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When we talk about static force fields, we use virtual particles to describe the interaction of such fields with other particles. These virtual particles are not real particles, they are a mathematical model.

I have read this question:

Let's say that the Sun moves away from us, this should cause a change in gravitational force, when will this change be noticed by us?

Does static gravity follow spacelike geodesics?

Now since we are describing force fields with them, and force fields are created by a real object (like an electron for EM, or a star for gravity, quarks for strong, weak force) the virtual particles somehow propagate (at least theoretically) from the object to where the near field reaches.

Now in theory, virtual particles are off mass shell, and do not obey SR, speed of light.

There are though three worldlines as per SR:

1.spacelike

2.timelike

3.lightlike

Now in the case of gravity, there is a center of gravity, usually an object (with stress-energy), and these virtual gravitons exist everywhere in the gravitational field.

In the case of EM, there is a center object, a charge, that creates the EM field.

These virtual particles at least theoretically must mediate the forces from the object to the positions in the field wherever the field interacts with a particle.

Does SR somehow describe these virtual particles in terms of speed or worldline?

Question:

  1. Now do we at least theoretically have a worldline for these virtual particles?
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    $\begingroup$ There are no worldlines for quantum particles, either real or virtual. $\endgroup$ – G. Smith Jul 26 at 16:18
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    $\begingroup$ @G.Smith do you mean that a photon is not following a geodesic? see here en.m.wikipedia.org/wiki/World_line , look for "quantum" $\endgroup$ – anna v Jul 26 at 17:36
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    $\begingroup$ @annav I should have said that a quantum particle does not follow a particular worldline. In the Feynman path integral formulation, it follows all possible worldlines. I do not think this what the OP had in mind. $\endgroup$ – G. Smith Jul 26 at 18:56
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No, virtual particles don't follow world lines, and I highly advise to stop thinking about anything in QFT in terms of virtual particles - except for the one thing they're actually useful for, which is computing scattering amplitudes.

  1. It is difficult to talk about "the" worldline for quantum objects to begin with, as unless there is a continuous position measurement on these particles, they will possess uncertainties in position. So the question of whether quantum objects "follow world lines" depends on how you measure them. It would probably be better to think of a thicker "smeared out" world line.

  2. Though it seems a pervasive thought, virtual particles are not real. (Every word in the preceding sentence is a link to a different answer where I say the same in different words) You can't measure them, you can't interact with them, and in any formulation of QFT that does not use perturbative Feynman diagrams there isn't even anything you could call "virtual particle". They're just lines in a diagram, encoding certain integrals, in a truly non-perturbative formulation of quantum field theory you wouldn't see them.

    Contrary to popular belief, Feynman diagrams do not depict a process in space and time. They are visualization of contributions to an amplitude, but they do not depict a quantum field theoretic process in any sense. Their axes have no units (there is no "time direction", only the marking of the external legs as "in" or "out" is relevant!), all that matters is their structure as a graph. The internal lines don't corresponding to any physical "entity" that we could measure or interact with. Virtual particles don't "follow world lines" because they are exactly as real as the lines for latitude and longitude we draw on maps. They are tools we use to describe nature, but they are not part of nature.

  3. Even if you absolutely insist in imbuing these lines with much more ontological weight than they deserve, in order to describe measuring a virtual particle, you would have to let it interact with some detector, so it would have to be the external line of some QFT process so that it can escape the process and end up at a detector and hence would by definition not be a virtual particle! What does it matter whether something whose position you can't even in prinicple measure "follows a world line"? What would that even mean?

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