A new approach for the vertical part of the contour in thermal field theories

A lot of work has been devoted in the past to understand the role of vertical branch of the time path in thermal field theories, and in particular to see how to deal with it in the real-time formalism. Unlike what is commonly believed, I emphasize on the fact that the vertical part of the path contributes to real-time Green's functions, and I prove that this contribution is taken into account simply by the substitution $n(\omega_{\boldsymbol k})\to n(|k_o|)$ in the real time Feynman rules. This new proof is based on very simple algebraic properties of the contour integration.