The effect of the vertical part of the path on the real time Feynman rules in finite temperature field theory 2-point functions and vacuum diagrams

Abstract

The effect of the contribution of the vertical part of the real time path is studied completely in the case of two points functions and vacuum diagrams. 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 functions 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 and vacuum diagrams, the contribution of the vertical part can be taken into account by the n(|k₀|) prescription in the usual RTF Feynman rules. The extra Feynman rule needed for vacuum diagrams is shown not to be related directly to the contribution of the vertical part of the path.