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This is said to amputate the diagrams by removing the external leg propagators and putting the external states [[on shell and off shell|on-shell]]. All other off-shell contributions from the correlation function vanish. After integrating the resulting delta functions, what will remain of the LSZ reduction formula is merely a [[Fourier transform|Fourier transformation]] operation where the integration is over the internal point positions <math>x</math> that the external leg propagators were attached to. In this form the reduction formula shows that the S-matrix is the Fourier transform of the amputated correlation functions with on-shell external states.
It is common to directly deal with the '''momentum space correlation function''' <math>\tilde G(q_1, \dots, q_n)</math>, defined through the Fourier transformation of the correlation function<ref>{{cite book|last=Năstase|first=H.|author-link=Horațiu Năstase|date=2019|title=Introduction to Quantum Field Theory|url=|doi=|___location=|publisher=Cambridge University Press|chapter=9|page=79|isbn=978-1108493994}}</ref>
<math display="block">
(2\pi)^4 \delta^{(4)}(q_1+\cdots + q_n) \tilde G_n(q_1, \dots, q_n) = \int d^4 x_1 \dots d^4 x_n \left(\prod^n_{i=1} e^{-i q_i x_i}\right) G_n(x_1, \dots, x_n),
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