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How can we understand the quantum field $$\phi(x) = \int \frac{d^3p}{(2\pi)^3\sqrt{2E}} ( a_p e^{-ipx}+ a^\dagger_p e^{ipx}) ,$$ where $ a_p$ and $a^\dagger_p$ are creation annihilation operators, in QFT, as a representation of the Poincaré Group?

As I understand it, you have to find a generator of one of the symmetries of the Poincaré group and exponentiate it in order to have a (unitary) representation of the Poincaré group. For example $U(t)= e^{-itH}$ for the representation of the time translation. So it has nothing to do with this quantum field.

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    $\begingroup$ What do you mean "understanding" the quantum field as a representation of the Poincaré group? Are you asking how the field transforms under a Poincaré transformation? Can you try to be more specific? Possibly related: physics.stackexchange.com/q/127989/50583, physics.stackexchange.com/q/174898/50583 $\endgroup$
    – ACuriousMind
    Dec 25, 2021 at 21:48
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    $\begingroup$ I am a beginner in QFT, my teacher said that if you build a state $\mid p >= \sqrt{2E} a^\dagger_p \mid 0>$ then $U(\Lambda) = \sqrt{2E} a^\dagger_p $ is a unitary representation of Lorentz. So first, I don't understand that. And he goes further by saying that $\phi(x)$ is a representation of Lorentz. (he said Lorentz, not Poincaré by the way, sorry). $\endgroup$ Dec 25, 2021 at 21:59
  • $\begingroup$ Closely related physics.stackexchange.com/q/799610/226902 $\endgroup$
    – Quillo
    Jan 31 at 13:06

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