On 7/06/2017 10:38 pm, Bruno Marchal wrote:
On 07 Jun 2017, at 11:42, Bruce Kellett wrote:
On 7/06/2017 7:09 pm, Bruno Marchal wrote:
On 06 Jun 2017, at 01:23, Bruce Kellett wrote:
I have been through this before. I looked at Price again this
morning and was frankly appalled at the stupidity of what I saw.
Let me summarize briefly what he did. He has a very cumbersome
notation, but I will attempt to simplify as far as is possible. I
will use '+' and '-' as spin states, rather than his 'left', 'right'.
He write the initial wave function as for the case when you and I
agree in advance to have aligned polarizers:
|psi_1> = }me, electrons,you> = |me>(|+-> - |-+>)|you>
= |me, +,-,you> - |me,-,+,you>
He says that at this point no measurements have been made, and
neither observer is split. But his fundamental mistake is already
present.
A little test for you: what is wrong with the above set of
equations from a no-collapse pov?
skipping some tedium, he then gets
|psi_3> = |me[+],+,-,you[-]> - |me[-],-,+,you[+]>
where the notation me[+] etc means I have measured '+', you[-]
means you have measured '-'.
He then claims that the QM results of perfect anticorrelation in
the case of parallel polarizers has been recovered without any
non-local interaction!
Spoiler -- in order to write the final line for |psi_1> he has
already assumed collapse, when I measure '+', you are presented
*only* with '-', so of course you get the right result -- he has
built that non-locality in from the start.
?
From the start shows that it is local.
Your failure to see the problem here is symptomatic of your complete
failure to understand EPR in the MWI.
I could say the same, but emphatic statements are not helping. My
feeling is that you interpret the singlet state above like if it
prepares Alice and Bob particles in the respective + and - states, but
that is not the case. The singlet state describe a multiverse where
Alice and Bob have all possible states, yet correlated.
The singlet state is rotationally invariant, yes, and can be expanded in
any basis of the 2-d complex Hilbert space. That has never been in doubt.
Then in absence of collapse, all interactions, and results are
obtained locally, and does not need to be correlated until they spread
at low speed up their partners.
That does not follow. Although there are an infinity of possible bases
for the singlet state, these are potential only, and do not exist in any
operative sense until the state interacts with something that sets a
direction. You appear to claim that A and B exist in separate worlds
corresponding to each of this infinity of bases. But that is a
misunderstanding. They are in superpositions in every base, sure, but
that does not mean that there are 'worlds' corresponding to each
possible base until some external interaction occurs. As you yourself
have said, a world is something that is closed to interaction. But
superpositions are not closed to interaction, they can interfere -- as
in the two slit experiment, and essentially every other application of QM.
So there are no separate worlds corresponding to every possible
orientation of the polarizers. Worlds can arise only after interaction
and decoherence has progressed so that the overlap between the branches
of the superposition is zero (FAPP if you like). It is only then that
the branches can no longer interfere (interact) and are closed to
interaction, and thus constitute different worlds.
The standard procedure in quantum mechanics when one is faced with a
superposition that interacts with something external, is to expand the
superposition in a base that corresponds to the external context. That
is what happens when an unpolarized spin meets a polarizer aligned in a
particular direction -- one expands the rotationally symmetric
unpolarized state in the basis matching the external context. That is
all that is happening with the singlet state above; when Alice comes to
measure the symmetric state, it is convenient to expand the singlet
state in a basis that corresponds to the orientation of Alice's
polarizer. Then the result of the interaction is easily calculated. If
one use some other basis, in some other direction, one would end up with
a superposition of states after measurement, and that superposition
would be exactly the same as the eigenstate obtained when one expanded
in the aligned basis. So using a different basis merely complicates the
calculation, it doesn't actually change anything. It is like trying to
drive from Melbourne to Sydney using a map based on an orthographic
projection based on Brisbane. You might manage it, but it would be
needlessly difficult.
I am sorry that I have had to spend so much time on this diversion into
Quantum Mechanics 101, but you seem determined to fail to understand the
application of the most fundamental of quantum principles.
So, in the measurement of the singlet state
|psi> = (|+>|-> - |->|+>),
the basis is arbitrary until someone wants to measure this state. If
Alice measures the state, we expand in Alice's basis and get the above;
Alice has a 50/50 chance of getting '+' or '-'. What is the state after
Alice makes her measurement? According to quantum mechanics, the
measurement reduces the state to the eigenvalue corresponding to the
measurement result. This is entirely local, and is necessary because of
the experimental fact that repeated measurements of the same state give
the same result. So if Alice got '+', the state reduces to |+>|->, and
if she got '-', the state reduces to |->|+>. This is fine for Alice
locally, she is actually measuring only the first part of the
superposition |psi>, the part corresponding to her particle. But the
second part of the state, the '|->' part in |+>|->, corresponds to the
particle that Bob has at his remote location. If everything is local,
then Alice's measurement cannot affect Bob's particle, so Bob must also
be presented with the original state |psi>. His situation is then
exactly like Alice's, we expand the symmetric singlet state in the basis
corresponding to Bob's polarizer, and find that he, too, has a 50/50
chance of getting '+', or '-'. It follows immediately that if the two
measurements are indeed independent, and they are both measuring the
same state unaffected by the other's measurement, both get a 50/50 mix
of the two possible results. And, crucially, their results will be
totally independent, there will be no correlation. Independent
measurements must lead to uncorrelated results, that is what
'independent' means.
But we know that, experimentally, Alice's and Bob 's results are
correlated, anything between -1 and +1, depending on the relative
orientation of their polarizers. So the measurements that Alice and Bob
make cannot be independent: Bob's measurement is affected, in some way
or another, by the measurement that Alice makes (or vice versa). That is
the origin of the claim of non-locality. Before Bell, one could imagine
that there was some hidden variable that carried an interaction from
Alice to Bob. That might have been reasonable if Alice and Bob had a
timelike separation, so that Bob's measurement was in Alice's forward
light cone. But experiment shows that the correlations are the same even
if Alice and Bob make their measurements at space-like separations, so
no sub-luminal hidden variable interaction could connect the two
measurements. That is non-locality.
The question then, is whether many worlds can provide a fully local
account of this situation. I claim, with most present day physicists,
that MWI does not provide any such local account.
After all this, we can go back to Price as above. He writes:
|psi_1> = |me, electrons,you> = |me>(|+-> - |-+>)|you> = |me, +,-,you>
- |me,-,+,you>.
His expansion of 'electrons' into the singlet state is correct, but he
then takes this to give:
|me>|+->|you> - |me>|-+>|you>.
So that if I measure '+', you are presented with the collapsed state
|+>|-> (in my basis). Similarly if I measure '-', you receive the
corresponding collapsed state. But the |+>|-> in my basis state
corresponds to a |+> polarization for my electron and a |-> polarization
for your electron -- and you and I are widely separate, possibly by
indefinitely large space-like distances! In other words, Price has built
the standard quantum mechanical non-local collapse into his account. Not
unnaturally, he gets the correct correlation results, but then he has
done nothing different from the standard non-local quantum account, so
it is no surprise that he gets the same answers.
Tipler does exactly the same thing with his account of measurements at
arbitrary polarizer angles, differing by theta. And I hope it will not
be necessary for me to go through this tedious analysis for that case
too -- it is exactly the same mistake, doing the standard QM calculation
and claiming that it is totally local.
Another argument is that the linear wave description is described by a
differential equation which imposes locality, and make the
non-locality only apparent in *all* branches (assuming the singlet
state to be 100% pure).
The argument from linearity fails because Schrödinger's equation is
linear only in configurations space, and the two-particles singlet state
is also defined only in configuration space -- each particle exists in
its own 3-subspace of the total configuration space. So while the
particles may be widely separated in ordinary physical 3-space, they are
in different subspaces of configuration space, and that might be
completely local! So it might be the case that linearity implies
locality in configuration space, but that does not carry over into
ordinary 3-space.
As an aside, on an historical note, apparently Schrödinger originally
envisaged his 'wave' as a physical wave in space-time, just like an
electromagnetic wave or some such, and that his equation governed the
local deterministic evolution of this wave in 3-space. When
Schrödinger's formalism was applied to two-body systems, such as the
hydrogen atom, it was realized that each of the two particles had to
exist in separate subspaces of configurations space. Schrödinger was
devastated by this finding, and apparently even went so far as to say
that he wished he had never invented that 'stupid equation' (or
something similar).
I agree it is weird that the "phase space is the real thing", but that
is where the quantum weirdness comes from. Yet, the MWI just abandon
the CFD, I don't see, in the Bell inequality violation any reason to
believe that a influence at a distance should be called for.
As I have said, this simply means that you have not understood it
properly. Incidentally, CFD is just a red herring -- nothing in either
Bell, CI, or MWI ever depends on the violation of CFD.
I can go through that in the sort of tedious detail that I have used
above if you really must, but I would prefer that you just accept normal
physical practice: which is that when faced with a superposition, a
detailed calculation on a typical member of the superposition is all
that is required. We then sum over the result for that typical
component, with weights appropriate for the weights of each component in
the superposition, in order to get the final result. So if there are
several terms in the superposition, there is no violation of
counterfactual definiteness, and one can calculate on just one typical
member. Once again, that is all that happens here, and it is just
standard quantum mechanics.
Bruce
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