The LSZ reduction formula is a fundamental result in quantum field theory that relates the scattering amplitudes of interacting particles to the vacuum expectation values of time-ordered products of fields. It was derived by physicists Joos-Weinberg and Lehmann-Symanzik-Zimmermann (hence the name LSZ) independently in the 1950s.
The LSZ reduction formula is significant for several reasons:
Connection between observables and the S-matrix: The LSZ formula provides a formal connection between experimentally measurable quantities (scattering amplitudes) and the underlying quantum field theory. It allows us to relate scattering experiments to the fundamental interactions encoded in the Lagrangian of the theory.
Derivation of Feynman rules: The LSZ reduction formula is a key ingredient in deriving Feynman rules, which are graphical representations of the terms contributing to scattering amplitudes in terms of Feynman diagrams. Feynman diagrams provide a powerful visual tool for calculating and understanding particle interactions in quantum field theory.
Perturbation theory: The LSZ reduction formula is crucial for implementing perturbation theory in quantum field theory. Perturbation theory allows us to calculate scattering amplitudes order by order in an expansion parameter, usually the coupling constant of the theory. LSZ formula plays a central role in connecting the interaction picture of the theory to the asymptotic in/out states.
Renormalization: The LSZ reduction formula is intimately linked to the process of renormalization, which is necessary to deal with divergences that arise in quantum field theory calculations. Renormalization involves redefining parameters and fields to absorb these divergences, and the LSZ reduction formula provides a way to relate the renormalized amplitudes to the bare amplitudes.
Overall, the LSZ reduction formula is a fundamental tool in quantum field theory that enables the calculation and interpretation of scattering amplitudes, Feynman diagrams, perturbation theory, and renormalization. It connects the formalism of the theory to experimental observables, making it an essential tool in high-energy physics.