Hamiltonian formalism and canonical quantization

In this book, we use the classical Hamiltonian formalism to write the energy of a system in a form that lends itself to canonical quantization. For this to work, we must be sure to identify pairs of conjugate canonical variables which, after quantization, will give pairs of operators with commutators equal to iħ. To achieve this, we write down Hamilton's equations for the relevant energy expression and check that they do indeed return the previously established dynamical equations. This is what was done in Chapter 4 for the electromagnetic field in vacuum. We expressed the electromagnetic energy in terms of the real and imaginary parts of normal variables, and showed that Hamilton's equations were equivalent to Maxwell's equations. In Chapter 6, we used an analogous approach for an interacting system of charges and fields, but we did not supply all the details of the calculations.

The aim in this complement is to show explicitly that the Hamiltonian for classical electrodynamics, expressed in the form given at the end of Section 6.1, does indeed lead back to the equations of motion of the particles under the effect of the fields, but also to the dynamical equations of the fields themselves in the presence of the charges, i.e. the Maxwell–Lorentz equations as given in Section 6.1.1. This is then used to confirm that we have indeed identified pairs of conjugate canonical variables.