# Confusion in understanding of quantum fluctuations and vacuum energy [duplicate]

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I'm having a bit of trouble understanding what exactly is meant by a quantum fluctuation of a quantum field and its relation to the vacuum energy attributed to such a field.

Is the point, that due to the Heisenberg uncertainty principle, one cannot precisely determine the value of the field at a given point and as such its standard deviation is non-zero. So by saying that a quantum field is fluctuating is simply a handwavy statement of the fact that its value (in vacuo) at a given point in space-time is not prescisely determined (i.e. it is not well-defined) and this results in a non-zero vacuum energy?!

Or are quantum fluctuations simply the name that is given to higher-order corrections to correlation functions (for example $\langle 0\lvert T\lbrace\hat{\phi}(x)\hat{\phi}(y)\rbrace\rvert 0\rangle$) in perturbation theory, due to the fact that these corrections are represented by loops in the corresponding Feynman diagrams, which can heuristically be thought of as virtual particles being produced by fluctuations in the vacuum state of the quantum field?

Apologies if I'm spouting nonsense, I just want to understand this concept and thought it helpful to put down my current thoughts about it.

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• This might be of help: profmattstrassler.com/articles-and-posts/… – user108787 May 20 '16 at 19:08
• @count_to_10 Thanks for the link. I've actually read that before, and it is a bit to much of a layman explanation for my liking. I'm hoping to gain a more technical understanding of whether vacuum energy is physical and if quantum fields are actually fluctuating in their ground state?! It seems to me that they are simply in their ground state, but due to the Heisenberg uncertainty principle a given field and its conjugate momentum don't have well-defined values, which is why people refer to them as fluctuating, whereas in reality they are simply in their ground state. – user35305 May 21 '16 at 15:14
• "Quantum fluctuation" is a weasel word people either use to make the fact that standard deviations of operators can be non-zero more mysterious, to evade having to state anything precise, or to mask that they have no idea what they're talking about. Physics would lose nothing if we all just agreed to stop using that word. – ACuriousMind May 23 '16 at 10:47
• @ACuriousMind I'm finding it very confusing trying to learn the subject with the fact that these kind of terms keep being used without any real clarification and especially as they seem to have no meaning (as you pointed out). As a consequence I'm finding it hard how to understand concepts such as vacuum polarisation, and ideas in semi-classical gravity where the cosmological constant pops up and is attributed to the presence of vacuum energy (which itself is commonly cited as being due to vacuum fluctuations of the matter fields, i.e. vacuum loop diagrams)?! – user35305 May 23 '16 at 10:57
• There is an extended discussion of this at physicsforums.com/threads/… – Arnold Neumaier May 30 '16 at 17:50

## 2 Answers

Almost everything from the wikipedia page you link is just false, or at best very misleading. IMHO, that page was written by someone that doesn't know anything about quantum mechanics beyond what one could find in TV documentaries. "Not even wrong" came into my mind many times as I was reading the article.

In quantum physics, a quantum fluctuation (or quantum vacuum fluctuation or vacuum fluctuation) is the temporary change in the amount of energy in a point in space, as explained in Werner Heisenberg's uncertainty principle.

No, not at all. In quantum mechanics, there are no fluctuations, at all. A fluctuation is a statistical concept; unfortunately there are many words in QM taken from statistical mechanics. For example, "correlation functions" in QM doesn't measure correlations. It is just misleading terminology.

The energy doesn't fluctuate from point to point or from time to time. Energy is locally conserved, it is exactly locally conserved, not on average. QM is not statistical mechanics. It is not like in classical mechanics, when studying a gas of particles, where you may have thermal fluctuations. The fact that, in QM, you cannot simultaneously measure the position and momentum of a particle is radically different from the fact that, in statistical mechanics, the velocity of the gas fluctuates about the average value.

Oh, and the HUP has nothing to do with energy.

According to one formulation of the principle, energy and time can be related by the relation $\Delta E\Delta t\ge\frac{1}{2}\hbar$. This allows the creation of particle-antiparticle pairs of virtual particles. The effects of these particles are measurable, for example, in the effective charge of the electron, different from its "naked" charge.

Nope. That's not true. Nothing here is true.

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. Vacuum energy may also be responsible for the current accelerated expansion of the universe (cosmological constant).

Not a single (theoretical nor experimental) reason to believe this is true.

A quantum fluctuation is the temporary appearance of energetic particles out of empty space, as allowed by the uncertainty principle. The uncertainty principle states that for a pair of conjugate variables such as position/momentum or energy/time, it is impossible to have a precisely determined value of each member of the pair at the same time. For example, a particle pair can pop out of the vacuum during a very short time interval.

Nope. (Almost) Everything here is false. Particles don't temporarily appear and disappear. And, if that were true, it would have nothing to do with the HUP. There is no logical relation in the different sentences from this paragraph. The "For example" part doesn't follow from what it is stated before.

An extension is applicable to the "uncertainty in time" and "uncertainty in energy"

No, it's not. Why would it?

In quantum field theory, fields undergo quantum fluctuations.

Nope. You won't find this assertion in any book on QFT. Only in pop-science.

Now, to your question.

So by saying that a quantum field is fluctuating is simply a handwavy statement of the fact that its value (in vacuo) at a given point in space-time is not prescisely determined (i.e. it is not well-defined) and this results in a non-zero vacuum energy?!

The value of any field at a point is not precisely determined, but it has nothing to do with fluctuations. It has to do with the fact that fields are distributions, and so to extract a number from them you have to integrate them over together with an spatially extended function: $$\langle\phi(x_0)\rangle_f=\int\mathrm dx\ f(x)\langle \phi(x)\rangle$$ where $f(x)\in L^2$ is a function that is highly localised near $x\approx x_0$.

Oh, and in QFT the vacuum energy is an irrelevant concept. It can have whichever value you want it to. You cannot measure it, and you may set it to zero if want to. This is by no means different from the fact that, in classical mechanics, the origin of energies is under-determined. I don't know why people think that in QFT this changes.

Or are quantum fluctuations simply the name that is given to higher-order corrections to correlation functions (for example $\langle 0\lvert T\lbrace\hat{\phi}(x)\hat{\phi}(y)\rbrace\rvert 0\rangle$) in perturbation theory, due to the fact that these corrections are represented by loops in the corresponding Feynman diagrams, which can heuristically be thought of as virtual particles being produced by fluctuations in the vacuum state of the quantum field?

AFAIK, no. But it is possible. The truth is, actual physicists don't speak of vacuum fluctuations. It is not an important concept. If I were you, I'd just forget about it. As far as we know, the cosmological constant has nothing to do with quantum mechanics.

• In "Quantum Field Theory for the gifted amateur" the author refers to quantum fluctuations as the quantum corrections to the classical solution, and quite a few papers on the arXiv seem to refer to quantum fluctuations in the context of higher-order (loop) corrections in perturbation theory (for example arxiv.org/pdf/1502.05296v1.pdf , c.f. page 3 onwards). Furthermore, several introductory books to QFT (including Peskin and Shroeder and QFT for the gifted amateur) refer to virtual (or "off-mass-shell") particles in perturbative calculations.... – user35305 May 24 '16 at 15:22
• ... All this leaves me very confused as to what's going on?! Are these simply artifacts of perturbative calculations that have the "nice" visualisation of virtual (exchange) particles in Feynman diagrams? Someone also told me that there are no "vacuum fluctuations" in the interaction picture since we are working with the vacuum state of the free theory, $\lvert 0\rangle$, hence it really is the state with zero particles, energy, etc. However, in the Heisenberg picture we work with the interaction vacuum, $\lvert\Omega\rangle$ which is much more complicated.... – user35305 May 24 '16 at 15:27
• ... and furthermore can have so-called "vacuum fluctuations" in which a given quantum field can act on $\lvert\Omega\rangle$ to produce off-mass-shell particle-antiparticle pairs?! – user35305 May 24 '16 at 15:29
• @user35305 1) using "quantum fluctuations" for "loop corrections" is not very standard: not many people say that. But I guess you can use that words if you want to (I dont). 2) virtual particles are artifacts, yes (see this answer of mine: virtual particles are not real). There are no vacuums fluctuations in QFT, but this has nothing to do with the picture you use (after all, physics are independent of the picture). And the interacting picture works with the interacting vacuum (1/2) – AccidentalFourierTransform May 24 '16 at 15:38
• (2/2) , not the free one (but they are related, so in the end you can work with the free one, bc its easier). 3) We dont have vacuum fluctuations! I thought I made that clear in my post :P Actually, unless you can define what a vacuum fluctuation is, I don't think I'll be able to convince you about this. Vacuum fluctuations are not a concept physicists use (physicists that work with QM). Its only used by pop-science. Unless you find someone who has a proper (mathematical) definition for VF, I don't think it makes sense to try to understand whether they are real or not. – AccidentalFourierTransform May 24 '16 at 15:43

Nina Arkani-Hamed lectures, David Tong lectures Sean Carrol lectures, World Science Festival, there many sources. https://youtu.be/zNVQfWC_evg It is an analogy and quantum phisics uses language to translate Phisics and mathematical into concepts that use rhetoric figures...