We have investigated the electron pair emission from a Cu(111) surface excited with a primary electron beam. We use the diffraction of electron pairs to identify the contribution of the Shockley surface state to the intensity. Exploring energy and momentum conservation laws makes it possible to select this state and to investigate the electron-electron interaction within this two-dimensional electronic system. We also find for this system the emergence of a zone of reduced intensity, which is the manifestation of the exchange-correlation hole. Furthermore, the region of reduced intensity displays an intensity pattern that reflects the symmetry of the surface.