Non-Perturbative QFT without Virtual Particles

[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.