# Seeking derivation of real scalar field Lagrangian

by snoopies622
Tags: derivation, field, lagrangian, real, scalar, seeking
 P: 611 Here and there I come across the following formula for the Lagrangian density of a real scalar field, but not a deriviation. $$\mathcal{L} = \frac {1}{2} [ \dot \phi ^2 - ( \nabla \phi ) ^2 - (m \phi )^2 ]$$ Could someone show me where this comes from? The m squared term in particular is a mystery.
 P: 611 Seeking derivation of real scalar field Lagrangian Yes, I bumped into it studying quantum field theory. It's supposed to have the same form as the Klein-Gordon Lagrangian, yet come about without any quantum assumptions. I assumed that there was some physical situation that corresponds to it, as $$L = \frac {1}{2}m \dot {x}^2 - \frac {1}{2}kx^2$$ corresponds to a mass on a spring obeying Hooke's law.
 P: 611 Oh I see — the first two terms are analogous to the Minkowski displacement vector with magnitude $ds^2 = (c dt) ^ 2 - dx ^2$ and the third term is the potential. Yes, that makes some sense. Thanks, Andrien! :)