## 3.3.2 Contour-Ordered GREEN's Function

To express the field operators in the interaction representation an operator is defined (see Appendix B.4) and applied for calculating the GREEN's functions as in Section 3.1.1. The time in (3.8) is taken over the interval . The state at is well defined as the ground-state of the non-interacting system . The interactions are turned on slowly. At the fully interacting ground state is . The state at must be defined carefully. If the interactions remain on, then this state is not well described by the non-interacting ground state. Alternatively, one could require that the interactions are turned off at large times, which returns the system to the ground-state .

SCHWINGER  suggested another method of handling the asymptotic limit . He proposed that the time integral in the operator has two parts; one goes from while the second goes from . The integration path is a contour, which starts and ends at . The advantage of this method is that one starts and ends the operator expansion with a known state . Instead of the time-ordering operator (B.21), a contour-ordering operator can be employed. The contour-ordering operator orders the time labels according to their order on the contour . Under equilibrium condition the contour-ordered method gives results that are identical to the time-ordered method described in Section 3.1.1. The main advantage of the contour-ordered method is in describing non-equilibrium phenomena using GREEN's functions. Non-equilibrium theory is entirely based upon this formalism, or equivalent methods.

Any operator in the HEISENBERG picture can be transformed into the interaction picture (see (B.13)) (3.19)

Analogous to the derivation of (B.24), it can be shown that the operator is given by (3.20)

where the operators are in the interaction representation. The ordinary time-ordering can also be written as ordering along contour branches and as depicted in Fig. 3.1 (3.21)

By combining two contour branches, , (3.19) can be rewritten as (3.22)

where, (3.23) In equation (3.17) describes the equilibrium state of the system before the external perturbation is turned on. Interactions , which are switched on adiabatically at , are present in . However, to apply WICK's theorem (Section 3.4.1), one has to work with non-interacting operators. A methodology similar to the MATSUBARA theory can be applied to express the many-particle density operator in terms of the single-particle density operator , see Appendix B.5. If the contour is chosen (Fig. 3.1), then (B.34) takes the form (3.24)

Therefore, (3.17) can be rewritten as (3.25)

Using the relations (3.22) and (3.25), the GREEN's function in (3.18) becomes (3.26)

The twofold expansion of the density operator and the field operators may conveniently be combined to a single expansion. The two contours and can be combined together, (Fig. 3.2), and a contour-ordering operator , which orders along , can be introduced. Hence, a point on is always earlier than a point on . Furthermore, we define an interaction representation with respect to on and with respect to on . Therefore, the GREEN's function in (3.18) is given by (3.27)

where represents the statistical average with respect to . From here we assume that all statistical averages are with respect to and drop the 0 from the brackets . M. Pourfath: Numerical Study of Quantum Transport in Carbon Nanotube-Based Transistors