### Ad Infinitum

Have you ever taken any functional derivatives? Because as of now, I can honestly say I have taken an infinite number of them. Multiple times. All you variational calculus jocks out there are laughing in mockery (“Have I ever taken a functional derivative? Please.”), but I’ve never messed with the things before this Quantum Field Theory course I’m taking (Path Integral Formalism rocks). Well, not messed with them to the extent I have now, anyway.

They play a little hard to get, functional derivatives do. But once you get the hang of them, they’re reasonably straightforward to cope with. I’ve known and loved various sorts of delta functions for quite some time; functional differentiation isn’t such a step from there. It’s the functional integration that gets a little tricky. As far as I’m concerned, defining and comparing measures is done by waving tiny little hands over yellow bits of paper and writing down an equal sign here or there.

I have never once thought, “I know enough math.”

Yeah, measure theory is a bitch.

Allegedly, it’s all (mostly) mathematically rigorous. But I was recently having a look at an integral with the space of all unitary matrices as the measure. What the hell does that mean?!?It means it gets results!Another good example of an unintuitive integral (to me, at least) is one over the space of Grassmann numbers.

Ah, good call — you mean the one that behaves exactly like differentiation? Where we should note that the differentiation is screwed to begin with?

Sometimes I wonder why we bother…

Indeed: Differentiation = Integration.

Sneaky and Clever it may be, but it just ain’t right.

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