JenniT said:
Purpose = I am trying to arrive at the heart of Bell's mathematics.
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BELL LOCALITY
Section II ("Formulation") of Bell's original paper begins with a summary of the EPR argument in terms of the example of Bohm and Aharonov (spin-½ particle-pair in "singlet" state).
In that section, without providing any mathematical details, Bell promptly arrives at the following conclusion:
... it follows that the result of any such measurement must actually be predetermined.
It is in connection with this (EPR-type) argument that the "Bell Locality" condition becomes relevant. That is, the "Bell Locality" condition is intended to provide a
mathematical basis by which one is able arrive at the above conclusion of "predetermined outcomes" – not
just for QM ... but for
any theory.
[In another thread, I have referred to this part of the Bell argument as "stage 1".]
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DEFINITION
"Bell Locality" (for the "Alice-and-Bob, spin-½" scenario) is defined by the following two symmetrical probability conditions:
(a) P(A|
a,
b,B,λ) = P(A|
a,λ)
(b) P(B|
a,
b,A,λ) = P(B|
b,λ) .
In the above:
A ≡ Alice's outcome (±1) ,
B ≡ Bob's outcome (±1) ,
a ≡ Alice's setting (some unit vector) ,
b ≡ Bob's setting (some unit vector) ,
and λ denotes a
complete specification of the "state" of the particle pair with respect to a spacelike hypersurface S having the following characteristic (see
diagram[/color]):
S intersects the (two) backward light-cones (associated with the measurements of Alice and Bob) in their regions of non-overlap.
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APPLICATION to "stage 1"
From conditions (a) and (b) in conjunction with the usual rule for conditional probabilities, it follows that
P(A,B|
a,
b,λ) = P(A|
a,λ) P(B|
b,λ) .
To this relation, we then apply the condition of "perfect anti-correlation" for equal settings, namely,
[1] P(A=s,B=s|
a=
n,
b=
n,λ) = 0 (arbitrary s and
n) ,
and arrive at
[2a] P(A|
a,λ) = 0 or 1 (never in-between)
and
[2b] P(B|
b,λ) = 0 or 1 (never in-between)
[for details see section III, page 6, of the following paper:
http://arxiv.org/PS_cache/quant-ph/pdf/0601/0601205v2.pdf ] .
From [2a] and [2b], it is seen quite clearly that we are dealing with "predetermined outcomes".
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CONCLUSION of "stage 1"
Any theory which satisfies both
(i) "Bell Locality"
and
(ii) "perfect anti-correlation" for equal settings
will
necessarily have "predetermined outcomes" with respect to
each (and
every)
complete specification λ.
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In that case, we may as well just write (in place of [2a] and [2b]):
A(
a,λ) = ±1 , B(
b,λ) = ±1 .
This is where the mathematics begins in Bell's original paper (equation (1) therein).
[In another thread, I have referred to this part of the Bell argument (i.e. from equation (1) and onward) as "stage 2".]
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JOINING "stage 1" to "stage 2"
Upon joining together the arguments of these two "stages", we arrive at the following:
Any theory which satisfies both
(i) "Bell Locality"
and
(ii) "perfect anti-correlation" for equal settings
will
necessarily satisfy "Bell's inequality".
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EVALUATING a specific CANDIDATE theory
Suppose we are given a
specific candidate theory for which we can calculate the various probabilities associated with the possible outcomes of Alice and Bob, and that (upon calculating) we find for some λ, A, B,
a, and
b that
P(A|
a,λ) P(B|
b,λ) ≠ P(A,B|
a,
b,λ) .
Then, the candidate theory in question does
NOT satisfy "Bell Locality".
For example, this is the case for QM:
P(A|
a,λ) P(B|
b,λ) = ½ ∙ ½ = ¼ ≠ P(A,B|
a,
b,λ) .
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MEANING of "Bell Locality" VIOLATION
... I am sorry, I have not yet fully sorted this matter out! ...
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