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A combinatoric procedure in QFT of reducing arbitrary products of creation and annihilation operators to sums of products of pairs of these operators. A string of such operators is rewritten as the normal-ordered product of the string, plus the normal-ordered product after all single contractions among operator pairs, plus all double contractions, etc., plus all full contractions.

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Why do we have $[\phi_1^+,:\phi_2\phi_3:]=:[\phi_1^+,\phi_2^-]\phi_3:+:\phi_2[\phi_1^+,\phi_...

How $$[\phi_1^+,:\phi_2\phi_3:]=:[\phi_1^+,\phi_2^-]\phi_3:+:\phi_2[\phi_1^+,\phi_3^-]:$$ with $\phi_i=\phi(x_i)$ field operators ($\phi_i^+$ is the annihilation part while $\phi_i^-$ is the creation …
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