Quantum Electro-Magnetic Field - Photons

To have a consistent theory of emission and absorption of electromagnetic ra- diation by quantum systems, not only should the particle system be quantized, but the electro-magnetic field as well. Hence we have to quantize Maxwell’s equations. We do this by analogy with the quantization of Classical Mechan- ics as we have done this in Section 4.1. This analogy suggests we have to put the classical electro-magnetic field theory, which is originally given in terms of Maxwell’s PDEs, into Hamiltonian form. As before we do this in two steps: by introducing the action principle, and performing a Legendre transform. Then we define the quantization map by associating with canonically conjugate classical fields the corresponding operators, and quantizing observables cor- respondingly. Since Maxwell’s equations are wave equations for vector fields with constraints, to provide the reader with a simpler guide, we first quantize the scalar wave equation (and the slightly more general Klein-Gordon equa- tion). The reader familiar with the quantization of the Klein-Gordon equation can proceed directly to the next section on quantization of the Maxwell equations.