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