The theory of non-equilibrium Green’s functions is based on the idea of deriving a closed
set of equations for the two-point correlation functionˆΨ(2) ˆΨ^{†}(1)which represents the
probability of adding a particle in state 1 = {r1, t_{1}, s_{1}} prior to removing one in state2.
Physically, this correlation function describes the propagation of a particle and therefore it
is called propagator. Similarly a hole propagatorˆΨ^{†}(2) ˆΨ(1)can be deﬁned, representing
a generalization of the single-particle density matrix.

In this chapter, the theory of NGF is outlined in a rather compact fashion. More detailed reviews on non-equilibrium Green’s functions can be found for example in Refs. [61, 75–

77].

**Quantum statistics**

For the discussion, a Hamiltonian of the form Hˆ =

Hˆ_{0}+ ˆH_{int}

+ ˆH_{ext}(t), (3.17)

is considered, including a non-interacting partHˆ_{0}an interacting partHˆ_{int}and an explicitly
time dependent perturbationHˆ_{ext}(t). Note, that the Hamiltonian (3.2) is of this form, with
Hˆ_{ext}(t)speciﬁed below.

The expectation value of an arbitrary operator in the Schrödinger picture is given by
Aˆ=ΦS(t)|Aˆ_{S}|ΦS(t) . (3.18)
After transformation into the Dirac picture one has

|ΦS(t)=S(t, t_{0}) |Φ_{0}(t_{0}) , (3.19)
Aˆ_{D}(t) =U^{†}(t, t_{0}) ˆA_{S}U(t, t_{0}). (3.20)
The temporal evolution of the states is determined by the external perturbation and that of
the operators given by the free and the interacting part of the Hamiltonian according to

S(t, t_{0}) =T e^{−}^{}^{i}

R_{t}

t0dtHˆext(t), (3.21)
U(t, t_{0}) =e^{−}^{}^{i}^{[ ˆ}^{H}^{0}^{+ ˆ}^{H}^{int}^{](t−t}^{0}^{)}. (3.22)

The operator T ensures chronological ordering of all operators from right to left. Now, Eq. (3.18) can be cast into

Aˆ=Φ_{0}|S(−∞,∞)T

S(∞,−∞) ˆA_{D}(t)

|Φ_{0} , (3.23)

when considering the semi-group propertyS(t_{1}, t_{2}) =S(t_{1}, t_{3})S(t_{3}, t_{2})of the time
evolu-tion operator and choosingt_{0} =−∞for the initial time. One ﬁnds the right part ordered
chronologically and the left part ordered anti-chronologically. To apply quantum ﬁeld
theo-retical methods like Feynman’s diagram rules [50, 51], a uniform time ordering is required.

This could be achieved by introducing the so-called Keldysh time contourC and the
corre-sponding Keldysh timet. On this contour the operators are, with respect to the real time,
chronologically ordered on the upper branch t_{+} and anti-chronologically ordered on the
lower brancht_{−}. This is schematically depicted in Fig. 3.1.

t_{0}

t_{+}

t_{−}

t → ∞

**Fig. 3.1: Keldysh time contour** C for the temporal evolution from the initial time t =
t_{0} to t = ∞ and back. With respect to the real time, the upper branch t_{+} is ordered
chronologically while the lower brancht_{−}is ordered anti-chronologically.

By introducing the Keldysh contour one artiﬁcially distinguishes between an external
per-turbation on the upper branch Hˆ_{ext}(t_{+}) and a perturbation on the lower branch Hˆ_{ext}(t_{−}).
Deﬁning the contour-ordered expectation value as

Aˆ(t)_{C} = Tr

ρ_{D}T_{C}

S_{C}Aˆ_{D}(t)

Tr{ρ_{D}S_{C}} , (3.24)

the original expectation value (3.23) is recovered in the physical limitHˆ_{ext}(t_{+}) = ˆH_{ext}(t_{−}).
The corresponding contour-ordered time-evolution operator contains the forward evolution
S_{+}on the upper branch and the backward evolutionS_{−}on the lower branch,

S_{C} =S_{+}S_{−} =T_{C} exp

−i

C

dτ Hˆ_{ext}(τ)

, (3.25)

which in the physical limit reduces toS_{C} = 1.

**Carrier Green’s function**

For the calculation of single particle expectation values it is sufﬁcient to know the single-particle Green’s function

G(1,2) =−i
n_{2}

Tr

ρ_{0}T_{C} S_{C} ˆΨ(2) ˆΨ^{†}(1)

Tr{ρ_{0}S_{C}} . (3.26)

Here, the argument 1 includes the branch index n_{1} ∈ {+1,−1} of the Keldysh time,
where +1 denotes the upper branch and denotes −1 the lower one. With respect to the
branch index, the Keldysh GF has the matrix structure

G(1,2) =

G(1+,2+) G(1+,2−)
G(1_{−},2_{+}) G(1_{−},2_{−})

. (3.27)

The diagonal elements are the chronologically and the anti-chronologically ordered GFs, whereas the non-diagonal elements are given by the propagators:

G(1+,2+) =−i

T ˆΨ (1+) ˆΨ^{†}(2+)

= G(1,2), (3.28)

G(1−,2−) =−i

T ˆΨ (1−) ˆΨ^{†}(2−)

= G(1,2), (3.29) G(1+,2−) = i

ˆΨ^{†}(2−) ˆΨ (1+) =−G^{<}(1,2), (3.30)
G(1_{−},2_{+}) =−i

ˆΨ (1−) ˆΨ^{†}(2_{+}) = G^{>}(1,2). (3.31)
From a linear combination of the propagators the retarded and advanced GF are deﬁned,

G^{R}(1,2) = Θ(t_{1}−t_{2}) G^{>}(1,2)−G^{<}(1,2)

, (3.32)

G^{A}(1,2) =−Θ(t_{2}−t_{1}) G^{>}(1,2)−G^{<}(1,2)

. (3.33)

Since also the (anti-)chronologically ordered GF can be expressed in terms of propagators according to

G(1,2) = Θ(t_{1}−t_{2})G^{>}(1,2) + Θ(t_{2} −t_{1})G^{<}(1,2), (3.34)
G(1,2) = Θ(t_{2}−t_{1})G^{>}(1,2) + Θ(t_{1} −t_{2})G^{<}(1,2), (3.35)
only two elements of the Keldysh matrix are independent. Frequently used properties of
these GFs are:

G^{≷}(1,2)_{†}

=−G^{≷}(2,1), (3.36)

G^{R}(1,2)_{†}

= G^{A}(2,1). (3.37)

**Schwinger functional derivative technique and Dyson equation**

To obtain a set of equations for a perturbation expansion of the contour-ordered GF, the functional derivative technique introduced by Schwinger is employed [61, 77]. Alterna-tively a unitary transformation might be used, which enables to use Wick’s theorem and to derive Dyson’s equation as in equilibrium theory [78, 79].

For the following derivation the Hamiltonian

Hˆ = ˆH_{ph}+ ˆH_{0}+ ˆH_{e-e}+ ˆH_{e-i} (3.38)
is considered. The single components in real-space representation and second quantization
read

Hˆ_{0}=

dx_{1} ˆΨ^{†}(1)h(1) ˆΨ(1), (3.39)

Hˆ_{e-e} = 1
2

dx_{1}

dx_{2} ˆΨ^{†}(1) ˆΨ^{†}(2)V(1,2) ˆΨ(2) ˆΨ(1), (3.40)
Hˆ_{e-i}=

dx_{1} V_{e-i}(1) ˆΨ^{†}(1) ˆΨ(1) (3.41)
withx∈ {r, s},

dx=

s

d^{3}rand

h(1) =− ^{2}

2m∇^{2}+**dE(r**1, t_{1}), (3.42)

V(1,2) = e^{2}
4πε_{0}

1

|r1−**r**2| δ(t_{1}−t_{2}), (3.43)
V_{e-i}(1) =−Ze^{2}

4πε_{0}

d**R** Nˆ_{0}(R) + Δ ˆN(R)

|r_{1}−**R|** . (3.44)

Note, that in the Hamiltonian all particles are considered explicitly and hence no
back-ground screening is included in the interaction potentials. According to the
Born-Oppenhei-mer approximation the ions are decoupled from the electrons. The kinetic part and the
ion-ion interaction are summarized in the phonon Hamiltonian Hˆ_{ph} which in harmonic
approximation is given by Eq. 3.4. The phonon Hamiltonian describes the equilibrium
positions of the ions, which can be determined from the minima of the Born-Oppenheimer
energy surface (cf. Refs. [62, 80]). However, their explicit values are not necessary for the
following discussion.

The ion density Nˆ = ˆN_{0} + Δ ˆN entering the electron-ion interaction potential V_{e-i} is
split into an equilibrium partNˆ_{0}and a ﬂuctuation partΔ ˆN. AsV_{e-i}represents an effective
single-particle potential for the electrons, the equilibrium part describes the lattice-periodic
potential which enters the band structure. The ﬂuctuation part gives rise to the
carrier-phonon interaction. As external perturbation we consider the Hamiltonian

Hˆ_{ext} =

d^{3}r

nˆ(r)−Nˆ(r)

U_{ext}(r, t) + ˆN(r)J_{ext}(r, t)

(3.45)

which includes the coupling of the total carrier density to an electrostatic potentialU_{ext}(r, t) =
e φ_{ext}(r, t)as well as an external source J_{ext}(r, t)that can be interpreted as a mechanical
force acting only on the ionic lattice. Here, the electron density operator is denoted as
ˆ

n(1) = ˆΨ^{†}(1) ˆΨ(1)and the total charge density is given byeρˆwithρˆ= ˆn −Nˆ.

The time dependence of the creation and annihilation operators for carriers is found using Heisenberg’s equation of motion. Applying the chain rule, the equation of motion for the Keldysh GF (3.26) with respect to the ﬁrst time is then given by

i ∂

∂t_{1}G(1,2) =δ(1−2) + [h(1)−U_{ext}(1)] G(1,2)

−i

d3n_{3}V(1,3)
ˆ

n(3)−Nˆ(3)

ψˆ(1) ˆψ^{†}(2)

, (3.46)

where a four-point correlation function appears which in addition to the two-particle carrier
GF also contains a term involving the ion density. Analogous, the equation of motion with
respect to the second time can be obtained. At this point, the many-body hierarchy problem
is explicitly present. The basic idea to deal with the many-body hierarchy is to consider the
response of the system to the external perturbationU_{ext} by means of applying Schwinger’s
functional derivative technique. In terms of response theory, the GF is considered as a
functionalG[U_{ext}(t)]whose derivative with respect to the external perturbation is given by

δG(1,2)

δU_{ext}(3) =−
ˆ

n(3)−Nˆ(3)

ψˆ(1) ˆψ^{†}(2)

+G(1,2) ˆ

n(3)−Nˆ(3)

. (3.47) Here, the GF itself acts as a generating functional for higher order correlation functions.

The second term on the right-hand side of Eq. (3.47) can be identiﬁed as a Hartree interac-tion which is lumped into an effective potential

U_{eff}(1) =U_{ext}(1)−in_{1}

d3V(1,3) ˆ

n(3)−Nˆ(3)

. (3.48)

Via the functional derivative the selfenergy is deﬁned as i

d3n_{3} V(1,3) δG(1,2)
δU_{ext}(3) ≡

d3 Σ(1,3)G(3,2). (3.49) Using this deﬁnition of the selfenergy, the structure of a Dyson equation equation

i ∂

∂t_{1} −h(1)−U_{eff}(1)

G(1,2)−

d3 Σ(1,3)G(3,2) = δ(1,2) (3.50) is obtained from the equation of motion (3.46). In the non-interacting case, i.e.Σ(1,2) = 0, the free inverse GF

G^{−1}_{0} (1,2) =

i ∂

∂t_{1} −h(1)−U_{eff}(1)

δ(1,2) (3.51)

can be identiﬁed. Successive application of the functional derivative then yields a funda-mental set of equations where the hierarchy problem is eliminated formally. In the follow-ing this somewhat lengthy calculation is skipped and only the basic steps are outlined. De-tails are presented for example in Refs. [61, 80]. Using the deﬁnition

d3G(1,3)G^{−1}(3,2) =
δ(1−2)of the inverse GF we may write

δG(1,2)
δU_{ext}(3) =

d4d5d6G(1,4) δG^{−1}(4,5)
δU_{eff}(6)

δU_{eff}(6)

δU_{ext}(3) G(5,2). (3.52)
The expression Eq. (3.49) for the selfenergy together with the deﬁnition of the vertex
func-tion

Γ(1,2,3) = δG^{−1}(1,2)

δU_{eff}(3) (3.53)

and the dielectric function
ε^{−1}(1,2) = δU_{eff}(1)

δU_{ext}(2) =δ(1−2) +

d3V(1,3) δˆn(3)−Nˆ(3)

δU_{ext}(2) (3.54)
yields the ﬁnal expression (3.59d) for the selfenergy. Note, that the screening of the bare
Coulomb interaction contains an electronic and an ionic contribution. Evaluating the
di-electric function further, we ﬁnd

ε^{−1}(1,2) =δ(1−2) +

d4d5ε^{−1}(5,3)P(4,5)V(4,1)

−

d4 δNˆ(4)

δU_{ext}(2) V(4,1), (3.55)
where the electronic polarization function

P(1,2) =−in_{1} δˆn(1)

δU_{eff}(2) (3.56)

is introduced. The evaluation of the functional derivative of the ion density yields

− δNˆ(2)

δU_{ext}(1) = δρˆ(1)

δJ_{ext}(2) =D(1,2) +

d3d4P(1,3)V(3,4) δρˆ(4)

δJ_{ext}(2) , (3.57)
which contains the ion density-density correlation function

iD(1,2) = Δ ˆN(1) Δ ˆN(2) . (3.58)
Solving Eq. (3.57) for _{δJ}^{δ}^{ρ}^{ˆ} and inserting into Eq. (3.55) yields the interaction given by
Eq. 3.59c, containing the usual screened Coulomb interaction (3.59b) and a carrier-phonon
contribution.

In summary, the following fundamental set of equations is found, whose diagrammatic representation is depicted in Fig. 3.2:

Dyson equation

G(1,2) = G_{0}(1,2) +

d3d4 G_{0}(1,3) Σ(3,4)G(4,2), (3.59a)
W_{e}(1,2) = V(1,2) +

d3d4 V(1,3)P(3,4)W_{e}(4,2) ; (3.59b)
Screened interaction

W(1,2) =W_{e}(1,2) +

d3d4W_{e}(1,3)D(3,4)W_{e}(4,2) ; (3.59c)
Selfenergy and polarization function

Σ(1,2) =−in_{1}

d3d4G(1,3) Γ(3,2,4)W(4,1), (3.59d)
P(1,2) = in_{1}

d3d4G(1,3) Γ(3,4,2)G(4,1) ; (3.59e) Vertex function

Γ(1,2,3) =−δ(1,2)δ(1,3) +

d4d5d6d7 δΣ(1,2)

δG(4,5) G(4,6) Γ(6,7,3)G(7,5). (3.59f)
In case ofD(1,2) = 0, corresponding to a rigid lattice, the equations reduce to the purely
electronic part. The inﬂuence of a lattice displacements enters the theory via the ion
density-density correlation function D(1,2) which is aN-body quantity as it depends on
the actual position of all ions. Note, that so far no equation for this correlation function
is given. Its evaluation in general can only be done approximately [80]. Considering
the Fröhlich Hamiltonian, the evaluation of the correlation function D(1,2)is shown in
Chap. 4.2, giving rise to so-called phonon interaction lines. Furthermore, even though the
vertex functionΓis formally the same as in the purely electronic case, additional
contribu-tions are included. The selfenergy (3.59d’) contains the full interactionW. Therefore, the
functional derivative_{δG}^{δΣ} also introduces mixed diagrams that include Coulomb and phonon
interaction lines, in addition to diagrams with only one type of interaction line.

1 2 = 1 2 + 1 3 2 Σ 4

(3.59a’)

1 2 = 1 2 + 1 3 2

P 4

(3.59b’)

1 2 = 1 2 + 1 3 2

D 4

(3.59c’)

1 Σ 2 = 1 2^{Γ}

4

3

(3.59d’)

1 P 2 = 1 Γ 2

3

4

(3.59e’)

Γ 3

1

2

=

-1 = 2 = 3+ ^{δΣ}_{δG} Γ

5 4

2 1

3 6

7

(3.59f’)

**Fig. 3.2: Diagrammatic representation of the fundamental set of equations.**