  We calculate the Coulomb scattering amplitude for two electrons injected with
opposite momenta in an interacting 2DEG. We include the effect of the Fermi
liquid background by solving the 2D Bethe-Salpeter equation for the
two-particle Green function vertex, in the ladder and random phase
approximations. This result is used to discuss the feasibility of producing
spin EPR pairs in a 2DEG by collecting electrons emerging from collisions at a
pi/2 scattering angle, where only the entangled spin-singlets avoid the
destructive interference resulting from quantum indistinguishability.
Furthermore, we study the effective 2D electron-electron interaction due to the
exchange of virtual acoustic and optical phonons, and compare it to the Coulomb
interaction. Finally, we show that the 2D Kohn-Luttinger pairing instability
for the scattering electrons is negligible in a GaAs 2DEG.
