Connection to S-matrix
TRANSITION PROBABILITY, ONCE AGAIN
In the previous disucssion, it was found that the transition probability per unit
volume of space-time is given by
Since in the notation used here the states are not normalised to one but to
their energies, see above,
this normalisation has to be taken into account, leading to
for the properly normalised transition probability. In scattering experiments in particle
physics, usually two particles (target and projectile) interact, producing many particles.
According to the definitions above, the cross section for such a 2→ n scattering reads
The integration over the final state particle states in the equation above just takes into
account that a sum over different final states has to be performed. Of course, at this
point certain constraints on this phase space may be applied, which usually depend on
experimental conditions and on the process under consideration. Note here that there is no
integration over the initial state particles, reflecting that they are usually well prepared,
i.e. in a "clean state" (momentum eigenstate). Finally, the relative velocity of beam and
target particle are cast into a Lorentz-invariant form (i.e. in a form that depends on
Lorentz-scalars such as the scalar products of four momenta only). To this end, note that
leading to the
FINAL EXPRESSION
Obviously, at this point, the expression for the cross section above is in units of
energy/momenta rather than in units of length. The connection of energy and length, however,
is given by
indicating that cross sections have units of inverse squares of energy.