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In my particle physics course notes, I see that the Lagrangian (density) for free scalars is given by

$$ \mathcal{L} = \frac{1}{2} g^{\mu\nu} \partial_\mu\phi \partial_\nu \phi - \frac{1}{2}m^2\phi^2 $$

I'm used to see $L = T-V$ from classical mechanics and from what I understand, the second term would be kind of an harmonic potential so the first term would be related to kinetic energy ($p^2$ being related to the derivatives by analogy with QM).

Looking at Schwartz Quantum Field Theory and Standard Model. In chapter 3 about classical fields, they say

"Consider the following example where $\mathcal{L}=\frac{1}{2}(\partial_\mu \phi)(\partial_\mu \phi) - \mathcal{V}[\phi]$..."

and then associate the temporal derivative of $\phi$ with kinetic energy and the rest with potential. After that, they use Euler-Lagrange equations and end up with Klein-Gordon equation.

For me, this Lagrangian density seems to appear out of nowhere. My question is: is there a way to derive this Lagrangian density from classical mechanics? Or do we just use it because it's convenient and it leads to KG equation? In that case, what is the relation with KG equation and free scalar and why is it the Lagrangian of free-scalar?

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  • $\begingroup$ In case you're asking why the Lagrangian of the classical non-interacting scalar field takes this form: In Physics from Symmetry. Jakob Schwichtenberg. Second Edition, Springer. chapter 6, among other things, the Lagrangian of a free scalar field is motivated. See also this related PSE post. $\endgroup$
    – Tobias Fünke
    Commented Jan 26, 2022 at 22:17
  • $\begingroup$ That's a bit more satisfying, they are trying a general lagrangian $\mathcal{L}=A + B\phi + C \phi^2 + D\partial_\mu\phi + E\partial_\mu\phi\partial^\mu\phi +F\phi \partial_\mu\phi $ and deduce that because we want the action to be Lorentz invariant, we should only retain $\mathcal{L} = C \phi^2 + E\partial_\mu\phi\partial^\mu\phi$. It's a better argument but not quite what I was searching for. Is there a derivation from classical mechanics instead of trying to guess a solution? $\endgroup$
    – Jérémy Talbot-Pâquet
    Commented Jan 28, 2022 at 4:59

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