Uncertainty vs. speed of light

[jkleid]

You have some quantum particle, and at time t_0 you know its position and momentum reasonably well. Next, you measure the particle's momentum with 100% precision, losing all information about its position.

But, after a time interval t, the farthest something could travel is c*t. So, given your initial knowledge of the particle's position, it must now be somewhere in the sphere of radius c*t (sufficiently extended to take into account your initial [small] uncertainty of the particle before the measurement).

If you take smaller and smaller t (the amount of time after the measurement), you can effectively shrink this sphere until it is the same size as it was originally, i.e. when your uncertainty of position was small. But now, in addition, you know it's momentum with 100% precision. This clearly violates the uncertainty principle.

Where is the flaw in this line of reasoning?

 

[Haelfix]

The flaw is not in the uncertainty principle, but in locality. The second you measure momentum 100%, the uncertainty in the position becomes infinite. Vanilla Quantum mechanics says it can be ANYWHERE, that means its position can be outside its own classical lightcone. Troubling eh? Einstein thought so too.

Its best to think of the whole thing as a field, infinite in scope, and not as a classical particle.

The really troubling thing is that the limit is hard to make sense off. But as far as we know, there is an instaneous collapse at t = 0 (the measurement), that is naively absent in the limit sending t --> 0.

 

[caribou]

I think maybe you can know the position even the tiniest fraction of a second after you know the momentum, and it's only knowing both at the exact same moment of time that is forbidden.

I'd guess the shrinking sphere never quite gets to be the same size.

 

[dextercioby]

Yes,true,you can know the position "even the tiniest fraction of a second after you know the momentum",but once u do,u'll have no idea of the momentum and how large the sphere may be...

Daniel.

 

[hellfire]

A short time ago I have been reading about this in Peskin and Schröder. For a free scalar field the propagator does not vanish outside the lightcone. This is the same problem that arises in quantum mechanics as described in the first post here. However, in QFT the commutator of the field operator for spacelike separated points vanishes. Does this mean that, although the particle may propagate outside its lightcone, this can never be actually measured?

 

[jkleid]

What about Exclusion Principle?

Vanilla Quantum mechanics says it can be ANYWHERE, that means its position can be outside its own classical lightcone. Troubling eh?

Yep, a bit.

Say the particle is an electron, and we measure its momentum with 100% precision. Now it is, in principle, equally likely to be anywhere in the universe. But because of the Pauli exclusion principle, it can not be in the same region as another electron with the same spin, right? This would not buy us much knowledge given the universe's current configuration, but if you could imagine an earlier time, when everything was more densely packed, then we might be able to rule out a significant percentage of possible locations. Or -- does uncertainty take precedence over Pauli's exclusion principle?