Bell Proof Against Hidden Variables in EPR

In summary, the conversation discusses John Bell's paper on hidden variable approaches in quantum mechanics. The speaker has a question about the logic behind a certain point in the paper and wonders why the probability function cannot be stationary at a certain point. The other person explains that this is due to the difference between b and c taking all sorts of small values, and that the CHSH inequalities were derived to address this issue. The violation of these inequalities requires b and c to be orthogonal, and the magnitude of their difference is equivalent to the square root of 2. The conversation also mentions that Bell's theorem is often seen as ruling out local hidden variables, but the speaker notes that this is a contemporary perspective and that Bell himself viewed it as a failure of
  • #1
msumm21
218
16
I have a question regarding the paper by John Bell (www.drchinese.com/David/Bell_Compact.pdf‎ ) in which he shows that a certain hidden variable approach cannot reproduce the expectation values predicted by QM for a pair of particles in the singlet state.

After eqn 15 on page 4, I don't understand the logic. Why can't ##P(b,c)## be stationary at the point ##b=c##? Seems like ##P## could have a minimum at ##b=c## and hence be a stationary point. How does ##P(b,c)## being the order of ##|b-c|## around ##b=c## prevent that? I guess I'm missing something big here.
 
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  • #2
Hi Actually saying Bell's theorem rules out local hidden variables is today's take. Bell viewed the violation of his inequalities as a failure of local causality of Special Relativity.

To me your question is just this: as b and c get close, then the difference b-c will take all sorts of small values and in Bell's words is not stationary.

I think this is a mathematical point: apply two colinear fields at b=c and the correlation will also be non stationary.

This point was addressed, if I recall, by CHSH who derived their inequalities to remove this point. However please note that this case (b=c) is very rare and the violation of the CHSH requires b and c to be orgononal, not colinear. Hence their magnitude is |b-c| = root(2)--look familiar.

hope this helps
 
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Related to Bell Proof Against Hidden Variables in EPR

1. What is the EPR paradox?

The EPR paradox is a thought experiment proposed by Albert Einstein, Boris Podolsky, and Nathan Rosen in 1935. It involves the entanglement of two particles, where measuring one particle's properties instantaneously affects the properties of the other particle, even if they are separated by large distances. This phenomenon challenged the principles of local realism, which states that objects have definite properties independent of observation.

2. What are hidden variables?

Hidden variables refer to the properties of particles that are not directly measurable or observable. In the context of the EPR paradox, hidden variables are proposed as a way to explain the instantaneous connection between entangled particles without violating the principles of local realism. These hidden variables would determine the outcomes of measurements on entangled particles, but they are not accessible to us through observation.

3. How does Bell's theorem relate to the EPR paradox?

Bell's theorem is a mathematical proof that shows that no local hidden variable theory can reproduce all the predictions of quantum mechanics. This means that if we assume that particles have definite properties that determine their behavior, then the observed correlations between entangled particles cannot be explained. Bell's theorem provides a way to test for the presence of hidden variables and has been experimentally confirmed.

4. What is Bell's inequality?

Bell's inequality is a mathematical expression that sets a limit on the correlations between measurements on entangled particles in a local hidden variable theory. If the observed correlations exceed this limit, then it can be concluded that the particles do not have definite properties and that there are no hidden variables determining their behavior. Bell's inequality was first proposed by physicist John Stewart Bell in 1964.

5. How does the Bell test prove that there are no hidden variables in the EPR paradox?

The Bell test involves performing measurements on entangled particles in a way that violates Bell's inequality. If the observed correlations between the particles exceed the limit set by Bell's inequality, then it can be concluded that there are no hidden variables determining the particles' behavior. This provides evidence for the non-existence of local hidden variables and supports the principles of quantum mechanics, which state that particles do not have definite properties until they are measured.

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