3.7.1 Real Time Formalism

The contour $C_K$ depicted in Fig. 3.3 consists of two two branches, $\mathrm{C_1}$ and $\mathrm{C_2}$. 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

$$G(t,t') \ = \ \left\{ \begin{array}{ll} G^\mathrm{>}(t,t') & t\... ...G_\mathrm{\tilde{t}}(t,t') & t,t'\ \in \ \mathrm{C_2} \end{array} \right. \ .$$

The greater ( $G^\mathrm{>}$), lesser ( $G^\mathrm{<}$), time-ordered ( $G_\mathrm{t}$), and anti-time-ordered ( $G_\mathrm{\tilde{t}}$) GREEN's functions can be defined as

$$\begin{array}{ll} G^\mathrm{>}(t,t') \ &\displaystyle = \ -i... ... (t,t') \ + \ \theta(t-t') G^\mathrm{<} (t,t') \ , \end{array}$$ (3.51)

where the time-ordering operator $T_\mathrm{t}$ is defined in (B.21). The anti-time-ordering operator $T_\mathrm{\tilde{t}}$ can be defined in a similar manner. Since $G_\mathrm{t} + G_\mathrm{\tilde{t}} = G^\mathrm{>} + G^\mathrm{<}$, 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 $G^\mathrm{\gtrless}$, and the retarded ( $G^\mathrm{r}$) and advanced ( $G^\mathrm{a}$) GREEN's functions defined as

$$\begin{array}{ll} G^\mathrm{r}(t,t')&\displaystyle = \ +\the... ...\ [ G^\mathrm{<}(t,t') - G^\mathrm{>}(t,t') ] \ . \end{array}$$ (3.52)

It is straightforward to show that $G^\mathrm{r}-G^\mathrm{a} = G^\mathrm{>}-G^\mathrm{<}$. M. Pourfath: Numerical Study of Quantum Transport in Carbon Nanotube-Based Transistors