198
13 Polarization Propagator
useful simplification is obtained by equating each two time arguments,
rs,r s (t, t
) = i R rs ,sr (t, t
; t
+
, t
+
)
(13.13)
and this expression, involving Eqs. (13.11), (13.12), constitutes the original definition
of the polarization propagator. As the reader should check, the definition (13.13)
does not depend on the time-ordering of the limits t 1 → t
1 = t and t 2 → t
2 = t
; the
particular ordering chosen on the right-hand side of Eq. (13.13) compares directly to
the form of Eq. (13.9).
It is important to note that, unlike with the electron propagator, there is no Dyson
equation approach to the polarization propagator. Of course, one might postulate a
Dyson-type equation according to
(ω) =
(0)
(ω) +
(0)
(ω) P(ω)(ω)
(13.14)
thereby defining the quantity P(ω) as an analogue to the self-energy in the Dyson
equation part (ω). However, in contrast to (ω), there is no direct and diagrambased approach to determine P(ω), which means that Eq. (13.14) is all but useless.
Only in a rudimentary form, obtained by restricting P(ω) to the constant first-order
approximation, Eq. (13.14) acquires a meaning: Here, it represents the defining equation of the random-phase approximation (RPA) to the polarization propagator, which
will be discussed in some detail in Chap. 15.
As an actual counterpart to the Dyson equation for the electron propagator one
may see the Bethe–Salpeter equation [2] for the p-h response function (13.12), which
in a shorthand notation can be written as
R 12,1 2 = G 12 G 21 − i G 1τ G σ1 K
ph
τσ ,στ R τ 2,σ 2
(13.15)
Here, repeated indices imply summation over one-particle states and integration
over time arguments; the quantity K
ph is referred to as the irreducible p-h vertex. Obviously, being an integral equation involving a fourfold time integration, the
Bethe–Salpeter equation is by far more complex than the Dyson equation.
13.2 Diagrammatic Perturbation Theory
for the Polarization Propagator
The diagrammatic perturbation theory formulated for the electron propagator in
Chaps. 4–7 can be transferred with minor adjustments to the case of the polarization
propagator. We recall the key steps as follows.
The starting point is the time representation of the polarization propagator
13 Polarization Propagator
useful simplification is obtained by equating each two time arguments,
rs,r s (t, t
) = i R rs ,sr (t, t
; t
+
, t
+
)
(13.13)
and this expression, involving Eqs. (13.11), (13.12), constitutes the original definition
of the polarization propagator. As the reader should check, the definition (13.13)
does not depend on the time-ordering of the limits t 1 → t
1 = t and t 2 → t
2 = t
; the
particular ordering chosen on the right-hand side of Eq. (13.13) compares directly to
the form of Eq. (13.9).
It is important to note that, unlike with the electron propagator, there is no Dyson
equation approach to the polarization propagator. Of course, one might postulate a
Dyson-type equation according to
(ω) =
(0)
(ω) +
(0)
(ω) P(ω)(ω)
(13.14)
thereby defining the quantity P(ω) as an analogue to the self-energy in the Dyson
equation part (ω). However, in contrast to (ω), there is no direct and diagrambased approach to determine P(ω), which means that Eq. (13.14) is all but useless.
Only in a rudimentary form, obtained by restricting P(ω) to the constant first-order
approximation, Eq. (13.14) acquires a meaning: Here, it represents the defining equation of the random-phase approximation (RPA) to the polarization propagator, which
will be discussed in some detail in Chap. 15.
As an actual counterpart to the Dyson equation for the electron propagator one
may see the Bethe–Salpeter equation [2] for the p-h response function (13.12), which
in a shorthand notation can be written as
R 12,1 2 = G 12 G 21 − i G 1τ G σ1 K
ph
τσ ,στ R τ 2,σ 2
(13.15)
Here, repeated indices imply summation over one-particle states and integration
over time arguments; the quantity K
ph is referred to as the irreducible p-h vertex. Obviously, being an integral equation involving a fourfold time integration, the
Bethe–Salpeter equation is by far more complex than the Dyson equation.
13.2 Diagrammatic Perturbation Theory
for the Polarization Propagator
The diagrammatic perturbation theory formulated for the electron propagator in
Chaps. 4–7 can be transferred with minor adjustments to the case of the polarization
propagator. We recall the key steps as follows.
The starting point is the time representation of the polarization propagator