3.7.1 Real Time Formalism

The contour depicted in Fig. 3.3 consists of two two branches, and . Each of the time arguments of the GREEN's function can reside either on the first or second part of the contour. Therefore, contour-ordered GREEN's function thus contains four different GREEN's functions

The

where the time-ordering operator is defined in (B.21). The anti-time-ordering operator can be defined in a similar manner. Since , there are only three linearly independent functions. The freedom of choice reflects itself in the literature, where a number of different conventions can be found. For our purpose the most suitable functions are the , and the retarded ( ) and advanced ( ) GREEN's functions defined as

It is straightforward to show that .