Quantum field theory of the free-electron laser

A simple N-electron free-electron-laser (FEL) Hamiltonian is seen to define a (1+1)-dimensional quantum field theory. For large N, the FEL dynamics is shown to be solved by a single-electron Schr\"odinger equation in a self-consistent field. The fluctuations around such a Schr\"odinger wave function are shown to be O(1/ {}N ) and computable by a perturbative strategy. A number of observations are also reported on the best strategy to solve the Schr\"odinger equation.