Can the energy of a particle ensemble in QFT be bounded over time?

[asimov42]

Hi all,

Question for which .I feel silly asking - but since I'm still learning:

A particle state in QFT is considered to be an asymptotic state with a well defined energy. Now, if I take an ensemble of particles after a very large number of interactions (say, e.g., a macroscopic object like a person's body, and perhaps since the birth of the universe), can one say that the ensemble has a bounded energy?

It would seem that the energy would have to be bounded at any time, even in an interacting picture (otherwise there would be no bound on spacetime curvature, etc., and yes, I realize I'm mixing in general relativity here). I think I'm basically asking about the spectrum of the Hamiltonian for the system - but since I'm still struggling to get through several texts, I'm unsure.

Also, given the above, there would be a maximum energy that a body could emit over a given time, correct? (If the above is correct)

Thanks!

 

[asimov42]

Ok, scratch the above as a very poorly formed question - I'll ask this instead:

Still talking about particles in QFT (subtleties aside) - although in an interacting picture a particle may not have a well-defined energy, this does not mean it may have any energy. What is the correct method to compute the distribution of possible energies (after a larger number of interactions)? And what would the support of this function be?

Thanks.

 

[asimov42]

Ok, one more time - hopefully this will be more clear, would really appreciate anyone's help to set me straight:

A particle can be considered as a wave packet. If we consider the momentum wave function, for any real particle is it not the case that the wave function should exactly vanish outside of a specified, finite, interval? Otherwise, if the momentum wave function is non-zero over ±∞, then the particle could, with some non-zero probability, have any momentum and hence any energy (arbitrarily close to infinite). Take, e..g., the Gaussian wave packet, for example, which has non-zero tails.

Note here that I'm not concerned with the expectation value of the energy, rather the probability that the energy of a particle is enormous (which depends on the particle of course - but should never go off to infinity). This is what I was trying to get at above.

Apologies in advance for not being more precise and for missing the obvious, I'm sure - lots to learn...

 

[PeterDonis]

for any real particle is it not the case that the wave function should exactly vanish outside of a specified, finite, interval?

No. But see below.

I'm not concerned with the expectation value of the energy, rather the probability that the energy of a particle is enormous (which depends on the particle of course - but should never go off to infinity)

There's a more general problem in QFT, which is that, if we just take the simple math at face value, even processes in which the actually measured energy is finite can contain virtual particle states with arbitrarily large energies. This is called "ultraviolet divergence". It is typically handled either by imposing an arbitrary cutoff or by renormalization.

 

[vanhees71]

A cutoff is just a (not too clever) way to regularize undefined infinite expressions called "proper vertex functions" as one step to get to the physically well defined expressions and measurable quantities, which are the S-matrix elements of renormalized perturbation theory.

 

[asimov42]

Thanks PeterDonis and vanhees71.

Ok, so let's consider the regularized and renormalized case - if particles going in are asymptotic states with well-defined energies, then after scattering, the particles coming out may not have well-defined energies immediately, but their total energy cannot exceed the input energy (conservation of energy)?

More generally - consider, say, the wave packet for, e.g., an electron that happens to be part of a desk in my office - and now the momentum wave function for this particle... if I understand PeterDonis, then there is some finite, non-zero probability (because the wave function is non-zero everywhere), that the particle in my desk has more momentum, and hence more energy, that the most energetic cosmic ray ever to hit the Earth...

The above is an extreme example, but illustrates the point. Again, speaking in the renormalized case, how can the above be possible?

 

[PeterDonis]

consider, say, the wave packet for, e.g., an electron that happens to be part of a desk in my office

Which is only an approximate model anyway, and you are simply pushing it into a regime where the approximation breaks down. Similar remarks apply to non-renormalized QFT; you can use it if you put in a cutoff, but that just amounts to admitting that you know you're just using an approximation and you need to avoid the regime where the approximation breaks down.

The fact that total energy is conserved in particle physics processes, however, is not an approximation, so we don't expect it to break down in the regime where the approximations you are using are no longer valid.

 

[asimov42]

Interesting - I didn't realize the wave packet was just an approximate model. Specifically in the renormalized case (i.e., using renormalized QFT), does one confront the same issue?

So, then, can I say that the momentum (and hence the energy) of the electron in my desk is bounded to a 'reasonable' value (or range, I should say, since the momentum is not well defined)? How does one determine what is 'reasonable', i.e., where the wave packet model breaks down?

 

[PeterDonis]

I didn't realize the wave packet was just an approximate model

It's approximate because it's non-relativistic. So it can't really be applied to trying to calculate the probability of an electron (for example) having an arbitrarily large momentum, because that would not be non-relativistic. One of the key things about any approximation is learning what not to use it for.

can I say that the momentum (and hence the energy) of the electron in my desk is bounded to a 'reasonable' value

Obviously, because electrons don't shoot out of your desk, they stay in it.

How does one determine what is 'reasonable'

There is no exact boundary of what is "reasonable". You have to look at each individual case and try to apply common sense.

 

[asimov42]

Ah, ok, thanks PeterDonis. Moving to the relativistic case (which would be relativistic QFT), then, it should be possible to assign an exact bound to the range of energies of the electron, no? Or the computation is done in such a way that this is not an issue...

Let's just consider the 'electron in my desk' case - in relativistic QFT, how would the energy of the particle be determined?

Thanks for the help by the way!

 

[PeterDonis]

Moving to the relativistic case (which would be relativistic QFT), then, it should be possible to assign an exact bound to the range of energies of the electron, no?

You don't need relativistic QFT to do that. You just need, as I said before, to know what questions not to ask your non-relativistic model. For most atoms the non-relativistic approximation gives pretty good predictions for the ionization energy of the atom (how much energy you have to add to an electron in the atom to take it out of the atom and make it a free electron); that energy is a good upper bound for the energies of the electrons in the atom. The fact that the non-relativistic model, mathematically, assigns non-zero values for the electron's wave function to points in momentum space that, classically speaking, are not consistent with that energy upper bound is one of the questions you should not ask the model, because that is getting into the range where the model's approximation breaks down.

 

[asimov42]

Ah, ok got it! Out of curiosity, if you did use relativistic QFT, could you get an exact bound (instead of the approximation)? (presumably yes? - and I should have thought about the relativistic case earlier and not conflated this with the QM solution)

Last question, promise

 

[PeterDonis]

could you get an exact bound

What do you mean by "an exact bound"? Do you mean a more accurate value for the ionization energy of the atom? Yes, in principle including relativistic corrections should give you that.

 

[vanhees71]

I think, here's a lot of confusion around.

First of all it is important to realize that in relativistic QT a proper particle interpretation is only possible for asymptotic free states, and of course you have wave-packets for single-particle states as the only proper, i.e., normalizable, single-particle states (plane waves are not representing states but generalized eigenfunctions of the momentum operator).

To understand this, read the section, where the S-matrix is introduced in the textbook by Peskin and Schroeder.

"Virtual particles" is slang for "propagators". It's represented by internal line of Feynman diagrams, and strictly speaking Feynman diagrams do not depict real processes (although on a heuristic level they indeed do) in nature but are a clever notation of formulae to evaluate the S-matrix in perturbation theory.

 

[LeandroMdO]

It's approximate because it's non-relativistic. So it can't really be applied to trying to calculate the probability of an electron (for example) having an arbitrarily large momentum, because that would not be non-relativistic. One of the key things about any approximation is learning what not to use it for.

A wave packet is just a specific linear combination of plane wave states. As vanhees71 stated, plane wave states are useful because we can construct physical states out of them (by way of Fourier transform), not because they are themselves physically relevant. These (normalizable) physical states, obtained as a linear combination of basis plane wave states, is what we call a wave packet. There's nothing non-relativistic about the concept. You can build wave packets out of solutions of the Schrödinger equation, the Klein-Gordon equation, Dirac, Rarita-Schwinger, whatever. It's just a Fourier transform.

There's a more general problem in QFT, which is that, if we just take the simple math at face value, even processes in which the actually measured energy is finite can contain virtual particle states with arbitrarily large energies. This is called "ultraviolet divergence".

What the OP was talking about is the fact that in, for example, a Gaussian wave packet, there is a nonzero amplitude for a measurement of momentum to give arbitrarily large values. Formal infinities in loop integrals are something altogether different.

It is typically handled either by imposing an arbitrary cutoff or by renormalization.

Renormalization is the art of teasing out measurable parameters from a theory written in a theorist's notebook in terms of some abstract object such as a Lagrangian. In principle it has nothing to do with the appearance of infinities: even wholly finite theories require renormalization. The procedure that parameterizes infinities so they are expressed in a form suitable for renormalization is called regularization. Cutoffs are one example of regularization, although one typically reserved for textbook toy models rather than realistic theories.

 

[vanhees71]

BTW, there is a way to formulate perturbative QFT such as to give finite results from the very beginning, the Epstein-Glaswer approach of "causal perturbation theory", but it's more complicated then the standard way. The infinities are due to the naive multiplication of operator valued distributions like field operators; strictly speaking already the Lagrangian or Hamiltonian are not well-defined as we write them naively down.

It leads to the same result as the hand-waving physicists' way but it's good to read about it to understand the troublesome infinities a bit better. A standard source is

G. Scharff, Finite Quantum Electrodynamics, Springer

 

[asimov42]

Hi all, thanks for your responses... I'm afraid now I'm a bit more confused.

There's nothing non-relativistic about the concept. You can build wave packets out of solutions of the Schrödinger equation, the Klein-Gordon equation, Dirac, Rarita-Schwinger, whatever. It's just a Fourier transform.

What the OP was talking about is the fact that in, for example, a Gaussian wave packet, there is a nonzero amplitude for a measurement of momentum to give arbitrarily large values.

This is in fact the question I'm wondering about - if you take into account relativistic effect, using e.g., relativistic QFT (although I realize there's no clear definition for a particle except as an asymptotic free state) do you still end up with wave packets where there is a nonzero amplitude for a measurement of momentum to give arbitrarily large values. This seems impossible to me.

BTW, there is a way to formulate perturbative QFT such as to give finite results from the very beginning, the Epstein-Glaswer approach of "causal perturbation theory", but it's more complicated then the standard way. The infinities are due to the naive multiplication of operator valued distributions like field operators; strictly speaking already the Lagrangian or Hamiltonian are not well-defined as we write them naively down.

It leads to the same result as the hand-waving physicists' way but it's good to read about it to understand the troublesome infinities a bit better. A standard source.

vanhees71, I'm not sure - above are you talking about finite results for momentum, so you get, e.g. proper bounded values for the momentum of particles (or field excitations ...) .

Basically, my original post was concerning the possibility of arbitrarily large measured momentums - my example was an electron in the desk in my office having some non-zero probability of having more momentum than an extraordinarily energetic cosmic ray...

 

[vanhees71]

Momenta are always finite. How could they be infinite in any experiment?

 

[asimov42]

Momenta are always finite. How could they be infinite in any experiment?

Yes, being finite in any experiment makes sense, of course, but this still doesn't answer the question about the values of measured momenta being bounded in the relativistic case - or maybe I'm just not understanding your answer.

E.g. (sorry for beating a dead horse here): I have a relativistic wave packet that describes an 'electron-like' field excitation (that is part of an everyday macroscopic object, so nothing exotic) - is the momentum wave function for the wave packet exactly zero outside of an envelope?

If not, I may measure a huge momentum, granted with very small probability, but still ... and this is what doesn't make sense to me. PeterDonis answered this above and I got it, but there seems to be some disagreement.

Thanks.

 

[PeterDonis]

is the momentum wave function for the wave packet exactly zero outside of an envelope?

I think it is important here to carefully distinguish the model from reality. In the model, the wave packet is not strictly zero anywhere. But that is because it's a lot easier to work with such wave packets (e.g., Gaussians) than to try to work with the messy things you get when you insist on the amplitude being strictly zero outside of some finite interval.

In reality, we don't measure energies to be unbounded in ordinary objects, so whatever thing in reality we are attempting to model with our wave packets presumably is zero outside of some finite interval (heuristically speaking). So the model is an approximation, as I said; we simply refrain from asking it what the probability is of measuring some arbitrarily large energy for an electron in an ordinary object, because we know the answer it gives us ("small but nonzero") doesn't match what we actually observe.

 

[asimov42]

Is the same true in the relativistic case? I.e., our momentum space wave function is non-zero everywhere? (in our model, that is)

Because of the mass increase with increasing momentum, I would assume that the wave packets derived using solutions to, e.g., the Klein-Gordon equation, would look much different from the non-relativistic case.

 

[PeterDonis]

Is the same true in the relativistic case? I.e., our momentum space wave function is non-zero everywhere?

In QFT we don't have "wave functions" in the sense you're used to. You can construct wave packets, yes, but they play a different role in the model.

Because of the mass increase with increasing momentum

"Relativistic mass" is not used in modern relativity, and is not a good heuristic here. The mass of a quantum field is a constant.

 

[asimov42]

Ah, ok, thanks PeterDonis, much appreciated - I'm definitely getting way out of my realm of understanding here. Let's just talk relativistic QM then, instead of QFT. Does this change the outcome? i.e. the momentum space wave function being zero at certain points?

PeterDonis, I think you already answered about a model vs. reality - but do we really use models that predict arbitrary momentum values just for convenience? Particularly, you mentioned non-relativistic models as approximations, and where you should avoid, 'listening to the model', but didn't mention the relativistic case.

 

[asimov42]

Momenta are always finite. How could they be infinite in any experiment?

vahhees71, can you comment on my post about the momenta being bounded? Obviously measuring an infinite momentum is not possible, but this doesn't say anything about measuring, say, an electron with an enormous momentum, which the Gaussian wave function clearly indicates is possible... (there's a clear distinction here between infinite and enormous)

 

[PeterDonis]

Let's just talk relativistic QM then, instead of QFT.

There is no such thing as relativistic QM instead of QFT. Relativistic QM is QFT.

do we really use models that predict arbitrary momentum values just for convenience?

Not for convenience so much as for being able to obtain solutions at all.

you mentioned non-relativistic models as approximations, and where you should avoid, 'listening to the model', but didn't mention the relativistic case

The same general caution applies to the relativistic case.

 

[asimov42]

Ok, so, in general, even in the relativistic case, 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.

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. 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?

 

[LeandroMdO]

vahhees71, can you comment on my post about the momenta being bounded? Obviously measuring an infinite momentum is not possible, but this doesn't say anything about measuring, say, an electron with an enormous momentum, which the Gaussian wave function clearly indicates is possible... (there's a clear distinction here between infinite and enormous)

I think what's missing from your understanding here is a bit of physics.

A wave packet in a realistic setting is not just some distribution handed from god. For simplicity, consider a classical electromagnetic wave: what does it mean to say that the wave has a Fourier component with an extremely high momentum? Well, it means that the source of the wave radiated that amount at the corresponding frequency. Energy, momentum, etc, are conserved, and they all come from the source, so if you find a Fourier component that carries quite a bit of energy, well, that's the reason.

In quantum mechanics the situation is a little stranger because of the question of measurement, but fundamentally it is the same: when you measure the momentum of some particle, you're also making an indirect (weak) measurement of whatever it is the particle entangled itself with in the past. So, for example, if an electron is accelerated by a pair of capacitor plates, the final state is entangled because of conservation of energy and momentum: a state in which the electron moves up has the plates moving down a bit (a very small amount if the plates are macroscopic, but in principle it's there). If, perchance, you measure the electron's momentum and get a very high amount, you're also indirectly measuring the momentum of the plates that accelerated it, getting a correspondingly huge amount, and so on through the entire history of interactions that led to this electron finding its way into your detector.

In short, if you measure a huge momentum in a wave packet, something put it there.

 

[asimov42]

I think what's missing from your understanding here is a bit of physics.

... If, perchance, you measure the electron's momentum and get a very high amount, you're also indirectly measuring the momentum of the plates that accelerated it, getting a correspondingly huge amount, and so on through the entire history of interactions that led to this electron finding its way into your detector.

In short, if you measure a huge momentum in a wave packet, something put it there.

LeandroMdO thanks - yes, this is exactly what I would expect from the physics of the situation, specifically the entanglement, and makes perfect sense - it's precisely the answer I expected.

But the math still indicates (non-zero tails in momentum space over the whole real line), at least from what I understand, that you might measure an enormous momentum value - the question is why is this a possibility? It is of course possible that whatever the particle was entangled with in the past imparted this enormous momentum but... Sigh, I know I'm beating a dead horse, and I apologize - but how can the momentum space wave function being non-zero over ±∞ possibly represent physical reality where there is (presumably) only a finite amount of energy.

 

[LeandroMdO]

LeandroMdO thanks - yes, this is exactly what I would expect from the physics of the situation, specifically the entanglement, and makes perfect sense - it's precisely the answer I expected.

But the math still indicates (non-zero tails in momentum space over the whole real line), at least from what I understand, that you might measure an enormous momentum value - the question is why is this a possibility? It is of course possible that whatever the particle was entangled with in the past imparted this enormous momentum but... Sigh, I know I'm beating a dead horse, and I apologize - but how can the momentum space wave function being non-zero over ±∞ possibly represent physical reality where there is (presumably) only a finite amount of energy.

Let's just put it this way. Take, for example, heights of children age 10. We often assume a normal distribution, say, with mean 1 m and standard deviation 0.1 m. Given this, what's the probability that a child will have a height
1. smaller than -0.5 m and 2. larger than 3 m?

The answer, of course, is that the heights aren't really normally distributed, but that approximation is often good enough for government work. We choose the normal distribution because of the central limit theorem. If a series of random processes add or remove energy from a particle, sampling from a fixed but unknown distribution, you'd expect the distribution of particle momenta to approach a normal distribution. It is, however, only an approximation, just like the normal distribution is an approximation to the real distribution describing the heights of school children.

 

[PeterDonis]

the math still indicates (non-zero tails in momentum space over the whole real line), at least from what I understand, that you might measure an enormous momentum value

As I've said: this is because the math is only an approximation, and you are trying to use it in a regime where the approximation breaks down. The correct answer is to not ask the math that question. It isn't to keep wondering why the math says something that doesn't make sense. Any approximation will tell you things that don't make sense if you apply it in a regime where it breaks down.

 

[asimov42]

Peter and LeandroMd0, thanks to you both for putting up with all of these questions. I can certainly understand the approximation, and I know the central limit theorem well. So this, in fact, makes total sense to me.

The reason I asked the last question is because, in another posting (where I tried to clear things up a bit but asked essentially the same question), mfb replied that:

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.

This quote seems to imply that we absolutely need to use functions with long (infinite) tails to represent reality, or somehow the math breaks and ruins physics. So again, I'm in a slightly confused state - are we dealing with approximations (makes 100% total sense)? And why would recognizing that we're dealing with approximations ruin physics (require new laws) as mbf suggests?

p.s. I'll have to send you both beers via mail for all your help with this

 

[PeterDonis]

This quote seems to imply that we absolutely need to use functions with long (infinite) tails to represent reality, or somehow the math breaks and ruins physics.

I think what @mfb was trying to say was that, mathematically, the functions with infinite tails are what our current mathematical formulations of physical laws give us. Is that because those functions with infinite tails actually "represent reality"? Or is it just because we haven't figured out yet how to write mathematical formulations of physical laws that make correct predictions but don't have the infinite tails? I don't think we know for sure; but @mfb also made the point that we have no way of testing the difference in practice, since the predicted probability, from the math that has functions with infinite tails, for measuring, say, the energy of an electron in the table in front of you to be large enough to send it flying out of the table, is so low that we would not expect to see such an event for a time much longer than the lifetime of the universe.

To me, treating the functions with infinite tails as approximations allows us to not have to worry about which of the above possibilities is actually true, because treating them as approximations means only using the functions in regimes where we know we can test their predictions.

 

[asimov42]

Whew ok - thanks again all. To be clear on the QM side of things - take the electron in my desk which I (surprisingly) measure with my apparatus to have an energy much larger than than a highly energetic cosmic ray. Now, I've made a measurement which had a low probability - and the energy of the particle was not precisely defined until I measured it... that energy had to come from somewhere, as @LeandroMdO noted. Until I make the measurement, the energy imparted by all of the entangled particles is also unknown - so the measurement causes wavefunction collapse...

There's no question here - just interesting to think that the energy imparted by the whole chain can only be determined after a measurement...

 

[asimov42]

The answer, of course, is that the heights aren't really normally distributed, but that approximation is often good enough for government work. We choose the normal distribution because of the central limit theorem. If a series of random processes add or remove energy from a particle, sampling from a fixed but unknown distribution, you'd expect the distribution of particle momenta to approach a normal distribution. It is, however, only an approximation, just like the normal distribution is an approximation to the real distribution describing the heights of school children.

Thanks @LeandroMdO - so you would say that the continuous momentum function is simply an approximation? But certainly provides accurate predictions over any range we can measure...

 

[vanhees71]

vahhees71, can you comment on my post about the momenta being bounded? Obviously measuring an infinite momentum is not possible, but this doesn't say anything about measuring, say, an electron with an enormous momentum, which the Gaussian wave function clearly indicates is possible... (there's a clear distinction here between infinite and enormous)

Perhaps, I misunderstood your question. I still do not understand your problem properly obviously. You just construct a single-particle state with a square-integrable wave function in momentum space by (for simplicity I consider a Klein-Gordon field)
$$|\phi \rangle=\int_{\mathbb{R}^3} \frac{\mathrm{d}^3 \vec{p}}{(2 \pi)^3} \frac{1}{\sqrt{2 E_{\vec{p}}}} \hat{a}^{\dagger}(\vec{p}) |\Omega \rangle.$$
Here, $|\Omega \rangle$ is the free-particle vacuum, $\hat{a}^{\dagger}(|\vec{p}|)$ the creation operator of the relativistically covariantly normalized plane-wave mode, i.e., fulfilling the commutator relation
$$[\hat{a}(\vec{p}),\hat{a}^{\dagger}(\vec{p}')]=(2 \pi)^3 2 E_{\vec{p}} \delta^{(3)}(\vec{p}-\vec{p}'),$$
and $E(\vec{p})=\sqrt{m^2+\vec{p}^2}$.

Then $|\phi \rangle$ defines a free-particle wave packet normalized to 1:
$$\langle \phi|\phi \rangle=\int_{\mathbb{R}^3} \frac{\mathrm{d}^3 \vec{p}}{(2 \pi)^3} |\phi(\vec{p})|^2=1$$