Concept Deep Dive: Green's Functions

Green's functions can be your best friends if you deal with derivation operators a lot, they have a tendency to be a bit cryptic otherwise. If you - like me - fall into the latter category, I have bad news for you: They are very much everywhere. They appear from very elementary electrics, to particle physics, to quantum field theory, and each time it makes sense to not want to solve them explicitly.

I'm not happy with how I was introduced to them, back in my second semester of university. I think they were one of the first problems that I just kinda gave up on originally, because integrals were never a real strong suite of mine, and solving a Green's function integral is brutal regardless. Still, that's what we're going to do today, and so much more.

So can we solve it? In short: Yes. Of course it's a little more complicated, or we wouldn't be doing this. The Green's function doesn't get defined explicitly, but in conjunction to a differential operator. Let's call that operator L, and the green's function G. Then, the Green's function is defined as: LG(x, s) = δ(x - s) with a fixed point s. The delta function δ evaluates to null everywhere but at the point where its argument is 0, where it evaluates to ∞. When we talk about solving the Green's function, we actually want to understand this construction here:

This is a little intimidating, and the non-confusing solution to this problem comes from differential equations, which aren't exactly my forte anyway. As the differential equation is with respect to x, it can be pulled out of the integral and on the right side of the equation the delta function in the integral moves f(s) to f(x). The equation could then be written as Lu(x) = f(x), where the remaining integral on the left side is rewritten into u(x). Solving the Green's function from here is mostly a question of what choices of L and f(x) are made. It's unfortunately rarely an easy computation.

Linear operators span a pretty wide field of commonly used operators. In fact, one of many such linear operators also happen to be differential operators, such as on smooth topologies. This makes the Green's function applicable to cases where L is a derivative. This of course gives that differential equation an oddly familiar form: pf(x) = f'(x), which is the differential wave equation. This elevates the Green's operator to the solution of a wave equation. For these purposes we begin with a generalized Schroedinger equation.

The derivations are the linear operator, and we have a function of (x, t) on the right side of equation. This justifies this following Green's equation.

The equation can be Fourier-transformed, and there is an integral form for the time delta function, which can be solved as a Gaussian integral.

This kind of problem is part of the standard differential equations and symmetrically decomposes into an advanced and retarded solution.

One such case is the d'Alembertian derivative ☐G(x-s) = δ(x - s). This is an interesting case, because the d'Alembertian is pretty important as a space-time derivative, but as such is also subject to the underlying metric. That means that it doesn't actually have a universal inverse operator, which would we very useful in many cases. The Green's function can help identify that inverse. In that process the d'Alembertian is written in terms of the Green's function (once that has been identified) and that result is then inverted.

The general form of the Green's function is

This suggests that a wave function, for example, decomposes into a "pure" component and a perturbative component. We set the pure component such that its d'Alembertian goes to zero. Of course the Green's function should, by definition of linear operators, be translation-invariant in all its variables, so we might as well set t' and x' to zero. Through complex analysis the Green's function can be narrowed down to the following form.

Usually it's just accepted as the proper solution, and proving the validity of this solution is done by straight forward computation of the derivatives. The general form of the Green's function then is

We've associated Green's functions with wave equations a lot the entire time, and while it's technically correct in the sense that it has the form of a wave equation, but what can we actually use this for? Well, with that last example of the Green's function, we understand that they might have a general form, even if relatively broad constraints are set for them from the outset. That implies that there might even be a way to use the Green's function as a sort of very complicated variable.

Quantum field theory does exactly that by associating particle propagators with Green's functions. In this case, the Green's function is the solution to the Klein-Gordon equation, the equation equivalent to the Schroedinger equation of "vanilla" quantum mechanics. Because of the way Feynman diagrams work, these Green's functions just become factors in the S-matrix component. In fact, the specific Green's function has a not-so-complicated solution, which almost trivializes the computation. The method of converting Feynman diagrams into S-matrix components isn't difficult, but maybe better placed in its own segment.