From: *Bruno Marchal* <[email protected] <mailto:[email protected]>>
On 11 Jun 2018, at 03:37, Bruce Kellett <[email protected]
<mailto:[email protected]>> wrote:
From: *Bruno Marchal* <[email protected] <mailto:[email protected]>
On 8 Jun 2018, at 14:55, Bruce Kellett <[email protected]
<mailto:[email protected]>> wrote:
The choice of basis makes all the difference in the world.
Everett prove the contrary, and he convinced me when I read it. I
found “his proof” used in many books on quantum computing, although
with different motivation. Thee result of an experiment, obviously
depend of what you measure, but when you embed the observer in the
wave, you get that what they find is independent of the choice of
the base used to describe the “observer” and the “observed”. If not,
the MW would already be refuted.
In that case, MW is refuted. Clearly, what the observer finds is
dependent on the basis in which he is described.
?
I disagree. The finding can depend on what the observer decide to
measure, which is akin to choosing a base,
I think there is more to it than this. We can choose a base in which to
describe the state, but we are not able to choose the base for the
actual measurement result. In a spin measurement of a spin half
particle, we can decide to orient the magnet at any angle to the
direction of motion. For convenience in understanding the dynamics, we
then expand the spin wave function in the basis corresponding to that
orientation; that is choosing what to measure. But the result of our
measurement in that orientation is that the particle emerges from the
S-G magnet on either the up or down trajectory. The actual measurement
is then made at screen downstream of the magnet. We do not have any
control over the basis for the resulting position measurement.
Decoherence decides that for us, and the stable basis for position
measurements is the set of delta functions at each point along the
spatial axis. Why do we not see the 'up' or 'down' result as a
superposition of a lot of different positions on the screen? We don't
because such superpositions are not stable under decoherence.
but the couple “observer + that chosen base” can be studied in any
base, and the same result will described in the memories of the
observer. I will search Everett proofs of this, as he is the one who
convinced me on this.
The situation as I understand it is the following. The original wave
function |psi> can be expanded in any basis that spans the corresponding
Hilbert space. So we can have
|psi> = Sum_i c_i |a_i>
= Sum_j d_j |b_j>,
where the |a_i> and |b_j> are sets of vectors which independently span
the space. The expansion coefficients are different, so in general the
c_i =/= d_j.
If we measure in one basis, say that of the |a_i>, the result is one of
the |a_i> with probability |c_i|^2. However, if the measurement
corresponds to the other basis (that will be a different operator in the
Hilbert space), we will, in general, get a different result, one of the
|b_j> with probability |d_j|^2. The results of these measurement will be
different. If we now decohere these states with the environment, in one
case the environment will be entangled with the |a_i> states, and in the
other case, the environment (including the observer) will be entangled
with the |b_j> states. So the observer will see different things
according to the basis in which he is working and the measurement made.
Now since the bases both relate to the same Hilbert space, and the same
original state is expanded, the basis vectors (and the expansion
coefficients) are related, so we can always express one set in terms of
the other:
|a_i> = Sum_j f_{ij} |b_j>.
So in this sense, the basis chosen does not matter in the overall
description. But once we take environmental decoherence into account,
only one basis will be stable, say the |a_i> basis, and if we describe
things in the |b_j> basis, those basis states are immediately decohered
into the corresponding |a_i> states. Consequently, any observer will
only ever see results, or be located in branches (worlds), corresponding
the the stable decohered base. Describing things in terms of another
base doesn't change the reality, just as describing the orbit of the
Moon in terms of a coordinate system based on Jupiter doesn't actually
change the orbit of the Moon, it just makes the description a lot more
complicated. However, in the absence of decoherence to a preferred
stable basis, the results in the }a_i> and }b_j> bases are different.
Or else experiments would not have definite results when described in
the laboratory from the 1p perspective.
I don’t see why.
Without decoherence, measurements in the |b_j> basis are superpositions
of the |a_i> states, and by construction, only the |a_i> states are
stable under decoherence, corresponding to definite results.
Even if you take the 'bird' view of the whole multiverse -- which is,
I agree, independent of the basis
OK. At least we agree on that.
in which it is described -- the view of any observer embedded in the
multiverse is totally basis-dependent.
Only in the sense that the biological brain has evolved through
decoherence with respect to some base, but as you say, that process
lead to the same result from the 1p perspective of those who have
chosen the base, or have the base imposed through decoherence and
evolution, say.
If not then QM would be inconsistent, and had to different laws of
physics for different observers.
That would be the result absent decoherence to the stable basis for any
measurement.
Bruce
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