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A closed-string scattering amplitude is a worldsheet correlation function of external string vertex operators, integrated over their unfixed positions and worldsheet moduli, with the necessary worldsheet ghost fields.

Crossing symmetry relates scattering amplitudes by analytic continuation exchanging incoming and outgoing particles. A symmetric four-point scalar amplitude has equivalent descriptions under permutations of the Mandelstam variables.

For a finite set $S$ of crystal sites, the lattice structure factor at momentum transfer $Q$ isIt multiplies the single-atom scattering amplitude.

A Dirac field theory invariant under $\psi \mapsto e^{i\alpha }\psi$, $\bar{\psi }\mapsto e^{-i\alpha }\bar{\psi }$ has a conserved global charge, subject to the quantum theory preserving the symmetry. In particle language this charge counts particles minus antiparticles. When the S-matrix commutes with it, a scattering amplitude between states of different charge vanishes. A neutral scalar and a Yukawa interaction preserve the charge, so a process with one incoming Dirac particle and only one outgoing Dirac antiparticle plus neutral particles is forbidden. This net charge is distinct from a sum of positive occupation numbers for all particle and antiparticle modes.

For a central potential and the convention $S_l=1+2i{f}_l=e^{2i{\delta }_l}$, the scattering amplitude isThe identity $|S_l|=1$ is equivalent to $\mathrm{Im}{f}_l=|f_l{|}^2$.