Why do we say that the electromagnetic interaction is mediated by photons (or to use http://en.wikipedia.org/wiki/Force_carrier" [Broken] phrase, that the photon is the force carrier for the interaction)?

(I understand "mediated" … it's "photons" I object to.)

Take the simplest-order Feynman diagram for Compton scattering: an "H"-shaped diagram in which a real electron interacts with a real photon by "exchanging" a virtual electron.

Here, the interaction is obviously mediated by the virtual electron.

More generally, simple counting (per vertex) shows that in any large Feynman diagram, the number of virtual electrons must be roughly twice the number of virtual photons.

Perhaps we should simply say that the photon is the vector (or boson?) field associated with the interaction, while the electron is the spinor (or fermion?) field associated with the interaction, but they both mediate it?

As Meir Achuz says, every gauge interaction has a gauge particle, which must be a vector particle …

… but where does the Higgs scalar boson come into this?

What they mean is that a self-consistent theory of electromagnetism requires the existence of real photons. "Virtual photons" are a mathematical trick that one shouldn't read too much into, but there is a real and deep connection between electric and magnetic fields and photons. You can't have either without the other.

Can I see if my understanding of the electromagnetic interaction is good.

Take two electrons approaching each other. We all know as the carry negative charge they will repel. This requires a change of momentum for each particle although overall momentum is conserved. The virtual photon is the mechanism by which this happens. I am not so sure virtual photons are a mathematical trick to make the theory work, other wise we can say a gravitational field is a mathematical construction to allow us to calculate gravitational field strengths. My understanding always was that these field are a real as anything can be.

In the case of the two approaching electrons one coming from the left and the other from the right. Virtual photons are continuously being created with the electric field around each electron. The wave function of these photons fill space with its momentum pointing in an undetermined direction. So when another electron approaches from the right there is a probability of an interaction occurring between the virtual photon and the other real electron. If that interaction occurs to impart the virtual photons momentum to this second electron it will cause a change in momentum, reversing it direction, deflecting it slowing it down... Attraction can work if the virtual photons interact with an opposing charge to push it towards the other charge. This is controlled by the wavefunctions of the two real particles which changes the probability of virtual photons momentum pointing one way or the other.

Is that a clear? I would just like to know if I think about the electromagnetic interaction and other field interactions in the right but simplified way.