On 11/05/2016 11:37 pm, Bruno Marchal wrote:
On 10 May 2016, at 02:10, Bruce Kellett wrote:
Bell's original argument didn't mention collapse, and the argument
that his theorem fails because he assumed definite outcomes from
measurements is actually without substance: no such assumption is
required by Bell.
Bell does not mention collapse, nor EPR, because it is the assumption
by default.
Bell didn't mention collapse because his argument is valid whether or
not you make this assumption (or the equivalent assumption of
countertfactual determinism). Let me try and explain it in a different way.
We have Alice with her measurement apparatus where she can make a
measurement x, say, and get result a. Similarly Bob can make a
measurement y and get result b. The Bell of CHSH inequalities apply to
the outcomes of experiments, so we calculate probabilities (expectation
values) for a sequence of joint outcomes for Alice and Bob. Since we are
dealing with correlations between outcomes, the data we are working with
come only after Alice and Bob have met and exchanged information. Let us
call the joint result of such a meeting in one world as (ab|xy) for
results a and b given settings x and y, respectively.
If there are two possible outcomes for a and b, there are four such
combinations in the superposition representing any run of the
experiment. If, in order to estimate correlations, we do N runs of the
experiment, there will be 4^N possible sequences of results for Alice
and Bob, represented by:
(ab|xy)_1, (ab|xy)_2, (ab|xy)_3,........(ab|xy)_N,
where the subscripts indicating run number actually apply to all of
a,b,x, and y; and a,b can each be + or -, giving the 4^N distinct
sequences. In the many worlds approach, all such sequences are realized
in one world or another.
All we have to do now is choose a typical world, any world, and look at
that particular sequence. The sequence consists of actual measurement
results, pointer setting, or whatever. And they occur after Alice and
Bob have met after each trial. So this is now just a sequence of results
that could have been obtained in a run of N trials in the collapse
model. There is absolutely no difference in the data sequences obtained
from either model.
The Bell inequalities are obtained by calculating the expectation values
for just such sequences: where the sequence came from -- a collapse
model or selection of one world from the universal wave function -- is
irrelevant for Bell's calculation. So Bell did not make explicit an
collapse assumption because no such assumption is needed for his derivation.
The proof of non-locality, even in a many worlds model, is immediate.
Since the sequence under consideration comes from a series of quantum
events it must violate the Bell inequalities. And Bell has shown that
these inequalities must hold for any local theory. Hence quantum
mechanics, even in the many worlds interpretation, is non-local.
Bruce
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