If one does not consider initial correlations, one can let
. Since we assume that the GREEN's function falls off sufficiently
rapidly as a function of the separation of its time arguments, one can neglect
the part of the contour extending from to
[197]. It has been shown that by explicitly taking
the initial correlations into account
[198,199,200,201] the neglect of this part of the
contour corresponds to the neglect of initial correlations. The initial
condition, that the system is assumed to be in equilibrium before the
external perturbation is turned on, can then be imposed directly on the
DYSON equation in integral form. This provides an independent demonstration
that, for cases where initial correlations can be neglected, one can discard
the contribution of the contour from to
. The contours
and become identical, as they both start and end at . They can be
extended beyond the largest time by considering that the time-evolution
operator is a unitary, and one then obtains the contour introduced by
KELDYSH [202].