For what I know, the Action should be Lorentz-invariant in accordance to the axioms of SR. For example for a free particle the Action would be the integral of the proper time, which of course is invariant. Now my doubt is the following: does that mean that the Lagrangian should be Lorentz-invariant as well? In some texts I've read it is often stated that the Lagrangian should be a Lorentz scalar, but how does this assures the Action to be Lorentz-invariant too? For example, once one gets the Lorentz-invariant Lagrangian for the EM field $\mathcal{L}=-\frac{1}{4}F^{\mu\nu}F_{\mu\nu}$, why should its action:$$\int -\frac{1}{4}F^{\mu\nu}F_{\mu\nu} d^4x$$ be Lorentz invariant too? Aren't we integrating a scalar over a specific set of coordinates which are not Lorentz-invariant?
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2$\begingroup$ Does this answer your question? Lorentz invariance of the integration measure $\endgroup$– EnthusiastiCCommented Mar 29, 2020 at 11:55
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$\begingroup$ The 4-dimensional volume-element $d^4x$ is invariant on Lorentz-transformations for a similar reason, why the 3-dimensional volume-element $d^3x$ is invariant on space rotations. $\endgroup$– Thomas FritschCommented Mar 29, 2020 at 12:03
2 Answers
It's true that the specific set of coordinates that we choose is not Lorentz invariant, but when you integrate over all space, it doesn't matter. One however has to check that the measure is invariant.
It's a nice excercise to show that, under a Lorentz transformation $\Lambda$ the integral transforms are $$ \int \mathrm{d}^4x\, f(x) \mapsto \int \mathrm{d}^4x' \,|\det\Lambda|\, f(\Lambda x')\,. $$ Since, by definition, Lorentz transformations have determinant $1$ or $-1$, the integral is invariant.
Action is constructed as Lorentz invariant dencity integrated over all Minkowski space. $\mathcal{L}=-\frac{1}{4}F^{\mu\nu}F_{\mu\nu}$ is Lorentz invariant dencity.
$d^4x$ is Lorentz invariant measure of integration in rectangular coordinate system.
If you wanna to choose some another coordinate system (curvilinear coordinate system), in general you need consider another measure of integration $d^4y \sqrt{-g}$, $g_{\mu\nu}$ is Minkowski metric in specific coordinates, that one can choose.
$$ g_{\mu\nu}(y) = \eta_{\rho\sigma}\frac{\partial x^{\rho}}{\partial y^{\mu}}\frac{\partial x^{\sigma}}{\partial y^{\nu}} $$
