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All quantum fields exhibit vacuum fluctuations. For the electron field this means that virtual pairs constantly pop in and out of existence. I know, a popular view, but so is that of a real particle moving through spacetime.

Now, virtual electron-positron pairs can influence the motion of real particles by exchanging virtual photons with the virtual pairs. But what about the EM vacuum field? Does it influence charged particles? If so, how? If there are no other other real particles and we don't pay attention to the virtual electron vacuum field, is it even meaningful to ask? Has EM vacuum field an existence by itself?

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This is a pretty good question. I am curious to see what other commenters say.

My first thought is this. It is true that the EM field fluctuates, and measuring the average value of $\vec{E}(x)$ in some region of space will include statistical randomness. While $\langle \vec{E}(x) \rangle = 0$, there is still uncertainty as $\langle \vec{E}(x)^2 \rangle \neq 0$.

Having said that, one has to be careful about how one pictures a charged particle. If you picture it as a classical point, you may conclude that its position will jitter in a Brownian motion style (which will influence the spread of its wavefunction).

However, in an interacting theory, one must be careful with what one means the definition of an electron. If you want to be as precise as possible, a definite momentum (plane-wave) electron state $| \vec{p} \rangle$ in an interacting theory is an exact energy eigenstate of the theory. In other words, the plane wave simply picks up a uniform phase at it evolves in $t$ and only depends on the energy of the electron. If one makes a wave packet state, with some roughly well localized position, these phases alone will govern how the wavepacket will spread out in time.

In other words, there won't be any special jittering.

I suppose one could imagine that the EM field fluctuations are already taken into account when one defines what exactly one means by an electron in an interacting theory,

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    $\begingroup$ Can you provide links for the EM vacuum fluctuations, and does this relate to $$\nabla^2 \vec{E} - \mu_0 \epsilon_0 \frac{\partial^2\vec{E}}{\partial t^2} = 0$$ $$\nabla^2 \vec{B} - \mu_0 \epsilon_0\frac{\partial^2\vec{B}}{\partial t^2} = 0 $$ as maxwells equations allow for non zero E,B fields in the presence of no charges or currents, as he electromagnetic field is not uniquely determined by charges and currents $\endgroup$
    – jensen paull
    Dec 9, 2021 at 22:03
  • $\begingroup$ Nice answer!Thanks! I wait a bit with accepting it. It's my feeling that the "Zitter" is associated with virtual electron-positron pairs. I just can't envision a vacuum photon field without electric charge being present. Don't photons need a charge to be created? Though reading the previous comment this should be possible in the classic case. Can virtual photons pop in and out of existence without charge? Is the virtual vacuum field determined by real charge, giving a means for interaction? That being said, the vacuum gluon field contains gluon-antigluon pairs, so it doesn't need real color. $\endgroup$
    – MatterGauge
    Dec 9, 2021 at 23:11
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Indeed, this is a good question.

But first, you need to understand that virtual particles don't exist at all. These are mathematical artifacts from the series expansion of the $S$ matrix: propagators as Feynman rules.

That being said, one says that there's an electron-positron pair in the vacuum when dealing with Feynman diagrams with one propagator alone: a graph with one link but with no vertex. One can imagine a loop consisting of just one photon propagator, but it turns out that this can't be linked alone with electrons or positrons in one unique vertex. It is possible with the $W^\pm$ bosons thanks to the $\gamma \gamma W^+ W^-$ vertex.

Now your question is a bit more general as it asks for the existence of an influence of the electromagnetic fluctuations on the charged particles. The answer is "Yes" and the associated phenomena are called "electron self-energy" and "anomalous magnetic moment of the electron", for the electron but you surely have heard about the anomalous magnetic moment of the muon for example, so these are general phenomena arising from loop corrections of the tree-level diagrams.

I insist on the fact that when we are talking about virtual particles one has to keep in mind that this is quantum field theoretical stuff and that if one wants to understand completely this stuff, learning about QFT is the best thing to do.

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  • $\begingroup$ The question if virtual particles are real or not is up for debate. Personally, I think they are pretty real, despite being called virrtual. Nevertheless, thanks for your answer! I'm not sure I understand though. Aren't the anomalous magnetic moment and seĺf-energy connected with virtual photons from and into the electron (which means that when you measure the electron's MM a component of the value of the electron in between emitting and absorbing contributes). Do closed photon loops exist? How do these interact with the electron? Is the vacuum photon field just a bunch of closed photon $\endgroup$
    – MatterGauge
    Dec 10, 2021 at 9:26
  • $\begingroup$ Can we envision the vacuum filled with closed electron-positron loops and photon loops? I can't imagine closed photon propagator, without a charge being attached. Closed electron-positron loops are somewhat easier to imagine. Though this can be caused by considering them real. I mean, considering them real you can interact with them by means of a virtual photon. A second thought though, an electron can interact with a closed photon loop by attaching directly to it. But then it aint no closed loop anymore, because then it starts and at the electron propagator. $\endgroup$
    – MatterGauge
    Dec 10, 2021 at 9:49
  • $\begingroup$ I know these lines though with a closed loop on them though. In Ryder they are used. Still, photons without charge seem strange. Like closed electron-positron loops. How can two charges appear from nothing? You can say it's only math, of course. $\endgroup$
    – MatterGauge
    Dec 10, 2021 at 9:53
  • $\begingroup$ @MatterGauge You can have a closed photon loop attached to the $W$ boson propagator, this contributes to its self-energy. Photon loops with no external lines "exist", at least mathematically, and are vacuum bubbles. So yes you can envision the vacuum as filled with loops if you want, this contributes to the zero-point energy. When you talk about electron/position pair, in fact, there's only one excitation, not two particles. Finally, you can't attach an electron line to a photon loop because of the conservation of charge. $\endgroup$
    – Jeanbaptiste Roux
    Dec 10, 2021 at 10:36
  • $\begingroup$ Very clear. But how can a closed photon loop be attached to a W? Aren't they charged too? $\endgroup$
    – MatterGauge
    Dec 10, 2021 at 11:51

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