trosten said:
This decoherence thing is that a thing that can be proved from QM? As far as I can tell from my books it isn't clear what causes the collapse of the wave function?
Decoherence is indeed the straightforward application of quantum theory without any collapse. The idea is the following: interactions cause entanglement between the interacting parts, and when you do a few straightforward calculations, you find that macroscopic bodies unavoidably interact with their environment (black body radiation of 3 kelvin, a few atoms in a high vacuum ...). It is btw amazing how FAST this entanglement takes place.
And now the trick is the following:
Suppose your "system" is in a superposed state (in what basis, you will ask ; I'll come to that):
|S> = a |Sa> + b |Sb>
And your environment is, well, in a state, |E0>
So the overall state of system + environment is a product state:
|S>|E0>
The interaction between system and environment will extremely rapidly lead to an entanglement:
|psi> = ( a |Sa> |Ea> + b |Sb> |Eb> )
and usually, because of the many degrees of freedom of the environment, |Ea> and |Eb> will be essentially orthogonal.
Now, there is a treatment in standard quantum statistics that demonstrates that ALL LOCAL OBSERVABLES on a system HAVE THEIR EXPECTATION VALUES FULLY DETERMINED BY THE LOCAL DENSITY MATRIX.
The local density matrix is the full density matrix in which we take the partial trace with respect to all the non-local states (here, the environment states).
You can easily work out that it is a diagonal matrix with |a|^2 and |b|^2 on the diagonal. The off-diagonal elements are 0, due to the near orthogonality of the environment states |Ea> and |Eb>.
This describes a STATISTICAL MIXTURE. So, when looking locally onto the system, we have a probabilistic mixture of the state |Sa> and of |Sb> and any interference has disappeared (the system decohered). It is as if we would have applied a collapse.
However, there's a hic in the above demonstration, in that we use density matrices, which themselves are based upon the Born rule. So although we do not need an explicit COLLAPSE anymore, somehow we keep the Born rule which turns coefficients of base vectors into probabilities.
So decoherence theory (which is nothing else but a rigorous application of quantum theory) gives only part of the answer : we still don't know where the Born rule comes from. But it gives already a lot of insight in "why a quantum world looks finally quite classical", even though it doesn't give a final answer.
Many World proponents try to derive the Born rule from the other postulates of quantum theory. I'm in the process of writing a publication on why I think this method is doomed to fail (but I can fail myself, of course
). In my opinion (and I think I have good arguments), you still need the Born rule somehow, and it is still a mystery. However, decoherence teaches us something else:
if you apply the Born rule late enough in the procedure, collapse or no collapse will not make any difference in the results.
This is at the same time a blessing and a curse. It is a blessing, because it means that in practice, there's no point: if you apply Born's rule "late enough" it doesn't matter where you apply it. It is a curse, because it means that the question of "where is the Born rule physically applied" is not open to experimental enquiry.
That's why, in other discussions here, like on "quantum erasers" I try to insist on applying the Born rule only at the end of your calculation. You have then much less conceptual problems with collapsing states at a distance, or predicting decisions in the future and so on. Of course you get other conceptual problems, related to accepting that the universe is one big messy entangled state...
But when it matters in doing calculations, you always get the right result if you "collapse only at the very end".
There are still issues. One is: WHAT is the basis of the system in which we have to apply this "decoherence" thing ? It turns out that this follows from the structure of the interactions with the environment ; there is still a discussion on what exact criterion should be used, but in practice, you always find about the same "preferred base states". For instance, for a macroscopic body, it is in most cases if not all, the position basis. This explains why we don't see macroscopic objects "in two places at once".
cheers,
Patrick.
. Whats stopping the socks (or electrons) from having a nonzero probability density of beeing at the same place. (this is also a serious question about the electrons in Helium 
