DA (x, t, x , t )

0

D= (5.41)

DR (x, t, x , t ) DK (x, t, x , t )

is real and symmetric, regarded as a matrix in all its arguments, i.e. including its

dynamical indices which at this level amounts to the interchange R ” A. This

symmetric form is the useful representation, the symmetric representation, needed

when functional methods are employed, as discussed in Chapters 9 and 10.

In condensed matter physics a representation in terms of triagonal matrices is

often used, originally introduced by Larkin and Ovchinnikov [12]. To obtain this

triagonal representation, the π/4-rotation in Schwinger“Keldysh space is performed

on the matrix Green™s function in Eq. (5.18)

G = L G L†

ˇ (5.42)

and the triagonal matrix is obtained6

GR GK

G= . (5.43)

GA

0

Not only are these representations economical, they are also appealing from a

physical point of view as GR and GK contain distinctly di¬erent information: the

spectral function has the information about the quantum states of a system, the

energy spectrum, and the kinetic Green™s function, GK , has the information about

5 The alternative not to work with matrices at this stage, but instead base the description on the

G-greater and G-lesser Green™s functions is discussed in detail in Section 5.7. This choice emerges

if one starts from the so-called imaginary-time formalism, as we shall discuss. We shall eventually

abandon the matrices and interpret diagrams directly in terms of the three types of Green™s functions

and two simple rules for their behavior at vertices, the real rules: the RAK-rules.

6 No confusion with the notation for the time-ordered Green™s function should arise.

5.3. Triagonal and symmetric representations 129

the occupation of these states for non-equilibrium situations as discussed in Section

3.4.

The identity in Eq. (5.39) is of the type that guarantees that vacuum diagrams

lead to vanishing contributions.

Exercise 5.1. Consider free phonons in thermal equilibrium at temperature T , and

show that their matrix Green™s function in the triagonal representation

R K

D0 D0

D(0) = (5.44)

A

0 D0

has components that in terms of the momentum and energy variables or equivalently

wave vector and frequency variables are speci¬ed by

’ωk2

(D0 (k, ω))—

R A

D0 (k, ω) = = (5.45)

’ (ω + iδ)2

2

ωk

and

ω

D0 (k, ω) = (DR (k, ω) ’ DA (k, ω)) coth

K

, (5.46)

2kT

where ωk = c |k| is the linear dispersion relation for the longitudinal phonons, c being

the longitudinal sound velocity.

5.3.1 Fermion“boson coupling

Let us consider what happens to the fermion“boson interaction or electron“phonon

interaction dynamical index vertices when transforming to the triagonal matrix rep-

resentation, i.e. let us ¬nd the tensors for the vertices. To obtain the coupling

matrices for the fermion“boson interaction in this representation we transform all

matrix Green™s functions according to

Gij = Lii Gi j L† j

(1) ˇ (1) (5.47)

j

and similarly for the phonon Green™s function, and inserting the identity according

to7

δij = L† Li j (5.48)

ii

the absorption vertex becomes

i

L† j L† k

k

ˇk

γij = Lii γi j = (5.49)

j k

k

j

and the emission vertex becomes

7 From this it immediately follows that the coupling matrix for a scalar ¬eld in the triagonal

representation is the unit matrix in Schwinger“Keldysh space.

130 5. Real-time formalism

i

L† j

˜k ˇk

γij = Lii Lkk γi j = (5.50)

j

k

j

and simple calculation gives for the vertices

1

γij = γij = √ δij

1

˜2 (5.51)

2

and

1 (1)

γij = γij = √ „ij ,

2

˜1 (5.52)

2

(1)

where „ij is the ¬rst Pauli matrix

01

„ (1) = . (5.53)

10

The fermion“boson vertices can be considered basic as two-particle interaction

can also be formulated in terms of them, as discussed in Section 5.3.2. The above

four types of vertices thus represent the additional dressing of vertices needed for

describing non-equilibrium situations. In Section 5.4 we shall describe the physical

signi¬cance of the dynamical index structure of the vertices in the symmetric or

triagonal representations.

The diagrammatic representation is the same irrespective of the matrix represen-

tation used, only the matrices and tensors vary. The diagram displayed in Figure 5.1

represents in the triagonal representation the string of matrices

(1) (0) (0) (0) (0)

Gij (1, 1 ) = ig 2 Gii (1, 3) — γi l Gl l (3, 2) Dkk (3, 2) γlj — Gj j (2, 1 ) ,

k

˜k (5.54)

where straight and wiggly lines represent the free fermion and boson matrix Green™s

functions, or the free electron and free phonon matrix Green™s functions, respec-

tively, in the triagonal representation, and the vertices are speci¬ed in Eq. (5.51) and

Eq. (5.52).

The virtues of the triagonal representation are that the coupling matrix for a

classical ¬eld is the unit matrix in Schwinger“Keldysh space, and both the matrix

Green™s function and matrix self-energies are triagonal matrices, as we show in Section

5.5,

ΣR ΣK

Σ= , (5.55)

ΣA

0

making operative the property that triagonal matrix structure is invariant with re-

spect to matrix multiplication.

5.3. Triagonal and symmetric representations 131

5.3.2 Two-particle interaction

Another important interaction we will encounter is the two-body or two-particle

interaction, say Coulomb electron“electron interaction. The ready-made form for

perturbative expansion of the contour ordered Green™s function becomes, for the

case of two-particle interaction,

†

G(1, 1 ) = Tr ρ0 TC S ψH0 (1) ψH0 (1 ) , (5.56)

where

† †

S = e’i d„1 dx1 d„2 dx2 ψH (x1 ,„1 ) ψH (x2 ,„2 )U(x2 ,„2 ;x1 ,„1 )ψH 0 (x2 ,„2 ) ψH 0 (x1 ,„1 )

C C 0 0

(5.57)

and for an instantaneous interaction

U (x2 , „2 ; x1 , „1 ) = V (x2 , x1 ) δ(„2 ’ „1 ) , (5.58)

where, V (x2 , x1 ) is for example the Coulomb interaction, and the contour delta

function of Eq. (4.45) appears. In the hat-representation, the two-body interaction

will thus get the matrix representation

ˆ

U (x2 , t2 ; x1 , t1 ) = „ (3) δ(t2 ’ t1 ) V (x2 , x1 ) . (5.59)

The basic vertex for two-particle interaction is thus the one depicted in Figure

5.2, where the wiggly line represents the matrix two-particle interaction speci¬ed in

Eq. (5.59).

Figure 5.2 Two-particle interaction vertex.

However, the basic vertex for two-particle interaction can be interpreted as two

separate vertices in terms of the action of the real-time dynamical indices, and can be

formulated identically to the case of electron“boson or electron“phonon interaction.

Although γe’ph , Eq. (5.28), of course is capable of coupling the upper and lower

ˆ

ˆ

branch it is of no importance since such terms vanish since U is diagonal. One is

thus free to choose either of the forms

(3)

γij ∝ δij „jk γij ∝ δij δjk

ˆk ˆk

or (5.60)

the former choice making the separated two-particle or electron“electron interaction

vertices identical in the dynamical indices to the case of fermion“boson or electron“

phonon interaction.

Exercise 5.2. The wavy line in Figure 5.2, representing the two-body interaction,

can be assigned an arbitrary direction, which then in turn can be put to use in ac-

counting for the momentum ¬‚ow in the Feynman diagrams for two-body interactions.

132 5. Real-time formalism

Assuming the interaction in Eq. (5.58) is translational invariant and instantaneous,

its Fourier transform becomes independent of the energy variable, U (q, ω) = V (q).

Show that, for the two-body interaction, the following Feynman rule applies in the

momentum-energy variables. At both vertices in the basic interaction appearing in

diagrams, Figure 5.2, the out-going electron momentum and energy variables equals

the in-coming electron variables plus, for the case of momentum, the amount carried

by the interaction line, counted with a plus or minus sign determined by convention

by the arbitrarily assigned direction of the interaction wavy line. As a result, of

course, the total out-going electron momenta and energies equals the in-coming ones

in Figure 5.2.

Exercise 5.3. Obtain the matrix equations corresponding to the two lowest-order

terms in the electron“electron interaction for the electron matrix Green™s function

corresponding to the diagrams in Figure 5.3.

± +

Figure 5.3 Lowest-order two-particle interaction diagrams.

These correspond to the following self-energies.

± +

Figure 5.4 Lowest-order two-particle interaction self-energy diagrams.

These are the Hartree and Fock terms.8

8 In order for all diagrams to appear with a plus sign it is customary to bury fermionic quantum

statistical minus signs in the Feynman rule: each closed loop of fermi propagators is assigned a

minus sign.

5.4. The real rules: the RAK-rules 133

Exercise 5.4. Apply Wick™s theorem to obtain the result that, to second order in the

electron“phonon interaction, the diagrams for the electron matrix Green™s function

are given by the diagrams corresponding to the ¬rst three self-energy diagrams in

Figure 5.5.

Exercise 5.5. Apply Wick™s theorem to obtain the connected diagrams for the

fermion matrix Green™s function to second order in the two-particle interaction cor-

responding to the self-energy diagrams depicted in Figure 5.5.

+ +

± ±

±

+ +

± ±

Figure 5.5 Second-order two-particle interaction self-energy diagrams.

5.4 The real rules: the RAK-rules

The matrix structure of the contour ordered Green™s function was studied in the

previous sections, and the proper choice of representation, that of Section 5.3, was

governed by the split of information carried by the various matrix components, spec-

tral properties and quantum statistics. The matrix structure of the basic interaction

vertices should also be interpreted and will give rise to e¬cient rules in terms of our

preferred labeling of propagators. Going through the functioning of the dynamical

indices of vertices and the various possibilities for propagator attachments, leads to

the observation that the diagrammatic rules signi¬cant for describing non-equilibrium

states need not be formulated in terms of the individual dynamical or Schwinger“

Keldysh indices of the vertices, but can with pro¬t be formulated in terms of the

labels of the three di¬erent types of propagators entering in the non-equilibrium

134 5. Real-time formalism

description R, A and K. Consider, for example, the basic fermion“boson diagram

depicted in Figure 5.6.

Figure 5.6 Basic fermion“boson diagram.

The boson propagator can be either DR , DA or DK , and the non-equilibrium

diagrammatic rules can now be stated as the following two rules, the real rules.

For the case of DA a change in the dynamical index for the fermion takes place

only at the Absorption vertex and vice versa for the case of DR .

For the case of DK no change in the dynamical fermion index takes place at either

of the vertices.

The e¬ect of the DK component is thus analogous to that of a Gaussian dis-

tributed classical ¬eld with DK as correlator, an observation we shall take advantage

of when discussing the dephasing properties of the electron“electron interaction on

the weak localization e¬ect in Section 11.3.2.

To analyze the dynamical index structure for the propagator given by the dia-

gram in Figure 5.6, we can for example use the fact that the G21 component for

the fermion matrix Green™s function vanishes, i.e. we use the triagonal representa-

tion, and one immediately scans the diagram by in addition using identities such as

GR (1, 1 ) DA (1, 1 ) = 0, and obtains for the corresponding self-energy components

(adapting here the Feynman rule of absorbing the factor ig 2 into the phonon propa-

gator)

1

ΣR (1, 1 ) = DR (1, 1 ) GK (1, 1 ) + DK (1, 1 ) GR (1, 1 ) (5.61)

2

and

1

ΣA (1, 1 ) = DA (1, 1 ) GK (1, 1 ) + DK (1, 1 ) GA (1, 1 ) (5.62)

2

and

1R

ΣK (1, 1 ) = (D (1, 1 ) GR (1, 1 ) + DA (1, 1 ) GA (1, 1 ) + DK (1, 1 ) GK (1, 1 ))

2

1

((GR (1, 1 ) ’ GA (1, 1 ))(DR (1, 1 ) ’ DA (1, 1 )))

=

2

1K

D (1, 1 ) GK (1, 1 ) .

+ (5.63)

2

5.5. Non-equilibrium Dyson equations 135

Equivalent to an external Gaussian distributed classical ¬eld, the DK component

does not sense the quantum statistics of the fermions for the case of retarded and

advanced quantities, but of course carries the information of the quantum statistics

of the bosons. Contrarily, the DR and DA components introduce the GK compo-

nent carrying the information of the quantum statistics of the fermions, the non-

equilibrium distribution of the fermions.

The choice of the arrow on the boson Green™s function in Figure 5.6 is of course ar-

bitrary, the opposite one corresponding to the interchange DR (1, 1 ) ’ (DA (1 , 1))— ,

the complex conjugation being irrelevant for a real boson ¬eld, say for phonons.

We have ¬nally arrived at a convenient and complete physical interpretation of

the dynamical index that re¬‚ects the need for doubling the degrees of freedom to

describe non-equilibrium states.

5.5 Non-equilibrium Dyson equations

The standard topological arguments for partial summation of Feynman diagrams,

as presented in Section 4.5.2, organizes them into one-particle irreducible sub-parts

and two-particle irreducible self-energy skeleton diagrams, and we arrived at the

Dyson equation, Eq. (4.141), where the self-energy is expressed in terms of the full

propagators. When the corresponding equation for contour ordered quantities are

lifted to the real time matrix representation we obtain the matrix Dyson equation

ˆ ˆ ˆ ˆ ˆ

G = G0 + G0 — „ (3) Σ „ (3) — G , (5.64)

where the „ (3) -matrices absorb the minus signs from the return part of the closed

time path, or equivalently

ˇ ˇ ˇ ˇ ˇ

G = G0 + G0 — Σ — G . (5.65)

In the triagonal representation, the three equations in the matrix Dyson equation

G = G0 + G0 — Σ — G (5.66)

take the forms

R(A) R(A)

— ΣR(A) — GR(A)

GR(A) = G0 + G0 (5.67)

and, for the kinetic Green™s function,

GK = GK + GR — ΣR — GK + GR — ΣK — GA + GK — ΣA — GA . (5.68)

0 0 0 0

The matrix self-energy, Σ, can in naive perturbation theory be described as the

sum of diagrams that can not be cut in two by cutting only one internal free prop-

agator line, and is from this point of view a functional of the free matrix Green™s

functions, Σ = Σ[G0 , D0 ]. As discussed in Section 4.5.2, the self-energy can also be

thought of as a functional of the full matrix Green™s function, Σ = Σ[G, D], and is