The exponential of the derivative translates a function 𝑓(𝑥) by an amount εpic.twitter.com/mwQ1aLpOHt
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No analyticity is needed. As any mathematical physicist knows, there's a self-adjoint operator i d/dx on L^2(R) and a theory of exponentiating self-adjoint operators that makes exp(t d/dx)(f) = f(x+t) true for all functions f in L^2(R). It's not just wishful thinking.
I’m quite familiar with the momentum operator id/dx but how does that factor of i save us from what @wtgowers was saying? is it the hermiticity?
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After quantum mechanics was invented, people like Stone, von Neumann, Weyl and Wigner proved a lot of theorems showing that it really makes sense: it's not just formal trickery and wishful thinking. Here what we need is Stone's theorem: https://en.wikipedia.org/wiki/Stone%27s_theorem_on_one-parameter_unitary_groups …
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