Imagine:
Setup A, we smash relativistic electron at a stationery positron.
Setup B, we smash electron and positron head on both moving at fraction of speed of light.
Setup C, we allow electron to catch up to positron both travelling in the same direction.
Since antimatter and matter would annihilate each other anyway, so I wonder if there is any difference for the above setup?
The interaction between an electron and positron depends on their relative velocity, but it does not depend on the frame in which you observe the collision.
For example consider an electron and positron colliding head on with equal and opposite velocities $\pm v$ as you describe in $B$. I could observe this from a reference frame travelling at $v$ or $-v$ in which case either the electron or positron would be stationary and the other particle moving at $2v$, as you describe in $A$. Obviously this doesn't affect the interaction since all that has changed is the speed of the observer.
If I now move at $\pm 2v$ I'd see one particle moving at $v$ and the other moving at $3v$ and catching it up as you describe in $C$. Again this isn't going to affect the interaction.
What you don't say in your question is whether the relative velocity is the same or different in your three scenarios. If the relative velocities are different then yes the interaction will be different since the interaction depends on the relative velocity. If the the relative velocities are all the same then there won't be any difference.
Yes. The transition amplitude and the differential cross section for the annihilation into, say, two photons depend on the kinematics and would be different for these three cases.
If you are taking a quantum field theory course, you might have calculated $A$ or $B$. You could try calculating the others.
