. 8
( 22)


we obtain for the inelastic di¬erential cross section for neutron scattering o¬ the
d2 σ m2 L6 p 2π (f) (i)
| p, ES |a δ(ˆn ’ RN )|p , ES |2
= r
dˆ d (2π )

(f) (i)
— δ(ES ’ ES ’ ω) (6.143)

which we can express as

d(t ’ t ) ’i(t’t )ω
d2 σ m2 a2 p 2π
dx dx e’
n (x’x )·(p’p )
= e
(2π )3 p
dˆ d 2π

(f) (i) (i) (f)
— ES |n(x, t)|ES ES |n(x , t )|ES , (6.144)

where n(x, t) is the density operator for the nuclei of the material in the Heisenberg
picture with respect to the substance Hamiltonian HS .

Exercise 6.3. Show that, for scattering o¬ a single heavy nucleus, M mn , we
have for the total cross section

d2 σ mn a 2
σ= dˆ d = 4π . (6.145)
2π 2
dˆ d
4π 0
176 6. Linear response theory

In the scattering experiment we know only the probability distribution for the
initial state of the material, which we shall assume to be the thermal equilibrium
|ES (») P (ES (»)) ES (»)| ,
ρS = (6.146)
e’ES (»)/kT
HS |ES (») = ES (») |ES (») .
P (ES (»)) = , (6.147)
For the transition rate weighted over the thermal mixture of initial states of the
substance we have

“fp P (ES (»)) “¬

d(t ’ t ) ’i(t’t )ω
m2 2
a p 2π
= P (ES (»)) dx dx e
(2π )3 p 2π
» ’∞

i (f) (f)
— ES |n(x, t)|ES (») ES (»)|n(x , t )|ES
(x’x )·(p’p )
and we obtain for the weighted di¬erential cross section (we use the same notation)
d2 σ d2 σ
= P (ES (»))
dˆ d dˆ d
p p

d(t ’ t ) ’i(t’t )ω
m2 a2 p 2π
= P (ES (»)) dx dx e
(2π )3 p 2π
» ’∞

— e’
i (f) (f)
ES |n(x, t)|ES (») ES (»)|n(x , t )|ES
(x’x )·(p’p )
. (6.149)
Furthermore, in the experiment the ¬nal state of the substance is not measured,
and we must sum over all possible ¬nal states of the substance, and we obtain ¬nally
for the observed di¬erential cross section (we use the same notation)

d(t ’ t ) i(t’t )ω
d2 σ m2 a2 p 2π
dx e’ (x’x )·(p ’p)
= dx e
(2π )3 p
dˆ d 2π

— n(x, t) n(x , t ) , (6.150)
where the bracket denotes the weighted trace with respect to the state of the sub-

n(x, t) n(x , t ) trS (ρS n(x, t) n(x , t )) . (6.151)
6.7. Scattering and correlation functions 177

We thus obtain the formula
d2 σ p m 2 a2
= V S(q, ω) (6.152)
p (2π )3 2
dˆ d

where S(q, ω) is the Fourier transform of the space-time density correlation function
S(x, t; x , t ) ≡ n(x, t) n(x , t ) , and q ≡ p ’ p and ω is the momentum and en-
ergy transfer from the neutron to the substance. We note that S(’q, ’ω) = S(q, ω).
This correlation function is often referred to as the dynamic structure factor.21 The
dynamic structure factor gives the number of density excitations of the system with
a given energy and momentum. A scattering experiment is thus a measurement of
the density correlation function.

Exercise 6.4. Show that, for a target consisting of a single nucleus of mass M in
the thermal state, the dynamic structure factor is given by

1 2πM ’ 2kM q 2 ω’
S(q, ω) = eT . (6.153)
kT q 2

For the di¬erential cross section of a Boltzmann gas of N non-interacting nuclei
we have according to Eq. (6.153)

d2 σ m2 a2 p 2π
=N n3 S(q, ω)
dˆ d (2π ) p

m2 a2 p 2π q2
dt e’iωt e’ 2M (t kT ’i t)
=N n3 . (6.154)
(2π ) p

Exercise 6.5. Show that the limiting behavior of the total cross section for a Boltz-
mann gas is

d2 σ
σ = dˆ d
dˆ d
4π 0
21 We here follow the conventional notation, although in the standard notation of this chapter
we have S(x, x , ω) = χ> (x, x , ω). According to the ¬‚uctuation“dissipation theorem, the structure
function is related to the density response function according to S(x, x , ω) = 2 n(ω) m χ(x, x , ω).
178 6. Linear response theory

§ Mp 2
mn a 2
⎪ 4πN 2
for 1

⎪ √
2π 2 2m2 kT
⎨ Mp2
mn n
π (1 + 2m 2 k T
= (6.155)

⎪ 4πN
mn a 2 Mp
for 1.
2π 2 2m2 kT
(1 + ) n

The divergent result for low energies is caused by the almost vanishing ¬‚ux of in-
coming neutrons being scattered by the moving nuclei in the gas, and in the opposite
limit we recover the result for scattering o¬ N free and non-interacting nuclei.

For a discussion of the liquid“gas transition, and the phenomenon of critical
opalescence we refer the reader to chapter 7 of reference [1].

6.8 Summary
The non-equilibrium states of a system which allows a description with su¬cient
accuracy by taking into account only the linear response occupies an especially simple
regime. In fact, the non-equilibrium properties of such states could be completely
understood in terms of the ¬‚uctuations characterizing the equilibrium state. Since
the equilibrium state possesses universal properties, so does the dissipative regime of
ever so slight perturbations, a feature with many important practical consequences.
In Chapter 11 we shall return to study the linear response functions, the transport
coe¬cients or conductivities. In particular we shall study the electrical conductivity
of a disordered conductor in the quantum regime and take into account nonlinear
e¬ects in an applied magnetic ¬eld. To discuss such intricacies we shall express
transport coe¬cients in terms of Green™s functions and thereby have the powerful
method of Feynman diagrams at our disposal. The density response function is
for a system of charged particles equivalent to the e¬ective interaction as density
¬‚uctuations are the source of the interaction. The e¬ective interaction in a disordered
conductor is discussed in Chapter 11. In the next two chapters, we shall study general
non-equilibrium states, and universal properties are in general completely lost.

Quantum kinetic equations

In this chapter, the quantum ¬eld theoretic method will be used to derive quantum
kinetic equations. The classical limit can be established, and quantum corrections
can be studied systematically. Of importance is the fact that the treatment allows
us to assess the validity regime of the kinetic equations by diagrammatic estimates.
The quasi-classical Green™s function technique is introduced. It will allow us to go
beyond classical kinetics and, for example, to discuss renormalization e¬ects due to
interactions in a controlled approximation. Thermo-electric e¬ects, being depending
on particle“hole asymmetry, are not tractable in the quasi-classical technique and
are dealt with on a separate basis.1

7.1 Left“right subtracted Dyson equation
In a non-equilibrium situation, the ¬‚uctuation“dissipation relation is no longer valid,
and the kinetic propagator, GK , is no longer speci¬ed by the spectral function and
the quantum statistics of the particles, as for example in Eq. (3.116). To derive
quantum kinetic equations the left and right matrix Dyson equation™s, Eq. (5.66)
and Eq. (5.69), are subtracted giving

[G’1 ’ Σ — G]’ = 0 ,
, (7.1)

the left“right subtracted Dyson equation. The reason behind this trick will soon
become clear. Here we have again used — to signify matrix multiplication in the spa-
tial and time variables, and introduced notation stressing the matrix multiplication
structure in these variables

[A — B]’ = A — B ’ B — A , [A — B]+ = A — B + B — A ,
, , (7.2)

the latter anti-commutation notation to be employed immediately also. The general
quantum kinetic equation is obtained by taking the kinetic or Keldysh component,
1 This chapter, as well and the following chapter, follows the exposition given in references [3]
and [9].

180 7. Quantum kinetic equations

the o¬-diagonal component, of equation Eq. (7.1) giving
i K— i—K
[G’1 ’ eΣ — GK ]’ ’ [ΣK — [Σ , A]+ ’
, , eG]’ = [“ , G ]+ (7.3)
2 2
where we have introduced the spectral weight function

A(1, 1 ) ≡ i(GR (1, 1 ) ’ GA (1, 1 )) (7.4)

eG(1, 1 ) ≡ (G (1, 1 ) + GA (1, 1 )) (7.5)
and similarly for the self-energies

“(1, 1 ) ≡ i(ΣR (1, 1 ) ’ ΣA (1, 1 )) (7.6)

eΣ(1, 1 ) ≡ (Σ (1, 1 ) + ΣA (1, 1 )) . (7.7)
The way we have grouped the self-energy combinations in Eq. (7.3) appears at the
moment rather arbitrary (compare this also with Section 5.7.4). Recall that A and “
can be expressed as A = i(G> ’G< ) and “ = i(Σ> ’Σ< ) and appear on the right side,
whereas eΣ and eG are of a di¬erent nature. We shall later understand the physics
involved in this di¬erence of appearance of the self-energies: those on the left describe
renormalization e¬ects, i.e. e¬ects of virtual processes, whereas those on the right
describe real dissipative collision processes. The presence of the self-energy entails
one having to deal with a complicated set of equations for an in¬nite hierarchy of
the correlation functions, the starting equation being the Dyson equation. Of course,
the general quantum kinetic equation is useless in practice unless an approximate
expression for the self-energy is available.
Notice that in equilibrium, say at temperature T , the exact quantum kinetic
equation is an empty statement since the Green™s functions are related according to
the ¬‚uctuation“dissipation relation, which for the case of fermions reads3
GR (E, p) ’ GA (E, p)
GK (E, p) = tanh (7.8)
and consequently
ΣR (E, p) ’ ΣA (E, p)
ΣK (E, p) = tanh . (7.9)
As a consequence, the two terms on the right in Eq. (7.3) cancel each other and the
terms on the left are trivially zero in an equilibrium state since the convolution — in
this case is commutative.
In a non-equilibrium situation, the ¬‚uctuation“dissipation relation is no longer
valid. Since a Green™s function is a traced quantity, a closed set of equations can
2 The choice of notation re¬‚ects that in the Wigner or mixed coordinates, A and eG will be
purely real functions as shown in Exercise 7.1 on page 182.
3 Displayed for simplicity for the case of a translational invariant state.
7.2. Wigner or mixed coordinates 181

not be obtained, and one gets complicated equations for an in¬nite hierarchy of the
correlation functions. If one, preferably by some controlled approximation, can break
the hierarchy, usually at most at the two-particle correlation level, one obtains quan-
tum kinetic equations, i.e. equations having the form of kinetic equations, but which
contain quantum features which are not included in the classical Boltzmann equation
[22]. One of the earliest applications of the non-equilibrium Green™s function tech-
nique was to derive such kinetic equation [10] [14], though owing to their complicated
structure they leave in general little progress in their solution by analytical means.
However, as we shall see, combined with the diagrammatic estimation technique,
the enterprise has the virtue of giving access to quantitative criteria for the validity
of the so prevalently used Boltzmann equation, and thus not just the unquanti¬ed
statement of lowest-order perturbation theory.
We now embark on the manipulations leading to a form of the quantum kinetic
equation resembling classical kinetic equations. This is done by introducing Wigner

7.2 Wigner or mixed coordinates
To derive quantum kinetic equations resembling the form of classical kinetic equa-
tions, we introduce the mixed or Wigner coordinates
x1 + x1
r = x1 ’ x1
R= , (7.10)
and time variables4
t1 + t1
, t = t 1 ’ t1
T= (7.11)
in order to separate the variables, (r, t), describing the microscopic properties, gov-
erned by the characteristics of the system, from the variables, (R, T ), describing the
macroscopic properties, governed by the non-equilibrium features of the state under
consideration, say as a result of the presence of an applied potential. To implement
this separation of variables, we Fourier transform all functions with respect to the
relative coordinates, say for a Green™s function

G(X, p) ≡ dx e’ipx G(X + x/2, X ’ x/2) (7.12)

where the abbreviated notation has been introduced

X = (T, R) , x = (t, r) (7.13)

xp = ’Et + p · r .
p = (E, p) , (7.14)
We then express the current and density in terms of the mixed variables. The
average charge density, Eq. (3.54), becomes (the factor of two is from the spin of the
4 No danger of confusion with the notation for the temperature should occur.
182 7. Quantum kinetic equations

particles, say electrons)

ρ(R, T ) = ’2ie dE G< (E, p, R, T ) (7.15)
(2π)3 ’∞

and the average electric current density in the presence of a vector potential A,
Eq. (3.57), becomes in terms of the mixed variables

e dp
j(R, T ) = ’ dE (p ’ eA(R, T )) G< (E, p, R, T ) . (7.16)
m ’∞

1K i
G< = G+A (7.17)
2 2
the current and density can also be expressed in terms of the kinetic Green™s function

e dp
j(R, T ) = ’ dE (p ’ eA(R, T )) GK (E, p, R, T ) (7.18)
m ’∞

and for the density (up to a state independent constant)

ρ(R, T ) = ’2ie dE GK (E, p, R, T ) . (7.19)
(2π)3 ’∞

Exercise 7.1. Show that, for an arbitrary non-equilibrium state, retarded and ad-
vanced Green™s functions in the mixed coordinates are related according to

(GR (R, T, p, E))— = GA (R, T, p, E) . (7.20)

As a consequence, the spectral function in the mixed coordinates is a real function,

(ΣR (R, T, p, E))— = ΣA (R, T, p, E) . (7.21)

Note that in the Wigner coordinates, the spectral function is twice the imaginary
part of the advanced Green™s function.
Exercise 7.2. Show for an arbitrary non-equilibrium state the spectral representa-
tion in the mixed coordinates, the Kramers“Kronig relations,

dE GR (X, p ) ’ GA (X, p )
G (X, p) =
’∞’2πi E ’ E (’) i0

dE A(X, p )
p ≡ (p, E ) .
= , (7.22)
’∞ 2π E ’ E (’) i0
7.2. Wigner or mixed coordinates 183

We now show that a convolution C = A — B in the mixed coordinates is given by
’‚p ‚X )
(A — B)(X, p) = e 2 (‚X ‚p A(X, p) B(X, p) , (7.23)

‚X = (’‚T , ∇R ) , ‚p = (’‚E , ∇p )
‚A ‚B ‚A ‚B
≡’ ·
‚X ‚p + (7.25)
‚T ‚E ‚R ‚p
and the upper index refers to the function operated on. Let us here for clarity
distinguish quantities in the mixed coordinates by a tilde
C(X, x) ≡ C(X + x/2, X ’ x/2) = C(x1 , x1 ) . (7.26)

Consider the convolution

C(x1 , x1 ) ≡ dx2 A(x1 , x2 ) B(x2 , x1 ) , (7.27)

which in mixed coordinates becomes

C(X, x) ≡ dx2 A(X + x/2, x2 ) B(x2 , X ’ x/2)

˜ (X + x/2 + x2 ), X + x/2 ’ x2
= dx2 A

˜ (x2 + X ’ x/2), x2 ’ (X ’ x/2)
B . (7.28)
Making the shift of variable

x2 ’ x2 ’ (X ’ x/2) (7.29)

eliminates the X-dependence in the variable at the relative coordinate place, giving

˜ ˜ ˜
dx2 A(X + x2 /2, x ’ x2 ) B(X ’ x/2 + x2 /2, x2 ) .
C(X, x) = (7.30)

In the mixed coordinates we have

dx e’ixp ˜ ˜
dx2 A(X + x2 /2, x ’ x2 ) B(X ’ x/2 + x2 /2, x2 )
C(X, p) =

dp ’ip (x’x2 )
dx e’ixp
= dx2 e A(X + x2 /2, p )

— B(X ’ x/2 + x2 /2, p ) ,
184 7. Quantum kinetic equations

where in the last equality the integrand has been expressed in the mixed coordinates.
Performing a Taylor expansion and partial integrations then leads to Eq. (7.23).
In particular, for the case of interest of slowly varying perturbations, which cor-
responds to the lowest-order Taylor expansion, the convolution becomes
(A — B)(X, p) = A(X, p) B(X, p) + (‚X A(X, p)) ‚p B(X, p)

’ (‚p A(X, p)) ‚X B(X, p) . (7.32)
In the mixed coordinates, the operator part of the inverse Green™s function, G’1
of Eq. (3.68), becomes a simple multiplicative factor

G’1 (E, p, R, T ) = E ’ ξp ’ V (R, T ) , (7.33)

where V (R, T ) is an applied potential, and ξp = p ’ μ is the single-particle energy
measured from the chemical potential, and for quadratic dispersion, such as the case
for the free electron model, p = p2 /2m.

7.3 Gradient approximation
To make progress towards an intelligible and tractable equation, one assumes that the
spatial and temporal inhomogeneity is weak, inducing only slow variations in Green™s
functions and self-energies.5 In the following we assume the non-equilibrium state is
induced by an applied potential, V (R, T ), which is a slowly varying function of its
variables compared to the characteristic scales of equilibrium Green™s functions and
self-energies.6 We shall, for example, have a degenerate Fermi system in mind, say
conduction electrons in a metal, where the characteristic scales are the Fermi energy
and momentum. This allows for the approximation where only lowest-order terms
in the variation is kept, the so-called gradient approximation. In this approximation
we thus have
[A — B]+ ’ 2A(X, p) B(X, p)
, (7.34)
’i[A — B]’ ’ [A, B]p ,
, (7.35)

= ‚X A ‚p B ’ ‚p A ‚X B
[A, B]p

‚E ‚T ’ ‚T ‚E ’ ∇A · ∇B + ∇A · ∇B A(X, p) B(X, p) ,
= p R R p

5 If
one is interested only in the linear response, such an assumption is not needed, but the gradient
approximation allows, in principle, inclusion of all the nonlinear e¬ects of a slightly inhomogeneous
6 The coupling to a vector potential will be handled in Section 7.6.
7.3. Gradient approximation 185

and the subscript p on the bracket signi¬es its resemblance to the Poisson bracket of
classical mechanics.
In the gradient approximation, the quantum kinetic equation, Eq. (7.3), becomes

[G’1 ’ eΣ, GK ]p ’ [ΣK — eG]p = iΣK A ’ i“ GK .
, (7.37)

The ¬rst term on the left-hand side becomes, in the gradient approximation,

[G’1 — GK ]’ ’ [G’1 — GK ]p
, ,
0 0

‚T GK (E, p, R, T ) + ‚E GK (E, p, R, T ) ‚T V (R, T )

∇R GK (E, p, R, T ) · ∇p ξp ’ ∇p GK (E, p, R, T ) · ∇R V (R, T ).

In fact, the ¬rst term is always exact, and so is the third term for the case of quadratic
dispersion.7 We note that they are identical in form to the driving terms in the
Boltzmann equation, whereas the last term on the right, which also appears in the
Boltzmann equation, here is valid only in the gradient approximation, i.e. the mag-
nitude of the characteristic wave vector of the potential, q, is small compared with
the characteristic wave vector of the system, which in the case of degenerate fermions
is the Fermi wave vector, q < kF (usually no restriction at all for transport situations
in degenerate Fermi systems). The second term on the right looks strange in the
Boltzmann context, but we shall soon integrate the equation over E, upon which
this term disappears.
Since in equilibrium a Poisson bracket vanishes, the kinetic equation reduces to

0 = ΣK (E, p) A(E, p) ’ “(E, p) GK (E, p) (7.39)

and this identity can be interpreted as the statement of determining the equilibrium
distribution function as the one for which the right-hand side, the collision integral,

7.3.1 Spectral weight function
To make further progress we study the spectral weight function. The equation of
motion for the spectral weight function is obtained by subtracting the diagonal com-
ponents of Eq. (7.1), giving

[G’1 ’ eΣ — A]’ ’ [“ —
, , eG]’ = 0 . (7.40)

In the gradient approximation, the non-equilibrium spectral function satis¬es
(according to Eq. (7.40)) the equation

[E ’ ξp ’ V (R, T ) ’ eΣR , A]p + [ eGR , “]p = 0 . (7.41)
7 The ¬rst term is not dependent on the gradient approximation, but as usual is exact, simply
owing to the equation being ¬rst order in time, and similarly for the second term for the case of
quadratic dispersion.
186 7. Quantum kinetic equations

We note that
“(E, p, R, T )
A(E, p, R, T ) = (7.42)
2 2
“(E,p,R,T )
E ’ ξp ’ V (R, T ) ’ eΣR (E, p, R, T ) + 2

solves Eq. (7.41) since, because [A, B]p = ’[B, A]p , and noting that
= E ’ ξp ’ V (R, T ) ’
e GR (E, p, R, T ) eΣR (E, p, R, T ) , (7.43)

the left-hand side of equation Eq. (7.41) can then be rewritten in the form
i i
R ’1 R ’1
’i ’ “, ’“
eG eG
2 2

i i
R ’1 R ’1
+i eG + “, eG +“ , (7.44)
2 2

which vanishes, since for any function F , we have [A, F (A)]p = 0. In the far past,
where the system is assumed undisturbed, i.e. V vanishes, the presented solution,
Eq. (7.42), reduces to the equilibrium spectral function

“(E, p)
A(E, p) = , (7.45)
E ’ ξp ’ eΣ(E, p) + (“(E, p)/2)

which in this case can be obtained directly from Eq. (7.40). The solution Eq. (7.42)
is therefore the sought solution since it satis¬es the correct initial condition.
Adding the left and right Dyson equations for the retarded non-equilibrium Green™s
function, and performing the expansion within the gradient approximation, Eq. (7.34),
we similarly obtain the result
GR (E, p, R, T ) =
G’1 (E, p, R, T ) ’ ΣR (E, p, R, T )

= , (7.46)
E ’ ξp ’ V (R, T ) ’ ΣR (E, p, R, T )
and similarly for the advanced Green™s function.

7.3.2 Quasi-particle approximation
If the interaction is weak the self-energies are small, and the spectral weight function
is a peaked function in the variable E, in fact in the absence of interactions according
to Eq. (7.42)
A(E, p, R, T ) = 2π δ(E ’ ξp ’ V (R, T )) (7.47)
7.3. Gradient approximation 187

and therefore is GK also a peaked function in the variable E. We ¬rst consider this
so-called quasi-particle approximation.8 In Section 7.5 we will consider the case of
strong electron“phonon interaction and the spectral weight can not be approximated
by a delta function, and a di¬erent approach to obtaining a kinetic equation must
be developed.
The reason for subtracting the left and right Dyson equations is that the term
linear in E in G’1 then disappears, thereby, in view of Eq. (7.47), allowing the
equation, Eq. (7.37), to be integrated with respect to this variable giving

∇p ξp · ∇R ’ ∇R V (R, T ) · ∇p ) h(p, R, T )
(‚T +

ΣK (E = ξp + V (R, T ), p, R, T )

’ “(E = ξp + V (R, T ), p, R, T ) h(p, R, T ) , (7.48)

where we have introduced the distribution function

dE K
h(p, R, T ) = ’ G (E, p, R, T ) . (7.49)
’∞ 2πi

The two self-energy terms on the left in Eq. (7.37) must be neglected in this ap-
proximation since they are by assumption small and in addition multiplied by the
characteristic frequency, ω0 , of the external potential which is small compared with
the characteristic frequency of the system, which in the case of degenerate fermions
is the Fermi energy, ω0 F . In the event that the left“right subtracted Dyson equa-
tion allows for integrating over E, equal time quantities appear, and the distribution
function is of the Wigner type, and is related similarly to densities and currents.9
In equilibrium the distribution function is for fermions given by
h0 (p) = tanh (7.50)
in which case the sum of the two terms on the right in Eq. (7.48) vanish. We shall
now focus on the terms on the right-hand side of equation Eq. (7.48), and realize
they describe collisions and dissipative e¬ects.
Since the equation for the Green™s function is not closed we will eventually have
to make an approximation that cuts o¬ the hierarchy of correlations. For states
not too far from equilibrium, this can be done at the level of self-energies if, for
example, vertex corrections can be shown to be small in some parameter, viz. the one
characterizing the equilibrium approximation. To this end we recall the usefulness
of the diagrammatic estimation technique.
8 This is of course a most unfortunate choice of labeling used in the literature. The physical
implication of the approximation simply being that in between collisions, the particle motion is that
of a free particle.
9 For a discussion of the Wigner function see chapter 4 of reference [1].
188 7. Quantum kinetic equations

7.4 Impurity scattering
We now start to consider interactions of relevance, and begin with the simplest
case; that of impurity scattering. In the clean limit where impurity scattering say of
electrons in a metal or semiconductor is weak, so that any tendency to localization in
,10 diagrams with crossing
a three-dimensional sample can be neglected, i.e. F „
of impurity lines can be neglected, and the impurity self-energy is11

≡ pE pE
Σ(E, p, R, T ) (7.51)

corresponding to the analytical expression for the real-time matrix self-energy
|Vimp (p ’ p )|2 G(p , E, R, T ) .
Σ(p, E, R, T ) = ni (7.52)
(2π )3
For the kinetic component of the self-energy we have
|Vimp (p ’ p )|2 GK (p , E, R, T )
ΣK (p, E, R, T ) = ni (7.53)

“(p, E, R, T ) = i(ΣR (p, E, R, T ) ’ ΣA (p, E, R, T ))

|Vimp (p ’ p )|2 A(p , E, R, T ) .
= ni (7.54)
Since we work to lowest order in the impurity concentration, ni , the spectral weight
should be replaced by the delta function expression, and we obtain

(‚T + ∇p ξp · ∇R ’ ∇R V (R, T ) · ∇p ) h(p, R, T ) = I (1) [f ] (7.55)

where the right side, the electron-impurity collision integral, is
I (1) [f ] = ’2πni |Vimp (p’p )|2 δ(ξp ’ξp )(h(p, R, T )’h(p , R, T )) . (7.56)
We have arrived at the classical kinetic equation describing the motion of a particle
in a weakly disordered system, the Boltzmann equation for a particle in a random
10 In a strictly one-dimensional sample localization is typically dominant and in a two-dimensional
sample it is important at low enough temperatures. The ¬rst quantum correction to this classical
limit, the weak localization e¬ect, is discussed in Chapter 11.
11 For a detailed description of the standard impurity average Green™s function technique and

diagrammatic estimation, we refer the reader to reference [1], where also inclusion of multiple
impurity scattering is shown to be equivalent to the considered Born approximation by inclusion of
the t-matrix.
7.4. Impurity scattering 189

potential. The derived equation is called a kinetic equation because the collision
integral is not a functional in time (or space), i.e. local in both the space and time
variables, and a functional only with respect to the momentum variable. The only
di¬erence signaling we are considering the degenerate electron gas is the quantum
statistics, which dictates the distribution function to respect the Pauli principle, i.e
the equilibrium distribution is speci¬ed by Eq. (7.50).
The weak-disorder kinetic equation for a particle in a random potential is of course
immediately obtained from classical mechanics, granted a stochastic treatment of the
impurity scattering, giving the collision integral
It [f ] = ’ {W (p , p)f (p, t) ’ W (p, p )f (p , t)} , (7.57)

where W (p , p) is the classical transition rate between momentum states, the classical
scattering cross section. In classical mechanics the distribution function concept is
unproblematic because we can simultaneously specify position and momentum, and
the terms on the left-hand side of Eq. (7.55) are simply the streaming terms in phase
space for the situation in question.
In the quantum case we have, in the Born approximation for the transition rate
between momentum states,
|Vimp (p ’ p )|2 δ( ’
W (p , p) = p)

ni V | p|Vimp (ˆ )|p |2 δ( ’
= p) . (7.58)
x p

We note that in the Born approximation we always have W (p , p) = W (p, p ).12
We note that the expression W (p , p) in Eq. (7.58) is Fermi™s Golden Rule ex-
pression for the transition probability per unit time from momentum state p to
momentum state p (or vice versa) caused by the scattering o¬ an impurity, times
the number of impurities. The two terms in the collision integral thus have a sim-
ple interpretation because they describe the scattering in and out of a momentum
state. For example, the ¬rst term in the collision integral of the Boltzmann equation,
Eq. (7.56), is a loss term, and gives the rate of change of occupation of a phase space
volume due to the scattering of an electron from momentum p to momentum p by
the random potential. The probability per unit time of being scattered out of the
phase space volume around p, and into a volume around p , is the product of three
probabilities: (the probability that an electron is in that phase space volume to be
available for scattering) — (the transition probability per unit time for the transition
from state p to p ) — (the probability that there is an impurity in the space volume
to scatter). Similarly we have the interpretation of the other term as a scattering-in
The obtained equation is a quasi-classical equation because, in between collisions
with impurities, the electrons move along straight lines just as in classical mechan-
12 In general, potential scattering is time-reversal invariant, and we always have W (p , p) =
W (’p, ’p ). If, in addition, the potential is invariant with respect to space inversion, we have
W (p , p) = W (’p , ’p), and thereby W (p , p) = W (p, p ).
190 7. Quantum kinetic equations

ics, but the scattering cross section is the quantum mechanical one.13 Besides the
inherent quantum statistics, this is the only quantum feature surviving in the weak
disorder limit, / F „ 1, where „ is the characteristic time scale for the dynamics,
the momentum relaxation time, soon to be discussed. The presented diagrammatic
method for deriving transport equations is capable of going beyond the Markov pro-
cess described by the classical kinetic equation, to include quantum e¬ects. One
can construct a kinetic equation determining the ¬rst quantum correction, the weak
localization e¬ect, but it is easier to employ linear response theory as described in
Chapter 11.
Let us study the simplest non-equilibrium situation where the distribution is out
of momentum equilibrium for only a single momentum state on the Fermi surface

fp (t) = f0 ( ) + δfp (t) δp,p (7.59)

and we assume no external ¬elds. The Boltzmann equation then reduces to

‚δfp (t) δfp
=’ (7.60)
‚t „p

whose solution describes the exponential relaxation to equilibrium

fp (t) = f0 ( p ) + δfp (t = 0) e’t/„p (7.61)

and the momentum relaxation time (which for the considered isotropic Fermi surface
does not depend on the direction of the momentum)
= Wp ,p (7.62)

p (=p)

is seen to be identical to the imaginary part of the retarded self-energy for E = F

1 1 dp
|Vimp (pF ’ p )|2 δ( ’
= = 2πni F) . (7.63)
„ „ ( F) (2π)

We noted above that the collision integral rendered the kinetic equation a stochas-
tic equation for the momentum, Pauli™s master equation. In the case where „ (p) can
be considered independent of the momentum p, „ is the phenomenological parameter
of the Drude theory of conduction, and ”t/„ (p) is, according to Eq. (7.61), the prob-
ability that an electron with momentum p in the time span ”t will su¬er a collision
with total loss of momentum direction memory. Such an assumption is not valid in
the quantum mechanical description as the scattering of a wave sets up correlations
that can not lead to a total memory loss in general, as we shall discuss in detail in
Chapter 11.
One might miss Pauli blocking factors in the expression for the collision integral,
Eq. (7.56), but they need not, as just shown, appear in the considered case of potential
13 If we go beyond the considered Born approximation and include multiple scattering, we en-
counter the exact cross section for scattering o¬ an impurity as expressed by the t-matrix. For a
discussion see chapter 3 in reference [1].
7.4. Impurity scattering 191

scattering. If one uses the Kadano¬“Baym form of the kinetic equation, Eq. (5.136),
Pauli blocking factors would then appear in intermediate results. Another lesson to
learn is that the form of the appearance of the quantum statistics, here the Fermi“
Dirac distribution function or other forms, depends on the type of Green™s functions
one employs; a case in question is our choice leading to the distribution function in
Eq. (7.49) and Eq. (7.50).
For the sole purpose of obtaining the weak-disorder kinetic equation, the use of
quantum ¬eld theoretic methods and Feynman diagrams is hardly necessary. How-
ever, it allows us in a simple way to assess the validity criterion for the classical kinetic
description, and to go beyond the classical limit and study quantum corrections. In
view of the neglected diagrams, the validity of the Boltzmann equation requires
/l, where l = vF „ is the mean free path.14 In
/ F„ 1, or equivalently pF
addition for the gradient approximation to be valid, the characteristic frequency and
wave vector of the perturbation must satisfy the weak restrictions ω < F , q < kF .
There can be some satisfaction in deriving the Boltzmann equation, in particular to
establish validity criteria, i.e. to establish the Landau criterion and not instead the
devastating for applications Peierls criterion, ω < kT , which an argument based
on a simple quasi-particle picture would suggest. But for the sake of deriving clas-
sical kinetic equations, the venture into quantum ¬eld theory is over-kill. The more
so, that in practice it is di¬cult to go beyond the linear regime systematically and
study nonlinear e¬ects. However, there exists a successful technique that leads to an
exception to this state of a¬airs, viz. the so-called quasi-classical Green™s function
technique. We consider this technique applied in the normal state in Section 7.5, and
its even more important application to superconductivity will be studied in Chapter

Exercise 7.3. Show that the continuity equation is obtained by integrating the
kinetic equation, Eq. (7.55), with respect to the momentum variable.

For a discussion of the classical Boltzmann transport coe¬cients for a degenerate
Fermi system, electrical and thermal conductivities, we refer the reader to chapter 5 of
reference [1]. Here we just note that, for the case of a time-independent electric ¬eld,
the solution to the Boltzmann equation, Eq. (7.56), to linear order is immediately
obtained giving for the conductivity, σ0 , the Boltzmann result
ne2 „tr
σ0 = (7.64)
where „tr ≡ „tr ( F ) is the transport relaxation time in the Born approximation
dˆ F
|Vimp (pF ’ pF )|2 (1 ’ pF · pF ) .
= 2πni N0 (7.65)
„tr ( F ) 4π
The appearance of the transport time expresses the simple fact that small angle
scattering is not e¬ective in degrading the current. For isotropic scattering the mo-
mentum and transport relaxation times are identical, as each scattering direction is
14 This so-called Landau criterion is not su¬cient for the applicability of the Boltzmann equation
in low-dimensional systems, d ¤ 2. This is a subject we shall discuss in detail in Chapter 11.
192 7. Quantum kinetic equations

weighted equally. The transport relaxation time is the characteristic time a particle
can travel before the direction of its velocity is randomized.

Exercise 7.4. Show that the retarded impurity self-energy, Eq. (7.51), in equilibrium
and for |E ’ F | F and |p ’ pF | pF just becomes the constant

ΣR (E, p) = ’i (7.66)
dˆ F
|Vimp (pF ’ pF )|2 .
= 2πni N0 (7.67)
„ 4π

For later use, we end this section on dynamics due to impurity scattering by
considering Boltzmannian motion and its large scale features, Brownian motion.

7.4.1 Boltzmannian motion in a random potential
In later chapters we shall discuss quantum corrections to classical transport. How-
ever, in many cases we often still need to know only the classical kinetics of the
particle motion. We therefore take this opportunity to discuss the Boltzmannian
motion of a particle scattered by impurities, although we shall not need these results
before we discuss destruction of phase coherence due to electron“phonon interaction
in Chapter 11. The Boltzmann theory is a stochastic description of the classical mo-
tion of a particle in a weakly disordered potential. At each instant the particle has
attributed a probability for a certain position and velocity (or momentum). In the
absence of external ¬elds the Boltzmann equation for a particle in a random potential
has the form
‚f (x, p, t) ‚f (x, p, t) dp
+ v· W (p, p ) [f (x, p, t)’f (x, p , t)] , (7.68)
(2π )3
‚t ‚x

where we have introduced the notation v = vp = p/m for the particle velocity.
The Boltzmann equation is ¬rst order in time (the state of a particle is completely
determined in classical mechanics by specifying its position and momentum), and
the solution for such a Markov process can be expressed in terms of the conditional
probability F for the particle to have position x and momentum p at time t given it
had position x and momentum p at time t

f (x, p, t) = dx F (x, p, t; x , p , t ) f (x , p , t ) . (7.69)

For elastic scattering only the direction of momentum can change, and consequently
we need only integrate over the direction of the momentum. In the absence of ex-
ternal ¬elds the motion in between scattering events is along straight lines, and the
conditional probability describes how the particle by impurity scattering, is thrown
between di¬erent straight-line segments, i.e. a Boltzmannian path.
7.4. Impurity scattering 193

We de¬ne the Boltzmann propagator as the conditional probability for the initial
condition that it vanishes for times t < t , the retarded Green™s function for the
Boltzmann equation. The equation obeyed by the Boltzmann propagator is thus,
assuming isotropic scattering,

‚ ‚ 1 1 dˆ
+ vp · F (p, x, t; p , x , t ) ’
+ F (p, x, t; p , x , t )
‚t ‚x „ „ 4π

ˆp ˆ
δ(ˆ ’ p ) δ(x ’ x ) δ(t ’ t ) ,
= (7.70)

where δ is the spherical delta function

dˆ ˆ
δ(ˆ ’ p ) f (p ) = f (p) .
pˆ (7.71)

The equation for the Boltzmann propagator is solved by Fourier transformation, and
we obtain
dq dω iq·(x’x )’iω(t’t )
F (p, x, t; p , x , t ) = e F (p, p ; q, ω) , (7.72)

1 1/„ ˆp ˆ
I(q, ω) + δ(ˆ ’ p )
F (p, p ; q, ω) =
’iω + p · q/m + 1/„ ’iω + p · q/m + 1/„
I(q, ω) = , (7.74)
ql ’ arctan ql/(1 ’ iω„ )
where l = v„ is the mean free path.
We note, by direct integration, the property

F (x, p, t; x , p , t ) = dx F (x, p, t; x , p , t ) F (x , p , t ; x , p , t )

the signature of a Markov process. This property will be utilized in Section 11.3.1
in the calculation of the dephasing rate in weak localization due to electron“phonon

7.4.2 Brownian motion
If we are interested only in the long-time and large-distance behavior of the particle
motion, |x ’ x | l, t ’ t „ , the wave vectors and frequencies of importance in
15 For a Markov process, the future is independent of the past when the present is known, i.e.
the causality principle of classical physics in the context of a stochastic dynamic system, here the
process in question is Boltzmannian motion.
194 7. Quantum kinetic equations

the Boltzmann propagator, Eq. (7.73), satisfy ql, ω„ 1, and we obtain the di¬usion
I(q, ω) , (7.76)
’iω + D0 q 2
where D0 = vl/3 is the di¬usion constant in the considered case of three dimensions
(and isotropic scattering). By Fourier transforming we ¬nd that, in the di¬usion
approximation, the dependence on the magnitude of the momentum (velocity) in the
momentum directional averaged Boltzmann propagator appears only through the
di¬usion constant, t > t ,

dqdω eiq·(x’x )’iω(t’t )
D(x, t; x, t ) ≡ F (p, x, t; p , x , t ) =
’iω + D0 q 2
(4π)2 (2π)4

e’(x’x ) /4D0 (t’t )

= . (7.77)
(4πD0 (t ’ t ))d/2
This di¬usion propagator describes the di¬usive or Brownian motion of the particle,
the conditional probability for the particle to di¬use from point x to x in time span
t ’ t , described by the one parameter, the di¬usion constant. The absence of the
explicit appearance of the magnitude of the velocity re¬‚ects the fact that the local
velocity is a meaningless quantity in Brownian motion.

Exercise 7.5. Show that
≡ dx x2 D(x, t; x , t ) = x + 2dD0 (t ’ t ) ,
x2 (7.78)
t,x ,t

where the d on the right-hand side is the spatial dimension.

If we are interested only in the long-time and large-distance behavior of the Boltz-
mannian motion we can, as noted above, get a simpli¬ed description of the classical
motion of a particle in a random potential. We are thus not interested in the zigzag
Boltzmannian trajectories, but only in the smooth large-scale behavior. It is instruc-
tive to relate the large-scale behavior to the velocity (or momentum) moments of the
distribution function, and the corresponding physical quantities, density and current
density. Expanding the distribution function on spherical harmonics

f (x, p, t) = f0 ( p , x, t) + p · f ( p , x, t) + · · · (7.79)

we have the particle current density given in terms of the ¬rst moment
1 dp 1 dp
p p · f ( p , x, t) = p2 f ( p , x, t)
j(x, t) = (7.80)
(2π )3 3
m 3m (2π )
and the density given in terms of the zeroth moment
n(x, t) = f0 ( p , x, t) . (7.81)
(2π )3
7.4. Impurity scattering 195

Taking the spherical average


... ... (7.82)

of the force-free Boltzmann equation, Eq. (7.68), we obtain the zeroth moment equa-
‚f0 ( p , x, t)
∇x · f ( p , x, t) = 0 .
+ (7.83)
‚t 3m
Integrating this equation with respect to momentum gives the continuity equation

‚n(x, t)
+ ∇x · j(x, t) = 0 . (7.84)
This result is of course independent of whether external ¬elds are present or not.
This is seen directly from the Boltzmann equation by integrating with respect to
momentum as we have the identity

Ix,p,t [f ] = 0 (7.85)

simply re¬‚ecting that the collision integral respects particle conservation.
Taking the ¬rst moment of the Boltzmann equation, p . . . ,

dˆ ‚f (x, p, t) ‚f (x, p, t)
+ vp · ’ Ix,p,t [f ] =0 (7.86)
4π ‚t ‚x

we obtain the ¬rst moment equation

p2 p2 ‚f0 (x, p, t)
‚ 1
+ f (x, p, t) + = 0, (7.87)
3 ‚t „ ( p ) 3m ‚x

where we have repeatedly used the angular average formulas

dˆ dˆ
p p
p± pβ = δ±β , p± pβ pγ = 0 . (7.88)
4π 3 4π

We have thus reduced the kinetic equation to a closed set of equations relating the
two lowest moments of the distribution function, f0 and f , and we get the equation
satis¬ed by the zeroth moment f0 :

‚ 1 ‚f0 (x, p, t)

+ x f0 (x, p, t) = 0. (7.89)
‚t „ ( p ) ‚t

In a metal the derivatives of the zeroth harmonic of the distribution function for
the conduction electrons, ‚t f0 ( p , x, t) and ”x f0 ( p , x, t), are peaked at the Fermi
energy, and we can use the approximations
dp dp
p2 ”x f0 ( p , x, t) p2 ”x f0 ( p , x, t) (7.90)
3 (2π )3
(2π )
196 7. Quantum kinetic equations

dp ‚ 1 ‚f0 ( p , x, t) ‚ 1 ‚n(x, t)
+ + , (7.91)
(2π )3 ‚t „ ( p ) ‚t ‚t „ ‚t
where as usual „ ≡ „ ( pF ). Assuming only low-frequency oscillations in the density,
ω„ 1,
‚2n 1 ‚n
‚t2 „ ‚t
and we obtain from Eq. (7.89) the continuity equation on di¬usive form

’ D0 n(x, t) = 0 . (7.93)
Since ∇x f0 ( p , x, t) is peaked at the Fermi energy, we can use the approximation
dp dp
p2 ∇x f0 ( p , x, t) ∇x f0 ( p , x, t)
p2 (7.94)
3 (2π )3
(2π )
and assuming only low-frequency current oscillations
‚j(x, t) 1
|j(x, t)| (7.95)
‚t „
we obtain from the ¬rst moment equation, Eq. (7.87), the di¬usion expression for
the current density
‚n(x, t)
j(x, t) = ’D0 . (7.96)
If we assume that the particle is absent prior to time t , at which time the particle
is created at point x , the di¬usion equation, Eq. (7.93), gets a source term, and we
obtain for the conditional probability or di¬usion propagator D(x, t; x , t )

n(x, t) = dx D(x, t; x , t ) n(x , t ) (7.97)

the equation

’ D0 D(x, t; x , t ) = δ(x ’ x ) δ(t ’ t ) (7.98)
with the initial condition
D(x, t; x , t ) = 0 , for t<t . (7.99)
We can solve the equation for the di¬usion propagator, the retarded Green™s function
for the di¬usion equation, by referring to the solution of the free particle Schr¨dinger
Green™s function equation, Eq. (C.24), and letting it ’ t, and /2m ’ D0 , and we
obtain 2 (x’x )

e 4D 0 (t ’t )
D(x, t; x , t ) = θ(t ’ t ) . (7.100)
(4πD0 (t ’ t ))d/2
7.4. Impurity scattering 197

Exercise 7.6. Show that the Di¬uson or di¬usion propagator has the path integral
xt =x xt =x
Dxt e’SE [xt¯] = Dxt e’ d t L E (x t )
¯ ™¯
D(x, t; x , t ) = (7.101)
¯ ¯

xt =x xt =x

where the Euclidean action SE [xt ] is speci¬ed by the Euclidean Lagrangian


LE (xt ) = . (7.102)
The probability density of di¬usive paths is therefore given by
’ t ¯
’SE [xt ] t
PD [xt ] ≡ ¯
e =e . (7.103)
4D 0

Note that the velocity entering the above Wiener measure is not the local velocity
but the velocity averaged over Boltzmannian paths.16
Exercise 7.7. Show that, for a di¬using particle, we have the Gaussian property for
the characteristic function

Dxt PD [xt ] eiq·(x(t)’x(t ))
¯ ¯
= e’D0 q |t’t |
iq·(x(t)’x(t ))
<e >D = . (7.104)
Dxt PD [xt ]
¯ ¯

Exercise 7.8. Consider the Di¬uson or di¬usion propagator speci¬ed by the ladder

p+ p+ p+ p+
E+ p+ E+ p+ E+ p+

+ ···
DE (q, ω) = + +
p’ p’ p’ p’
Ep’ Ep’ Ep’


⎜ E+ p+

u⎜ 1
= +

⎝ A

16 Interms of diagrams the Di¬uson is given by the impurity ladder diagrams; see Exercise 7.8
and chapter 8 of reference [1].
198 7. Quantum kinetic equations


E+ p+ E+ p+

+ · · ·⎟ .
+ (7.105)

Ep’ Ep’

Show that for ql, ω„ 1, E F, the Di¬uson exhibits the di¬usion pole
D(q, ω) ≡ „ u’2 DE (q, ω) = , (7.106)
’iω + D0 q 2
where D0 = vF „ /d is the di¬usion constant in d dimensions.

7.5 Quasi-classical Green™s function technique
When particles interact there can be strong dependence of the self-energy on the
energy variable E, as in the case of electron“phonon interaction in strong coupling
materials, say as in a metal such as lead, which is the type of system we for exam-
ple shall have in mind in this section. The employed quasi-particle approximation
Eq. (7.47) is not valid and the structure in the spectral weight, Eq. (7.45), must be
respected, leaving no chance of simplicity by integrating over the energy variable E,
i.e. of obtaining equations involving only equal-time Green™s functions.
There exists a consistent and self-contained approximation scheme for a degen-
erate Fermi system, valid for a wide range of phenomena, that does not employ the
restrictive quasi-particle approximation. It is called the quasi-classical approxima-
tion.17 The electron“phonon interaction can lead to an important structure in the
self-energy, i.e. in eΣ and “, as a function of the variable E. In contrast, as noted
by Migdal, the momentum dependence is very weak as a consequence of the phonon
energy being small compared with the Fermi energy [25]. The spectral weight func-
tion thus becomes a peaked function of the momentum, and we shall exploit this
peaked character.
The left“right subtraction trick dismissed the strong linear E-dependence in the
inverse propagator G’1 , and similarly its strong momentum dependence, its ξp -
dependence, ξp = p ’ μ. It therefore allows, when there is only weak momentum
dependence of the self-energy, i.e. short-range e¬ective interaction, which is typi-
cally the case for electronic interactions, integration over the variable ξp , so-called
ξ-integration. The peaked character of the spectral weight in the variable ξ will, in
conjunction with multiplying other quantities, restrict their momentum dependence
to the Fermi surface. We shall therefore consider the ξ-integrated Green™s function
17 This scheme was ¬rst applied by Prange and Kadano¬ in their treatment of transport phenomena
in the electron“phonon system [23]. It was later extended to describe transport in super¬‚uid systems
by Eilenberger [24], the topic of the next chapter.
7.5. Quasi-classical Green™s function technique 199

or quasi-classical Green™s function18
g(R, p, t1 , t1 ) = dξ G(R, p, t1 , t1 ) . (7.107)
We note that care should be exercised with respect to ξ-integration, since the in-
tegrand is not well behaved for large values of ξ, falling o¬ only as 1/ξ. The ξ-
integration should be understood in the following sense of deforming the integration
contour as depicted in Figure 7.1: the ξ-integration is split into a low- and high-
energy contribution, and only the low-energy contribution is important in the kinetic
equation since high-energy contributions do not contribute.

= 2 2

Figure 7.1 Splitting in high- and low-energy contributions.

The semicircles are speci¬ed by a cut-o¬ energy Ec , which is chosen much larger than
the Fermi energy. The remaining high-energy contribution to the Green™s function
does not depend on the non-equilibrium state, i.e. it is a constant, and therefore
drops out of the left“right subtracted Dyson equation. We immediately return to
this point again when expressing physical quantities, such as average currents and
densities in terms of the quasi-classical Green™s function, and later in Section 8.3 to
provide a less formal and more physical understanding of the quasi-classical Green™s
Let us ¬rst determine measurable quantities in terms of the quasi-classical Green™s
function, say density and currents, in the presence of an electromagnetic ¬eld (A, •).
The charge density becomes, in terms of the quasi-classical Green™s function,
1 dˆ
ρ(R, T ) = ’ eN0 dE g K (E, p, R, T ) ’ 2e2 N0 •(R, T ) ,
ˆ (7.108)


. 8
( 22)