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We present a compact and simplified proof of a generalized Wick theorem tocalculate the Green's function of bosonic and fermionic systems in an arbitraryinitial state. It is shown that the decomposition of the non-interacting$n$-particle Green's function is equivalent to solving a boundary problem forthe Martin-Schwinger hierarchy; for non-correlated initial states a one-lineproof of the standard Wick theorem is given. Our result leads to newself-energy diagrams and an elegant relation with those of the imaginary-timeformalism is derived. The theorem is easy to use and can be combined with anyground-state numerical technique to calculate time-dependent properties.

Wick Theorem for General Initial States