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The time evolution operator is $$\hat U(t)=e^{-i\hat H t}$$ where $\hat H$ is the Hamiltonian, $\hbar=1$, and the state is at time $t=0$. The time translation operator is defined similarly, but with opposite sign $$\mathcal{\hat U} =e^{i\hat H t}$$ Why is it that these two operators have different sign? I am going through the book Quantum Field Theory for The Gifted Amateur by Lancaster and Blundell, and they derive the spatial evolution operator to be $$\hat U_a=e^{i\hat pa}$$ where $\hat P$ is the momentum and $a$ is the amount evolved by. The book then says that this is not the operator that is needed, and randomly changes the sign to give the spatial translation operator $$\mathcal {\hat U_a}= e^{-i\hat p a}$$ and I have no idea where the sign change comes from. Any help would be greatly appreciated.

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    $\begingroup$ Sounds like bad writing. Consider a better book. $\endgroup$
    – DanielSank
    Jan 20 '21 at 20:47
  • $\begingroup$ It's an alright introduction, but it's just that, an introduction $\endgroup$ Jan 20 '21 at 20:48
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The difference is likely a result of viewing transformations as being passive, as opposed to active. The difference can be a little subtle. In an active transformation, we actually move the state to a new position. In a passive transformation, we instead move the coordinate system in the opposite direction.

In this case, there are two possible ways to define the time translation operator. If we want to think about active transformations, we would keep the same coordinate system and evolve the system using $U(t_0)=e^{-iHt_0}$. If we want to think about a passive transformation, we would switch to a new coordinate system $t'=t-t_0$. In this new coordinate system, $|\psi(t)\rangle$ becomes $|\psi(t')\rangle=|\psi(t-t_0)\rangle=e^{iHt_0}|\psi(t)\rangle$.

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  • $\begingroup$ There was a sidenote in the book that said it was due to thinking of it as an active vs a passive transformation. I was wondering mostly about why the minus sign changes things from passive to active. $\endgroup$ Jan 21 '21 at 2:47
  • $\begingroup$ My book says that time evolution is passive and translation is active. I think that you are probably right, but I'm very confused. $\endgroup$ Jan 22 '21 at 17:47
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    $\begingroup$ The choice of viewing transformations as active or passive arbitrary. In any given problem, you can choose to use either perspective. $\endgroup$
    – Yachsut
    Jan 22 '21 at 19:30
  • $\begingroup$ Yeah, you're right there! I came to basically the same conclusion after struggling with it for a bit. $\endgroup$ Jan 22 '21 at 20:16

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