That's the point that perplexed physicists for 15 years. The quantities calculated from Dirac's field theory were infinite. Furthermore, there were other infinities. For instance, real (not virtual) photons exhibit the behavior that, as you look at the number of photons being radiated by an accelerating electron, for example, the number of photons goes to infinity as you look at lower and lower frequencies. That should be no problem, as long as the sum of the photon energies is finite...but when you do the calculation, it isn't!
Then, in the late 1940's, three men independently noticed that if you perform all of the calculations properly and put everything together, the infinite sum of the virtual corrections--all those infinite electron-positron pairs, plus the virtual photons of the field)--almost exactly cancels the infinite sum of all the real photons. What's left is a residual quantity that agrees with the experimental results to more than 10 decimal places.
Feynman, Schwinger and Tomonaga received the 1965 Nobel Prize for this discovery (the "renormalization" of quantum electrodynamics). All of the quantum field theories of particle physics exploit the renormalization principle; for a theory to be considered calculable, it must be renormalizable.
[Geek alert: Gravity is a spin-2 field, meaning that each momentum-carrying quantum of the field (i.e. a graviton) carries two h-bar units of angular momentum. The problem that has plagued quantum theories of gravitation is that the theories are not renormalizable. It turns out that there are only two spaces in which a spin-2 field can be renormalized: one of them has 26 dimensions; the other has 11.]
So let me get this straight, any virtual particle pair can be expressed as an electron-positron pair plus photon pairs?