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.
Answer
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.
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