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I only see it mentioned that we want Lorentz invariant Lagrangians in quantum field theory, but I would expect that we additionally also need translational invariance, i.e. Poincare invariance. After all we expect the laws of physics to be invariant under translation...

Am I mistaken to think we require the Lagragnian to exhibit Poincare invariance in QFT?

If yes, I would be very thankful if someone would be so kind as to elucidate why.

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You are not mistaken, but compared to Lorentz invariance, translation invariance is usually trivial because there are usually no fields with non-trivial behavior under translations - you have $\phi(x)\mapsto \phi(x+a)$ for a translation by $a$, but $\phi(x) \mapsto \rho(\Lambda)\phi(\Lambda x)$ under a Lorentz transformation $\Lambda$, where $\rho(\Lambda) = 1$ only if $\phi$ is a scalar.

So in order to build a Lagrangian that's a Lorentz scalar you need to be a bit careful since you have fields that aren't scalars themselves, but they're all "scalars" with respect to translations and so there isn't really anything to pay attention to when trying to build a translation-invariant Lagrangian in the vast majority of cases.

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