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I'm studying the canonical quantization of the real scalar field. I've managed to condense the Hamiltonian and momentum operators in the 4-momentum operator $P^{\mu}$ and have shown that its commutator with the field is \begin{equation} [P^{\mu},\phi(x)]=-i\partial^{\mu}\phi(x).\tag{1} \end{equation} Now I'm being asked to show that $(1)$ is "consistent, at an infinitesimal level," with $\phi(x)=e^{iPx}\phi(0)e^{-iPx}$, but I don't get what this means. Does anyone know what this question could possibly mean?

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    $\begingroup$ What's the context of you being asked this? E.g. does that wording come from a book? $\endgroup$ – David Z Dec 10 '17 at 4:16
  • $\begingroup$ @DavidZ no, it comes from a problem set. I tried looking for that wording in Peskin & Schroeder but couldn't find it. This could be just my teacher's idiosyncratic wording, since I haven't seen it elsewhere. $\endgroup$ – Drarp Dec 10 '17 at 4:29
  • $\begingroup$ Yeah, I would imagine so. In that case nobody except your teacher can really answer this with 100% certainty, although you can take the answer(s) as reasonable guesses. $\endgroup$ – David Z Dec 10 '17 at 4:38
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I think they mean for infinitesimal $x$. In other words, Taylor expand $\phi(x) \approx \phi(0) + x^\mu \partial_\mu \phi(0)$ and $e^{i P x} \approx I + i P x$. Shouldn't be too hard from there.

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