When does the wavefunction propagate at a finite speed, and when instantaneously?
What do you mean by propagate ? Propagate where to ?
It propagates instantaneosly when you assume objective wave function collapse. Otherwise, it always propagates with a finite speed.
Does it mean that the propagation depends on our mind? No, but human answers to human questions depend on human minds.
The "speed of propagation" is the hamiltonian !
Note that the wavefunction doesn't live in real space, and hence you cannot define such a thing as "speed of propagation" of the wavefunction in things like meters per second or so. The wavefunction lives in hilbert space.
Are you saying there is no mapping between the evolution in configuration space and the probability of finding a possible pattern in real space when a measurement is done with regards to c?
If the hamiltonian defines a local dynamics (which it does in QFT, and which it doesn't in NRQM), then each kind of local "field operator expectation value", which WILL define a field in spacetime, will indeed propagate at less than or equal c, if that is what you hint at. But these "field operator expectation values" are not necessarily "physical quantities in spacetime", and are certainly not identical to the wavefunction itself.
But under which circumstances do you think it will propagate less than c (and obviously I do not mean the average velocity).
True, and I am sorry you misunderstood that I did make such a claim.
When the field operator is the one of a massive field, for instance...
Or when we have a stationary solution !
Well forgive my ignorance but how do we know?
For instance are you saying that the amplitude for a mass particle to travel at c is zero for an abritrary short path?
And how can we exclude the possibility that a mass particle has an average velocity of < c but actually moves at the speed of c in different directions?
If your hamiltonian has been constructed that way, yes of course ! And if it isn't constructed that way, then you can give your "particle" (which I consider here, in QFT speak, as a "blob" in an expectation value of a field operator) any speed you like, even infinite speed. It depends on your time evolution of the wavefunction - which is given by the hamiltonian.
However, because of the lorentz-invariance of the lagrangian formulation in QFT, we get, indeed, that the blobs move at less than c. This is simply due to the lorentz-invariance of the green functions, which remain 0 outside of the lightcone.
Well, "a localised particle" is not well defined in a field-theoretical setting ; its best approximation would be a blob in some expectation value of a field operator. And as said above, you can do what you want. It all depends on exactly how you set up your theory hamiltonian, and exactly at what kind of quantity you look.
But all this are *consequences* of the "motion" of the state vector in hilbert space, to which it is hard to give a "velocity".
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