5 - Real-time formalism

Summary

The contour-ordered Green's function considered in the previous chapter was ideal for discussing general closed time path properties such as the perturbative diagrammatic structure for non-equilibrium states. However, the contour-ordered Green's function lacks physical transparency and does not appeal to intuition. We need a different approach, which brings quantities back to real time. To accomplish this we introduce a representation where forward and return parts of the closed time path are ordered by numbers, specifying the position of a contour time by two indices, i = 1, 2. Next is the diagrammatic perturbation theory in the real-time technique then formulated in a fashion where the aspects of non-equilibrium states emerge in the physically most appealing way. In particular, we shall construct the representation where spectral properties and quantum statistics show up on a different footing in the diagrams. Lastly, we consider the connection to the imaginary-time treatment of non-equilibrium states, and establish its equivalence to the real-time approach propounded in this chapter.

Real-time matrix representation

To let our physical intuition come into play; we need to get from contour times back to real times. This is achieved by labeling the forward and return contours of the closed time path, depicted in Figure 4.5, by numbers, specifying the position of a contour time by an index. The forward contour we therefore label c₁ and the return contour c₂, i.e. a contour time variable gets tagged by the label 1 or 2 specifying its belonging to forward or return contour, respectively.