Lecture notes prepared towards credit in the course "The Green's function method for many-body theory" under Prof. K. Kikoin, fall 2008.

1. An Introductory Lecture to the Keldysh Technique
for Non-equilibrium Green's Functions
Submitted by Inon Sharony
October 23, 2008
Towards credit in the course
The Green's Function Method for Many-Body Theory
Given in the spring semester, 2008 by
Professor K. Kikoin

2. 1 Motivation
Physical systems out of equilibrium usually represent one or more of the following behaviors:
• Relaxation: Towards the ground state.
• Dephasing: From a coherent state to a non-coherent state.
• Steady-state: A time-stationary state which is not the ground state. e.g., a system with a source
& drain, in which exists a constant current.
And so on...
In the case of relaxation, we deal with a system initially prepared (at time t = −∞) in some excited
state, and we explore its return to the ground state. Alternatively, the system begins in some ground
state, and an external eld begins to be applied. If the external eld is weak enough, we should recover
from our theory of non-equilibrium Green's functions (NEGF), the theory of linear response.
The NEGF method proposes a quantum mechanical formulation of the classical Boltzmann equation.
1.1 Non-equilibrium Dynamics [7]
We begin with a short denition of non-equilibrium. This is not a rigorous proof, but a mathematical
introduction to the term non-equilibrium with respect to physical problems.
1.1.1 Master Equation Solutions for Equilibrium and Non-equilibrium Cases
Looking at a single point in conguration space, C ≡ ({xi} , {pi}), we nd that writing a master equation
for a non-equilibrium process will consist of summing on all the processes that increase the probability
to nd the system with such a conguration at time t, and subtracting all the processes that decrease
that probability.
∂tP (C, t) =
C
[R (C ← C ) P (C , t) − R (C ← C) P (C, t)]
∂tP = −LP
Where R (C ← C ) represent the incoming and R (C ← C) the outgoing rates (assumed constant).
This master equation seems similar to the Hamilton equation, but diers from it in some respects
(e.g., all P's are non-negative Reals and not Complex as in the case of Ψ).
One class of solutions consists of all the time independent solutions, P (C). The dierence between
the equilibrium and non-equilibrium solutions lies in that the rates R related to the equilibrium solutions
obey detailed balance, whereas in the non-equilibrium case they do not.
1.1.2 Detailed Balance
Take a series of congurations:
C1, C2, . . . , Cn−1, Cn.
Consider the products of the rates around the cycle
Π+ ≡ R (C1 ← Cn) R (Cn ← Cn−1) · · · R (C3 ← C2) R (C2 ← C1)
Π− ≡ R (C1 ← C2) R (C2 ← C3) · · · R (Cn−1 ← Cn) R (Cn ← C1)
Detailed balance entails that Π+ = Π−for all cycles.
Writing
R (C ← C )
R (C ← C)
≡ eA
× . . .
Detailed balance becomes
× A = 0
A can be written as a gradient of some scalar (e.g. −H/kBT) so that
P (C) ∝ e−β·H(C)

3. and for all pairs C and C we have
R (C ← C ) P (C ) − R (C ← C) P (C) = 0 (1)
i.e., zero current everywhere (e.g., electrostatics).
By contrast, if the rates violate detailed balance, we will still have time-independent solutions P (C),
but this time the left-hand-side (LHS) of 1 will not equal zero, and we will get non-trivial current loops
in our system (e.g., magnetostatics). These are called non-equilibrium steady states. Examples for when
we can get these could be when coupling to two energy reservoirs or two electrodes. In general, we can
expect these if F = − V.
1.2 Technical Dierences
As opposed to the Matsubara GF method, where an analytical continuation was applied (to relate
discrete Imaginary frequencies with temperature), the Keldysh technique we will discuss here uses only
continuous Real frequencies. This makes the Keldysh technique more appealing even in equilibrium
cases, where the Matsubara method can be more complex. An alternative approach, by Kadanoff
and Baym [10] may be more exhaustive, and will not be discussed here.
In most cases we will deal with, we are interested in a cannonic or grand-cannonic ensemble (i.e., the
energy isn't known exactly, but the temperature is). Therefore the averaging now takes on a dierent
meaning from the quantum expectation value (in equilibrium with respect to some ground state), but
rather a quantum ensemble average. Therefore
ˆO ≡
1
Z
Tr ˆρ ˆO
Z ≡
1
Z
Tr [ˆρ]
This carries some importance as pertains to the initial state of the system, as dened by ˆρ (t = 0), but
we will assume that the system was initially prepared in a pure state. For a more generalized approach,
see [6].
2 Non-equilibrium Green's Functions
We remember that the Green's function method reveals to be a pertubative technique through the
scattering matrix, S ≡ S (−∞, ∞) ≡ ˆT exp −i
∞
−∞
V (t) dt 1, where V is the time-dependent part of
the Hamiltonian and where the time-ordering operator, ˆT, to the right as follows:
ˆTA (1) B (2) =
A (1) B (2) t1 t2
A (1) B (2) t1 t2
where we employed the shorthand notation (1) ≡ (r1, t1). We likewise dene the anti-time-ordering
operator,
ˆ
T, which operates to the right in the opposite way.
Now, as in the equilibrium case
G (1, 2) ≡ −i S−1 ˆTΨ (1) Ψ†
(2) S
S−1
≡
ˆ
Te+i
R ∞
−∞
V(t)dt
Since no adiabatic switching-on can be assumed in a non-equilibrium situation, we cannot assume
that the state of the system at time t = −∞ diers from its state at time t = ∞ by a phase factor
only (Gell-Man Low theorem) . Therefore, as in the Matsubara method, we will need to concern
ourselves with a situation where we cannot factorize out the term S−1
.
The second quantization of the interaction (for a system under an external eld) can be expressed as:
V (t) ≡ Ψ (r, t) U (r, t) Ψ†
(r, t) dr
1Throughout this work we will take = 1.

4. We can expand the GF as a pertubative series by taking the S-matrix term-by-term in accordance
with the series expansion of the exponent.
S = S(0) + S(1) + . . .
S(n) ≡
(−i)
n
n!
ˆT
∞
−∞
∞
−∞
. . .
∞
−∞
V (t) V (t ) . . . V t(n)
dtdt . . . dt(n)
G (1, 2) = G0 (1, 2) + G1 (1, 2) + G2 (1, 2) + . . .
G0 (1, 2) = −i S−1
(0)
ˆTΨ (1) Ψ†
(2) S(0) = −i ˆTΨ (1) Ψ†
(2)
G1 (1, 2) = −i S−1
(0)
ˆTΨ (1) Ψ†
(2) S(1) − i S−1
(1)
ˆTΨ (1) Ψ†
(2) S(0)
= −i ˆTΨ (1) Ψ†
(2) S(1) − i S−1
(1)
ˆTΨ (1) Ψ†
(2)
= −i ˆT Ψ (1) Ψ†
(2)
∞
−∞
Ψ (3) U (3) Ψ†
(3) dr3 − i
ˆ
T
∞
−∞
Ψ (3) U (3) Ψ†
(3) dr3 × ˆT Ψ (1) Ψ†
(2)(2)
Obviously, the second and third terms on the RHS are not equivalent due to the dierent time
ordering of the component eld operators. This leads us to the concept of contour integration.
2.1 Contour Integration
The Kadanoff-Baym approach rst introduced the concept of contour integration in relation to GF
by switching the time-ordering operator by a closed-time-path integration contour.
This can be extended to thermodynamic systems through an interaction contour by including the
negative imaginary time axis.
In the case where the initial state of the system is described at t0 = −∞ as some pure state (all
initial correlations are zero), we can ignore the hook part of the contour (t0 − iβ ≤ t ≤ t0). This is
correct assuming the GF falls o fast enough with respect to the dierence of its two time arguments.
Thus, the closed time-path contour and the interaction contour are identical.
2.2 The Keldysh Contour [2]2
Keldysh showed that the contour can be extended beyond the largest time argument, so that it extends
from −∞ to +∞ (this is called the C1 contour) then crosses the real time axis innitesimally and returns
from +∞ to −∞ (C2). This is named the Keldysh contour. Clearly, the C1 part of the contour is time-
ordered, and the C2 part is anti-time-ordered. Thus, we can change our perspective from time-ordering
to contour-ordering, which we will discuss in the next sections.3
2For a review see [5]
3We will take this time to point out that a generalisation of the Green's functions technique to include also thermody-
namic information and correlations has been proposed in [6].

5. We now need to dene Green's functions in accordance to the relative location of their two time
arguments along the Keldysh contour.
2.3 Introducing: The Non-equilibrium Green's Functions (NEGF)
The eld operators in each correction term to the GF are connected in pairs (as in equilibrium methods).
Each pair is averaged over separately as a zero-order GF, G0, where each of the two eld operators can
be either part of the time-ordered part or anti-time-ordered part.
2.3.1 (Massive) Particle NEGF
The greater and lesser GF The greater (and lesser) GF correspond to the cases where the
rst time argument lies on the lower (upper) part of the contour, and the second time argument lies on
the opposite part.4
G
(1, 1 ) ≡ −i ΨH (1) Ψ†
H (1 )
G
(1, 1 ) ≡ ±i Ψ†
H (1 ) ΨH (1)
where in the last line, the positive sign corresponds to a Fermion GF, and the negative sign to a
Boson GF.
The relation to the density matrix is evident, when taking coinciding time arguments
G
(r1, t ; r1 , t) =
N
V
ˆρ (r1, t ; r1 , t) (3)
Due to the commutation relations of the eld operators, we have another identity for GF with
coinciding time arguments:
G
(r1, t ; r1 , t) − G
(r1, t ; r1 , t) = iδ (r1 − r1 )
The time-ordered and anti-time-ordered GF The causal equilibrium GF is now renamed
the time-ordered NEGF, and corresponds to cases where both time arguments lie on the upper part of
the contour (the part which is directed in the positive time direction):
Gc
(1, 1 ) ≡ −i ˆTΨH (1) Ψ†
H (1 )
≡
G
(1, 1 ) t1 t1
G
(1, 1 ) t1 t1
We similarly dene an anti-time-ordered NEGF, Gec
, using the anti-time-ordering operator, and
corresponding to cases where both time arguments lie on the lower part of the contour. Since the time
ordering operator (or anti time ordering operator) is invoked, the type of NEGF chosen does not depend
on the relative order of the two time arguments, but only that they are on the same part of the contour
(as opposed to the greater and lesser NEGF).
Thus, it is clear that the greater GF is applied when the rst time argument is greater than the
second, and vice-versa for the lesser GF.
The advanced and retarded GF The advanced and retarded GF are dened as in equilib-
rium GF techniques
Gr
(1, 1 ) ≡



−i ΨH (1) , Ψ†
H (1 )
±
t1 t1
0 t1 t1
Ga
(1, 1 ) ≡



0 t1 t1
+i ΨH (1) , Ψ†
H (1 )
±
t1 t1
4The greater GF is sometimes termed the particle propagator and the lesser GF the hole propagator.

6. These last two GF, as in the equilibrium case, have no cuts and are analytic in the upper (retarded)
or lower (advanced) half-planes of the complex time coordinate. Physically, they may represent the
system's magnetic susceptibility or dielectric constant.
The contour-ordered GF The contour-ordered NEGF is now dened as:
G (1, 1 ) ≡ −i ˆTCΨH (1) Ψ†
H (1 )
≡



Gc
(1, 1 ) t1, t1 ∈ C1
G
(1, 1 ) t1 ∈ C2, t1 ∈ C1
G
(1, 1 ) t1 ∈ C1, t1 ∈ C2
Gec
(1, 1 ) t1, t1 ∈ C2
this can be written in a matrix notation:
G ≡
Gc
G
G
Gec =
G−−
G−+
G+−
G++ (4)
Where the Gαβ
notation signies on which of the contours each of the time arguments lie. Confusingly,
the right sign corresponds to the rst time argument and the left sign corresponds to the second time
argument. If a time argument lies on the top contour (C1) it is marked with a negative sign, and
vice-versa.
2.3.2 (Massless) Interaction Quanta NEGF
In order to calculate the interaction of particles via quanta such as phonons we need to dene NEGF for
such Bosons.
We begin by dening the gauge transformation of the Complex Boson eld operator, χ, into the Real
scalar Boson eld operator, Φ:
Φ ≡ χ + χ†
Since Φ is Real, we dene
D
(1, 1 ) ≡ −i ΦH (1) ΦH (1 )
D
(1, 1 ) ≡ −i ΦH (1 ) ΦH (1)
we note that
D
(1, 1 ) = D
(1 , 1)
and that they are both anti-Hermitian.
Once again,
Dc
(1, 1 ) ≡ −i ˆTΦH (1) ΦH (1 )
and likewise for
Dec
(1, 1 ) ≡ −i
ˆ
TΦH (1) ΦH (1 )
For further details see [3].
2.4 Mathematical Relations of the NEGF
2.4.1 Conjugation Relations
Ga
(1, 1 ) = Gr
(1 , 1)
∗
(5)
Gc
(1, 1 ) = −Gec
(1 , 1)
∗
G
(1, 1 ) = −G
(1 , 1)
∗
G
(1, 1 ) = −G
(1 , 1)
∗

7. 2.4.2 Linear Co-dependence
Ga
= Gc
− G
= Gec
− G
Gr
= Gc
− G
= Gec
− G
Gc
+ Gec
= G
+ G
This last relation can be used to remove an excess GF equation, using the Keldysh GF
Gk
≡ Gec
− Gc
= G
− G
(6)
2.5 The Fourier Transformed NEGF (FT-NEGF)
In the case of homogeneous space and time, we expand the eld operators in plane-waves (Fourier
transformation)
Ψ (r1, t1) ≡
1
√
V p
apei(p·r1−ξpt1)
ξp ≡ εp − µ
Using the correlations of the creation and annihilation operators in momentum space
a†
pap = npδ (p − p )
apa†
p =
[1 − np] δ (p − p ) Fermions
[1 + np] δ (p − p ) Bosons
where only at equilibrium is np equal to the Fermi-Dirac (or Bose-Einstein) distribution function,
and otherwise is just the non-equilibrium momentum distribution function.
G
(r1, t1 ; r1 , t1 ) = G
(r, t ; 0, 0)
= ±
i
V
npei(p·r−ξpt) V d3
p
(2π)
3
inserting a Dirac delta function in frequency space allows a further Fourier transformation in
frequency:
G
(r, t ; 0, 0) = ±2πi npei(p·r−ωt) d3
p
(2π)
3 δ (ω − ξp)
dω
2π
≡ ei(p·r−ωt)
G
(p, ω)
d4
p
(2π)
4
and we identify
G
(p, ω) ≡ ±2πinpδ (ω − ξp)
G
(p, ω) ≡ ±2πi [1 − np] δ (ω − ξp)
and we can use previous identities together with the Sokhatsky-Weierstrass theorem to get
Gc
(p, ω) ≡ PP
1
ω − ξp
+ iπ [±2np − 1] δ (ω − ξp)

8. 2.5.1 Spectral Representation [9]
We dene the spectral function A (p, ω) as follows:
Gr,a
(p, ω) ≡ lim
η→0+
A (p, ω )
ω − ω ± iη
dω
2π
and we note the following resultant identities
A = 2Im [Ga
] = −2Im [Gr
]
making use of the Lehmann expansion for the equilibrium case we nd
G
(p, ω) = inpA (p, ω)
G
(p, ω) = −inpA (p, −ω)
Gk
(p, ω) = − [±2np − 1] A (p, ω)
2.5.2 Correspondence to Equilibrium Green's Functions
At T=0, we take the averaging over the ground state,
. . . → 0 . . . 0 ≡ 0 |. . .| 0 .
Thus, the Fermi-Dirac distribution function is replaced by a Heaviside function,
np → Θ (|p| − pF ) ≡



1 |p| pF
1
2 |p| = pF
0 |p| pF
.
This will cause one of the rst order terms to drop, and likewise for higher orders, reverting from the
NEGF to the equilibrium GF.
G
T =0 (p, ω) = 0 ⇐ |p| pF
G
T =0 (p, ω) = 0 ⇐ |p| pF
3 (Feynman) Diagrammatics
We can form a diagrammatic technique for the NEGF which is almost identical to the equilibrium
Feynman technique used at T=0. Again, free particle propagators (G0) will be represented by solid
arrows, and interactions (U) through dashed lines. The relevant form of Wick's Theorem for the non-
equilibrium case can be proposed via a generalisation of the equilibrium Wick's Theorem however we
will not elaborate this here. We just assume that the average over any number of NEGF with their
dierent orderings can be simplied to a product of averages over connected pairs of NEGF, with some
coecients. More information on these points may be found elsewhere (For instance, [6]).
We return to the Gαβ
notation and reiterate that each propagator in the diagram will be assigned
two signs (±) according to the time-ordering with which its operators are related (time-ordered operators
get a negative index). 5 In accordance to the matrix representation of the NEGF, we will have four
such terms, G±±
. For each such element we will draw the diagrams complying with each term in the
pertubative expansion. Each element diers from the others through the rst and last of its sign indeces.
Let us observe an example.
3.1 First Order
In accordance with the equation for the rst order correction to the GF in 2 we have two diagrammatic
terms for G−−
1 .
5The external eld has two operators, but the dashed line associated with it is only connected to one vertex. Therefore
it is only associated with one sign index.

9. Note that the rst and last signs are negative, as should be for a term in the expansion of G−−
, and
the vertex 3 is integrated over as in the equilibrium Feynman technique (integration over space and
time, and summation over spin indeces).
G−−
1 (1, 2) = G−−
0 (1, 3) G−−
0 (3, 2) [−U (3)] + G−+
0 (1, 3) G+−
0 (3, 2) [U (3)]
The negative sign of the interaction in the rst term is due to the sign with which it is associated, as
a term in S.
The same equation may be written for the other elements, Gαβ
1 .
3.2 Second Order
We can extend the denitions to second order diagrams for the four terms of the G−−
element, G−−
2 .
(Note that we dropped the vertex numbering, or space time arguments)
3.3 First Order Two-particle Interaction
These diagrams are formed, as in the equilibrium case, by dierent combinations of two external-eld
diagrams, where the dashed line now represents an interaction quantum. Notice that the propagator of
the interaction quanta begins and ends with the same sign.
4 Self-energies and the Dyson Equation
We write the Dyson equation as for G−−
, similarly to the equilibrium case
G−−
(1, 2) = G−−
0 (1, 2) +
G−−
0 (1, 4) Σ−−
(4, 3) G−−
(3, 2) +G−
0 (1, 4) Σ−+
(4, 3) G+−
(3, 2) +
G−+
0 (1, 4) Σ+−
(4, 3) G−−
(3, 2) +G−+
0 (1, 4) Σ++
(4, 3) G+−
(3, 2)
We can write this same equation with a more general self-energy (note that the coordinate indeces
have also been changed)
We deal with a 2 × 2 matrix of the indeces corresponding to the dierent NEGF (±± ≡ {, , c, c})
and we can write the Dyson equation in matrix notation:
G = G0 [1 + ΣG]
We can draw diagrams for the dierent elements of dierent order terms of the self energy

10. where the top line is comprised of terms of the Σ−−
element, and the bottom line of the Σ−+
element.
5 RAK Representation
Using a unitary rotation matrix and 6 we can write 4 as
G ≡
0 Ga
Gr
Gk
Σ ≡
Σk
Σr
Σa
0
With
Σa
≡ Σ−−
+ Σ+−
Σr
≡ Σ++
+ Σ−+
Σk
≡ Σ++
+ Σ−−
Three kinetic equations are formed:
Ga
(1, 2) = Ga
0 (1, 2) + Gc
0 (1, 4) Σa
(4, 3) Ga
(3, 2) d4
X4
And likewise we have second the equation for Gr
(1, 2), however it is usually simpler just to use the
relations between the advanced and retarded GFs 5.
The third equation is:
Gk
(1, 2) = −ˆρ (1, 2) + Gr
0 (1, 4) Σk
(4, 3) Ga
(3, 2) + Σr
(4, 3) Gk
(3, 2) + Gk
0 (1, 4) Σa
(4, 3) Ga
(3, 2) d4
X3d4
X4
We can apply the operator [G (0, 1)]
−1
from the left of both sides of the equation, where
G−1
(0, 1) = i
∂
∂t1
+
2
r1
2m
+ µ
G−1
(0, 1) G (1, 2) = δ (r1 − r2)
G−1
(0, 1) Gk
0 (1, 2)
!
= 0
because G (1, 2) has poles at r1 − r2, but Gk
(1, 2) has no poles!
We get
G−1
(0, 1) Gk
(1, 2) = Σk
(1, 3) Ga
(3, 2) + Σr
(1, 3) Gk
(3, 2) d4
X3 (7)
An alternative derivation may be formulated via the Kadanoff-Baym technique and using the
Langreth theorem.6
6See [4, 8].

11. 6 Appendix: The Semi-classical NEGF Approximation The
Transport Equation
6.1 Equivalence to the Boltzmann Equation
For a free gas of classical hard spheres with some distribution function, n (r, p)
dn
dt
=
∂n
∂t
+
∂n
∂r
∂r
∂t
=
∂n
∂t
+
∂n
∂r
p
m
the LHS of the equation is equal to all the contributions from scattering in dierent collisions. This
may be expressed in integral form as the collision integral (Stosszahl), C (n).
If we add an external eld via the force it aects, F
∂n
∂t
+
p
m
∂n
∂r
+ F
∂n
∂p
= C (n)
∂
∂t
+
p
m
∂
∂r
+ F
∂
∂p
n = C (n)
which is quite similar7 to the Dyson equation 7
i
∂
∂t1
+
2
r1
2m
+ µ Gk
(1, 2) = C (n)
G−1
(0, 1) Gk
(1, 2) = C (n)
where we take the quantum equivalent of the classical distribution function n (r, p) → ˆρ (r, p), which
satises 3.
6.2 The Mixed Representation the Wigner Distribution
We combine two G−1
operators to form a 2-particle operator
ˆR ≡ G−1
(0, 1)
†
− G−1
(0, 2) = −i
∂
∂t1
+
∂
∂t2
−
1
2m
2
r1
− 2
r2
we separate X1 and X2 into a center-of-mass coordinate and a dierence coordinate
R ≡
r1 + r2
2
r ≡ r1 − r2
t ≡
t1 + t2
2
τ ≡ t1 − t2
and write the Wigner distribution function (a mixed representation)
ˆρ R +
1
2
r, t ; R −
1
2
r, t =
V
N
eip·r
n (R, p, t)
d3
p
(2π)
3
in reference to the NEGF, this representation gives (in 4-dimensional notation: X ≡ (R, t) , Ξ ≡ (r, τ))
G
(X, P) = eiP Ξ
G
X +
1
2
Ξ, X −
1
2
Ξ
d4
Ξ
(2π)
4
which, for simultaneous time arguments in the GF gives the identity
n (R, p, t) = −i G
(X, P)
dω
2π
7For a more rigorous comparison, see [4, 8].

12. Notice, also that ˆR in the new coordinates (at simultaneous times) is
ˆR (R, r, t) = −i
∂
∂t
−
i
m
R r
The RHS is exactly the RHS of the Boltzmann equation with no external eld, up to a factor of i.
We can generalize this result to a system under an external eld through the introduction of the vector
potential up to rst order in |A|
i r −→ i r −
e
c
A
The result will be the Boltzmann equation with an external force F = qE.
6.3 The Collision Integral
We operate using ˆR on G and use the mixed representation to disregard all short range spatial variations
(neglect dependence on r) nally arriving at a NEGF formulation for the collision integral:
C (n) ∼ −Σ−+
(X, P) G+−
(X, P) + Σ+−
(X, P) G−+
(X, P)
dω
2π
= iΣ−+
(ξp, p ; r, t) · [1 − n (r, p, t)] + iΣ+−
(ξp, p ; r, t) · n (r, p, t)
Where the rst term on the RHS of the last line corresponds to a gain of particles, and the second
term to a loss.
Some more specic approximations are needed to get results for specic models, however we can
note that this result is independent of the same-sign-index self-energies, Σ++
and Σ−−
. Therefore, the
rst order correction terms to the participating self energy elements are gotten from the second order
diagrams of type (here, for Σ−+
)
where (from conservation of 4-momentum) we have P1 = P + P1 − P , and the analytic expression
for this self energy element is
Σ−+
(P) = −i G−+
(P ) G+−
(P1) G−+
(P1) U2
(P1 − P ) δ (P + P1 − P − P1)
d4
P1d4
P
(2π)
8
References
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