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I have seen in some books that the simplest Lagrangian density of a massless scalar field is $$\mathscr{L}=\dfrac{1}{2}\partial^\mu\phi\partial_\mu\phi=\dfrac{1}{2}\left(\partial_\mu\phi\right)^2.$$ This may be a silly question, but: Where does this equation come from? I could not find a demonstration.

P.S. Also a demonstration for the Lagrangian density of a massive real scalar field, $$\mathscr{L}=\dfrac{1}{2}\left(\partial_\mu\phi\right)^2-\dfrac{1}{2}m^2\phi^2,$$ would be appreciated.

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    $\begingroup$ Are you asking for how to derive the massive/massless Klein-Gordon equation, given the above Lagrangians? Or are you asking how to derive the above Lagrangians, given the Klein-Gordon equation? $\endgroup$ Commented Mar 7, 2018 at 14:05
  • $\begingroup$ I'm asking how to reach those Lagrangians. As you say, I presume they come from Klein-Gordon equation (The book doesn't specify it). $\endgroup$ Commented Mar 8, 2018 at 11:23

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There are different kinds of possible answers, depending on where you're coming from. I'll give a few.

  1. Consider all possible Lagrangian densities that we can write down, and that are Lorentz-invariant, are non-trivial and have a stable vacuum state. Then the formulas you gave are quite literally the simplest possible options:

    • $\mathcal L$ must be a function of $\phi$ and $\partial_\mu\phi$.
    • $\mathcal L$ must contain $\partial_\mu\phi$ somewhere, otherwise the theory would be trivial because $\phi$ would not be dynamical.
    • The simplest Lorentz scalar we can construct with $\partial_\mu \phi$ is $\partial_\mu\phi \partial^\mu\phi$.
    • Going beyond that, the next-simplest option might be $\partial_\mu\phi \partial^\mu\phi + c \phi$, but this theory does not have a stable vacuum / ground state, because the potential $V(\phi) = -c\phi$ is not bounded from below.
    • Therefore the next-simplest option is $\partial_\mu\phi \partial^\mu\phi - m^2 \phi^2$.
  2. You want to describe a field that satisfies the Klein-Gordon equation $\partial_\mu\partial^\mu \phi = 0$. You write down the Lagrangian density $\mathcal L \sim \partial_\mu\phi \partial^\mu\phi$ and see that the correct equation of motion follows.

  3. You start with a number of coupled oscillators and write down their action like in (relativistic) classical mechanics. If there are many oscillators close together, they can be approximately described as a classical field. Performing this approximation yields an expression for the action in terms of the Lagrangian density $\mathcal L \sim \partial_\mu\phi \partial^\mu\phi - m^2\phi^2$.

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