Excess radiative energy and switching work

The stationary baseline need not have finite total radiation energy. Accordingly, we define finite excess energy from the change of the two-point kernel, not by subtracting two divergent total expectations.

The branch incoming amplitude is $-\mathrm{i}sg{k}_F$. Stationary scattering maps it to

The added occupation kernel is the rank-one kernel of this amplitude. The sign does not affect its energy.