Retarded Propagator Representation of Out-of-Equilibrium Thermal Field Theories

We represent out of equilibrium thermal field theories with finite time path in terms of retarded propagators exclusively. For the particle number, defined as the equal time limit of the Keldysh propagator, the time ordering of the diagrams contributing is particularly simple: all external end-points of propagators have maximal time, there are no internal vertices with locally maximal time, the property which guaranties causality), there is, at least one “sink” vertex (vertex with locally minimal time). The diagram looks like fisher net hanging on external vertices. At the “sink” vertices energy is not conserved, thus establishing realisation of uncertainty relations in out of equilibrium TFT. Even more, at the equal-time limit, the terms conserving energy at “sink” vertices vanish. This fact eliminates pinching problem and enables safe time→∞ limit. The retarded propagator in higher orders is regularized only as a part of of the diagram connected to equal time limit of multi point Green function representing expectation value of the product of number operators. These properties indicate clear advantage of finite time path, in large time limit over the use of Keldysh time path.