What is the difference which leads to attraction in e+e- scattering and repulsion in e-e- scattering in QFT What is the theoretical difference between the physical elementary interaction that causes an e+ to attract an e− when they exchange a virtual photon? Why is this exchange different from an e-e- scattering (which produces repulsion)?
This question is related to but more specific than the recently posted The exchange of photons gives rise to the electromagnetic force.  It has not been answered there.   It would be good to see the answer somewhere.
 A: Short answer: Calculate the scattering amplitude (matrix element) for electron-electron or electron-positron scattering. They will differ by a minus sign. This means that the electron electron interaction will have a positive sign in front of the 1/r Coulomb potential which corresponds to a positive (repulsive) force. The same procedure for electron-positron scattering will lead to a negative 1/r Coulomb potential and hence a negative (attractive) force.
Longer answer: To decide whether any scattering process is attractive or repulsive you calculate the scattering amplitude (matrix element). Then compare your answer to the Born approximation for nonrelativistic QM scattering to see what classical potential your QFT result corresponds to. If the potential increases as you separate the particles the corresponding classical force will be attractive and vice versa. The minus signs in the scattering calculation can arise from many sources (anticommuting fermion fields, vertex rules, propagator terms, etc.). When you choose different incoming or outgoing states, like electron-electron or electron-positron, the calculation results in an extra negative sign. Interestingly, for the Coulomb force, one of the deciding minus signs comes from the photon vector propagator. Any vector propagator will cause like fermion charges to repel and unlike fermion charges to attract. Fermions don't always have to repel other fermions of the same type. If fermions exchange a scalar particle (instead of a photon) they can attract another fermion of the same charge! Something else to consider is that the Coulomb potential 1/r you arrive at is only for first order Feynman diagrams. If you include higher order terms you get corrections to the usual Coulomb potential/force.
Note: Using a Born approximation from nonrelativistic QM is an admittedly weird way of going about answering this question. It seems strange that we would take a result from our more fundamental theory (QFT) and then have to translate it into a less fundamental theory like nonrelativistic QM. But by asking questions posed in classical or nonrelativistic language (e.g. is this interaction force attractive or repulsive), maybe we are forcing ourselves to use a classical or nonrelativistic approximation/theory. 
Check out Peskin pg 125-126 for more on deriving the Coulomb potential/force from QED.
