The effect of the vertical part of the path on the real time Feynman   rules in finite temperature field theory
  The effect of the contribution of the vertical part of the real time path is studied completely in the case of two points functions. Indeed, this vertical part generally contributes in the calculation of a given graph. Moreover, this contribution is essential in order to have a consistent equilibrium theory: thanks to this contribution, the Green function are effectively invariant by time translation, as they should be. As a by product, it is shown that the perturbative calculations give a result which does not depend on the initial time $t_I$ and final time $t_F$ of the path. The property of independence with respect to $t_I$ is closely related to the KMS conditions, i.e. to the fact the system is in thermal equilibrium. In the case of two point functions, the contribution of the vertical part can be taken into account by the $n(|k_0|)$ prescription in the usual RTF Feynman rules. 
Fran\c{c}ois Gelis