Lagragian density of a massless scalar field 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.
 A: There are different kinds of possible answers, depending on where you're coming from. I'll give a few.


*

*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 $\partial_\mu\phi \partial^\mu\phi - m^2 \phi^2$.


*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.

*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 $\mathcal L \sim \partial_\mu\phi \partial^\mu\phi - m^2\phi^2$.
