What is the Relationship Between Vacuum Energy and Fluctuations?

[Haroon1]

I was reading about this effect called the 'Casimir Effect' which provides evidence for the existence of Quantum Vacuum. What is Quantum Vacuum, and how is it different from the ordinary Vacuum?

 

[Bob_for_short]

An ordinary vacuum is a space without particles.

Quantum vacuum is the lowest energetic (ground) state of a quantum system. There is no quantum excitations (also called "particles" or "quasi-particles") in the quantum vacuum.

The Casimir effect is an interaction of neutralized charges at long distances akin to atom-atomic interaction at long distances (Van der Waals force).

 

[Born2bwire]

In the quantum vacuum, despite the fact that no photons are present, there are still fluctuating electromagnetic fields. However, the fields fluctuate about an expectation value of zero. In addition, the energy of the quantum vacuum state is divergent, infinite if you assume no bound to the frequencies of the vacuum (but this is usually truncated for various reasons). There are a lot of insights into the necessity of the vacuum fields, Peter Milonni has a good book that covers this in detail. For example, fluctuation-dissipation theorem can be used to help explain it though it is of course derivable from QED for fields in free-space. As for Casimir force providing evidence for its existence, Jaffe has an article that he wrote where he feels that this is not something that can be claimed. He feels that since we can characterize the Casimir force in a number of ways, one of which does not use the quantum vacuum at all, then it cannot be used as proof. Jaffe has been coauthor on some good Casimir force papers so I do not dismiss his assertions though I would have liked to have seen him get the paper published to read some responses to it.

http://arxiv.org/abs/hep-th/0503158

 

[Haroon1]

Thanks for the replies :)

 

[Halcyon-on]

In few words, the Casimir effect is not the (definitive) evidence that the vacuum energy exists. http://arxiv.org/abs/hep-th/0503158

But physicists are too busy to study unpredictable things to spend time for this concrete and fundamental issues! After all it is easier for them to publish hundreds of papers in topics that will be check eventually in some centuries.

 

[Freeman Dyson]

In the quantum vacuum, despite the fact that no photons are present, there are still fluctuating electromagnetic fields. However, the fields fluctuate about an expectation value of zero. In addition, the energy of the quantum vacuum state is divergent, infinite if you assume no bound to the frequencies of the vacuum (but this is usually truncated for various reasons). There are a lot of insights into the necessity of the vacuum fields, Peter Milonni has a good book that covers this in detail. For example, fluctuation-dissipation theorem can be used to help explain it though it is of course derivable from QED for fields in free-space. As for Casimir force providing evidence for its existence, Jaffe has an article that he wrote where he feels that this is not something that can be claimed. He feels that since we can characterize the Casimir force in a number of ways, one of which does not use the quantum vacuum at all, then it cannot be used as proof. Jaffe has been coauthor on some good Casimir force papers so I do not dismiss his assertions though I would have liked to have seen him get the paper published to read some responses to it.

http://arxiv.org/abs/hep-th/0503158

I am still trying to get my head around this question. I am not a physicist so bear with me. Isn't a quantum vacuum within an electromagentic field? How can there be a electromagnetic field with no photons or anything else? What is producing it? Where did it come from?

Where did the particles come from that are popping in and out of existence? Did they come from another space? Are they simply displaced? Were they anywhere before this? Does the same particle disappear one place and reappear in another? Are they really coming in and out of existence or just jumping around? Do virtual particles actually exist or are they placeholders? Thanks

 

[Born2bwire]

A quantum vacuum is simply a fancy name for the ground state. That is, it is the lowest energy state of the system. The interesting thing about the electric and magnetic fields in quantum electrodynamics is that their ground state is represented by zero photons. However, their ground state is not zero energy. In fact, in a completely empty space, the quantum vacuum can have an infinite number of frequencies of fluctuating fields occurring, a continuous spectrum. Each frequency represents a mode, a possible excitation of the fields in the system, and each mode has a certain discrete energy density. So the quantum vacuum has infinite energy if we do not restrict the possible frequencies of electric and magnetic fields. One way to think of this is that in quantum electrodynamics, we think of the photons as being the energy packets (quanta) that occur when we excite the electromagnetic waves. Each energy level of the electric and magnetic fields represents an additional photon being excited. These photons "come" from the vacuum state. Since the vacuum state has infinite energy, it has infinite photons. Everytime we add energy into the electromagnetic fields, we just pull a photon out of the vacuum state. It's an interesting idea, I recall I think it was Dirac who mentioned it.

Where this energy comes from we do not say. All we know is that in quantum mechanics, we often get systems where the energy cannot go to zero. Since we have an energy "bath" that we can draw upon, it forces fluctuations in the system (this is an idea from the fluctuation-dissipation that I mentioned earlier). For example, let's say I have a system that draws energy from a heat bath that surrounds it. It is constantly drawing energy from the bath but it cannot put energy back in. We find out that this stipulates that the system must have fluctuations. In the same way, we must have fluctuations in the vacuum state as well. But since these fluctuations are about a mean of zero, they are not measurable in the macroscopic world. So we never see truly see them. Sure we can get non-zero measurements should we attempt them but statistically we will only get a zero measurement in the long run.

So again, we can't say where the energy comes from, it's a definition of the quantum system. The fluctuations of the field can be explained in a few ways. We an show taht it must occur via mathematical rigor of quantum mechanics. The closest "physical" reason I have found is that the vacuum energy is an energy bath that couples with the electric and magnetic fields. Because of this, the fields must have fluctuations as shown by statistical mechanics. Photons are nothing more than the energy quanta of the electric and magnetic fields. We can think of them as being drawn out of the energy of the vacuum state. When they are created they come from the vacuum and when annihilated they return. Of course this may not be a truly physical picture. Anytime we add energy to the fields we create photons. Since they are nothing more than massless particles of energy/momentum, it is hard to say what they are created of. So if I dump energy into the fields using an antenna, then am I drawing the photons up from the vacuum or just creating them from the energy injection from my antenna.

As for virtual particles, they are not real. It is hard to say what they are but I have not heard of them as being any physically real object. They can be useful calculation tools though in Feynman diagrams. In the quantum vacuum, we can represent the vacuum fluctuations as virtual photons. The idea is that we momentarily create the photon let it interact and then destroy it. In the end, because we created and destroyed the particle we add no energy to the fields, but by allowing the particle to interact it is the same as allowing the field fluctuations to have interacted. For example, in the Casimir force, we can calculate it from the force induced by the fluctuation fields or we could calculate it as the "radiation" pressure force of the equivalent virtual photons. The results are identical.

 

[italiano vero]

I have some questions:

these processes broke the conservation of energy?

Some physicists claims that Casmir effec is not a valid proof of the existence of vacuum fluctuations (http://arxiv.org/abs/hep-th/0503158). Are they right? But the quanta of foundamental interactions are VIRTUAL bosons, so how can virtual particles don't exist?

How is considered quantum vacuum in loop quantum gravity? What LQG says about vacuum fluctuations and virtual particles?

 

[Freeman Dyson]

Thank you for taking the time with that answer, Born2bwire.

So it is incorrect to say that these quantum flucuations come from "nothing"? Physicists just calculate and don't worry about where fluctuations come from. They just deal with them.

And what is the difference between vacuum energy and dark energy?

 

[Fredrik]

I'd like to recommend this really cool article about vacuum to those who know a lot of math already. (DarMM recommended it to me in another thread).

 

[Freeman Dyson]

I'd like to recommend this really cool article about vacuum to those who know a lot of math already. (DarMM recommended it to me in another thread).

But for those who don't know a lot of math...this article touches on another thing I was wondering. Where is spacetime in all this? The gravitational field? Is the gravitational field the background in all of this?

 

[Born2bwire]

I have some questions:

these processes broke the conservation of energy?

Some physicists claims that Casmir effec is not a valid proof of the existence of vacuum fluctuations (http://arxiv.org/abs/hep-th/0503158). Are they right? But the quanta of foundamental interactions are VIRTUAL bosons, so how can virtual particles don't exist?

How is considered quantum vacuum in loop quantum gravity? What LQG says about vacuum fluctuations and virtual particles?

I do not see how they are breaking energy conservation. As stated previously, one way of thinking about it that I have read, Milonni reiterates this in his text though I do not know to what extent this idea is encouraged, is that the photons are "drawn" out of the vacuum. But since the vacuum has "infinite" energy, there is a limitless supply of photons. So there is not implicit creation of annihilation of energy. It's special relativity, the energy of the fields are realized as quanta. In the case of electric and magnetic fields, the quanta are photons. For matter fields, they can be electrons or etc.

EDIT: I will also note that some consequences of the vacuum (ignoring Jaffe's complaints for the moment) is the Casimir force. However, I have yet to hear of a true way of extracting energy by its use. Yes, the Casimir force does work and thus extracts energy, say from the vacuum, however, it does so conservatively. It is a static force, if I were to use the Casimir force to draw close a lever and thus do work, I would inject the energy back into the vacuum when I physically separate the lever again. No different from raising and dropping a ball in a gravity field. There have been some people saying they have some ideas on how to extract small amounts of energy but I have yet to hear of any actual physical results. So this aspect seems to be safe though I do not know what it would mean if we could extract energy. Be an interesting development though I do not think anyone expects to extract any meaningfull amounts of power given the weakness of the Casimir force over all but the minuteness of distances.

I have read Jaffe's paper and find it very interesting. I think he has a point but I would like to have seen it published to read repsonses from others. A lot of times I have felt that methods of calculating the Casimir force, while mathematically and conceptually different, are physically equivalent. For example, I stated earlier that we can calculate the force directly from a path integral of the field fluctuations. However, we can also calculate it using the radiation pressure of the virtual photons. But physically this is an equivalent process, since all we are doing is using a virtual photon to represent the equivalent fluctuating field.

A virtual particle isn't real because it just isn't a real particle. It's a tool of perturbation theory in quantum field theory. Quantum field theory differs from normal quantum mechanics because it incorporates (and to the best of our knowledge it does so correctly) special relativity into quantum mechanics. One of the greatest consequences of this is from the equivalence principle. Energy and mass are equivalent and this now allows for the creation and annihilation of particles. The virtual particles are representatives of momentary energies that can be represented as particles. We are never really saying that a particle is actually created or that a Feynman diagram is a true physical representation of a process. It is more of a mathematical tool and place marker. In the quantum vacuum, the energy of the ground state causes photons to come out and back into the vacuum state. These photons never persist though, and thus they do not raise the system from the ground state to an excited state. But they can interact, via their representation as a field fluctuation. But since we cannot directly measure these photons (because if we could then it would mean that the system was truly excited, every photon represents an excitation of the field) they are not considered to be real.

I am not familiar with loop quantum gravity theory.

 

[Born2bwire]

Thank you for taking the time with that answer, Born2bwire.

So it is incorrect to say that these quantum flucuations come from "nothing"? Physicists just calculate and don't worry about where fluctuations come from. They just deal with them.

And what is the difference between vacuum energy and dark energy?

Yeah I guess that is one way to look at it. Like a lot of quantum theory, we are dealing with some of the lowest levels of physics and as such we are at areas where there is no more underlying physics to use an explanation. No matter how complicated physics becomes, go down far enough and you will always reach a level where we can no longer offer an explanation and it just must be taken as the correct model. As far as I can remember, I have not been able to find a very straight-forward physical explanation for the vacuum fluctuations. In a way, it requires you to explain why the quantum harmonic oscillator has a non-zero ground state. The electric and magnetic field quantum representations are harmonic oscillators, which is why their ground state is non-zero. Though this is a bit of an aside because we can always renormalize the vacuum energy to be zero. But, even if we do we always get the vacuum fluctuations, they do not disappear. They come about from the homogeneous solution to the field equations for the vacuum and cannot be removed by renormalization of the vacuum energy. There may be more insight from statistical mechanics.

I am not terribly familiar with astrophysics outside some of the more interesting electromagnetic phenomenon that it encompasses. However, I distinctly remember dark energy to be one way of explaining the difference in the mass in the Universe from current theory. In this respect, it differs from the quantum vacuum because the quantum vacuum does not create persistent particles. First, as I stated above, we do talk about the energy bath being able to give rise to virtual photons. These photons are created and annihilated before they could be "observed" but they give rise to the fluctuations in the fields. So yes, the energy of the vacuum does create particles but these particles will not represent a persistant, measurable, or macroscopic property. In addition, photons are massless and the quantum vacuum is only related to photons. One would need to analyze the "vacuum" for matter waves to see if say the electron field has a non-zero ground state and a fluctuating electron vacuum. In this case, the electron vacuum could give rise to massive virtual particles though I believe that again by virtue of being a virtual particle they are of no consequence to the macroscopic world (and I am sure that you would probably find the electron vacuum to be devoid of fluctuations and non-zero ground energy or it would be a very very interesting vacuum to study in its own right).

But for those who don't know a lot of math...this article touches on another thing I was wondering. Where is spacetime in all this? The gravitational field? Is the gravitational field the background in all of this?

I have not read the aforementioned article. However, quantum field theory only incorporates special relativity. They have not fully integrated general relativity into a verified quantum theory (string theory is one attempt at this I believe). So a lot of times I believe gravity is ignored. It's aspects are not a topic of interest to my research so I have not taken any time to delve into any connection it may have with quantum field theory and the quantum vacuum. I do not have access to my resources currently so I cannot say whether or not such a topic is dealt with.

 

[italiano vero]

Thanks Born2bwire for the answers.

Where is spacetime in all this?

This is a question I'd like to ask. Vacuum without space is possible? Vacuum is the space itself? The infinite energy of vacuum is the energy of spacetime? Vacuum energy is the spacetime itself?

 

[Halcyon-on]

As for Casimir force providing evidence for its existence, Jaffe has an article that he wrote where he feels that this is not something that can be claimed. He feels that since we can characterize the Casimir force in a number of ways, one of which does not use the quantum vacuum at all, then it cannot be used as proof.
http://arxiv.org/abs/hep-th/0503158

At this point, assuming that Jaffe is right - that is that the Casimir effect can be obtained without any reference to the vacuum energy - and considering that the Higgs mechanics has not yet been inquired, there exist other evidences for the existence of the vacuum energy?

 

[italiano vero]

I think that there are some effects that can be explained through vacuum fluctuations, but there is not a real proof, if we do
not consider the casimir effect.

 

[Halcyon-on]

If there are no proofs, they are not explanations but conjectures. Would this means that the vacuum energy is a conjecture?

 

[Born2bwire]

If there are no proofs, they are not explanations but conjectures. Would this means that the vacuum energy is a conjecture?

There are several phenomenon that require the vacuum fluctuations to exist, the Lamb shift is one example. However, it seems that this is a bit independent of the question of the vacuum energy. As stated previously, the vacuum energy can be renormalized to be zero, but this does not remove the vacuum fluctuations. Rereading Jaffe's abstract it seems I should have mentioned that he is more specifically focusing on the proof of the vacuum energy, not the existence of both fluctuations and energy. I do not know if there is anything that conclusively supports the vacuum energy.

Thanks Born2bwire for the answers.



This is a question I'd like to ask. Vacuum without space is possible? Vacuum is the space itself? The infinite energy of vacuum is the energy of spacetime? Vacuum energy is the spacetime itself?

What do you mean by space though? The vacuum fields require four-space. A common method of describing the vacuum is to assume that we have a cavity with periodic boundary conditions. The true vacuum state is assumed to be the modes of this cavity when we take the limit of the cavity's size to infininty. In this way, we have assumed an infinite Minkowski space. Well, Minkowski space may be a bit of a stretch as I believe that most derivations do not strictly assume Minkowski space but obviously the vacuum can be expressed relativistically.

I am not sure what you mean by energy of spacetime or energy being spacetime. These are concepts I have never heard of.

 

[Halcyon-on]

There are several phenomenon that require the vacuum fluctuations to exist, the Lamb shift is one example. However, it seems that this is a bit independent of the question of the vacuum energy. As stated previously, the vacuum energy can be renormalized to be zero, but this does not remove the vacuum fluctuations. Rereading Jaffe's abstract it seems I should have mentioned that he is more specifically focusing on the proof of the vacuum energy, not the existence of both fluctuations and energy. I do not know if there is anything that conclusively supports the vacuum energy.

The lamb shift is given by the two point function and not by the zero point function, so it is not related to the vacuum energy but to corrections to the tree level EM propagator.
The vacuum fluctuations and the vacuum energy in my view are two different things. The former can be easily interpreted classically because are variations of the energy with respect a energy reference, there is no problem about that. The vacuum energy is not classical because it is an absolute value of the energy. So Jaffe seems to say that this peculiar aspect of quantum mechanics could be a myth.

 

[Born2bwire]

I did not say that the Lamb shift is related to the vacuum energies, but the fluctuations. There is coupling with the vacuum fluctuations that is required for the correct Lamb shift to be calculated. And I have yet to find any classical interpretation for the vacuum fluctuations. They can only arise by the quantization of the electric and magnetic fields.

 

[Bob_for_short]

I did not say that the Lamb shift is related to the vacuum energies, but the fluctuations. There is coupling with the vacuum fluctuations that is required for the correct Lamb shift to be calculated.

T. Welton (Phys. Rev. 1948) showed that it is the electron position fluctuations that modify the potential energy and cause the Lamb shift. And the electron position fluctuations are due to electron's being affected with the fields of quantum oscillators. He added the field of quantum oscillators as an external filed to the electron equation of motion and obtained an additional "electron smearing".

In fact, it is rather difficult to imagine that the quantum oscillators are not affected with the electron itself, that their weak fields are so strong that are "external" to the electron. It is much more natural to say that these oscillators are internal degrees of freedom in a compound system (electron + quantized EMF). That is why they are "decoupled" - they are just separated variables of a compound system. Then the electron in this compound system is not point-like but smeared (being a part of oscilltors) and this gives the Lamb shift.

Each charge capable of radiating is thus a compound system with its own internal degrees of freedom (its own oscillators).

And I have yet to find any classical interpretation for the vacuum fluctuations.

See the QM model described above.

They can only arise by the quantization of the electric and magnetic fields.

It is not the fields who are directly quantized but the oscillator energies. Fields flictuate and they are just usual "fluctuations" in a compound system like an electron position in an atom (it's uncertain there).

 

[italiano vero]

Is there anyone that has the patience to explain me the relation between vacuum energy and vacuum fluctuations? Aren't fluctuations the fluctuation of vacuum energy? So how can be proved fluctuations without proving energy?
Besides, as "Bob for short" says, there is not a final proof of the existence of quantum vacuum and its virtual particles, because there is not an evidence of the existence of vacuum energy and vacuum fluctations, or not? I know that weak, strong and electromagnetic forces are explained at the foundamental level through virtual bosons (photons, W and Z, gluons): so, if virtual particles do not exist, what are the other theories about the foundamental interactions?

 

[Freeman Dyson]

There are several phenomenon that require the vacuum fluctuations to exist, the Lamb shift is one example. However, it seems that this is a bit independent of the question of the vacuum energy. As stated previously, the vacuum energy can be renormalized to be zero, but this does not remove the vacuum fluctuations. Rereading Jaffe's abstract it seems I should have mentioned that he is more specifically focusing on the proof of the vacuum energy, not the existence of both fluctuations and energy. I do not know if there is anything that conclusively supports the vacuum energy.



What do you mean by space though? The vacuum fields require four-space. A common method of describing the vacuum is to assume that we have a cavity with periodic boundary conditions. The true vacuum state is assumed to be the modes of this cavity when we take the limit of the cavity's size to infininty. In this way, we have assumed an infinite Minkowski space. Well, Minkowski space may be a bit of a stretch as I believe that most derivations do not strictly assume Minkowski space but obviously the vacuum can be expressed relativistically.

I am not sure what you mean by energy of spacetime or energy being spacetime. These are concepts I have never heard of.

Ok. What I meant by space was spacetime/the gravitational field. By 4 space you mean they need spacetime? What I am wondering is where is spacetime/thegravitational field in all of this. Is spacetime the "carpet" which is underneath this vaccum? What is the vacuum "sitting" on. What is the background? Where is the gravitational field?

 

[Bob_for_short]

Is there anyone that has the patience to explain me the relation between vacuum energy and vacuum fluctuations?

There is no such a relation. Vacuum energy cannot be used so its value may be taken to be zero.

Aren't fluctuations the fluctuation of vacuum energy? So how can be proved fluctuations without proving energy?

Vacuum energy is certain, it does not fluctuate. It is zero. But it can be considered as a sum of "potential" and "kinetic" energies that vary in time. Their sum is constant and is minimum possible. This is the definition of quantum vacuum for a specific system. Think of atom as a model.

Instead of "virtual particle exchange" you may safely say "interaction with help of Coulomb or another potential".