If virtual particles are considered to be just mathematical constructs, does this imply there are no corresponding physical processes? Some people consider all elementary particles to be non-existent. They tell us that they are changes in the field configurations that last which are called "real particles", while those that not last long enough are called "virtual particles".
But field configurations refer to wavefunctions that are a part of the entire field. Seen in this light, particles do exist, with an associated wavefunction. And so do virtual particles, though they don't last long enough to be "real". They find themselves (only by saying this already gives some "realness" to their existence) not on mass-shell and appear in the interaction between real particles, which do find themselves on mass-shell (or in the field configuration of the vacuum).
In the interaction case (as well as in the vacuum field configuration case or, for that matter, even in the case of changing field configurations, representing real particles), they are taught to be just mathematical constructions.
One can read [here][2] (a nice exchange of views, questions, and answers):

However, it appears to be a very good question whether virtual particles actually "exist" i.e. if they play a causal role. There is the common talk about virtual particles being exchanged and thus mediating interactions. This view was criticized e.g. by Mario Bunge (1970) in his „Virtual Processes and Virtual Particles: Real or Fictitious?”, International Journal for Theoretical Physics, Vol. 3, No. 6, 507-508. A rather forceful argument to the same effect was given by Weingard in his „Virtual Particles and the Interpretation of Quantum Field Theory“, in: H. R. Brown und R. Harré (Hrsg.) „Philosophical Foundations of Quantum Field Theory” 1988, Oxford: OUP. Weingard's bottom line is, that virtual particles are an artifact of perturbation theory only.

It's also possible to execute a Feynman diagram without the use of virtual particles, i.e., without the use of particles that not lay on their mass shell, but I didn't take a "deep" look into that possibility. It can be found here.
My question is very simple: If it is said that virtual particles are just mathematical constructs (just as real particles), does that imply that there are no real correspondences with the real world? It seems to me that something real must happen in the reaction between two particles. Or that elementary particles must be real particles and not mathematical constructions (changes in field configurations). As I said, changes in the field configurations correspond to wavefunctions which in turn correspond to real particles. These can be point-like or have some other form (I'm not referring to string theory).
 A: Virtual particles can influence physical processes.
The Casimir effect predicts an attractive force between massive objects in a vacuum, due to a destructive self-interference of virtual particles confined between them. There is some experimental support for this claim, although not everybody accepts the evidence as valid.
Hawking's model of black holes suggests that they should evaporate through a mechanism involving virtual particles. When a virtual pair appears very close to the event horizon, one may "steal" energy from the other and escape as a real particle, while the other is drawn back in and disappears - effectively stealing the energy from the black hole. I am not sure to what extent the suggestion has been verified, but it is certainly very widely acclaimed.
Static magnetic and electric fields are also sometimes described as comprising virtual photons, being continuously emitted and re-absorbed by the charge concerned. However I have never come across any underlying theory that ties together quantum properties and classical field equations in the way that QFT does for ordinary photons.
A: If you look at some problem in the Classical Electrodynamics (CED), say, at scattering of two charges, you will have two equations - one for each particle, containing a retarded field from the other one. These retarded fields are virtual particles. They have "sources" and "sinks"; the latter is the other particle for which it serves as an external field.
