Non-Perturbative QFT without Virtual Particles

[rogerl]

Look in the literature at a proof of any of these statements, or even a more precise description of what is meant with them, and you won't find anything. This sort of discourse is good for story-telling, but for nothing else.

Let's say Virtual Particles are only used for story telling and really just mathematical artifacts of perturbation theory. How about quantum fluctuations? It is said that "Quantum fluctuations may have been very important in the origin of the structure of the universe: according to the model of inflation the ones that existed when inflation began were amplified and formed the seed of all current observed structure". But quantum fluctuations are related to virtual particles. If quantum fluctuations are also mathematical artifacts of perturbation theory. How come they have observable effect in that they form the seeds of galaxies, nebalae, etc.?!

Without quantum fluctuations (and maybe virtual particles). We may not even exist. So we owe them our lives and must regard them to the highest degrees.

 

[Demystifier]

How about quantum fluctuations?

They are real, or at least much more real than virtual particles.

But quantum fluctuations are related to virtual particles.

Not directly. In principle, and sometimes even in practice, you can calculate the fluctuations without using virtual particles.

If quantum fluctuations are also mathematical artifacts of perturbation theory.

Quantum fluctuations are NOT mathematical artifacts of perturbation theory.

 

[A. Neumaier]

Let's say Virtual Particles are only used for story telling and really just mathematical artifacts of perturbation theory. How about quantum fluctuations?

Fluctuations have a much better ontological status than virtual particles. Their properties are indeed computable nonperturbatively, hence are properties of the system under study and (unlike virtual particles) not of the approximation method used.

But they are not what conventional story-telling claims they are: They are not changes in time. Instead, quantum fluctuations describe uncertainties about what one gets when one tries to measure something. It's the fluctuations in the measurements when one repeats them under identical conditions - not fluctuations in what is measured.

Thus quantum fluctuations reflect something about the limits of measurement processes, not something about rapid changes in time.

It is said that "Quantum fluctuations may have been very important in the origin of the structure of the universe: according to the model of inflation the ones that existed when inflation began were amplified and formed the seed of all current observed structure". But quantum fluctuations are related to virtual particles. If quantum fluctuations are also mathematical artifacts of perturbation theory. How come they have observable effect in that they form the seeds of galaxies, nebalae, etc.?!

They don't form a seed in any dynamical sense (as a real seed - that changes in due time into a real plant).

More in the entry ''Does the vacuum fluctuate?'' in Chapter A7 of my theoretical physics FAQ at http://arnold-neumaier.at/physfaq/physics-faq.html#vacfluc

 

[dm4b]

But they are not what conventional story-telling claims they are: They are not changes in time. Instead, quantum fluctuations describe uncertainties about what one gets when one tries to measure something. It's the fluctuations in the measurements when one repeats them under identical conditions - not fluctuations in what is measured.

Thus quantum fluctuations reflect something about the limits of measurement processes, not something about rapid changes in time.

This way of explaining quantum fluctuations seems to make them dependent upon someone measuring them.

Do quantum fluctuations exist, if nobody makes a measurement?

 

[A. Neumaier]

This way of explaining quantum fluctuations seems to make them dependent upon someone measuring them.

Do quantum fluctuations exist, if nobody makes a measurement?

They are properties of the system, whether or not somebody measures it. In this sense ithey exist independent of measurement, like a tree exists no matter whether someone looks at it.

But the meaning of the quantum fluctuation of a quantity Q is not the value of a measurement of Q but the intrinsic uncertainty of the measurement result in any attempt to measure Q. (There may be additional uncertainty due to limitations of the particular equipment used - but this is not a property of the system but of the equipment.)

 

[dm4b]

They are properties of the system, whether or not somebody measures it. In this sense ithey exist independent of measurement, like a tree exists no matter whether someone looks at it.

But the meaning of the quantum fluctuation of a quantity Q is not the value of a measurement of Q but the intrinsic uncertainty of the measurement result in any attempt to measure Q. (There may be additional uncertainty due to limitations of the particular equipment used - but this is not a property of the system but of the equipment.)

Is this different than say measuring the position of the electron in a hydrogen atom? We'll get a different position each time we measure but, after many repeated measurements, on identically prepared systems, we'll notice that we obtain the probability distribution predicted by the Schrodinger Equation.

Is this essentially what's going on with quantum fluctuations?

Or, is it something more akin to the HUP? At small scales in size, the energy will fluctuate rapidly.

I guess, put simply, what the heck is a quantum fluctuation, exactly? ;-)

I guess I always took it as more of the latter. And, since in SR E=m, those fluctuations in energy can give rise to particles (the dreaded virtual particles mentioned above)?

 

[A. Neumaier]

Is this different than say measuring the position of the electron in a hydrogen atom? We'll get a different position each time we measure but, after many repeated measurements, on identically prepared systems, we'll notice that we obtain the probability distribution predicted by the Schrodinger Equation.

Is this essentially what's going on with quantum fluctuations?

Yes, precisely. Please read the FAQ entry mentioned in posting #93

Or, is it something more akin to the HUP? At small scales in size, the energy will fluctuate rapidly.

I guess, put simply, what the heck is a quantum fluctuation, exactly? ;-)

I guess I always took it as more of the latter.

It is _not_ the latter.
Fluctuations are neither objects nor energy but system properties, and have _nothing_ to do with changes in time.

What a quantum fluctuation is, exactly, is spelled out in the FAQ. The FAQ exists because I don't want to explain the same thing over and over again. So please read it before asking further questions.

 

[dm4b]

Hi A. Neumaier,

I read your FAQ and overall it seemed like a good description. I specifically liked your analogy with the 1D Harmonic Oscillator., which was helpful. I've read several areas of your FAQ, even the parts on Christianity/Religion, much of which I found interesting - thanks.

But, I am still having the following problems with the vacuum fluctutations.

(1) I'm not sure that everybody out there agrees with the description in your FAQ. It seems some say the HUP goes beyond just an observer measurement. That a particle cannot come to rest, because then you would have a perfectly defined position AND momentum, and that cannot be, whether somebody measures it or not. Hence, this gives rise to a ground state, or a Zero-Point Energy (ZPE). (I guess another way to look at it, is that the entity in question is not just a particle, but also partly a wave, which can never be assigned to a particular point in space). This seems to also be the common explanation of why you can never reach absolute zero. Do you feel this viewpoint is incorrect? If so, how?

(2) A non-trivial zero-point energy is established, as mentioned in your FAQ. Since E=m, what stops the creation of particles from this ZPE - whether real or virtual? And, if vacuum fluctuations are not a process in time, then wouldn't the ZPE have a constant value over space and time?

(3) I'm having a hard time picturing vacuum fluctuations having a physical effect on anything. Specifically, under inflation, aren't they supposed to provide the "seeds" for galaxy formations. How can something inherit to a measurement process, which presumably should include an observer, be able to kick off galaxy formation, when presumably no observers were around? Also, if the above is correct, and the ZPE is a constant value in all space, how can it "seed" anything within a particular spot in space?

Thanks for any feedback you can give on this.

 

[A. Neumaier]

(1) I'm not sure that everybody out there agrees with the description in your FAQ.

I am sure that not everybody out there agrees with the description in your FAQ. The FAQ is there to correct poor opinion. if everyone agreed, there were no need to discuss many of these questions.

It seems some say the HUP goes beyond just an observer measurement.

The Heisenberg uncertainty principle (HUP) just states that the product of the variances of p and q is bounded below by a small number. It doesn't say what the variances represent.

The general consensus is that the variance represents an ensemble average - i.e., the result of a statistics over many independent measurements on identically prepared systems.

In order to take the variance as a time average one needs to invoke an ergodic theorem stating that the time average equals the ensemble average. However such an ergodic theorem makes sense only semiclassically, and is valid only for very simple systems. Most systems are far from ergodic.

That a particle cannot come to rest, because then you would have a perfectly defined position AND momentum, and that cannot be, whether somebody measures it or not.

Here you assume a semiclassical picture. One cannot measure whether a microscopic particle is at rest - and apart from such a measurement the statement about the rest of a particle is meaningless.

Hence, this gives rise to a ground state, or a Zero-Point Energy (ZPE).

Only energy differences matter; the zero-point energy is completely spurious.

(I guess another way to look at it, is that the entity in question is not just a particle, but also partly a wave, which can never be assigned to a particular point in space).

This is another way to say that talking of rest is meaningless. When is a wave at rest?
But waves have real energy, not an unobservable ZPE.

This seems to also be the common explanation of why you can never reach absolute zero. Do you feel this viewpoint is incorrect? If so, how?

This is the explanation in classical mechanics. In quantum mechanics, zero absolute temperature is equivalent to being in the ground state, and this is very well possible for a single hydrogen atom, but impossible for a macroscopic body.

(2) A non-trivial zero-point energy is established, as mentioned in your FAQ.

Where did I mention this?

Since E=m, what stops the creation of particles from this ZPE - whether real or virtual?

Only energy differences can be exploited for the creation of anything.

(3) I'm having a hard time picturing vacuum fluctuations having a physical effect on anything.

Vacuum fluctuations cause nothing, hence have no effect. Their presence in the equations has some observable consequences.

Specifically, under inflation, aren't they supposed to provide the "seeds" for galaxy formations.

To discuss this, please provide a reference that says more specifically how vacuum fluctuations provide the "seeds" for galaxy formation. (In the present vagueness this is just another instance of modern mystic story telling.)

 

[dm4b]

Only energy differences matter; the zero-point energy is completely spurious.

I'm not sure this is true in GR, is it? The presence of energy alone has potential gravitational effects, does it not? Isn't that the whole cosmological constant problem?

Where did I mention this?

Here:

"On the other hand, the fact that sigma^2 resp. sigma^2(x) and similar
expectations do not vanish shows in nontrivial physics, for example,
a nontrivial zero-point energy."

Only energy differences can be exploited for the creation of anything.

okay, that makes sense.

Vacuum fluctuations cause nothing, hence have no effect. Their presence in the equations has some observable consequences.

How can something that causes nothing and has no effect, have observable consequences? That makes no sense to me.

To discuss this, please provide a reference that says more specifically how vacuum fluctuations provide the "seeds" for galaxy formation. (In the present vagueness this is just another instance of modern mystic story telling.)

C'mon, a reference? This is so commonly stated you must have heard it before.

If you think it's a myth, please provide details on the real mechanism on how this really works. Otherwise, folks have no reason to not believe in the "myth".

 

[A. Neumaier]

I'm not sure this is true in GR, is it? The presence of energy alone has potential gravitational effects, does it not? Isn't that the whole cosmological constant problem?

A zero-point energy is not energy present. The cosmological constant problem has a different origin.
It is a nontrivial term in the action.

Here:

"On the other hand, the fact that sigma^2 resp. sigma^2(x) and similar
expectations do not vanish shows in nontrivial physics, for example,
a nontrivial zero-point energy."

Yes. The context is that one compares the energy with another energy in perturbation theory, and since the reference energy also makes physical sense one has an energy difference. Nevertheless, since only one of the two systems is realized, this energy difference is not physically utilizable.

How can something that causes nothing and has no effect, have observable consequences? That makes no sense to me.

Immediately after the above quote, I wrote:
''The zero-point energy can often, but
not always be neglected. It can be utilized for the derivation of
observable consequences. One of them is the Casimir effect.
But the Casimir effect can also be derived without reference to
the zero-point energy.''
The last sentence shows why the zero-point energy cannot be regarded as a cause, though one can use it to derive the effect.

C'mon, a reference? This is so commonly stated you must have heard it before.

I want to have an online reference as a clear staring point for a discussion.

 

[dm4b]

A zero-point energy is not energy present. The cosmological constant problem has a different origin.
It is a nontrivial term in the action.

Well, Sean Carrol would seem to disagree with this. The nontrivial term in the action, which contains the Cosmological Constant, can be related to a vacuum energy density. In fact, see page 172, of his GR text, where he does this by decomposing the energy-momentum tensor into a matter piece and a vacuum peice. The action is then defined and he goes on to claim the terms "Cosmological Constant" and "Vacuum Energy" are essentially interchangeable.

Also, to parapharse other parts of the text on page 171:

"A characteristic feature of general relativity is that the source for the gravitational field is the entire energy-momentum tensor. In nongravitational physics only changes in energy from one state to another are measurable ... In gravitation, however, the actual value of the energy matters, not just the differences between states".

Is there any reason to expect the vacuum energy is zero? Quantum fluctuations change the zero-point energy from our classical expectation.

More on all this below.

Immediately after the above quote, I wrote:
''The zero-point energy can often, but not always be neglected. It can be utilized for the derivation of observable consequences. One of them is the Casimir effect. But the Casimir effect can also be derived without reference to the zero-point energy.''

Ah, okay, I see where you're coming from now. I agree up to this extent - that is, for the Casimir Effect.

However, as mentioned in the paper (http://lanl.arxiv.org/abs/hep-th/0503158) by R. L. Jaffe, "The object of this paper is to point out that the Casimir effect gives no more (or less) support for the “reality” of the vacuum energy of fluctuating quantum fields than any other one-loop effect in quantum electrodynamics ... ".

In other words, the analysis on the Casimir effect, to date, does not definitively determine whether the vacuum energy is real or not. It only shows that the ZPE has been incorrectly claimed as the cause of the Casimir Effect, when in actuality it is the Van Der Waals force between the plates.

The last sentence shows why the zero-point energy cannot be regarded as a cause, though one can use it to derive the effect.

Although apparently true for the Casmir effect, we don't know if this is true, in general.

Once again, as stated by R. L. Jaffe in his conclusion: "The deeper question remains: Do the zero point energies of quantum fields contribute to the energy density of the vacuum and ... to the cosmological constant?" Also, see Carroll above.

The jury is still out on this one, I believe. It seems to me, it may be a thornier one to deal with than the Casimir Effect, as well.

I want to have an online reference as a clear staring point for a discussion

Well, how about we just go back to Carrol again, which has the most sophisticated treatment on this topic that I have read, which sure isn't saying much. Indeed, Carrol says it is beyond the scope of his GR book, which is no surprise. Admittedly, I never fully understood what he was talking about anyhow.

On page 371, he mentions that that there are fluctuations in the inflation field phi, corresponding to a Gibbons-Hawking temperature, which is the tempature of a vacuum state of an accelerating Universe. Paraphrasing Carrol again: "Since the potential is by hypothesis nearly flat, the fluctuations in phi lead to small fluctuations in energy density ... Inflation therefore produces density perturbations ... which may be the origin of the CMB temperature anistropies and the large-scale sturcture in galaxies we observe today."

These fluctuations sure sound like they have a dynamical effect, but I sure don't understand the details. So, what am I missing?

 

[A. Neumaier]

Well, Sean Carrol would seem to disagree with this. The nontrivial term in the action, which contains the Cosmological Constant, can be related to a vacuum energy density. In fact, see page 172, of his GR text, where he does this by decomposing the energy-momentum tensor into a matter piece and a vacuum peice. The action is then defined and he goes on to claim the terms "Cosmological Constant" and "Vacuum Energy" are essentially interchangeable.

Also, to parapharse other parts of the text on page 171:

"A characteristic feature of general relativity is that the source for the gravitational field is the entire energy-momentum tensor. In nongravitational physics only changes in energy from one state to another are measurable ... In gravitation, however, the actual value of the energy matters, not just the differences between states".

Is there any reason to expect the vacuum energy is zero? Quantum fluctuations change the zero-point energy from our classical expectation.
[...]
Once again, as stated by R. L. Jaffe in his conclusion: "The deeper question remains: Do the zero point energies of quantum fields contribute to the energy density of the vacuum and ... to the cosmological constant?" Also, see Carroll above.
[...]
Well, how about we just go back to Carrol again, which has the most sophisticated treatment on this topic that I have read, which sure isn't saying much. Indeed, Carrol says it is beyond the scope of his GR book, which is no surprise. Admittedly, I never fully understood what he was talking about anyhow.

On page 371, he mentions that that there are fluctuations in the inflation field phi, corresponding to a Gibbons-Hawking temperature, which is the tempature of a vacuum state of an accelerating Universe. Paraphrasing Carrol again: "Since the potential is by hypothesis nearly flat, the fluctuations in phi lead to small fluctuations in energy density ... Inflation therefore produces density perturbations ... which may be the origin of the CMB temperature anistropies and the large-scale sturcture in galaxies we observe today."

These fluctuations sure sound like they have a dynamical effect, but I sure don't understand the details. So, what am I missing?

I am not an expert in quantum gravity, so maybe I am missing something. In any case, what I said is true for all observationally verified QM and QFT, including the the standard model.

I believe that in case of QG, the resolution of the problem is in the renormalization procedure. There is a difference between the energy-momentum tensor and the Hamiltonian. The latter is _formally_ the integral of the e/m tensor over all space. But if the e/m tensor contains an additive constant then this contribute infinity to the Hamiltonian. This infinity is removed by renormalization, where normal ordering moves the zero point energy to exactly zero.

I don't have the book by Carrol, and I need formulas, not mere words, to discuss the issue further. So if you are interested in my analysis of the situation, please provide a public online source that we can take as a formal starting point.

 

[dm4b]

I don't have the book by Carrol, and I need formulas, not mere words, to discuss the issue further. So if you are interested in my analysis of the situation, please provide a public online source that we can take as a formal starting point.

Yes, I would be interested. I believe Carroll's notes are online. I'll see if I can't find them later (or some other source) and will post back if I do.