it is genuinely astonishing how often this works. physicists will literally perform taylor series expansions in a parameter that is like ~1 and somehow this doesn't produce total garbage. completely mysterious to me
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the most based calculation i've ever seen in physics is a statmech calculation that works in "4 - epsilon" dimensions where epsilon is "assumed to be small" (for taylor expansions etc) and then set equal to 1, as a way to guess what happens in 3 dimensions. incredible
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my advisor told me about this last week, honestly the epsilon part isn't what got me (as far as i can tell you're just making an integer-valued variable real-valued, it's not that weird). the weird part is that the *dimensional analysis gives the wrong answer*
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i'm still fuckin strapped over that, he didn't explain the whole thing and i don't have time to dive into a research hole right now so i don't even understand how that's possible
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this is explained in david tong's notes i linked to! ctrl+F for "dimensional analysis". i haven't gone through this argument carefully but tldr dimensional analysis is about scaling behavior and there's an extra scaling when you renormalize
