In quantum field theory, the Lehmann–Symanzik–Zimmermann (LSZ) reduction formula is a method to calculate S-matrix elements (the scattering amplitudes) from the time-ordered correlation functions of a quantum field theory. It is a step of the path that starts from the Lagrangian of some quantum field theory and leads to prediction of measurable quantities. It is named after the three German physicists Harry Lehmann, Kurt Symanzik and Wolfhart Zimmermann. Although the LSZ reduction formula cannot handle bound states, massless particles and topological solitons, it can be generalized to cover bound states, by use of composite fields which are often nonlocal. Furthermore, the method, or variants thereof, have turned out to be also fruitful in other fields of theoretical physics. For example, in statistical physics they can be used to get a particularly general formulation of the fluctuation-dissipation theorem.
In and out fields S-matrix elements are probability amplitudes of transitions between in states and out states. An in state | { p } i n ⟩ {\displaystyle |\{p\}\ \mathrm {in} \rangle } describes the state of a system of particles which, in a far away past, before interacting, were moving freely with definite momenta {p}, and, conversely, an out state | { p } o u t ⟩ {\displaystyle |\{p\}\ \mathrm {out} \rangle } describes the state of a system of particles which, long after interaction, will be moving freely with definite momenta {p}. In and out states are states in Heisenberg picture so they should not be thought to describe particles at a definite time, but rather to describe the system of particles in its entire evolution, so that the S-matrix element:
S f i = ⟨ { q } o u t | { p } i n ⟩ {\displaystyle S_{\rm {fi}}=\langle \{q\}\ \mathrm {out} |\{p\}\ \mathrm {in} \rangle }
is the probability amplitude for a set of particles which were prepared with definite momenta {p} to interact and be measured later as a new set of particles with momenta {q}. The easy way to build in and out states is to seek appropriate field operators that provide the right creation and annihilation operators. These fields are called respectively in and out fields: Just to fix ideas, suppose we deal with a Klein–Gordon field that interacts in some way:
L = 1 2 ∂ μ φ ∂ μ φ − 1 2 m 0 2 φ 2 + L i n t {\displaystyle {\mathcal {L}}={\frac {1}{2}}\partial _{\mu }\varphi \partial ^{\mu }\varphi -{\frac {1}{2}}m_{0}^{2}\varphi ^{2}+{\mathcal {L}}_{\mathrm {int} }}
L i n t {\displaystyle {\mathcal {L}}_{\mathrm {int} }} may contain a self interaction gφ3 or interaction with other fields, like a Yukawa interaction g φ ψ ¯ ψ {\displaystyle g\ \varphi {\bar {\psi }}\psi } . From this Lagrangian, using Euler–Lagrange equations, the equation of motion follows:
( ∂ 2 + m 0 2 ) φ ( x ) = j 0 ( x ) {\displaystyle \left(\partial ^{2}+m_{0}^{2}\right)\varphi (x)=j_{0}(x)}
where, if L i n t {\displaystyle {\mathcal {L}}_{\mathrm {int} }} does not contain derivative couplings:
j 0 = ∂ L i n t ∂ φ {\displaystyle j_{0}={\frac {\partial {\mathcal {L}}_{\mathrm {int} }}{\partial \varphi }}}
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