Classical Field Theory
Lagrangian densities, Klein–Gordon equation, and the bridge to QFT.
Promote $L(q,\dot q) \to \mathcal L(\phi, \partial_\mu \phi)$ — a Lagrangian density depending on a field $\phi(x)$ and its spacetime derivatives. The action is $S = \int d^4x\, \mathcal L$. Euler–Lagrange for fields:
$$\partial_\mu \frac{\partial \mathcal L}{\partial (\partial_\mu \phi)} - \frac{\partial \mathcal L}{\partial \phi} = 0.$$The simplest relativistic scalar:
$$\mathcal L = \tfrac{1}{2}(\partial_\mu \phi)(\partial^\mu \phi) - \tfrac{1}{2}m^2 \phi^2$$gives the Klein–Gordon equation $(\Box + m^2)\phi = 0$. Plane-wave solutions $\phi = e^{-ik\cdot x}$ have dispersion $\omega^2 = \mathbf k^2 + m^2$ — relativistic energy of a particle of mass $m$.
Interactive: scalar field oscillation
Click to drop a Gaussian perturbation; watch dispersive Klein–Gordon waves propagate.