quantum field theory - Are virtual particles only a fictive tool in equations?

There is no "action at a distance" in nature. Attraction of a piece of iron by a magnet, attraction between distant electric charges of opposite sign, have to be mediated by something. The virtual particles are proposed as an explanation.

If they have an observable effect, it seems like they must exist. If so, why do we call them virtual? It can't be that they have short lifetimes, as there are short-lived particles (resonances) that we don't call "virtual."

On the other hand, some people say that the so-called virtual particles only exist on paper, or that they are just a technique to help us handle our equations.

So what is the meaning of the word "virtual" here? Do these particles really exist, or not?

Answer

Virtual particles refer to actual, nonzero features in the quantum fields of real objects, but they are features that are not particles in many ways so you should not expect anything from their being named "particle".

Basically, the idea of virtual particles was invented as a device for when you want to hold on to the particle picture while doing quantum physics. Keep in mind, we know that actually nothing is really a particle, but rather quantum fields are the fundamental objects. We can derive particle-like motions in fields but fields also show other behaviours. If you insist on everything being somehow a particle then these other behaviours need to be recognised and treated with care. Someone decided they shall be called "virtual particles".

For example take a hydrogen atom, a bound proton and electron. There is for sure a real electromagnetic field inside the atom, holding it together. This electromagnetic field is certainly not a particle in any classical sense. With quantum fields we can choose to represent the electromagnetic field in terms of photons (i.e., using a photon-like basis of states to describe the field). But in doing so we see that the photons inside of a hydrogen atom are not like familiar radiating photons in free space but rather something else, virtual somehow.

Again, the only thing that is fundamental and real is the quantum field, which does not care about any distinctions we choose to make between real and virtual particles, or between particles and waves. Nevertheless we like to come up with funny names to help ourselves come to grips with the reality. But perhaps the term "virtual particle" is more misleading than it is helpful.