3.2.2 MATSUBARA GREEN's Function

The single particle MATSUBARA GREEN's function is defined as

 \mathcal{G}({\bf {r}},\tau;{\...
 {\mathrm{Tr}[e^{-\beta\hat{K}}]} \ .
 \end{array}\end{displaymath} (3.13)

The GREEN's function now may be rewritten in the interaction picture

 \mathcal{G}({\bf {r}},\tau;{\...
...eta\hat{K}_0}\mathcal{\hat{S}}(\beta\hbar,0)]} \ ,
 \end{array}\end{displaymath} (3.14)

where (B.34) is employed for the transition from the first to the second line and (B.27) for the transition from the second to third line. Equation (3.14) has precisely the structure analyzed in (3.8). The operator $ \mathcal{\hat{S}}$ can be expanded as (see (B.33))

 \mathcal{G}({\bf {r}},\tau;{\b...
..._1)\ldots\hat{K}^\mathrm{int}(\tau_n)\}\right]}\ ,
 \end{array}\end{displaymath} (3.15)

where the denominator is just the perturbation expansion of the grand partition function. However, it serves to eliminate all disconnected diagrams, exactly as in the zero-temperature formalism. It is apparent that the perturbation expansion of the MATSUBARA GREEN's function (3.15) is very similar to that of the zero temperature GREEN's function (3.31). MATSUBARA [193] has proved that there exists a generalized WICK theorem (see Section 3.4.1) that deals only with the ensemble average of operators and relies on the detailed form of the statistical operator $ e^{\beta\hat{K}_0}$. M. Pourfath: Numerical Study of Quantum Transport in Carbon Nanotube-Based Transistors