ABSTRACT
This Chapter is devoted to the description of scattering processes based on reduction formulae of Lehman, Szymanzik and Zimmermann (LSZ). Within this scheme, a link is established between the q-point Green's function and the S-matrix elements yielding the amplitude of a process involving p initial-state particles and (q − p) final-state particles. The formalism is based on use of asymptotic “in” and “out” states, describing the system in the https://www.w3.org/1998/Math/MathML" display="inline"> t → ± ∞ limit, to obtain the scattering amplitudes. The properties of the sets of “in" and “out" states, notably the completeness implied by the asymptotic hypothesis and the expression of the S-matrix elements as a scalar product of “in" and “out" states are discussed. The derivation of the explicit form of the LSZ formulae is illustrated considering the case of particles described by a neutral scalar field in the presence of a https://www.w3.org/1998/Math/MathML" display="inline"> λ ϕ 4 interaction.
The calculation of the cross section of a process involving two particles in both the initial and final states is described in detail.
