Experimentally, the dynamics of physical systems are studied by preparing an initial state and then following its time evolution in the laboratory. In the broadest terms, it is practical to prepare only two types of initial states: those consisting of a single particle (or a bound state which behaves like a single particle) and those consisting of a beam of two particles (or two bound states) directed toward each other so that a collision is possible. The other logical possibilities, which include the direction of three or more beams at each other so as to produce a three or more body collision, are impractical, except in the most exceptional cases. Hence, our experimental studies are more or less limited to the following types of reactions:
one particle two particles many particles many particles
In the first instance, if the single particle remains a single particle, then we may measure its mass (or, in some cases, the frequency with which it oscillates into another single particle), while if it decays into two or more particles, we can study the decay rate or, when decay into two or more channels occurs, the branching fraction. In the second case, we measure the cross section. Hence, decay rates and cross sections are very important; they are among the very few physical quantities which can be measured. We now turn to a discussion of how decay rates and cross sections are calculated from the ¿"-matrix.
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