3.3.2 ContourOrdered 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 groundstate of the
noninteracting 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 noninteracting ground state. Alternatively, one could require
that the interactions are turned off at large times, which returns the system
to the groundstate
.
SCHWINGER [92] 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 timeordering
operator (B.21), a contourordering operator can be employed. The
contourordering operator
orders the time labels according to
their order on the contour . Under equilibrium condition the
contourordered method gives results that are identical to the timeordered
method described in Section 3.1.1. The main advantage of the
contourordered method is in describing nonequilibrium phenomena using
GREEN's functions. Nonequilibrium 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
timeordering 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) 
Figure 3.1:
The contour
runs
on the real axis, but for clarity its two branches and are
shown slightly away from the real axis. The contour runs from
to
.

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
noninteracting operators. A methodology similar to the MATSUBARA theory can
be applied to express the manyparticle density operator
in
terms of the singleparticle 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 [196]

(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 contourordering 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
.
Figure 3.2:
The contour
, runs
from to and from to
.

M. Pourfath: Numerical Study of Quantum Transport in Carbon NanotubeBased Transistors