By applying LANGRETH's rules to the DYSON equation (3.44) one obtains

(3.66) |

For convenience a notation where a product of two terms is interpreted as a matrix product in the internal variables (space, time, etc.) has been used. One can proceed by iteration with respect to . Iterating once, and regrouping the terms one obtains

The form of (3.67) suggests that infinite order iterations results in [185]

Equation (3.68) is equivalent to KELDYSH's results. In the original work, however, it was written for another function, . This difference is only of minor significance [185].

The first term on the right hand-side of (3.68) accounts for the initial conditions. One can show that this term vanishes for steady-state systems, if the system was in a non-interacting state in the infinite past [185]. Thus, in many applications it is sufficient to only keep the second term.

Similar steps can be followed to obtain the kinetic equation for . In integral form these equations can be written as

The relation between the KELDYSH equation and the KADANOFF-BAYM equation is analogous to the relation between an ordinary differential equation plus a boundary condition and the corresponding integral equation.