The Quantized Electromagnetic Field

To quantize a classical system means that we pass from a classical description of the system to the quantum description. The classical description involves a classical Lagrangian or Hamiltonian, and a set of classical coordinates describing the dynamical state, that is, a set of q’s and q̇’s or q’s and p’s. Classical observables are functions of these coordinates. In the quantum description, the q’s and p’s become reinterpreted as operators that act on a state space or Hilbert space of some kind. Since quantum mechanics has more physics in it than classical mechanics, the process of quantization can never be completely deductive. Instead, the classical mechanics offers at best a guide to guessing the structure of the quantum system. Ultimately the correctness of the quantum system we have obtained must be checked experimentally. Nevertheless, the procedure that is known to work for mechanical systems is to replace the q’s and p’s of the classical system by operators that satisfy the canonical commutation relations,