In other words, within the normal ordering symbol we can commute a and a. Since zero-point energy is intimately connected to the non-commutativity of a and a, the normal ordering procedure eliminates any contribution from the zero-point field. This is especially reasonable in the case of the field Hamiltonian, since the zero-point term merely adds a constant energy which can be eliminated by a simple redefinition for the zero of energy. Moreover, this constant energy in the Hamiltonian obviously commutes with a and a and so cannot have any effect on the quantum dynamics described by the Heisenberg equations of motion.