Classical field theory

Before we embark on quantizing an interacting theory, we will take a diversion into classical field theory and classical perturbation theory and see how far we can get. The reader is expected to be familiar with the Hamiltonian and Lagrangian formalisms of classical mechanics for a finite collection of degrees of freedom qi, and for the Hamiltonian formalism, their conjugate momenta pi. In the Lagrangian formalism we define a functional L({q̇i}, {qi}) of the coordinates and their time derivatives. From this functional we define an action S that defines the equations of motion via a minimization over paths {qi(t)} with a minimization condition: