LSZ

LSZ refers to the Lehmann–Symanzik–Zimmermann reduction formula, a foundational result in quantum field theory. Named after Harry Lehmann, Kurt Symanzik, and Wolfhart Zimmermann, it provides a direct link between the Green functions of an interacting field theory and the S-matrix elements that describe observable scattering processes.

The core idea is that the amplitudes for external particles appearing in a collision can be obtained from the theory’s n-point Green functions by “amputating” the external legs and placing those legs on their physical mass shells. In practical terms, each external particle contributes a factor derived from the residue of the full propagator at its mass, multiplied by the corresponding amputated Green function. This creates a bridge between calculable correlation functions in the theory and measurable scattering amplitudes in experiments.

The LSZ reduction formula is central to perturbative quantum field theory and underpins many standard techniques in particle physics. It allows theorists to compute S-matrix elements using Feynman diagrams and renormalized Green functions, while ensuring consistency with asymptotic states and relativistic unitarity.

Conditions and limitations include the need for well-defined asymptotic particle states (typically stable particles with a mass pole), a local and relativistic quantum field theory, and a framework where fields can be renormalized. The formula is most straightforwardly applied to stable particles and to theories where infrared issues and confinement do not obstruct the definition of asymptotic states; extensions exist for certain cases but require careful handling of masses, gauge bosons, and nonperturbative effects.