Originally posted by twhitehead
Any interaction.
[b]Given that entangled states between particle pairs can exist over large distances I've got to ask what you mean by a box?
Any set of particles/states grouped together. For it to be useful, it makes sense to use collections that are co-dependent in some way, so poison + cat for example.
Come to that what do you mean by "fl ...[text shortened]... consciously aware of it or not, then the wave function has collapsed as far as he is concerned.
Any interaction is too wide. I can justify that statement, but this post is huge so I'll leave it unless you are going to challenge it again.
You have the wavefunction collapsed by the computer, but not as far as the scientist is concerned. Either you have the wavefunction as something non-physical, you're interpreting it as some sort of Bayesian knowledge thing (this is a known interpretation), or this is a variant of the many universes interpretation.
The slight problem with your set up is you have continuous observation of the cat. Suppose the computer checks on the cat after one half-life of the atom which determines moggies fate.
Let AC(x, t) be the wave function of the Alive Cat at time t, DC(x, t) is the wave function of a dead one. D(x, t) is the wave function of the computer (D for detector) up until it makes a measurement, with AD(x, t) and DD(x, t) being wave functions of the computer when it has registered an alive or dead cat. E(x, t) is the wave function of the (conscious) experimenter, with AE(x, t) and DE(x, t) having the same meaning as for the cat.
We set up the experiment, at this point the wavefunction is:
W(x, t0) = E(x, t0)D(x, t0)AC(x, t0)
The cat is definitely alive at this point so we can use the wavefunction for a living cat. We wait for one half-life and the cat's wavefunction evolves into 1/sqrt(2) (AC(x, t1) + DC(x, t1)). I'll drop the factor of root 2 from now on as it doesn't add much. Just before our cat mortality detector does it's thing the overall wave function (up to normalization) is:
W(x,t1) = E(x, t1)D(x, t1)(AC(x, t1) + DC(x, t1))
if the wavefunction collapses at this stage (lets say to alive) it will be:
Wcat(x, t1) = E(x, t1)D(x, t1)AC(x, t1)
Wcat(x, t1) just means the wavefunction if wavefunction collapse happens now.
The detector does it's thing, the uncollapsed wavefunction is now:
W(x, t2) = E(x, t2)(AD(x, t2)AC(x, t2) + DD(x, t2)DC(x, t2))
If the detector collapses the wavefunction it would be:
Wdet(x, t2) = E(x, t2)AD(x, t2)AC(x, t2)
had the wavefunction collapse happened earlier the system would have evolved continuously into:
Wcat(x, t2) = E(x, t2)AD(x, t2)AC(x, t2)
Note that Wcat(x, t2) = Wexp(x, t2).
The experimenter comes along and we get:
W(x, t3) = AE(x, t3)AD(x, t3)AC(x, t3) + DE(x, t3)DD(x, t3)DC(x, t3)
Where the first term corresponds to a universe where the cats been lucky and the second term corresponds to a universe where the cat's been unlucky. The experimenter is in a linear superposition, but isn't conscious of it, in fact the experimenter has been copied with the only difference being whether his brain is in a state of mourning or relief. This is the basis of Everetts many universes interpretation.
If the experimenter collapses the wavefunction we would have:
Wexp(x, t3) = AE(x, t3)AD(x, t3)AC(x, t3)
and had the collapses happened earlier we have:
Wcat(x, t3) = Wdet(x, t3) = Wexp(x, t3)
So by now the outcome of "early" and "late" collapse is
indistinguishable even at the level of wavefunctions.
You seem to have Wdet(x, t3) but there is no reason to prefer it to Wcat or Wexp. The Everett interpretation does have a different wavefunction, but it is:
W(x, t3) = AWexp(x, t3) + DWexp(x, t3)
There may be a way to distinguish this from the collapse type interpretations, but I don't know what it is.
The key point I'm trying to make is that there is no way on God's green Earth that you can distinguish the collapse happening early and the collapse happening late, and you can't experimentally demonstrate that it is definitely not because of the consciousness of the the experimenter. In a dualist outlook the experimenters body (including his brain) can be in a linear superposition of states right up until the point that his mind collapses the wavefunction. There is no experimental way of ruling this out.