[PAOLO137 (OP)]

I have read all that was possible to find in the field (especially Stephen Hawking and Richard Feynman). Nobody has been
able to help me how to figure out what the verb "mediate" must be understood. I have confidence in physicist, but this
appears as a trick to get out from the quicksand of the concept of "field" . I remember having read that a certain Newton
notwithstanding his discovery of gravitation, felt himself not at ease thinking about a Sun exerting a force on the Earth
across such distant and empy space.

[Soul Surfer]

Experiments show clearly that all the elementary units in our universe can be observed to behave both as particles (individual localised objects) and waves (periodic variations in the value of a medium or field).

Quantum field theory is the most accurate of any scientific theories and is the mathematical technique used to describe these elementary units and the way in which they interact with each other.

The uncertainty principle defines a region of precision within which we cannot look so all interactions between particle are such that two elementary units (say particles)  with defined properties go into the uncertainty box to interact and one or more particles come out with properties defined by the way they come out and the conservation laws that apply.

These interactions can always be described in terms of the exchange of other particles (elementary units) with defined properties from within the full set of such units that exist.

These elementary units that only exist inside the box and are not seen outside are said to mediate the interaction and could be described as the force between the particles but they can do much more than change the direction of motion of a particle.  They can change its mass and electrical or other charge as well.

[Soul Surfer]

let me give you an example of what I mean.  A high energy electron can interact with a proton and be deflected by it (inside the uncertainty box) and a deflected electron and a recoiling proton come out it could have exchanged one photon of electromagnetic energy to do this but it could have exchanged several less energetic photons to do exactly the same thing we can never know the result is just the same.  This is what Feynman diagrams are all about.

On the other hand it can also go into the same interaction box and the electron together with an electron - positron pair can come out as well as the original proton.  Here, some of the energy has been turned into mass.   

The probabilities of these interactions are all defined precisely by quantum field theory.

[PAOLO137 (OP)]

 :XAt last!  Since I like very much to explain scientific or technical subjects to people (who unfortunately after five minutes begin to talk between them about the last trousers by Armani) , for this reason I was so frustrated being unable to understand the meaning of these "mediator" particles that for some reason were not taught to physics students in the fifties (I got my degree in 1960 and worked in low temperature field studying properties of liquid Helium 3).
I  think that this subject gives some problems to many. I have recently read a book by K.W. Ford (The quantum world.Quantum physics for everyone). Well, at the beginning he promises to explain later those mediating particles, so I was eager to arrive at that point. It never came. Now let me annoy you with a detail : have these particles been observed? I suppose yes since the Higgs boson which has been on the newspapers these days should belong to that family. But how happens the detection of those particles with respect to the others? Are there special problems?
Thanks a lot for your patience. Make the BBC people happy : my daughter and I watch The World News every evening
after dinner! Sincerely Yours, Paolo, Rome.

[Soul Surfer]

We cannot know what happens inside the uncertainty box and therefore cannot observe the mediator particles inside the reactions. 

Quantum field theory in effect considers every possibility,  there are an infinite number of them,  and adds them up to get a set of probabilities for all the possible results that can come out of the box.  That includes the creation or destruction of particles and multiple photon interactions. With a bit of cannie mathematics called renormalisation the infinite series all add up to finite numbers.

We can see some of the mediator particles involved because they come out of the box and last long enough to be observed.  Photons are very well known Individual quarks are not seen because detaching them from (say) a proton requires so much energy that you always create an antiquark to go with them when you knock them off and make a meson.  The mediator particle of this interaction called gluons are also never seen alone for similar reasons.

In some cases where complex interactions and very short lived particles are involved.  The particles themselves are not detected but can be inferred when you measure their predicted decay products.
THe W and Z particles which are the weak interaction mediators are observed this way.

It is hoped that when the LHC is doing big collisions with lead nucleii in one of its phases some shadows of the gluons in the form of glueballs and other exotic particles may be detected but these collisions produce extremely complex outputs and some of the events may be very rare.

You must also remember that the detectors can in general only measure electrically charged particles.  Neutral particles also have to be inferred by their decay products and creation points.