einstein was intellectually superior to Bohr and that Einstein was probably correct about a hidden variable. He didn't however feel strongly either way. He just wanted some more clarity. Bell was able to spend some time on the problem when he spent a year at SLAC at Stanford University.

Bell started with the locality assumption. There was no reason to assume this was not correct. And up until then no one had questioned it. When Newton couldn't come up with a mechanism for his gravity force he simply responded, "I frame no hypothesis". And up until Bell all interactions were framed in terms of the four forces - gravity, electromagnetic, strong, and weak. Two things in common with these four forces are they all decrease with distance from the source of the fields and they all travel at the speed of light at best.

If non-local forces existed they would not decay with distance since nothing crosses the intervening space. They act as strongly at a millimeter as ten thousand kilometers. And they would act instantaneously.

Let's tackle Bell's theorem. It surprisingly is very easy to understand and it goes shockingly quick. So pay attention. We start with the same experimental setup as in the EPR thought experiment. Bell however considers another measurement. Something called polarization correlation. We set the green polarizer device at some angle A and the blue polarizer device at some angle B. If we set off two photons at these devices then blue will measure a hit or miss and green will measure a hit or miss. If they both hit or both miss then we define PC(A-B) as a match (where PC stands for polarization correlation). If they are different then we define it as a miss. Simple. Nothing hard yet. We can keep conducting this experiment and get the percentages of matches and misses for any given set of angles for the two polarizers.

At PC(0) we know we get 100% matches and 0% misses. There's no angular difference between the devices and we've already stated that the photons are phase entangled and thus always record the same polarization - they always match. At PC(90) we get 0% matches and 100% misses.

It turns out that the only thing that matters in this experiment is the delta between the two angles, not the actual angles of the polarizer. Makes sense since the light is unpolarized in every direction - what would make it prefer one angle over another.

Since we know the two limiting cases we can presuppose that PC() of anything in between 0 and 90 gives something larger than 0% and less than 100% for the matches. This is born out by experiment. We might get something like this for an angle in between - H(Hit) and M(Miss)

Blue measurement
Green measurement

Some of these are matches and some are not. We conduct enough trials and we have a statistically significant sample and we can calculate the value of PC() for that particular angle. Again this is very straight forward.

Now at some angle (or rather at some angular difference between the polarizer device settings) we will get PC(some angle)=75%. 75% of the trials result in a match between the two polarizers and 25% of the trials result in misses. On average 1 in 4 tries does not match. Let's call this angle, J.

Align the polarizer devices at 0 degrees. We observe the matches are 100% - hits and misses match.

Move the green polarizer by J degrees. We observe the matches are 75%. Move it back.

Move the blue polarizer by J degrees. We observe the matches are 75% again. No surprise.

This occurs in the lab and it occurs if the devices are one billion miles apart.

Ready? Now we demonstrate the proof.

Let's assume locality. Turning the blue device can change only the blue reading. If you don't believe this think about the polarizers being a billion miles apart and the device moves into place just before the photon arrives. If the blue device changes the green reading then it had to travel 2 billion miles in the blink of an eye.

Now turn the blue detector by J degrees. At the same time turn the green detector by J degrees in the opposite direction. They are now 2J degrees apart. What's match rate? Let's hypothesize. Since only turning blue by J degrees puts one miss for every 4 attempts (on average, 75% match) and only turning green by J degrees puts one miss for every 4 attempts we might naively guess that when we turn both detectors we get two misses every 4 attempts on average - or two matches per 4 attempts. However in some cases the misses will lead to matches. Blue will miss 1 in 4 but green could 'miss' in the same way on that same attempt. But let's not figure out by how much - it's not important. Let's just content ourselves that some of these misses will lead to matches. [read that again and understand it, we're done by the next two sentences.]

Therefore we would more correctly assume that the match rate is greater than 2 matches per 4 attempts. Thus PC(2J) must be greater than 50%.

So how do the experiments come out? PC(2J) = 25%. We get 3 misses for every 4 attempts on average. Go back and look at what Bell assumed. He assumed locality. That's all he assumed. Locality is wrong. The world is non-local. Assuming locality and all that crap I wrote above about "if you don't believe this think..." is WRONG. It does act instantaneously two billion miles apart.

There is one interesting thing about Bell's Inequality that you may have missed. It doesn't depend on quantum theory at all. It was entirely developed with logic and facts. Suppose quantum theory becomes replaced by a more accurate theory eventually. That will not have any effect on Bell's findings. His result stands. Non-locality is a reality of the world.

What's truly amazing here is that anyone even found out about this. We can't detect how nonlocality works. There's no force to measure. There's no experiment to see nonlocality in direct action. There's only Bell's Inequality that comes out of some very basic experimental facts and some logic. But to go out and show someone non-locality in action and say 'watch this' is impossible. It's only inferred from experimental facts.

Well at first this seems to clearly support Bohr in our previous discussion. But Bohr doesn't really like the idea of non-locality. At the same time you might think Einstein hits the skids at this point. But ironically we now have a mechanism for our pilot wave (our hidden attribute model) to act instantaneously and without detection on our quons. But Einstein doesn't really like the idea of non-locality either. So in the end we really can't place Bohr or Einstein as the winner of this argument. And since non-locality is so against the grain of what science believed at the time, it just left everyone with a bad taste in their mouths. So be it. They got over it.

That's it. I'm done. I'm going to go read Dr. Seuss now to unwind.

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1 comment:

Anonymous said...

Just thought I would post my solution to Bell's Inequality here, would really love feedback on this from that study Bell's Inequality