LSZ reduction formula

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The LSZ reduction formula (after its discoverers, the German physicists Harry Lehmann , Kurt Symanzik and Wolfhart Zimmermann ) is a method to calculate the S-matrix elements of the scattering amplitude from the time-ordered correlation functions of a quantum field theory . It is an intermediate step in the prediction of measurement results from the Lagrangian function of the theory.

The reduction formula is schematic

Here is the S matrix. Their matrix elements are the scattering amplitudes, the indices and denote the incoming or outgoing particles.

The reduction formula says that the scattering amplitudes are given by the corresponding vertex functions .

The right side of the LSZ formula is often written as a correlation function of fields, from which the outer propagators are then explicitly cut off. These external propagators contain the exact self-energy and stand for the incoming and outgoing particles. Cutting off the propagators leads to the (not 1-particle irreducible ) vertex function.

A formal derivation of the LSZ formula with operators and states in the Fock space is a bit cumbersome. An alternative to this is a derivation within the framework of the path integral representation of quantum field theory.


  • H. Lehmann, K. Symanzik and W. Zimmermann: For the formulation of quantized field theories. , Nuov. Cim. 1 (1955) 205.
  • H. van Hees: Introduction to Relativistic Quantum Field Theory , (2016).