# Moller scattering and identical particles

Moller scattering is often described with a t-channel and u-channel. The difference lies in the assignment whether $p_1$ changes to $p_3$ or $p_4$. Why is such an assignment valid?

Aren't the two incoming electrons identical particles so that the two 'classical' combinations are actually just one in quantum mechanical sense?

Aren't the two incoming electrons identical particles so that the two 'classical' combinations are actually just one in quantum mechanical sense?

Not necessarily. If nothing else, the particles can be distinguished by their momenta; something like "the electron moving to the left" and "the electron moving to the right" (given a previously chosen reference frame) is a perfectly valid way of distinguishing the two electrons.

When setting up the calculation, you have to decide that, for example, among the initial-state electrons you will choose the leftward-moving one to be the upper line (top left of the diagram, and you'll probably label that momentum $p_1$) and the rightward-moving one to be the lower line (bottom left, $p_2$), and among the final-state electrons you will choose the rightward-moving one to be the upper line (top right, $p_3$) and the leftward-moving one to be the lower line (bottom right, $p_4$). It doesn't matter which position (upper/lower) or number (1/2 or 3/4) you assign to which actual particle, but you do have to make a decision and stick to it. The whole point of using the numerical subscripts is to help you stick to the decision you made at the beginning.

Personally, when I can get away with it, I like to forego the numbers and use descriptive subscripts, like $p_{\text{initial,left-moving}}$, or more likely $p_{i,\leftarrow}$, and then $p_{i,\rightarrow}$, $p_{f,\leftarrow}$, and $p_{f,\rightarrow}$. They're just labels so this makes no difference to the calculation, but it helps you remember what distinguishes one particle from another.

Given that, you can see how there are two separate ways the process could proceed:

• either the electrons bounce off each other, so the one that started out moving left winds up moving to the right and vice versa,
• or the electrons pass by each other and just "nudge" each other with a low-momentum photon exchange, in which case the one that started out moving left continues moving left and likewise for the right-moving one.

Depending on how you chose your labels and positions, one of these will correspond to the t-channel diagram and the other will correspond to the u-channel diagram. (Of course once you move beyond leading order it gets more complicated, but I'm ignoring that for this answer.)