From: *Bruno Marchal* <[email protected] <mailto:[email protected]>>
You do seem to have got yourself into a bit of a tangle, Bruno.
What I still do not understand in your view is how can you interpret
|psi> = (|u>|d> - |d>|u>)/sqrt(2)
as a unique superposition. It seems to me you can do that because
Alice and Bob have prepared that state, but that state represents also
another superposition, like |psi> = (|u'>|d'> - |d'>|u'>)/sqrt(2). Why
would |psi> denotes a superposition of |u>|d> and |d>|u> and not
|u'>|d'> and |d'>|u’>. It seems to me that you choose a particular
base, when Everett makes clear that this would lead to nonsense. The
physical state represent by |psi> must be the same whatever base is
chosen.
Of course. When Have I ever said otherwise? Let me spell it out for you.
The basis vectors |u> and |d> are just examples -- place holders if you
like -- for whatever basis vectors are most convenient for your
purposes. Given the expression in terms of |u> and |d>, we can always
rotate to another basis by applying the formulae for the rotation of
spinors that I gave in my paper:
|u> = cos(theta/2)|u'> + sin(theta/2)|d'>,
|d> = -sin(theta/2)|u'> + cos(theta/2)|d'>.
Substitute these expressions in the above, and you will get the state in
terms of the rotated spinors:
psi = (|u>|d> - |d>|u>)/sqrt(2) = (|u'>|d'> - |d'>|u'>)/sqrt(2).
That is what is meant by rotational symmetry, and that is all there is
to it. Nothing could be simpler.
That leads to considering that psi describes not one superposition,
but many superposition. That get worse with GHZ and n-particles state,
and that is why I have often (in this list or on the FOR list of
Deutsch) explained why the multiverse is a multi-multi-multi-…
multi-verse. I don’t insist too much because more careful analysis
would require a quantum theory of space-time, and the Everett theory
will certainly needs some improvement.
|psi> does not describe just one superposition -- by rotation the
spinors we can go to any basis whatsoever. You certainly do not get many
superpositions, one for each possible basis. Let me spell out in detail
how superpositions arise in Everettian quantum mechanics. You start with
a state |psi>, which is just a vector in the appropriate Hilbert space.
The Hilbert space is spanned by a complete set of orthonormal
eigenvectors for each operator in that space. Again, the basis is not
unique, so we can perform an arbitrary rotation in the Hilbert space to
any other set of vectors that span the space. But this is rather beside
the point, because we usually choose a basis because it is useful, not
just because it is possible. (This relates to the preferred basis
problem, which I prefer not to got into at the moment.)
Given a basis, the state |psi> can be expanded in terms of these basis
vectors:
|psi> = Sum_i c_i |a_i>,
where we the basis we have chosen is the set of eigenvectors for some
operator A and labelled them by the corresponding eigenvalues. This
expansion is the basis of the superposition, and of the formation of
relative states (or parallel universes, or the many worlds of Everett.)
We operate on the vector |psi> with the operator A, which gives
A|psi> = A(Sum_i c_i|a_i>) = Sum_i c_i a_i |a_i>
where |c_i|^2 is the probability that we will be in the state relative
to the observed eigenvalue, a_i. We could continue the operation of the
Schrödinger equation to include the apparatus for operator A, the
observer, and the rest of the environment, but you should be able to do
this for yourself.
The point is that this is the only way in which superpositions can be
formed, and from those superpositions, the many worlds or relative
states of Everett develop by normal Schrödinger evolution. But you do
not have such a superposition for the different possible orientations of
eigenvectors for the singlet state. Nor do you have an operator in some
Hilbert space that picks out the angle in which a measurement is to be
made. So the "many superpositions" that you posit are entirely
arbitrary, pulled out of the air without any justification. They clearly
do not form any part of standard quantum mechanics, because the account
of superpositions and Schrödinger evolution to many worlds that I have
given above is the only way in which these can be formed in quantum
mechanics.
It is also why I prefer to describe the “many-worlds” as a many
relative states, (or even many histories), and you are right, they are
not all reflecting simply the superpositions, but different partitions
of the multiverse. When Alice choses a direction for measuring her
particle’s spin, she choose the partition, and enforced “her” Bob,
that is the Bob she can meet in the future, to belong to that
partition, wth the corresponding spin. But she could have used another
direction, and they both would be described (before the measurement),
by a different (locally) superposition, despite it describing the same
state.
That is where you are completely wrong. This is not how the singlet
state is treated in quantum mechanics. Alice could measure the same
state in a different direction simply by rotating her basis to the new
angle at which her magnet is set. Nothing mysterious, no waving of your
magical Everettian wand to produce more superpositions from thin air.
(You were right that it is different from the position of the electron
in the orbital).
So if you can clarify your view of the MW-description of the equality
between (|u>|d> - |d>|u>)/sqrt(2) and (|u'>|d'> - |d'>|u'>)/sqrt(2),
it could be helpful.
Done above.
It seems to me that the “many-worlds” are not dependent of the choice
of the base |u>,|d> or |u'>,|d’>. I mean, unlike Deustch (initially)
and some many-worlders, the whole multiverse has to be the same
physical object whatever base we are using.
Sure, physics has to be independent of the base. But that does not mean
that some bases are not more useful that others. Or that one cannot pick
out a typical basis vector, or branch of the wave superposition, to
argue that anything proved in that branch, and does not depend on the
particular branch chosen, must apply equally to every branch, and so to
the superposition as a whole. This is the basis of my proof that quantum
mechanics is non-local, even in the many-worlds interpretation. Bell's
theorem proves non-locality in a typical branch. That non-locality does
not depend on the eigenvalue measured in that branch (in other words,
the proof is branch-independent), so non-locality is proved for the
singlet state as a whole. And since for the singlet state, for the
universal wave function as a whole.
Dreaming up multiple imaginary superpositions or multi-multi-verses is
not going to change this result.
Bruce
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