Fermi's golden rule
/ FAIR-mee /
Suppose you know how strongly a system is nudged from one state toward another, and you want to know how fast the change actually happens. There is a remarkably simple recipe that ties these together, named Fermi's golden rule after the physicist Enrico Fermi (though Dirac wrote it earlier). In plain words it says: the rate at which a process happens is set by two things multiplied together — how strong the underlying interaction is, and how many final states are available for the system to end up in.
More precisely, the rate is proportional to the magnitude of the amplitude squared, times the density of available final states (the phase space). The first factor is the dynamics — the physics of the interaction itself, what the Feynman diagrams compute. The second factor is purely about counting room — how many ways the outgoing particles can share the available energy and momentum. A weak interaction can still give a fast process if there are vast numbers of final states to fall into, and a strong interaction can be slow if there is hardly any room. The golden rule cleanly separates these two influences.
This rule is the conceptual hinge between an abstract amplitude and a number you can actually measure, like a decay rate or a scattering cross section. Whenever a physicist turns a computed M into a predicted lifetime or reaction rate, the golden rule (or its relativistic cousin) is doing the work in the background. It is the reason the same calculated amplitude can be reshaped into many different observable quantities, just by changing what counts as the available final states.
A heavy particle that can decay into many light particles often decays faster than one that can only decay into a couple, even if the interaction strength is similar. The extra speed comes entirely from the second factor in the golden rule: more available final states means more phase space to fall into.
More room to land in means a faster process.
The golden rule splits a rate into dynamics (the amplitude) times kinematics (phase space). A process being fast or slow does not by itself tell you whether the interaction is strong — you must separate out the available final states.