3.8.1 The KADANOFFBAYM Formulation
The starting point of the derivation is the differential form of the
DYSON equation. By assuming that
,
equation (3.44) and (3.45) can be rewritten as [203]

(3.60) 

(3.61) 
Note that the singular part of the selfenergy on the contour, which
corresponds to the HARTREE selfenergy (Section 3.6), does not
appear explicitly in the kinetic equations, but is included in the potential
energy of the singleparticle HAMILTONian , see (F.7).
Using the LANGRETH rules (Table 3.1) and fixing the time
arguments of the GREEN's functions in (3.60)
and (3.61) at opposite sides of the contour, one
obtains the KADANOFFBAYM equations [93,203]

(3.62) 

(3.63) 
One should note that the deltafunction term in (3.60) and
(3.61) vanishes identically, because the timelabels required
in the construction of and are, by the definition on different
branches of the contour.
The KADANOFFBAYM equations determine the time evolution of the GREEN's
functions, but they do not determine the consistent initial values. This
information is contained in the original DYSON equations
(3.44) and (3.45), and lost in the
derivation. To have a closed set of equations, the
KADANOFFBAYM equations must be supplemented with DYSON equations for
and
. By subtracting (3.63)
from (3.62), one finds the equation satisfied by
[203]

(3.64) 

(3.65) 
Similar relations hold for the advanced GREEN's functions.
M. Pourfath: Numerical Study of Quantum Transport in Carbon NanotubeBased Transistors