we obtain for the inelastic di¬erential cross section for neutron scattering o¬ the

substance

d2 σ m2 L6 p 2π (f) (i)

ˆ

| p, ES |a δ(ˆn ’ RN )|p , ES |2

n

= r

3p

dˆ d (2π )

p

N

(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’

i

n (x’x )·(p’p )

= e

(2π )3 p

dˆ d 2π

p

’∞

(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

ˆ

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)

p

2π 2

dˆ d

p

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

state

|ES (») P (ES (»)) ES (»)| ,

ρS = (6.146)

»

where

e’ES (»)/kT

HS |ES (») = ES (») |ES (») .

P (ES (»)) = , (6.147)

ZS

For the transition rate weighted over the thermal mixture of initial states of the

substance we have

(i)

≡

“fp P (ES (»)) “¬

»

∞

d(t ’ t ) ’i(t’t )ω

m2 2

a p 2π

n

= 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 )

(6.148)

and we obtain for the weighted di¬erential cross section (we use the same notation)

d2 σ d2 σ

(i)

= P (ES (»))

dˆ d dˆ d

p p

»

∞

d(t ’ t ) ’i(t’t )ω

m2 a2 p 2π

n

= 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)

i

n

= dx e

(2π )3 p

dˆ d 2π

p

’∞

— n(x, t) n(x , t ) , (6.150)

where the bracket denotes the weighted trace with respect to the state of the sub-

stance

≡

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

n

= V S(q, ω) (6.152)

p (2π )3 2

dˆ d

p

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

q2

1 2πM ’ 2kM q 2 ω’

S(q, ω) = eT . (6.153)

2M

kT q 2

V

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

p

∞

m2 a2 p 2π q2

dt e’iωt e’ 2M (t kT ’i t)

2

=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

p

dˆ d

p

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

)2

mn n

π (1 + 2m 2 k T

M

n

= (6.155)

⎪

⎪

⎪ 4πN

©

2

mn a 2 Mp

1

for 1.

2

2π 2 2m2 kT

mn

(1 + ) n

M

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.

7

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)

0

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].

179

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)

0

2 2

where we have introduced the spectral weight function

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

and2

1R

eG(1, 1 ) ≡ (G (1, 1 ) + GA (1, 1 )) (7.5)

2

and similarly for the self-energies

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

and

1R

eΣ(1, 1 ) ≡ (Σ (1, 1 ) + ΣA (1, 1 )) . (7.7)

2

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

E

GR (E, p) ’ GA (E, p)

GK (E, p) = tanh (7.8)

2kT

and consequently

E

ΣR (E, p) ’ ΣA (E, p)

ΣK (E, p) = tanh . (7.9)

2kT

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

coordinates.

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)

2

and time variables4

t1 + t1

, t = t 1 ’ t1

T= (7.11)

2

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)

and

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)

∞

dp

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)

(2π)3

m ’∞

Since

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)

(2π)3

m ’∞

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

∞

dp

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,

and

(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 )

R(A)

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 AB

i

(A — B)(X, p) = e 2 (‚X ‚p A(X, p) B(X, p) , (7.23)

where

‚X = (’‚T , ∇R ) , ‚p = (’‚E , ∇p )

A A

(7.24)

and

‚A ‚B ‚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)

1

˜ (X + x/2 + x2 ), X + x/2 ’ x2

= dx2 A

2

1

˜ (x2 + X ’ x/2), x2 ’ (X ’ x/2)

B . (7.28)

2

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 )

(2π)4

dp

e’ip

— B(X ’ x/2 + x2 /2, p ) ,

x2

(7.31)

4

(2π)

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

i

(A — B)(X, p) = A(X, p) B(X, p) + (‚X A(X, p)) ‚p B(X, p)

2

i

’ (‚p A(X, p)) ‚X B(X, p) . (7.32)

2

In the mixed coordinates, the operator part of the inverse Green™s function, G’1

0

of Eq. (3.68), becomes a simple multiplicative factor

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

0

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)

and

’i[A — B]’ ’ [A, B]p ,

, (7.35)

where

= ‚X A ‚p B ’ ‚p A ‚X B

A B A B

[A, B]p

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

AB AB

= p R R p

(7.36)

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

perturbation.

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)

0

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 ).

+

(7.38)

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,

vanishes.

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)

0

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

’1

= 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

’1

i i

R ’1 R ’1

’i ’ “, ’“

eG eG

2 2

p

’1

i i

R ’1 R ’1

+i eG + “, eG +“ , (7.44)

2 2

p

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)

2

2

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

1

GR (E, p, R, T ) =

G’1 (E, p, R, T ) ’ ΣR (E, p, R, T )

0

1

= , (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

0

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

ξp

h0 (p) = tanh (7.50)

2kT

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)

p ERT

corresponding to the analytical expression for the real-time matrix self-energy

dp

|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

dp

|Vimp (p ’ p )|2 GK (p , E, R, T )

ΣK (p, E, R, T ) = ni (7.53)

3

(2π)

and

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

dp

|Vimp (p ’ p )|2 A(p , E, R, T ) .

= ni (7.54)

(2π)3

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

dp

I (1) [f ] = ’2πni |Vimp (p’p )|2 δ(ξp ’ξp )(h(p, R, T )’h(p , R, T )) . (7.56)

3

(2π)

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

(1)

It [f ] = ’ {W (p , p)f (p, t) ’ W (p, p )f (p , t)} , (7.57)

p

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,

2πni

|Vimp (p ’ p )|2 δ( ’

W (p , p) = p)

p

V

2π

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

term.

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)

p

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)

1

= 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)

p

3

„ „ ( 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

8.

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)

m

where „tr ≡ „tr ( F ) is the transport relaxation time in the Born approximation

dˆ F

p

|Vimp (pF ’ pF )|2 (1 ’ pF · pF ) .

ˆˆ

= 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)

2„

where

dˆ F

p

|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

dˆ

p

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

4π

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

p

+ 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)

ˆ

where δ is the spherical delta function

dˆ ˆ

p

δ(ˆ ’ p ) f (p ) = f (p) .

pˆ (7.71)

4π

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)

(2π)4

where

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/„

(7.73)

and

ql

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

dˆ

p

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

4π

(7.75)

15

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

interaction.

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

approximation

1/„

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 )

dˆdˆ

pp

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 )

2

= . (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

2

≡ 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

dp

n(x, t) = f0 ( p , x, t) . (7.81)

(2π )3

7.4. Impurity scattering 195

Taking the spherical average

dˆ

p

≡

... ... (7.82)

4π

of the force-free Boltzmann equation, Eq. (7.68), we obtain the zeroth moment equa-

tion

p2

‚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)

‚t

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

dˆ

p

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

4π

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)

p

+ vp · ’ Ix,p,t [f ] =0 (7.86)

p

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

p2

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 :

p2

‚ 1 ‚f0 (x, p, t)

’

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

3m2

‚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)

F

3 (2π )3

(2π )

196 7. Quantum kinetic equations

and

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

(7.92)

‚t2 „ ‚t

and we obtain from Eq. (7.89) the continuity equation on di¬usive form

‚

’ D0 n(x, t) = 0 . (7.93)

x

‚t

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)

F

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)

‚x

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)

x

‚t

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

o

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

representation

xt =x xt =x

t

Dxt e’SE [xt¯] = Dxt e’ d t L E (x t )

¯ ™¯

D(x, t; x , t ) = (7.101)

t

¯ ¯

xt =x xt =x

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

¯

x2

™t

™

LE (xt ) = . (7.102)

4D0

The probability density of di¬usive paths is therefore given by

x2

™¯

’ t ¯

’SE [xt ] t

dt

PD [xt ] ≡ ¯

e =e . (7.103)

4D 0

t

¯

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 |

2

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

diagrams

R R R

p+ p+ p+ p+

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

+ ···

DE (q, ω) = + +

A A A

p’ p’ p’ p’

Ep’ Ep’ Ep’

⎛

R

p+

⎜

⎜ E+ p+

⎜

2⎜

u⎜ 1

= +

⎜

⎜

⎝ A

p’

Ep’

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

⎞

R R

p+

⎟

⎟

E+ p+ E+ p+

⎟

⎟

+ · · ·⎟ .

+ (7.105)

⎟

⎟

⎠

A A

p’

Ep’ Ep’

Show that for ql, ω„ 1, E F, the Di¬uson exhibits the di¬usion pole

1

D(q, ω) ≡ „ u’2 DE (q, ω) = , (7.106)

’iω + D0 q 2

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 -

0

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

i

ˆ

g(R, p, t1 , t1 ) = dξ G(R, p, t1 , t1 ) . (7.107)

π

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.

1

+1

= 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

function.

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

p

ρ(R, T ) = ’ eN0 dE g K (E, p, R, T ) ’ 2e2 N0 •(R, T ) ,

ˆ (7.108)