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I have taken it granted that an action in the special relativity must be a lorentz scalar. However is there a fundamental reason for this requirement? I cannot think of a plausible reason for this question.

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Because in QFT you impose a Poincaré invariance and Poincaré includes Lorentz, this comes from the fact that physics must not depend from the frame you choose and so has to be rotation/translation(and boost for a Minkowskian metric) invariant, so your Lagrangian has to be what we call "Lorentz scalar" ie invariant under any Lorentz transfo.

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  • $\begingroup$ OP didn't say anything about QFT though... $\endgroup$ Commented Mar 29, 2018 at 17:39

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