**5.1 Example: Scattering process**

Let us consider, however, another simple example: the electromagnetic scattering of two protons with the exchange of a single virtual photon. To fix the ideas, we choose a definite energy of the two protons as seen in their center of mass system, for instance *E*=10-13 J 1 MeV. (Magnitude orders are important in these considerations, in order to estimate the wavelength and the number of the exchanged particles, as we shall see below in the case of gravitons.) In this reference system the exchanged energy is zero and the exchanged momentum is of the order of <sup>20</sup> 2 10 *m Ep* kg m/s (non-relativistic approximation).

Fig. 8. Proton-proton scattering through the exchange of a single virtual photon, as seen from the center of mass reference system.

Suppose that this momentum is carried by one single virtual photon . The photon is offshell, with imaginary mass *m* 2<0: *m* 2*c*2=*E* 2-*p* <sup>2</sup>*c*<sup>2</sup> *m* 2=-*p* 2/*c*2. The virtual photon energy and momentum are exactly defined and their ratio *E*/*p* is exactly zero in this reference system (it is not Lorentz-invariant). The wavelength of the photon, defined as =*h*/*p* , is of the order of 10-14 m. One can estimate, classically, that the minimum distance reached by the protons is of the order of 10-16 m. If the virtual photon is emitted at this point, its wavefunction can clearly not be regarded as a plane wave. Its propagation velocity *v* is hardly observable and relation (41) appears to suggest that *v* is very large; if we assume *v*=*c*, it is only by analogy with the familiar retarded classical effects.

The situation appears, in conclusion, to be very different from the previous example. It seems reasonable to draw a clear distinction between a scattering process, which can be described as the exchange of a single virtual particle, and the inter-particle force in static or quasi-static conditions, which is in general equivalent to the exchange of a large number of virtual particles.
