LSZ reduction formula

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

${\displaystyle \langle o|S|i\rangle =S_{o,i}=\Gamma _{o,i}.}$

Here is the S matrix. Their matrix elements are the scattering amplitudes, the indices and denote the incoming or outgoing particles. ${\displaystyle S}$ ${\displaystyle S_{o,i}}$ ${\displaystyle i}$ ${\displaystyle o}$

The reduction formula says that the scattering amplitudes are given by the corresponding vertex functions . ${\displaystyle \Gamma _{o,i}}$

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.

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• 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).