Message from discussion
Many Worlds
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From: r...@maths.tcd.ie
Newsgroups: sci.physics.research
Subject: Re: Many Worlds
Date: Sat, 22 May 2004 09:48:42 +0000 (UTC)
Organization: Dept. of Maths, Trinity College, Dublin, Ireland.
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da...@atc-nycorp.com (Daryl McCullough) writes:
>r...@maths.tcd.ie says...
>>>What I mean by "local states" is this: You partition the universe into
>>>a countable number of regions of finite size. Then the state
>>>of the universe is determined by giving the state in each partition,
>>>together with how the partitions are connected. Any classical theory
>>>has this property: There is no nonlocal state information; if you know
>>>what's happening in each region of spacetime, then you know all there
>>>is to know about the universe.
>>
>>There is a "state of knowledge" versus "state of the system" subtlety
>>here. If we talk about states of knowledge, then there can be entangled
>>states in classical systems. For example, I might know that my pen
>>is in one of my two pockets, but not know which pocket. This is
>>nonlocal information because finding out what's in one pocket would
>>tell me something about what's in the other.
>Yes, I agree. But in classical physics, nonlocal states of knowledge
>can be interpreted as imperfect information about an underlying local
>*physical* state. Such an interpretation of quantum mechanics is not
>possible (according to Bell's theorem).
I can see this can only be settled with a rendition of the theorem,
which I provide below.
>>They are a set of conditions that
>>any local theory must satisfy, where "local" refers to the experimental
>>predictions of the theory (in such and such an experiment you will
>>get such and such a result) and not to the mathematical formalism,
>>which is where ideas like "nonlocal states" reside.
>I disagree. The notion of "local" in Bell's theorem necessarily
>involves the notion of local states.
Ok, so here's the version of the theorem which I find the easiest
to fit inside my head all at once (if anybody knows a simpler
one I'd be pleased to hear about it):
There are two experimenters separated by a distance so large that
a light signal cannot reach one experimenter from the other during
the time in which they are performing measurements. One could
imagine that each can see the other on a television screen, and
are broadcasting while they experiment, so that each experimenter
performs his measurements, makes his notes, and then watches the
television to see the other experimenter doing the same procedure,
knowing that the television signals which he is now watching were
travelling towards him across the vastness of space while he was
doing his experiment a few moments before.
Each experimenter recieves N particles and performs one of
three experiments, A, B or C, on each particle. Each of
the experiments gives either +1 or -1 as an answer. The
experimenter decides for himself which experiment he will
perform on which particle, and deliberately doesn't decide
which to do until just before the particle arrives.
We have fairly good reasons to believe that, if the two
experimenters perform the same experiment (A, B or C), they will
get opposite results (one will get +1 and the other will get -1).
The reasons are both theoretical (conservation laws) and
experimental (whenever it has been tried it has in fact turned
out to be true, and it has been tried millions of times).
The condition of locality which is used here is the condition
that, whatever the results of a given experiment depends upon,
it does not depend upon the choice made by the distant experimenter
about which experiment (A, B, or C) he will perform. That is,
locality means that neither experimenter should find that the results
which he gets have been affected by the choices of the other
experimenter, which he sees on the television a few minutes after
he does the experiment.
We say that an experimental result, X, "depended upon" a choice
from the set {A, B, C} by an experimenter if it can be shown (or
statistically inferred) that, had the experimenter made a different
choice, X would have had a different value. In such a circumstance
we can draw up a table like this:
choice | A B C
|---------
value of X | +1 -1 -1
or at least we could do so if we knew what the various results
would have been in the various circumstances.
When we say that X did not depend on a choice, it means that X would
have had the same value that it actually did regardless of what
choice was made, and we can draw a table that looks like:
choice | A B C
|---------
value of X | +1 +1 +1
where it has been assumed that the result was +1. Drawing
such a table isn't very interesting or worthwhile, but it
is important to realise that the statement that a
result doesn't depend on a choice means that such a table
can be drawn up and is accurate.
Now, back to the experimenters. If experimenter 1 is to declare
that one of his results depended on his own choice of which
experiment to do, but did not depend on the experimenter 2's
choice of which to do, then that means that he can draw
up a table that looks like:
choice of experimenter 1 | A B C
---------------------------|---------
choice A| +1 -1 -1
of B| +1 -1 -1
experimenter 2 C| +1 -1 -1
or at least he could if he knew what the other results would have
been had he made the other choices. Since the choice of experimenter
2 doesn't affect the result, the table can be considered to be
specified by any one of its rows.
Now we add in the bit about the experimenters getting opposite
results when they choose the same experiment, and we realise that
if we suppose that the result of an experimenter depends on
his own choice of what experiment to do, but doesn't depend on the
choice of the other experimenter, then we can draw a pair of tables
like:
1's choice| A B C 2's choice| A B C
|--------- |---------
result | +1 -1 -1 result | -1 +1 +1
or at least we could etc. At least, the statements above about
what depends on what are equivalent to the statement that such
tables can be drawn up.
Then we can categorize the N particles according to the
results that would be obtained upon a measurement of
each of A, B and C, by experimenter 1, knowing that that
fixes the result that experimenter 2 would get. For an
individual particle, each experimenter can, having watched
the other experimenter on the television, partially
reconstruct the tables for each particle. That is, suppose
experimenter 1 performed experiment A and got +1 and
experimenter 2 performed experiment B and got +1. Then
the tables can be reconstructed thus far:
1's choice| A B C 2's choice| A B C
|--------- |---------
result | +1 -1 ? result | -1 +1 ?
That is, experimenter 1 can say to himself: "Well he
did experiment B and got +1, so if I had done experiment
B I would have gotten -1." For him to be wrong on this
point, it would have to be the case that, had he
done experiment B instead of A, he would have gotten
+1 (obviously). That would mean that, had he done experiment B,
experimenter 2 would have to have gotten -1, because the two
experimenters have to get opposite results when they do the same
experiment.
So, either experimenter 1 is correct in drawing
up his table (the table on the left), *or* the result of
experimenter 2 depended on the choice of experimenter 1.
Similarly, experimenter 2 has the same justification
for drawing up the table on the right.
So the two experimenters draw up partially reconstructed
tables for each pair of particles. Now, we don't know
what was in column C in the tables above, but what we
do know is that it was +1 in one table and -1 in the
other. That is, the full table for experimenter 1 was
either:
1's choice| A B C
|---------
result | +1 -1 +1
or
1's choice| A B C
|---------
result | +1 -1 -1
There were N pairs of particles altogether. We can call the number
of pairs for which the first table above was correct N_3, and draw
a table which assigns similar symbols to the numbers of pairs of
particles with each table:
1's choice| A B C 2's choice| A B C
|--------- |---------
N_1 | +1 +1 +1 | -1 -1 -1
N_2 | +1 +1 -1 | -1 -1 +1
N_3 | +1 -1 +1 | -1 +1 -1
N_4 | +1 -1 -1 | -1 +1 +1
N_5 | -1 +1 +1 | +1 -1 -1
N_6 | -1 +1 -1 | +1 -1 +1
N_7 | -1 -1 +1 | +1 +1 -1
N_8 | -1 -1 -1 | +1 +1 +1
where N_1 + ... + N_8 = N
The example given above, where experimenter 1 performed experiment
A and got +1 for a result, and experimenter 2 performed experiment B
and got +1 as a result, would be a contribution to either N_3 or N_4 (we
don't know which because we don't know what result would have been
obtained upon a measurement of C, though we know it would have been
either +1 or -1).
Bell's inequality is: N_3+N_4 <= N1+N3 + N4+N7
which is trivial because all of these are non negative integers,
denoting as they do numbers of particles involved in the
experiments.
The numbers N_3 + N_4 and so on, cannot be measured directly,
since each experimenter can only make one measurement on the
particles which arrive, but they can be estimated in the following
way. Take the number of particles for which the
situation above arose (A +1, B +1), and divide it by the fraction
of experiments for which experiments A and B were performed by
experimenters 1 and 2 respectively. If the choice of which
experiment to do does not affect which table is associated with
that particle, then this will provide a fair sampling - that
is, if the pair of experiments (A,B) was done one nineth of the
time, and 500 of those pairs of experiments yielded +1 for
both, then we infer that, had the pair (A,B) been done all
of the time, there would have been about 9*500 incidents, and
so N_3 + N_4 is about 9*500.
We can make estimates of N1+N3 in a similar way by counting
how often experimenter 1 performs experiment A and gets +1
while experimenter 2 performs C and gets -1. Something similar
works for N4+N7.
Notice that this inference can only systematically fail (as
opposed to failing because of a statistical fluctuation) if
the choice of experimenter B affects the appropriate table
for experimenter A, since the choice of experimenter A is
already taken into account in his table. That is, it can
only fail if experimenter B's choice affects the result
of experimenter A, or vice versa.
For the sake of completeness, I'll add that some people
have historically objected to the use of "counterfactuals",
claiming that this argument makes too much use of "What
would have happened if such and such an experimenter had chosen
something else." In each case, where such "facts" were used in the
proof, there was a good reason. For example, it was supposed that,
had C been performed, either +1 or -1 would have been obtained.
This statement was made because the experiment C has been done many
times before and either +1 or -1 have been obtained in each case.
Such inferences are the stuff of science. That is, this objection,
while self-consistent, is rather too powerful since it rejects
the kind of reasoning used in science all the time. For example,
a person who refuses to accept arguments involving counterfactuals
would not accept as evidence of faster than light signalling an
experiment in which the message "This is a faster than light signal"
was sent at one event and received at a distant destination, because
they would have to object that it is not known, or reasonable to
discuss, what would have been received if a different message had
been sent, and so could not accept the conclusion that the contents
of the sent message "caused" the contents of the received message.
Also notice that during the entire proof of the inequalities, the
things referred to were the experimenters, their choices and
experimental results, they tables they draw up and so on. No
mention is made of any theoretical framework or even of any
theory making the predictions. Essentially, the theorem is not
about theories at all, but says that if certain experimental
results are obtained, namely counts of +1's and -1's which
violate the inequalities, then the choices of one experimenter
affect the results of a distant one.
>There is no demonstration that the setting
>of one instrument can affect the results obtained on a distant one.
>That is an *interpretation* of the experimental results.
I think this is where I say that this appears to be incorrect,
unless you're talking about a loophole.
The experiments were done to look for a evidence that the
setting of one device can affect the results obtained on
a distant one, and the evidence was certainly found. I'll
mention quantum mechanics for the first time in this post
to point out that the role it had was to suggest what the
experiments A, B and C should be.
R.
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