The momentum nothing determines

Every amplitude in this course so far has been a tree: a diagram you could cut into separate pieces by snipping internal lines, with no closed path anywhere in it. That was never the whole answer to anything. It was the first term of a series, and this part is about the next one.

The difference is easy to see and hard to unsee. On a tree, momentum conservation at the vertices determines everything. Fix what comes in, and every internal line's momentum is forced — it is some sum and difference of the external momenta, and there is nothing left to choose.

Now close a loop. Take a photon and let it split into an electron and a positron that meet again. Momentum conservation still holds at both vertices, exactly as before. But it no longer determines anything: add $\ell$ to the momentum going one way around the loop and subtract it from the other, and both vertices still balance perfectly. Every value of $\ell$ describes the same diagram with the same external particles.

The rules do not let you pick one. A momentum that is not determined is a momentum you sum over, and for a continuous variable summing means integrating ¹:

over all of momentum space, in every direction, out to infinity.

That integral is the subject of this entire part. And the first thing to know about it is that, taken at face value, it is very often infinite.