Does a charged particle propagating in free space have a 'self-energy' like term due to it’s interaction with the fluctuations of the quantum vacuum? Does a charged particle propagating in free space have a 'self-energy' like term due to it’s interaction with the fluctuations of the quantum vacuum? (particle-antiparticle pairs popping into and out of existence)
Or is it the case the the fluctuations of the quantum vacuum don't exist long enough (from the uncertainty principle in order to conserve energy) for the electron to 'notice' them.
Thanks!
 A: If you want to think of a free on-shell particle as a mode that propagates like a soliton far from interactions then you can think of it as a thing that interacts with the vacuum so as to make more of itself.
But really you should stick to asking questions the formalism is designed to answer.
The vacuum is not a thing like a house that provides space for particles to appear and disappear. And if it were it doesn't stand there with a stopwatch enforcing an uncertainty principle.
What you have is a single electron-positron field whose modes encompess every possible allowed way of having any number of electrons and/or positrons. And this single field has values at every point in space and the values aren't real numbers (or complex numbers or vectors) they are operators.
And there is a different field for photons and another for neutrinos and another for W and another for Z and one for all the up quarks and ones for all the other quarks and they include the modes for the antiparticles as well and there is one for the gluons.
And the vacuum isn't a separate thing. It is just a state with fewer electrons and such. So few that the mathematical operations that normally remove electrons instead act on the vacuum as if you just didn't act with a mathematical operation at all (and I don't mean doing nothing, I mean as if you didn't use any operation, or you can just think it gives zero but that doesn't mean it gives the vacuum the vacuum and zero are different).
So the vacuum is just a state of the field like other states it isn't a separate thing. You describe states of having electrons by having a mathematical operation act on the vacuum
So having something act again would be redundant, the whole idea is confused. It would be like if we defined cookies as taking flour and sugar and cooking them (a bit of an over simplification) for 10 minutes. And then you asked if we need to say how cookies interact with flour and sugar and heat. Cookies are a state of flour and sugar.
An electron isn't distinct from a vacuum. The vacuum state is one possible state of the electron-positron field and the state of one electron is another and the state of two electrons is another state. They are all states of the field. So you never have a vacuum and an electron (well, there are states that are superpositions of states with different numbers of electrons).
But the point is the state of having an electron is already a modification of the vacuum state, making a different modification would just be not having that state of the electron.
