13.2 Diagrammatic Perturbation Theory for the Polarization Propagator
201
Here one has to deal with altogether 24 different contraction schemes (see Exercise 13.1). Of these schemes, 12 represent disjoint and/or unlinked contributions
and can be discarded from the outset. Another 8 are of the form G
0 G
(1) , which, as
discussed in Sect. 6.2, can be disregarded if the PT is based on a HF representation
and Møller–Plesset partitioning of the hamiltonian. The remaining four contraction
schemes can be subsumed within the two first-order Feynman diagrams shown in
Fig. 13.2. The corresponding analytical expressions read
(1)
rs,r s (t, t
) =
∞
−∞
dt 1 (V rs r s − V rs sr ) G
0
r (t, t 1 )G
0
s (t 1 , t)G
0
r (t 1 , t
)G
0
s (t
, t 1 )
(13.21)
where the free Green’s functions are assumed to be diagonal (cf. Eq. 6.9).
The diagram rules (F1)–(F4) for the electron propagator stated in Sect. 6.1 can
serve as a template for the case of the polarization propagator. Apart from some selfexplanatory adjustments, there are two specific amendments concerning the rules (F1)
and (F4). The first amendment, (F1’), reflects the subtraction term in the defining
Eq. (13.16); the second, (F4’), is an additional phase rule applying to a continuous
fermion line, which can be established analogously to the considerations before and
after Remark 5 in Sect. 6.1. The HF one-particle representation will be supposed in the
following. As discussed in Sect. 6.2, this considerably simplifies the diagrammatic
PT expansions. The findings of Sect. 6.2 can directly be transferred to the case of the
polarization propagator.
Now, the Feynman diagram rules for the polarization propagator can be formulated
as follows:
Feynman Diagram Rules for the Polarization Propagator
(F1) To generate the nth-order contribution to the polarization propagator, draw
all topologically distinct connected diagrams with n wiggly interaction lines
and 2n + 2 directed free fermion or G
0 -lines, which start and end, respectively, at the external vertices (r
, s
; t
) and (r, s; t) with a pair of upwards
and downwards directed free fermion lines (labeled (r
, s
) and (r, s)).
(F1’) Skip all disjoint diagrams, i.e., diagrams of the structure shown in Fig. 13.1.
Such diagrams contribute to the subtraction term −i G rs (t, t
+
)G s r (t
, t
+
).
(F2) To evaluate a given diagram, assign one-particle indices and time arguments
to the interaction lines (inner vertices), thereby defining the one-particle
indices and time arguments of the free fermion lines. The arrows specify
order of the one-particle indices and time arguments in the G
0 -functions.
Replace the graphical symbols by the corresponding analytical expressions,
V ouvw and G
0
u (t i , t j ) (supposing diagonal G
0 functions). In the case of a G
0 -
function with equal time arguments, the limit G
0
(t i , t
+
i ) applies according to
Remark 1 in Sect. 5.2.
(F3) Sum over indices and integrate over time arguments of the inner vertices.
201
Here one has to deal with altogether 24 different contraction schemes (see Exercise 13.1). Of these schemes, 12 represent disjoint and/or unlinked contributions
and can be discarded from the outset. Another 8 are of the form G
0 G
(1) , which, as
discussed in Sect. 6.2, can be disregarded if the PT is based on a HF representation
and Møller–Plesset partitioning of the hamiltonian. The remaining four contraction
schemes can be subsumed within the two first-order Feynman diagrams shown in
Fig. 13.2. The corresponding analytical expressions read
(1)
rs,r s (t, t
) =
∞
−∞
dt 1 (V rs r s − V rs sr ) G
0
r (t, t 1 )G
0
s (t 1 , t)G
0
r (t 1 , t
)G
0
s (t
, t 1 )
(13.21)
where the free Green’s functions are assumed to be diagonal (cf. Eq. 6.9).
The diagram rules (F1)–(F4) for the electron propagator stated in Sect. 6.1 can
serve as a template for the case of the polarization propagator. Apart from some selfexplanatory adjustments, there are two specific amendments concerning the rules (F1)
and (F4). The first amendment, (F1’), reflects the subtraction term in the defining
Eq. (13.16); the second, (F4’), is an additional phase rule applying to a continuous
fermion line, which can be established analogously to the considerations before and
after Remark 5 in Sect. 6.1. The HF one-particle representation will be supposed in the
following. As discussed in Sect. 6.2, this considerably simplifies the diagrammatic
PT expansions. The findings of Sect. 6.2 can directly be transferred to the case of the
polarization propagator.
Now, the Feynman diagram rules for the polarization propagator can be formulated
as follows:
Feynman Diagram Rules for the Polarization Propagator
(F1) To generate the nth-order contribution to the polarization propagator, draw
all topologically distinct connected diagrams with n wiggly interaction lines
and 2n + 2 directed free fermion or G
0 -lines, which start and end, respectively, at the external vertices (r
, s
; t
) and (r, s; t) with a pair of upwards
and downwards directed free fermion lines (labeled (r
, s
) and (r, s)).
(F1’) Skip all disjoint diagrams, i.e., diagrams of the structure shown in Fig. 13.1.
Such diagrams contribute to the subtraction term −i G rs (t, t
+
)G s r (t
, t
+
).
(F2) To evaluate a given diagram, assign one-particle indices and time arguments
to the interaction lines (inner vertices), thereby defining the one-particle
indices and time arguments of the free fermion lines. The arrows specify
order of the one-particle indices and time arguments in the G
0 -functions.
Replace the graphical symbols by the corresponding analytical expressions,
V ouvw and G
0
u (t i , t j ) (supposing diagonal G
0 functions). In the case of a G
0 -
function with equal time arguments, the limit G
0
(t i , t
+
i ) applies according to
Remark 1 in Sect. 5.2.
(F3) Sum over indices and integrate over time arguments of the inner vertices.