On the generalized golden rule for transition probabilities

We show that the usual expression found in the literature, which tries to generalize the 'Fermi golden rule' beyond second order in perturbation, is (surprisingly) incorrect. After identifying the weak steps of the two usual derivations, we derive a new expression of this generalized golden rule, intrinsically very different from the previous one, even though its form may look similar. We show that this new result already affects the next nonzero term of the transition probability expansion. From a direct comparison of this new result with its exact value-as given by the exact evolution operator-we also show that the usual approaches are inherently quite questionable beyond the second order in perturbation.