Two notes come due

Two promises were made earlier in this course. Neither has been kept.

The first was in lesson 1.2. Writing down the propagator, we replaced $q^2 - m^2$ by $q^2 - m^2 + i\varepsilon$, said that this selects the time-ordered Green's function, and then said something else: we carry the $+i\varepsilon$ silently from here on. You have carried it through every amplitude in Part 2. Nothing has asked what it does.

The second was one lesson ago. Dimensional regularization arrived as an instruction to work in $d = 4 - 2\varepsilon$ dimensions, and the scene was careful to insist this is an analytic continuation, not a claim about spacetime. That sentence is carrying the whole method. Nothing has said what a continuation is, or why continuing a function should give one answer rather than many.

They are the same subject.

Both are statements about a function of a complex variable — where it blows up, what happens to an integral when you move its path around, and how far a function defined by an integral can be extended past where the integral converges. That subject has a name and a small set of theorems, and this lesson is those theorems, in the order the physics needs them.

At the end of it the $+i\varepsilon$ stops being a decoration on a denominator and becomes a statement about which side of the real axis a pole sits on. Which is, it turns out, the thing that licenses the rotation lesson 3.3 needs.