## Is the wavefunction ontological or epistemological?

###
Part 3: Bell
Theorem

There are several ways to understand Bell ’s
theorem. Let’s start with a simple but relevant coin game. Suppose two people,
say Alice and Bob want to play the following game: each of them has a coin and
they go in separate rooms. Then each of them flip their coin and record the
outcome. Also they try to guess their partner’s experimental outcome. They win
if both of them guessed right. Suppose this
game is played N times and suppose Alice and Bob agree ahead of time on a
common strategy to maximize their odds of winning this games, but they cannot
communicate during the actual games. What would be the best possible odds of
winning the game? A natural first guess would be 25% because the coin flips are
independent and each can guess right 50% of the time. However, there are two
strategies of reaching 50% overall (try to figure out the solution). The 50%

*is Bell’s limit (or the*__correlation limit__*limit) in this particular case.*__local hidden variable__
There are many Bell
inequalities and they establish the maximum possible correlations of classical
separated systems unable to send signals. However, using Bell ’s
theorem is that it provides a way to settle the issue of the existence of local
hidden variable theories by experimental means. One way to understand why
quantum mechanics can achieve higher correlations is by invoking quantum
superposition. But the actual root cause of superposition is the existence of
Hilbert spaces which demands that the logic of nature is the logic of
projections, and not the usual Boolean logic. In particular the logical
operators “OR” and “NOT” are behaving strangely in quantum mechanics. To see
how this happens, we need to start with the usual Venn diagrams of classical
sets.

*one can achieve higher correlations. This can be checked experimentally and the major merit of*__quantum mechanics resources__
In particular, one Bell
inequality (based on an idea by d’Espagnat) can be nicely explained pictorially
using three overlapping circles corresponding to three sets: A, B, and C:

From the picture one can easily see that [A and not B] + [B
and not C] >= [A and not C]

This is my favorite example of Bell
inequality and the interesting fact is that in a particular case, Nature (and
quantum mechanics) does violate it.

So how can we translate this in an actual physical system?
We’ll follow one of Bell ’s papers
from his famous book:

*Speakable and Unspeakable in Quantum Mechanics*(http://www.amazon.com/Speakable-Unspeakable-Mechanics-Collected-philosophy/dp/0521368693):*Bertlmann’s socks and the nature of reality:*
Suppose we have a source of pair particles in total spin
zero as in the EPR -B experimental setting.
Also suppose we measure the spin on 0, 45, and 90 degrees angles. From the
inequality above:

The number of particles that could pass at 0 and not at 45

+

The number of particles that could pass at 45 and not at 90

>=

The number of particles that could pass at 0 and not at 90

Since in an EPR -B pair
the spins are always anticorrelated, it is impossible to detect the left and
the right particles spin oriented in the same direction. Hence we can switch
the “not part” for one particle with measuring the other particle on that
particular direction.

Now the inequality becomes:

The probability of one particle passing at 0 and the other
at 45

+

The probability of one particle passing at 45 and the other
at 90

>=

The probability of one particle passing at 0 and the other
at 90

It is time to plug in quantum mechanics’ mathematical
formalism and compute those probabilities. For an angle alpha between the two
measurement directions, the probability is ½ [sin(alpha/2)] ^2

Plugging this in the Bell
inequality yields:

½ sin(22.5)^2 + ½ sin(22.5)^2 >= ½ sin(45)^2

or

0.1464 >= 0.2500
IMPOSSIBLE!!!

And experiments confirm the violation of Bell ’s
inequalities precisely as quantum mechanics predicts. The explanation for
nature’s strange correlations cannot be explained by any local causal mechanism.
Regarding correlations, nature is nonlocal. For some time people got spooked by
this, but now the emphasis is on engineering: exploiting this correlation nonlocality
for practical applications.

In the next post I’ll attempt to introduce an intuition on
all this and illustrate in a geometric way the Hilbert space (responsible for superposition
and strange correlations).

Florin,

ReplyDeleteInteresting post.

Did you read my fqxi essay this year (well supported and 2nd in community scoring).

I show that although Bells theorem is correct, and relativity cannot be entirely deterministic, we can resolve the EPR paradox without superluminal communication.

The solution is geometric.

I can see the problem and constraint of your assumptions, which is not recognising nature's reality of (Godel) n-values between 0 and +1, and 0 and -1.

So setting constraints of cardinalised +1/-1 is not modelling nature, which is what Bell was trying to do. Bells belief in no boundary between SR and QM is also verified.

I do hope you'll read the essay and comment.

Best wishes

Peter Jackson

Thanks Peter,

DeleteI'll put your essay on my "to read" list for today. I don't quite get your statement: "we can resolve the EPR paradox" In your opinion, what exactly is the paradox there? (many people have may opinions about this)