Momentum and energy in QM and QFT

[asimov42]

Hi all - apologies, I'm starting a new thread here for something buried at the end of another thread - but I think the topic of that thread had changed sufficiently to warrant a more succinct top-level post. Thanks very much to PeterDonis for his very useful answers in the previous thread.

Here's the scenario and question - consider the momentum wave function:

In general, even in a relativistic setting, one may measure an arbitrarily large momentum with some (very tiny) non-zero probability, assuming we use a wave function / wave packet formulation that allows for a solution (yes, I realize that wave functions as such are not used in QFT).

Here's what really bothers me: take QFT, and run a scattering experiment where at t = -∞ the particles going in are asymptotic free states with well-defined energies. Likewise, at t = +∞ one ends up with particles again with well defined energies, and energy is conserved. So there is no possibility here for arbitrarily large momentums - i.e., exactly zero probability that at any point the energy can be larger than the input energy, the energy is bounded.

Now, in some sense (and please correct me if I'm wrong), the entire universe is one large scattering experiment with a fixed amount of energy. Then in no case should there ever be a situation in which the momentum wave function for a particle (or whatever this translates to in terms of field excitations in QFT) can have an arbitrarily large momentum (e.g., one should never use a Gaussian with non-zero tails if one truly want real answers... granted the Gaussian makes a great approximation if you basically ignore the tails, which is the case if I understand correctly).

Am I correct with the above? If so, what gives? Why use wave packets with non-zero tails that imply some (very small but still) non-zero probability of enormous momentums or energies, when this cannot possibly be reflected in physical reality?

Thanks.

 

[mfb]

For the universe to be a scattering experiment, you need a final state with no further interactions. You could argue that the accelerated expansion will lead to such a final state in the very distant future, but then you end up in a universe where you cannot even measure the energy of particles.

All this is irrelevant for practical measurements. You simply do not care about things with 10^-1000 probability, although the mathematics requires them to be there. Removing these odd things artificially would need new physical laws, and there is no evidence for such a change.

 

[asimov42]

Thanks mfb. But doesn't the mere existence of things with 10^-1000 probability imply unphysical situations? Yes, you're incredibly unlikely to measure such a something in such a range, but the possibility is still there in the maths.

If I were somehow able to conduct enough experiments of regular, vanilla electrons, for example, in everyday objects (nothing exotic), don't the maths say I would eventually measure one with a momentum larger than that of an energetic cosmic ray?

 

[mfb]

But doesn't the mere existence of things with 10^-1000 probability imply unphysical situations?

Why?

If I were somehow able to conduct enough experiments of regular, vanilla electrons, for example, in everyday objects (nothing exotic), don't the maths say I would eventually measure one with a momentum larger than that of an energetic cosmic ray?

That is about as likely as your detector getting destroyed from random thermal fluctuations. Yes, things malfunction once in a while, although you can make it so unlikely that it won't occur within the lifetime of the universe.. An unphysical high energy measurement is in the same class of things.

 

[asimov42]

Right - and yep, got it about the likelihood of the detector being destroyed being in the same class.

Ok, so not unphysical - but let's say, after this extraordinarily large number of measurements of electrons, I do happen to measure one with such an enormous energy. Where did the energy come from? Presumably from entanglement with everything leading up to the measurement in my detector ... ok... so... next paragraph.

Lastly, given that the amount of energy in the universe is finite (I think) - the existence of a non-zero momentum space wave function that goes off to infinity implies that, with an infinitesimal probability, I could measure a particle energy greater than the total amount of energy in the universe. Yes, I realize you'd need a detector, etc. etc., and it might take an amount of time approaching eternity. But this seems to be a paradox - how can this be possible? (if, as you say, the mathematics require those long tails to be there...)

Or is it that the infinite support of the momentum wave function is simply an approximation to what's really going on?

 

[mfb]

Where did the energy come from?

The source must have had a lot of energy - again as random fluctuation.

A measurement larger than the total energy in the universe doesn't mean your particle actually had that energy. And it won't happen anyway. We often use approximations in calculations - approximations that are perfectly fine to some number of digits, but not necessarily millions of digits. That should be sufficient for all practical purposes.
There is no exactly Gaussian beam. Using a Gauss function 10¹⁰⁰ standard deviations away from the mean doesn't make sense.

 

[asimov42]

Ah - exactly what I was wondering, thanks @mfb.

Just to make sure I'm clear - if I were to measure an 'average' electron, and find it to have an 'average' amount of momentum (say it's just a room-temperature electron doing its thing) then, this measurement should change the shape of the wave packet, such that it's less likely (at least temporarily) to then make a second measurement with the enormous energy I was speaking of earlier.

 

[mfb]

If you measure the momentum, and then measure it again, you will get the same result.

But even that is an approximation, because you cannot make exact momentum measurements. At least not in a finite time.

 

[A. Neumaier]

in some sense (and please correct me if I'm wrong), the entire universe is one large scattering experiment

No. The time in the universe cannot be taken to minus infinity, according to most current models.

 

[RockyMarciano]

All this is irrelevant for practical measurements. You simply do not care about things with 10^-1000 probability, although the mathematics requires them to be there. Removing these odd things artificially would need new physical laws, and there is no evidence for such a change.

But you insist here and all along this thread that all these things in the math are basically irrelevant and nonsensical FAPP(although when asked if they are unphysical you answer "why?"), in what sense removing it from the math would need new physical laws? It would seem that it would lead to physics more in agreement with the practical measurements, no? Perhaps you mean that it would need a deep theoretical change?