114
8 Self-Energy and the Dyson Equation
The geometrical series for
pq (t, t
) in Fig. 8.3 may be used in the representation
of the electron propagator according to Fig. 8.1 or Eq. (8.2). The result, shown
graphically in Fig. 8.4, can be written as follows:
G pq (t, t
) = G
0
pq (t, t
) +
dt 1 dt
1 G
0
p (t, t 1 )) pq (t 1 , t
1 )G
0
q (t
1 , t
)+
r
· · ·
dt 1 dt
1 dt 2 dt
2 G
0
p (t, t 1 )) pr (t 1 , t
1 )G
0
r (t
1 , t 2 )) rq (t 2 , t
2 )G
0
q (t
2 , t
) + · · ·
(8.7)
The latter equation can be cast into a more compact, albeit implicit form:
G pq (t, t
) = G
0
pq (t, t
) +
r
dt 1 dt 2 G
0
p (t, t 1 )) pr (t 1 , t 2 )G rq (t 2 , t
) (8.8)
This is the famous Dyson equation [1], relating the electron propagator G pq (t, t
) to
the self-energy part pq (t, t
). Note that the explicit expansion of Eq. (8.7) results
by solving the Dyson equation iteratively for G pq (t, t
). A graphical representation
of the Dyson equation is given in Fig. 8.5.
Using the energy representation,
pq (ω) =
∞
−∞
e
iω(t−t
)
pq (t, t
) d(t − t
)
(8.9)
and an obvious matrix notation, the Dyson equation can be cast in the compact form
G(ω) = G
0
(ω) + G
0
(ω)(ω)G(ω)
(8.10)
The formal solution for G(ω) is readily obtained, reading
Fig. 8.5 Schematic
representation of the Dyson
equation