Main Question or Discussion Point
I've been working through a qft book by Sadovskii (while I wait for my Peskin book to come in) and I've used some later chapters of Griffith's Into to Elementary Particles as an introduction to some qft. My issue with both of these is that, where in classical mechanics we have the Lagrangian defined as the difference in potential and kinetic energy, we have no explicit definition of the Lagrangian density in qft. Everything I've seen has been, "so here's the basic Lagrangian density we want for this situation" with no derivation or even justification beyond, "we use this form because it gives us what we want later". This hardly seems like a legitimate method to me, and they have the tendency to imply that the additional terms in the density used ensure invariance are just happened upon, as though someone just kind of noticed it worked out if they added it on. I was wondering if there's a way to derive the Lagrangian density for, say, the free Klein-Gordon equation and how the additional terms are methodically sought out?