Herbert's Proof of Bell's theorem: A Logical Restatement
Posted Apr5-12 at 11:21 PM by lugita15
Here is Nick Herbert's version of the proof of Bell's theorem. It is, I believe, the simplest known proof of Bell, and the particular numerical example used by Herbert was in fact the one that Bell used in his talks to popular audiences.
I should say, however, that Herbert states the conclusion of his argument a bit too strongly. He claims that his reasoning implies that experiments have definitively disproven the hypothesis that reality is local. But that's not quite true, because there are various experimental loopholes. So I would rather state the conclusion in this way:
"The predictions of quantum mechanics cannot all be completely correct in a local deterministic universe." And without further ado, here is my restatement of Herbert's argument to get to that conclusion:
1. Entangled photons behave identically at identical polarizer settings.
2. The believer in local hidden variables says that the polarizer angles the photons will and won't go through are agreed upon in advanced by the two entangled photons.
3. In order for the agreed-upon instructions (to go through or not go through) at -30 and 30 to be different, either the instructions at -30 and 0 are different or the instructions at 0 and 30 are different.
4. The probability for the instructions at -30 and 30 to be different is less than or equal to the probability for the instruction at -30 and 0 to be different plus the probability for the instructions at 0 and 30 to be different.
The last statement is what is known as a Bell inequality, and it is in direct contradiction with the predictions of quantum mechanics, which in this case says that the probability for the results to be different at -30 and 30 is actually GREATER than the probability for the results to be different at -30 and 0 plus the probability for the results to be different at 0 and 30.
I should say, however, that Herbert states the conclusion of his argument a bit too strongly. He claims that his reasoning implies that experiments have definitively disproven the hypothesis that reality is local. But that's not quite true, because there are various experimental loopholes. So I would rather state the conclusion in this way:
"The predictions of quantum mechanics cannot all be completely correct in a local deterministic universe." And without further ado, here is my restatement of Herbert's argument to get to that conclusion:
1. Entangled photons behave identically at identical polarizer settings.
2. The believer in local hidden variables says that the polarizer angles the photons will and won't go through are agreed upon in advanced by the two entangled photons.
3. In order for the agreed-upon instructions (to go through or not go through) at -30 and 30 to be different, either the instructions at -30 and 0 are different or the instructions at 0 and 30 are different.
4. The probability for the instructions at -30 and 30 to be different is less than or equal to the probability for the instruction at -30 and 0 to be different plus the probability for the instructions at 0 and 30 to be different.
The last statement is what is known as a Bell inequality, and it is in direct contradiction with the predictions of quantum mechanics, which in this case says that the probability for the results to be different at -30 and 30 is actually GREATER than the probability for the results to be different at -30 and 0 plus the probability for the results to be different at 0 and 30.
Total Comments 0
