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