[From Bill Powers (2005.04.03.14129 MDT)]
So proof of violations of Bell’s
Inequality seem to be based on not doing experiments. If anybody had ever
just gone ahead and done the experiments and written down the results in
a table, and then analyzed the table, Bell’s Inequality would have been
shown to be true.
Unfortunately, many people have done the experiments, using different
kinds of entities, incuding photons, electrons, and ions. And the
inequality has been shown always to fail.
Let me backtrack. When you say the inequality has been shown to fail, has
this been done by making measurements and filling in a table like the one
shown in Rae on page 37, and then counting the occurrances of up and down
spins in the table? If so then there is something extremely wrong,
because as Rae says, and I can’t contradict him, there is NO WAY those
relationships can violate the basic inequality. That inequality will hold
for ANY arrangement of ups and downs in the table, including
experimentally determined arrangements.
To show me I am wrong, all you have to do is present a table showing the
measured spins in the three columns, and demonstrate that the numbers of
combinations of +'s and -'s do not obey Bell’s Inequality.
Perhaps the problem is that in fact it is impossible to fill in the table
that Rae describes. Maybe that’s what he’s saying that I interpreted to
mean that the experiments weren’t done. But the way he described it, it
would certainly be possible.
New point. On pages 39-40 where he describes the experiments, he seems to
be describing experiments in which only two of the three spins can be
obtained at any given time. Thus in any table of results, one of the
entries for each triplet would be blank. That’s different, of course,
from saying that all three relationships hold at the same time – the two
on the left of the inequality and the relationship between their sum and
the quantity on the right. The counts that Rae talks about couldn’t be
done if there were blanks in the table.
Just to make it clear what we’re talking about, here is the table I’ve
been talking about, with up and down spins being indicated by + and
-:
Condition
1 2 3
+ + -
- - +
- + -
and so on, in any random arrangment.
The Bell Inequality is then simply
n[1+,2+] + n[2-,3+] >= n[1+,3+],
where n[1+,2+] (for example) means number of rows in which
spin is up in both condition 1 and condition 2.
Since this inequality holds for any arrangement of +'s and -'s
whatsoever, it is not false for any arrangement. Therefore if this table
is filled in with experimental determinations, whether done
simultaneously or in any sequence, the inequality must hold true. Rae’s
description of how to obtain the numbers precludes any dependence on
assumptions or experimental conditions or anything else. The inequality
is simply a mathematical fact about tables of this kind and has nothing
to do with reality or experiments. Is that a misunderstanding on my part?
If so, Rae seems to be under the same impression.
That is so obvious it makes me
think that something is missing from these explanations that everybody
knows but nobody bothers to mention.
Do you think the same is true in the quantum eraser experiment?
As I read Harrison’s paper about Bell’s theorem in the link, it says that
if there are three classes of something, with possible properties A, B,
and C, the classes being Class X (those that have A and not B), Y (those
that have B and not C), and Z (those that have A and not C), the total
number of the members of Classes X and Y is at least as great as the
number of class Z. This is consistent with classical logic. The problem
is that in the actual experiment, the “logically necessary”
result doesn’t hold.
Martin, try to answer this simple question. How are the results of the
experiments tabulated? Are they in a table showing triples of results, as
in the Rae table above? If they are, then the inequality holds. There is
simply no way it can not hold, because this is a property of the table
and has nothing to do with how the table was constructed. It is true of
all possible tables of this kind.
The only guess I can make is that such tables have never been constructed
– that the results have been presented in some other way, and never
written down in a table of triplets of + and - spins.
Harrison says:
We have made two assumptions in the proof. These are:
lLogic is a valid way to reason.lParameters exist whether they are measured or not. For example, when
we collected the terms Number(A, not B, not C) + Number(A, B, not
C) to get Number(A, not C), we assumed that either not
B or B is true for every member.
Why do we have to assume anything? Aren’t these three quantities EACH the
results of experimental determinations? In other words, we measure A, B,
and C, noting down whether the spins are + or - after each determination.
That completes one row of the table. Then we repeat as many times as we
like, and finally sum up the numbers of + and - marks for each condition.
Isn’t that the way it’s done?
You may call it spooky, perhaps,
but if you accept the experimental observations, I think you have to deal
with the apparent spookiness.
Since we’re talking about perceptions here and not external reality, the
spookiness is relative to what we expect. The spookiness is in the
explanation we offer, not in nature. I’m not yet convinced that we don’t
have an internal contradiction here, which is the only kind of spookiness
we have to disallow unconditionally.
Best,
Bill P.