ńņš. 6 |

and the ļ¬rst order term for the matrix ij-component becomes

ā

Ė (1) Ė (0) Ė (0)

Gij (1, 1 ) = dx2 dt2 Gi1 (1, 2) V (2) G1j (2, 1 )

ā’ā

ā

Ė (0) Ė (0)

ā’ dx2 dt2 Gi2 (1, 2) V (2) G2j (2, 1 ) . (5.10)

ā’ā

Introducing in Schwingerā“Keldysh or dynamical index space the matrix

(3)

Ė

Vij (1) = V (1) Ļ„ij (5.11)

proportional to the third Pauli-matrix

10

Ļ„ (3) = (5.12)

0 ā’1

124 5. Real-time formalism

we have

ā

Ė (1) Ė (0) Ė (0)

Ė

Gij (1, 1 ) = dx2 dt2 Gik (1, 2) Vkk (2) Gk j (2, 1 ) , (5.13)

ā’ā

where summation over repeated Schwingerā“Keldysh or dynamical indices are implied.

Instead of treating individual indexed components, the condensed matrix notation is

applied and the matrix equation becomes

Ė Ė ĖĖ ĖĖ Ė

G(1) = G(0) ā— V G(0) = G(0) V ā— G(0) , (5.14)

where ā— signiļ¬es matrix multiplication in the spatial variable (as well as possible

internal degrees of freedom) and the real time, for the latter integration from minus

to plus inļ¬nity of times. For the components of the free equilibrium matrix Greenā™s

Ė

function, G(0) , we have, according to Section 3.4, explicit expressions.

We introduce a diagrammatic notation for this real-time matrix Greenā™s function

contribution

Ė

G(1) (x1 , t1 ; x1 , t1 ) = (5.15)

x1 t1 x2 t2 x1 t1

The diagram has the same form as the one depicted in Eq. (4.111) for the contour

Greenā™s function, but is now interpreted as an equation for the matrix propagator in

Schwingerā“Keldysh space: each line now represents the free matrix Greenā™s function,

Ė

G(0) , and the cross represents the matrix for the potential coupling, Eq. (5.11). We

get the extra Feynman rule characterizing the non-equilibrium technique: matrix

multiplication over internal dynamical indices is implied.

For the coupling to the scalar potential, the higher-order diagrams are just repe-

titions of the basic ļ¬rst-order diagram, and we can immediately write down the ex-

pression for the matrix propagator for a diagram of arbitrary order. Re-summation of

diagrams to get the Dyson equation, as discussed in Section 4.5, is trivial for coupling

to external classical ļ¬elds, giving

Ė Ė Ė ĖĖ Ė Ė ĖĖ Ė

G = G(0) + G(0) ā— V G G = G(0) + G V ā— G(0) ,

, (5.16)

where the potential can be placed on either side of the convolution symbol.

According to Eq. (3.65), Eq. (3.66) and Eq. (3.67), the free equilibrium matrix

Ė

Greenā™s function, G(0) , satisļ¬es

Gā’1 (1) G0 (1, 1 ) = Ļ„ (3) Ī“(1 ā’ 1 ) ,

Ė (5.17)

0

where Gā’1 (1) is given by Eq. (3.69) for the case of coupling to both a scalar and

0

a vector potential. Since we want the inverse matrix Greenā™s function operating

on the free equilibrium matrix Greenā™s function to produce the unit matrix in all

variables including the dynamical index, it can be accomplished by either of the

objects carrying the third Pauli matrix, Ļ„ (3) . For example, introducing the matrix

representation

Ė Ė

G11 (1, 1 ) G12 (1, 1 )

Ė Ė

G(1, 1 ) ā” Ļ„ (3) G(1, 1 ) = (5.18)

Ė Ė

ā’G21 (1, 1 ) ā’G22 (1, 1 )

5.2. Real-time diagrammatics 125

we then have

Gā’1 (1) G0 (1, 1 ) = 1 Ī“(1 ā’ 1 ) ,

Ė (5.19)

0

where the unit matrix 1 in the dynamical index space will often be denoted by 1 and

often left out when operating on a matrix in the dynamical index space. Introducing

the inverse free matrix Greenā™s function

Gā’1 (1, 1 ) = Gā’1 (1) Ī“(1 ā’ 1 ) 1 (5.20)

0 0

we have

(Gā’1 ā— G0 )(1, 1 ) = 1 Ī“(1 ā’ 1 ) = (G0 ā— Gā’1 )(1, 1 ) .

Ė Ė (5.21)

0 0

We can therefore rewrite the Dyson equations for real-time matrix Greenā™s function

in the forms

((Gā’1 ā’ V ) ā— G)(1, 1 ) = 1 Ī“(1 ā’ 1 )

Ė (5.22)

0

and

(G ā— (Gā’1 ā’ V ))(1, 1 ) = 1 Ī“(1 ā’ 1 ) .

Ė (5.23)

0

We note that in the matrix representation, Eq. (5.18), the coupling to a scalar

potential is a scalar, i.e. proportional to the unit matrix in Schwingerā“Keldysh space

Ė Ė ĖĖ

G(1) = G(0) ā— V G(0) , (5.24)

where

Ė Ė

Vij (1) = V (1) Ī“ij , V (1) = V (1) 1 . (5.25)

The matrix representation introduced in Eq. (5.18) serves the purpose of absorb-

ing the minus signs associated with the return contour into the third Pauli matrix.

5.2.2 Feynman rules for interacting bosons and fermions

For a three-line type vertex, such as in the case of fermionā“boson interaction or

electronā“phonon interaction, more complicated coupling matrices appear in the dy-

namical index or Schwingerā“Keldysh space than for the case of coupling to an external

ļ¬eld. For illustration of the matrix structure in the dynamical index space it suļ¬ces

to consider the generic bosonā“fermion coupling in Eq. (2.71). As noted in Section

2.4.3, this is also equivalent to considering the electronā“phonon interaction in the

jellium model where the electrons couple only to longitudinal compressional charge

conļ¬gurations of the ionic lattice, the longitudinal phonons. Our interest is to dis-

play the dynamical index structure of propagators and vertices; later these can be

sprinkled with whatever additional indices they deserve to be dressed with: species

index, spin, color, ļ¬‚avor, Minkowski, phonon branch, etc.

In the expression for the lowest-order perturbative contribution to the contour

ordered Greenā™s function, Eq. (4.127), we parameterize the two real-time contours

according to Eq. (5.9). In Schwingerā“Keldysh space this term then becomes

Ė (0) Ė k Ė (0) Ė (0) Ė k

Ė (1) Ė (0)

Gij = ig 2 Gii ā— Ī³i l Gl l Dkk Ī³ lj ā— Gj j

Ė (5.26)

126 5. Real-time formalism

or equivalently for the components of the lowest order self-energy matrix components

(0) Ė (0) Ė k

Ė (1) Ėk Ė

Ī£ij = ig 2 Ī³il Gl l Dkk Ī³ lj ,

Ė (5.27)

where the third rank tensors representing the phonon absorption and emission ver-

tices are identical

Ėk

(3)

Ėk

Ī³ij = Ī“ij Ļ„jk = Ī³ ij .

Ė (5.28)

The third rank vertex tensors vanish unless electron and phonon indices are identical,

reļ¬‚ecting the fact that the ļ¬elds in a vertex correspond to the same moment in

contour time. The presence of the imaginary unit in Eq. (5.26) is the result of

one lacking factor of ā’i for our convention of Greenā™s functions: two factors of ā’i

are provided by the interaction and one provided externally in the deļ¬nition of the

Greenā™s functions. Such features are collected in oneā™s own private choice of Feynman

rules.

In the present representation, Eq. (5.1), instead of thinking in terms of the dia-

grammatic matrix representation one can visualize the components diagrammatically,

and we would have diagrams with Greenā™s functions attached to either of the forward

or return parts of the contour. It can be useful once to draw these kind of diagrams,

but eventually we shall develop a form of diagram representation without reference

to the contour but instead to the distinct diļ¬erent physical properties represented

by the retarded and kinetic Greenā™s functions of Section 3.3.2.

The vertices, Eq. (5.28), are diagonal in the fermion, i.e. lower Schwingerā“Keldysh

indices since the two fermi ļ¬eld operators carry the same time variable. The boson

ļ¬eld attached to that vertex has of course the same time variable, but the other

bose ļ¬eld it is paired with can have a time variable residing on either the forward or

backward path, giving the possibilities of Ā±1 as reļ¬‚ected in the matrix elements of

the third Pauli matrix.

The diagrammatic representation of the matrix Greenā™s function, Eq. (5.26), is

displayed in Figure 5.1, where straight and wiggly lines represent fermion and bo-

son matrix Greenā™s functions, or the free electron and free phonon matrix Greenā™s

functions, respectively, and the vertices represent the third rank tensors speciļ¬ed in

Eq. (5.28).

1 3 2 1

Figure 5.1 Diagrammatic representation of the matrix Greenā™s function G for

fermionā“boson interaction.

5.3. Triagonal and symmetric representations 127

In the matrix representation speciļ¬ed by Eq. (5.18), the diagram represents (using

(3) (3)

Ī“ij = Ļ„ik Ļ„kj ),

Ė (0) Ė k Ė (0) Ė (0) Ė k

Ė (1) Ė (0)

Gij = ig 2 Gii ā— Ī³i l Gl l Dkk Ī³ lj ā— Gj j ,

Ė (5.29)

where the absorption vertex is

i

(3) (3) (3)

Ėk Ėk

= Ī³ij = Ī³ii Ļ„i j Ļ„k k = Ī“ij Ļ„jk (5.30)

k

j

and the emission vertex is

i

Ėk (3)

Ėk

= Ī³ ij

Ė = Ī³ ii Ļ„i j

Ė = Ī“ij Ī“jk . (5.31)

k

j

In this representation the absorption and emission vertices thus diļ¬er.

In terms of the lowest order matrix self-energy, Eq. (5.29) becomes

Ė Ė Ė Ė

G(1) = G(0) ā— Ī£(1) ā— G(0) . (5.32)

5.3 Triagonal and symmetric representations

Since only two components of the matrix Greenā™s function, Eq. (5.1), are independent

it can be economical to remove part of this redundancy. In the original article of

Keldysh [10], one component was eliminated by the linear transformation, the Ļ/4-

rotation in Schwingerā“Keldysh space,

G ā’ L G Lā ,

Ė Ė (5.33)

where the orthogonal matrix, Lā = L, is

ā’1

1 1 1

L = ā (1 ā’ iĻ„ (2) ) = ā (5.34)

1 1

2 2

i.e. 1 denotes the 2 Ć— 2 unit matrix and Ļ„ (2) is the second Pauli matrix

ā’i

0

Ļ„ (2) = . (5.35)

i 0

Using the following identities (recall Section 3.3)

Ė Ė Ė Ė

GR (1, 1 ) = G11 (1, 1 ) ā’ G12 (1, 1 ) = G21 (1, 1 ) ā’ G22 (1, 1 ) (5.36)

128 5. Real-time formalism

and

Ė Ė Ė Ė

GA (1, 1 ) = G11 (1, 1 ) ā’ G21 (1, 1 ) = G12 (1, 1 ) ā’ G22 (1, 1 ) (5.37)

and

Ė Ė Ė Ė

GK (1, 1 ) = G21 (1, 1 ) + G12 (1, 1 ) = G11 (1, 1 ) + G22 (1, 1 ) (5.38)

and

Ė Ė Ė Ė

0 = G11 (1, 1 ) ā’ G12 (1, 1 ) ā’ G21 (1, 1 ) + G22 (1, 1 ) (5.39)

the linear transformation, the Ļ/4-rotation in Schwingerā“Keldysh space Eq. (5.33),

amounts to5

Ė Ė GA

G11 G12 0

ā’ (5.40)

Ė Ė GR GK

G21 G22

where the retarded, advanced and the Keldysh or kinetic Greenā™s functions all were

introduced in Section 3.3.2.

For real bosons or phonons, the matrix

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 ā” 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 Ļ/4-rotation in Schwingerā“Keldysh space is performed

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

G = L G Lā

Ė (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 diļ¬erent information: the

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

energy spectrum, and the kinetic Greenā™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 Greenā™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 Greenā™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 Greenā™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 Greenā™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 speciļ¬ed by

ā’Ļk2

(D0 (k, Ļ))ā—

R A

D0 (k, Ļ) = = (5.45)

ā’ (Ļ + iĪ“)2

2

Ļk

and

Ļ

D0 (k, Ļ) = (DR (k, Ļ) ā’ DA (k, Ļ)) coth

K

, (5.46)

2kT

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

the longitudinal sound velocity.

5.3.1 Fermionā“boson coupling

Let us consider what happens to the fermionā“boson interaction or electronā“phonon

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

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

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

matrix Greenā™s functions according to

Gij = Lii Gi j Lā j

(1) Ė (1) (5.47)

j

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

to7

Ī“ij = Lā Li j (5.48)

ii

the absorption vertex becomes

i

Lā j Lā k

k

Ėk

Ī³ij = Lii Ī³i 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 ļ¬eld in the triagonal

representation is the unit matrix in Schwingerā“Keldysh space.

130 5. Real-time formalism

i

Lā j

Ėk Ėk

Ī³ij = Lii Lkk Ī³i j = (5.50)

j

k

j

and simple calculation gives for the vertices

1

Ī³ij = Ī³ij = ā Ī“ij

1

Ė2 (5.51)

2

and

1 (1)

Ī³ij = Ī³ij = ā Ļ„ij ,

2

Ė1 (5.52)

2

(1)

where Ļ„ij is the ļ¬rst Pauli matrix

01

Ļ„ (1) = . (5.53)

10

The fermionā“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

signiļ¬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) ā— Ī³i l Gl l (3, 2) Dkk (3, 2) Ī³lj ā— Gj j (2, 1 ) ,

k

Ėk (5.54)

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

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

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

Eq. (5.52).

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

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

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

5.5,

Ī£R Ī£K

Ī£= , (5.55)

Ī£A

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 electronā“electron interaction. The ready-made form for

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

case of two-particle interaction,

ā

G(1, 1 ) = Tr Ļ0 TC S ĻH0 (1) ĻH0 (1 ) , (5.56)

where

ā ā

S = eā’i dĻ„1 dx1 dĻ„2 dx2 ĻH (x1 ,Ļ„1 ) ĻH (x2 ,Ļ„2 )U(x2 ,Ļ„2 ;x1 ,Ļ„1 )ĻH 0 (x2 ,Ļ„2 ) ĻH 0 (x1 ,Ļ„1 )

C C 0 0

(5.57)

and for an instantaneous interaction

U (x2 , Ļ„2 ; x1 , Ļ„1 ) = V (x2 , x1 ) Ī“(Ļ„2 ā’ Ļ„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

Ė

U (x2 , t2 ; x1 , t1 ) = Ļ„ (3) Ī“(t2 ā’ 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 speciļ¬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 electronā“boson or electronā“phonon interaction.

Although Ī³eā’ph , Eq. (5.28), of course is capable of coupling the upper and lower

Ė

Ė

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)

Ī³ij ā Ī“ij Ļ„jk Ī³ij ā Ī“ij Ī“jk

Ėk Ėk

or (5.60)

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

vertices identical in the dynamical indices to the case of fermionā“boson or electronā“

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 ļ¬‚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, Ļ) = 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 electronā“electron interaction for the electron matrix Greenā™s function

corresponding to the diagrams in Figure 5.3.

Ā± +

Figure 5.3 Lowest-order two-particle interaction diagrams.

These correspond to the following self-energies.

Ā± +

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 Wickā™s theorem to obtain the result that, to second order in the

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

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

Figure 5.5.

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

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

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

+ +

Ā± Ā±

Ā±

+ +

Ā± Ā±

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 Greenā™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 eļ¬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 signiļ¬cant for describing non-equilibrium

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

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

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

134 5. Real-time formalism

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

depicted in Figure 5.6.

Figure 5.6 Basic fermionā“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 eļ¬ect of the DK component is thus analogous to that of a Gaussian dis-

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

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

the weak localization eļ¬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 Greenā™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

Ī£R (1, 1 ) = DR (1, 1 ) GK (1, 1 ) + DK (1, 1 ) GR (1, 1 ) (5.61)

2

and

1

Ī£A (1, 1 ) = DA (1, 1 ) GK (1, 1 ) + DK (1, 1 ) GA (1, 1 ) (5.62)

2

and

1R

Ī£K (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 ) ā’ GA (1, 1 ))(DR (1, 1 ) ā’ 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 ļ¬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 Greenā™s function in Figure 5.6 is of course ar-

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

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

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

the dynamical index that reļ¬‚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

Ė Ė Ė Ė Ė

G = G0 + G0 ā— Ļ„ (3) Ī£ Ļ„ (3) ā— G , (5.64)

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

time path, or equivalently

Ė Ė Ė Ė Ė

G = G0 + G0 ā— Ī£ ā— G . (5.65)

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

G = G0 + G0 ā— Ī£ ā— G (5.66)

take the forms

R(A) R(A)

ā— Ī£R(A) ā— GR(A)

GR(A) = G0 + G0 (5.67)

and, for the kinetic Greenā™s function,

GK = GK + GR ā— Ī£R ā— GK + GR ā— Ī£K ā— GA + GK ā— Ī£A ā— GA . (5.68)

0 0 0 0

The matrix self-energy, Ī£, 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 Greenā™s

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

thought of as a functional of the full matrix Greenā™s function, Ī£ = Ī£[G, D], and is

then the sum of all the skeleton self-energy diagrams, i.e. the diagrams which can not

136 5. Real-time formalism

be cut in two be cutting only two full propagator lines. It is the latter representation

that is useful in the Dyson equation.

Equivalently, by iterating from the left gives the matrix Dyson equation

G = G0 + G ā— Ī£ ā— G0 . (5.69)

For an equilibrium state the two equations are redundant, since time convolutions

by Fourier transformation become simple products for which the order of factors is

irrelevant. However, in a non-equilibrium state, the two matrix equations contain

diļ¬erent information and subtracting them is a useful way of expressing the non-

equilibrium dynamics, and we shall exploit this in Chapters 7 and 8.

Since the transformation of the real-time matrix self-energy is identical to the one

for the matrix Greenā™s function we get, analogously to the equations from Eq. (5.36)

to Eq. (5.39), and therefore for the components of the self-energy matrix,

Ė Ė Ė Ė

Ī£R = Ī£11 ā’ Ī£12 = Ī£21 ā’ Ī£22 (5.70)

Ė Ė Ė Ė

Ī£A = Ī£11 ā’ Ī£21 = Ī£12 ā’ Ī£22 (5.71)

Ė Ė Ė Ė

Ī£K = Ī£11 + Ī£22 = Ī£12 + Ī£21 (5.72)

Ė Ė Ė Ė

0 = Ī£11 ā’ Ī£12 + ā’ Ī£21 + Ī£22 . (5.73)

By construction

Ė

Ī£12 (x1 , t1 , x1 , t1 ) t1 > t1

Ė

Ī£11 (x1 , t1 , x1 , t1 ) = (5.74)

Ė

Ī£21 (x1 , t1 , x1 , t1 ) t1 > t1

and

Ė

Ī£21 (x1 , t1 , x1 , t1 ) t1 > t1

Ė

Ī£22 (x1 , t1 , x1 , t1 ) = (5.75)

Ė

Ī£12 (x1 , t1 , x1 , t1 ) t1 > t1

and the matrix self-energy has in the triagonal representation the same triagonal

form as the matrix Greenā™s function

Ī£R Ī£K

Ī£= . (5.76)

Ī£A

0

Exercise 5.6. Introducing

Ė

Ī£< (x1 , t1 , x1 , t1 ) = Ī£12 (x1 , t1 , x1 , t1 ) (5.77)

and

Ė

Ī£> (x1 , t1 , x1 , t1 ) = Ī£21 (x1 , t1 , x1 , t1 ) (5.78)

show that we have, identically to the relationships for the Greenā™s functions, the

relation for the retarded self-energy

Ī£R (x, t, x , t ) = Īø(t ā’ t ) Ī£> (x, t, x , t ) ā’ Ī£< (x, t, x , t ) (5.79)

and advanced self-energy

Ī£A (x, t, x , t ) = ā’Īø(t ā’ t) Ī£> (x, t, x , t ) ā’ Ī£< (x, t, x , t ) (5.80)

5.5. Non-equilibrium Dyson equations 137

and for the kinetic component

Ī£K (x, t, x , t ) = Ī£> (x, t, x , t ) + Ī£< (x, t, x , t ) . (5.81)

Show that the components of the self-energy matrix satisļ¬es

ā—

Ī£A (x, t, x , t ) = Ī£R (x , t , x, t) (5.82)

and

ā—

Ī£K (x, t, x , t ) = Ī£K (x , t , x, t) . (5.83)

Exercise 5.7. Show that in the case where the matrix Greenā™s function is represented

in symmetric form

GA

0

G= (5.84)

GR GK

the matrix self-energy has the form

Ī£K Ī£R

Ī£= . (5.85)

Ī£A 0

We shall not at present take the diagrammatics beyond the self-energy to higher-

order vertices, since in the following chapters only the Dyson equation is needed. In

Chapter 9 we shall study diagrammatics in their full glory.

From the equation of motion for the free Greenā™s function (or ļ¬elds) we then get,

for the matrix Greenā™s function, the equations of motion

(iā‚t1 ā’ h(1))G(1, 1 ) = Ī“(1 ā’ 1 ) + (Ī£ ā— G)(1, 1 ) (5.86)

and

(iā‚t1 ā’ hā— (1 ))G(1, 1 ) = Ī“(1 ā’ 1 ) + (G ā— Ī£)(1, 1 ) (5.87)

or introducing the inverse free Greenā™s function

Gā’1 (1, 1 ) = (iā‚t1 ā’ h(1)) Ī“(1 ā’ 1 ) (5.88)

0

the two equations can be expressed through operating with the inverse free matrix

Greenā™s function from the left

(Gā’1 ā’ Ī£) ā— G = Ī“(1 ā’ 1 ) (5.89)

0

and from the right

G ā— (Gā’1 ā’ Ī£) = Ī“(1 ā’ 1 ) . (5.90)

0

These two non-equilibrium Dyson equations will prove useful in Chapter 7 where

quantum kinetic equations are considered.

The matrix equation, Eq. (5.89), comprises the three coupled equations for GR,A,K

(Gā’1 ā’ Ī£R(A) ) ā— GR(A) = Ī“(1 ā’ 1 ) (5.91)

0

138 5. Real-time formalism

and

Gā’1 ā— GK = Ī£R ā— GK + Ī£K ā— GA . (5.92)

0

Analogously, from Eq. (5.90), we obtain

GR(A) ā— (Gā’1 ā’ Ī£R(A) ) = Ī“(1 ā’ 1 ) (5.93)

0

and

GK ā— Gā’1 = GR ā— Ī£K + GK ā— Ī£A . (5.94)

0

Exercise 5.8. Show that, subtracting the left and right Dyson equations for GK ,

the resulting equation can be written in the form

1R 1

[Gā’1 ā— GK ]ā’ [Ī£ + Ī£A ā— GK ]ā’ ā’ [Ī£K ā— GR + GA ]ā’

ā’

, , ,

0

2 2

1 1

ā’ [Ī£K ā— (GR ā’ GA )]+ + [(Ī£R ā’ Ī£A ) ā— GK ]+ . (5.95)

= , ,

2 2

If at the end of the day, one makes the lowest-order approximation for the self-

energy (as often done!), introducing the Greenā™s function formalism and diagrammat-

ics is of course ridiculous as ļ¬nal results follow from Fermiā™s Golden Rule.9 A virtue

of the real-time formalism and its associated Feynman diagrams is that nontrivial

approximations can be established using the diagrammatic estimation technique, and

higher-order correlations studied systematically, as we shall consider in the following

chapters, not least in chapter 10.10

Before studying applications of the real-time technique we shall make obsolete

one version of the imaginary-time formalism, viz. the too pervasive Matsubara tech-

nique. The general imaginary-time formalism has virtues for special Euclidean ļ¬eld

theory purposes as well as for expedient proofs establishing conserving approxima-

tions. After the discussion of the equilibrium Dyson equation in the next section,

we demonstrate the equivalence of the imaginary-time formalism to the closed time

path formulation and the real-time technique introduced in this chapter.

5.6 Equilibrium Dyson equation

In equilibrium all quantities depend only on time diļ¬erences, and for translational

invariant situations also only on spatial diļ¬erences, and convolutions are by Fourier

transformation turned into products. In terms of the self-energy we therefore have

for the retarded Greenā™s function the equilibrium Dyson equation11

GR (p, E) = GR (p, E) + GR (p, E) Ī£R (E, p) GR (p, E) (5.96)

0 0

9 In the same vein, if one employs a mean-ļ¬eld approximation, introducing the formalism of

quantum ļ¬eld theory seems excessive. This point of view was taken in references [1] and [13].

10 For a discussion of the diagrammatic estimation technique see chapter 3 of reference [1].

11 We recall the result of Section 3.4, that in thermal equilibrium all the various Greenā™s functions

can be expressed in terms of, for example, the (imaginary part of the) retarded Greenā™s function.

5.6. Equilibrium Dyson equation 139

which we immediately solve to get

1 1

GR (p, E) = = . (5.97)

Gā’1 (p, E) ā’ Ī£R (E, p) Eā’ p ā’ Ī£ (E, p)

R

0

The retarded self-energy determines the analytic structure of the retarded Greenā™s

function, i.e. the location of the poles of the analytically continued retarded Greenā™s

function onto the second Riemann sheet through the branch cut along the real axis

(recall Section 3.4), the generic situation being that of a simple pole. For given

momentum value the simple pole is located at E = E1 + iE2 , determined by E1 =

p + e Ī£(E, p ) and E2 = m Ī£(E, p ), and as

ā

dE eā’iEt GR (p, E)

R

G (p, t) = (5.98)

ā’ā

the imaginary part of the self-energy thereby determines the temporal exponential

decay of the Greenā™s function, i.e. the lifetime of (in the present case) momentum

states. The eļ¬ect of interactions are clearly to give momentum states a ļ¬nite lifetime.

For the Fourier transform of Eq. (5.97) we get (in three spatial dimensions for

the prefactor to be correct)

ā

|xā’x | 2m(Eā’Ī£R (E, pE p))

i

ā’m e Ė

GE (x ā’ x ) =

R

(5.99)

|x ā’ x |

2Ļ 2

where pE is the solution of the equation pE = 2m(E ā’ Ī£R (E, pE p)). Interactions

Ė

will thus provide a ļ¬nite spatial and temporal range of the Greenā™s function.

For the case of electrons, say in a metal, the advanced Greenā™s function likewise

describes the attenuation of the holes.

Exercise 5.9. Show that the spectral function in equilibrium is given by (using now

the grand canonical ensemble)

Ī“(E, p)

A(E, p) = (5.100)

2 2

Ī“(E,p)

E ā’ Ī¾p ā’ eĪ£R (E, p) + 2

where

1R

eĪ£(E, p) ā” Ī£ (E, p) + Ī£A (E, p) (5.101)

2

and

Ī“(E, p) ā” i Ī£R (E, p) ā’ Ī£A (E, p) . (5.102)

We note that the sum-rule satisļ¬ed by the spectral weight function, Eq. (3.89), sets

limitation on the dependence of the self-energy on the energy variable. The general

features of interaction is to broaden the peak in the spectral weight function and to

shift, renormalize, energies.

140 5. Real-time formalism

Exercise 5.10. Show that for bosons in equilibrium at temperature T , their self-

energy components satisfy the ļ¬‚uctuationā“dissipation relations

E

Ī£R (E, p) ā’ Ī£A (E, p)

Ī£K (E, p) = coth (5.103)

2kT

and for fermions

E

Ī£R (E, p) ā’ Ī£A (E, p)

Ī£K (E, p) = tanh . (5.104)

2kT

5.7 Real-time versus imaginary-time formalism

Although we shall mainly use the real-time technique presented in this chapter

throughout, it is useful to be familiar with the equivalent imaginary-time formalism

in view of the vast amount of literature where this method has been employed. Or

more importantly to realize the link between the imaginary-time formalism and the

Martinā“Schwingerā“Abrikosovā“Gorkovā“Dzyaloshinskiā“Eliashbergā“Kadanoļ¬ ā“Baymā“

Langreth analytical continuation procedure. In the classic textbooks of Kadanoļ¬

and Baym [14] and Abrikosov, Gorkov and Dzyaloshinski [15] on non-equilibrium

statistical mechanics, the imaginary-time formalism introduced by Matsubara [16]

and Fradkin [17] and Martin and Schwinger [18] was used. Being then a Euclidean

ļ¬eld theory it possesses nice convergence properties. However, it lacks appeal to

intuition.

5.7.1 Imaginary-time formalism

The workings of the imaginary-time formalism are based on the mathematical formal

resemblance of the Boltzmann statistical weighting factor in the equilibrium statis-

tical operator Ļ ā eā’H/kT and the evolution operator U ā eā’iHt/ for an isolated

Ė

system. The imaginary time Greenā™s function

H ā’Ī¼ N

G(x, Ļ„ ; x , Ļ„ ) ā” ā’Tr eā’ Ė

TĻ„ (Ļ(x, Ļ„ ) Ļ(x , Ļ„ )) (5.105)

kT

is deļ¬ned in terms of ļ¬eld operators depending on imaginary time according to (we

suppress all other degrees of freedom than space)

Ļ(x) eā’

1 1

Ļ„ (Hā’Ī¼N ) Ļ„ (Hā’Ī¼N )

Ļ(x, Ļ„ ) = e (5.106)

and

Ļ(x, Ļ„ ) = e Ļ„ (Hā’Ī¼N ) Ļ ā (x) eā’ Ļ„ (Hā’Ī¼N ) ,

1 1

Ė (5.107)

where Ļ(x) is the ļ¬eld operator in the SchrĀØdinger picture, and TĻ„ provides the

o

imaginary time ordering (with the usual minus sign involved for an odd number

of interchanges of fermi ļ¬elds). The Ļ„ s involved are real variables, the use of the

word imaginary refers to the transformation t ā’ ā’iĻ„ in which case the time-ordered

real-time Greenā™s function, Eq. (4.10), transforms into the imaginary-time Greenā™s

Ė

function (more about this shortly). Note that Ļ(x, Ļ„ ) and Ļ(x, Ļ„ ) are not each others

5.7. Real-time versus imaginary-time formalism 141

adjoints. Knowledge of the imaginary-time Greenā™s function allows the calculation

of thermodynamic average values.

The imaginary-time single-particle Greenā™s function respects the Kuboā“Martinā“

Schwinger boundary conditions, for example

G(x, Ļ„ ; x , 0) = Ā± G(x, Ļ„ ; x , Ī²) , (5.108)

owing to the cyclic invariance property of the trace (the notation Ī² = /kT is used).

The periodic boundary condition is for bosons, and the anti-periodic boundary con-

dition is for fermions (the identical consideration in connection with the ļ¬‚uctuationā“

dissipation theorem was discussed in Section 3.4, and is further discussed in Section

6.5). We note the crucial role of the (grand) canonical ensemble as elaborated in

Section 3.4.

In its simple equilibrium applications in statistical mechanics, thermodynamics, or

in linear response theory, the involved imaginary-time Greenā™s functions are expressed

in terms of a single so-called Matsubara frequency

1

eā’iĻn (Ļ„ ā’Ļ„ ) G(x, x ; Ļn ) ,

G(x, Ļ„ ; x , Ļ„ ) = (5.109)

Ī² Ļn

where Ļn = 2nĻ/Ī² for bosons and Ļn = (2n + 1)Ļ/Ī² for fermions, respectively,

n = 0, Ā±1, Ā±2, . . . . Equilibrium or thermodynamic properties and linear transport

coeļ¬cients can therefore be expressed in terms of only one Matsubara frequency,

and the analytical continuation to obtain them from the imaginary-time Greenā™s

functions is trivial, say the retarded Greenā™s function is obtained by GR (x, x ; Ļ) =

G(x, x ; iĻn ā’ Ļ + i0+ ) as the two functions coincide according to GR (iĻn ) = G(Ļn )

for Ļn > 0.

The imaginary-time Greenā™s functions can also be used to study non-equilibrium

states by letting the external potential depend on the imaginary time. The Matsub-

ara technique is then a bit cumbersome, but can be used to derive exact equations,

say, the Dyson equation for real-time Greenā™s functions. In fact this was the method

used originally to study non-equilibrium superconductivity in the quasi-classical ap-

proximation [19].12 However, for general non-equilibrium situations, the necessary

analytical continuation in arbitrarily many Matsubara frequencies becomes nontrivial

(and are usually left out of textbooks), and are more involved than using the real-

time technique. Furthermore, when approximations are made, the real-time results

obtained upon analytical continuation can be spurious. However, the main disad-

vantage of the imaginary-time formalism is that it lacks physical transparency. We

shall therefore not discuss it further in the way it is usually done in textbooks, but

use a contour formulation to show its equivalence to the real-time formalism.13

12 Amazingly, the non-equilibrium theory of superconductivity was originally obtained using the

Matsubara technique [19], as, I guess, the imaginary-time formalism was in rule at the Landau

Institute. A plethora of papers and textbooks have perpetrated the use of the imaginary-time

formalism. It is the contestant to be the most important frozen accident in the evolution of non-

equilibrium theory. Letā™s iron out unfortunate ļ¬‚uctuations of the past! Its proliferation also testiļ¬es

to the fact that idiosyncratically written papers, such as the seminal paper of Schwinger [5], can be

a long time in germination.

13 The imaginary-time formalism can be useful for special purpose applications such as diagram-

142 5. Real-time formalism

5.7.2 Imaginary-time Greenā™s functions

The imaginary-time Greenā™s functions are proļ¬tably interpreted as contour-ordered

Greenā™s function, viz. on an imaginary-time contour. First we note, that the times

entering the imaginary-time Greenā™s function can be interpreted as contour times.

Choosing the times in the time ordered Greenā™s function in Eq. (3.61), instead to lie

on the contour starting at, say, t0 and ending down in the lower complex time plane

at t0 ā’ iĪ², the appendix contour ca in Figure 4.4, turns the expression Eq. (3.61)

into the equation for the imaginary-time Greenā™s function, Eq. (5.105). This observa-

tion, by the way, gives the standard Feynman diagrammatics for the imaginary-time

Greenā™s function since Wickā™s theorem involving the appendix contour is a trivial

corollary of the general Wickā™s theorem of Section 4.3.3. We can thus, for example,

immediately write down the non-equilibrium Dyson equation for the imaginary-time

Greenā™s function, t1 and t1 lying on the appendix contour ca . Considering the case

where the non-equilibrium situation is the result of a time-dependent potential, V ,

the Dyson equation for the imaginary-time Greenā™s functions or appendix contour-

ordered Greenā™s function is

G(1, 1 ) = G0 (1, 1 ) + dx3 dĻ„3 dx2 dĻ„2 G0 (1, 3) Ī£(3, 2) G(2, 1 )

ca ca

Ļ3 Ļ2

+ dx2 dĻ„2 G0 (1, 2) V (2) G(2, 1 ) . (5.110)

ca

Ļ2

The appendix contour interpretation of the imaginary-time Greenā™s function is in

certain situations more expedient than the real-time formulation when it for example

comes to diagrammatically proving exact relationships, since it has fewer diagrams

than the real-time approach if unfolding its matrix structure is needed. It should

thus be used in such situations, but then it is preferable not to use the Matsubara

frequency technique, but instead stick to the appendix contour formalism.

In practice we need to know how to analytically continue imaginary-time quanti-

ties to real-time functions, say for the imaginary-time Dyson equation, or for terms

appearing in perturbative expansions of imaginary-time quantities. Instead of turn-

ing to standard textbook imaginary-time formalism, the Matsubara technique, it is in

view of the above preferable to go to the appendix contour ordered Greenā™s functions,

and perform the analytical continuation from there. In fact, this analytical contin-

uation procedure becomes equivalent to the analytical continuation of the contour

ordered functions in the general contour formalism, for example the transition from

the contour-ordered Greenā™s function to real-time Greenā™s functions, which we now

turn to discuss. We illustrate this in the next section by performing the proceduce

for the Dyson equation.

The Boltzmann factor can guarantee analyticity of the Greenā™s function for times

on the appendix contour ca and indeed in the whole strip corresponding to translating

the real time t0 (recall Exercise 3.21 on page 74). The appendix contour can therefore

matic proofs of conservation laws, i.e. for proving exact relations. The imaginary-time formalism is

seen to be a simple corollary of the closed time path formalism.

5.7. Real-time versus imaginary-time formalism 143

be deformed into the contour depicted in Figure 4.4 on page 91, landing the original

times on the imaginary appendix contour onto the real axis producing the contour-

ordered Greenā™s function. This provides the analytical continuation from imaginary

times to the real times of interest. We now turn to discuss the procedure in detail.

5.7.3 Analytical continuation procedure

By using the closed time path approach, the analytical continuation procedure is

automated, and the equations of motion for the real-time correlation functions are

obtained without the irrelevant detour into Matsubara frequencies.

The procedure to obtain the real time Dyson equation from either the imaginary-

time formalism, i.e. from Eq. (5.110), or from the general contour formalism Dyson

equation, Eq. (4.141), is in fact the same. In, for example, the perturbative expansion

of the contour-ordered Greenā™s function or in the Dyson equation, we encounter

objects integrated over the contour depicted in Figure 4.4 on page 91, and we need

to obtain the corresponding formulae in terms of real-time functions, and as we

demonstrate now this is equivalent to analytical continuation.

Since space (and spin) and contour-time variables play diļ¬erent roles in the fol-

lowing argument, in fact only the contour-time variables play a role, we separate

space and contour-time matrix notations

(A Ć— B)(1, 1 ) ā” dx2 A(1, 2) B(2, 1 ) (5.111)

Ļ2

and

(A2B)(1, 1 ) ā” dĻ„2 A(1, 2) B(2, 1 ) . (5.112)

c

Here the contour-time integration could refer to the imaginary time appendix contour

ca , or the contour ci , stretching from t0 through t1 and t1 and back again to t0 (or

all the way to inļ¬nity and back) and ļ¬nally along the appendix contour to t0 ā’ iĪ²,

the contour depicted in Figure 4.4. The latter contour is obtained from the appendix

contour by the allowed analytical continuation procedure as discussed at the end

of the previous section. We shall not be interested in initial correlations, and can

therefore let the initial time protrude to the far past, t0 ā’ ā’ā, and the contour

in Figure 4.4 becomes the real-time closed contour, C, depicted in Figure 4.5. In

the case of analytical continuation from the imaginary-time appendix contour we

shall also eventually let the real-time t0 protrude to the far past. Everything in the

following, however, would be equally correct if we stick to the general contour, ci ,

depicted in Figure 4.4, allowing treating general initial states and therefore including

the completely general non-equilibrium problem. We would then just in addition

to integrations over the closed time path, have terms with integrations over the

imaginary time appendix contour ca .

Consider the case where the non-equilibrium situation is the result of a time-

dependent potential, V . The Dyson equation for the imaginary-time Greenā™s func-

tion is then given in Eq. (5.110), and for the contour-ordered Greenā™s function we

144 5. Real-time formalism

analogously have the equation

(0) (0)

GC (1, 1 ) = GC (1, 1 ) + dx3 dĻ„3 dx2 dĻ„2 GC (1, 3) Ī£(3, 2) GC (2, 1 )

Ļ3 Ļ2

C C

(0)

+ dx2 dĻ„2 GC (1, 2) V (2) GC (2, 1 ) , (5.113)

Ļ2 C

which can be written on the form (dropping the contour reminder subscript)

G + G(0) V

G = G0 + G0 Ī£ G, (5.114)

where Ī£ denotes the self-energy for the problem of interest.

We thus encounter explicitly contour matrix-multiplication, or multiplication in

series, a term of the form

C(Ļ„1 , Ļ„1 ) = dĻ„ A(Ļ„1 , Ļ„ ) B(Ļ„, Ļ„1 ) = (A2B)(Ļ„1 , Ļ„1 ) , (5.115)

c

where A and B are functions of the contour variable, and the involved contour could

be any of the three one can encounter as discussed above. Degrees of freedom are

suppressed since they play no role in the following demonstration that contour inte-

grations can be turned into integrations over the real-time axis. To accomplish this

we recall that the functions C < (Ļ„1 , Ļ„1 ) and C > (Ļ„1 , Ļ„1 ) are analytic functions in the

strips 0 < m(Ļ„1 ā’ Ļ„1 ) < Ī² and ā’Ī² < m(Ļ„1 ā’ Ļ„1 ) < 0, respectively.

Let us demonstrate the analytical continuation procedure for the case of C < . A

lesser quantity means by the general prescription, Eq. (4.41), that the contour time

Ļ„1 appears earlier than the contour time Ļ„1 , whatever contour is involved, i.e. we

<

have chosen the relationship Ļ„1 c Ļ„1 (irrespective of the numerical relationship of

their corresponding real time values in the case of the real-time contour). Exploiting

analyticity, the contour C or ci or the imaginary time contour ca is deformed into

the contour c1 + c1 depicted in Figure 5.7.14

t0 Ļ„1 Ļ„1

c1

c1

Figure 5.7 Deforming either of the contours C or ci or ca into the contour built by

the contours c1 and c1 .

14 Starting the ascent to real times, we essentially follow Langreth [20].

5.7. Real-time versus imaginary-time formalism 145

The expression in Eq. (5.115), for the chosen contour ordering, therefore becomes

C < (1, 1 ) = dĻ„ A(Ļ„1 , Ļ„ ) B(Ļ„, Ļ„1 ) + dĻ„ A(Ļ„1 , Ļ„ ) B(Ļ„, Ļ„1 )

c1 c1

dĻ„ A(Ļ„1 , Ļ„ ) B < (Ļ„, Ļ„1 ) + dĻ„ A< (Ļ„1 , Ļ„ ) B(Ļ„, Ļ„1 )

= (5.116)

c1 c1

<

and in the last equality, we have used the fact that on contour c1 we have Ļ„ c Ļ„1 ,

>

and on contour c1 we have Ļ„ c Ļ„1 . In the event of including initial correlations, or

rather staying with the general exact equation, the additional term with integration

over the appendix contour should be retained in the above equation.

Splitting in forward and return contour parts we have (as a consequence of the

contour positioning of the times on the contour parts in question as indicated to the

right)

ā’

ā’ : Ļ„ < Ļ„1

C < (1, 1 ) dĻ„ A> (Ļ„1 , Ļ„ ) B < (Ļ„, Ļ„1 ) c

= c1

ā’

ā’

c1

ā : Ļ„ > Ļ„1

ā’

dĻ„ A< (Ļ„1 , Ļ„ ) B < (Ļ„, Ļ„1 ) c

+ c1

ā

ā’

c1

ā’ : Ļ„ < Ļ„1

ā’

dĻ„ A< (Ļ„1 , Ļ„ ) B < (Ļ„, Ļ„1 ) c

+ c1

ā’

ā’

c1

cā’

ā : Ļ„ > Ļ„1 .

dĻ„ A< (Ļ„1 , Ļ„ ) B > (Ļ„, Ļ„1 ) c

+ (5.117)

1

cā’

ā1

Parameterizing the contours in terms of the real time variable, as in Eq. (4.35), and

noting that the external contour variables, Ļ„1 and Ļ„1 , now can be identiļ¬ed by their

corresponding values on the real time axis, gives (t0 ā’ ā’ā)

t1

dt (A> (t1 , t) ā’ A< (t1 , t)) B < (t, t1 )

<

C (1, 1 ) =

ā’ā

t1

dt A< (t1 , t)(B < (t, t1 ) ā’ B > (t, t1 ))

+ (5.118)

ā’ā

146 5. Real-time formalism

and thereby

ā

dt Īø(t1 ā’ t)(A> (t1 , t) ā’ A< (t1 , t)) B < (t, t1 )

C < (1, 1 ) =

ā’ā

ā

dt A< (t1 , t) Īø(t1 ā’ t)(B < (t, t1 ) ā’ B > (t, t1 )) .

+ (5.119)

ā’ā

Introducing the retarded function

AR (1, 1 ) = Īø(t1 ā’ t1 ) (A> (1, 1 ) ā’ A< (1, 1 )) (5.120)

and the advanced function

AA (1, 1 ) = Īø(t1 ā’ t1 ) (A> (1, 1 ) ā’ A< (1, 1 )) (5.121)

we have the real-time rule for multiplication in series

AR ā—¦ B < + A< ā—¦ B A ,

C< = (5.122)

where ā—¦ symbolizes matrix multiplication in real time, i.e. integration over the inter-

nal real-time variable from minus inļ¬nity to plus inļ¬nity of times.

Analogously one shows

AR ā—¦ B > + A> ā—¦ B A .

C> = (5.123)

We shall also need an expression for C R , and from Eq. (5.122) and Eq. (5.123)

we get

Īø(t1 ā’ t1 ) (AR ā—¦ B > )(t1 , t1 ) + (A> ā—¦ B A )(t1 , t1 )

C R (t1 , t1 ) =

ā’ (AR ā—¦ B < )(t1 , t1 ) + (A< ā—¦ B A )(t1 , t1 )

Īø(t1 ā’ t1 )((AR ā—¦ (B > ā’ B < ))(t1 , t1 ) + ((A> ā’ A< ) ā—¦ B A )(t1 , t1 )).

=

(5.124)

Expressing retarded and advanced functions according to Eq. (5.120) and Eq. (5.121)

gives

āt

1

Īø(t1 ā’ t1 ) ā dt (A> (t1 , t) ā’ A< (t1 , t)) (B > (t, t1 ) ā’ B < (t, t1 ))

R

C (t1 , t1 ) =

ā’ā

ā

t1

dt (A> (t1 , t) ā’ A< (t1 , t)) (B > (t, t1 ) ā’ B < (t, t1 ))ā

+

ā’ā

5.7. Real-time versus imaginary-time formalism 147

t1

Īø(t1 ā’ t1 ) dt (A> (t1 , t) ā’ A< (t1 , t)) (B > (t, t1 ) ā’ B < (t, t1 )) .

=

t1

(5.125)

Using the fact that t1 < t < t1 (as otherwise both left- and right-hand sides vanish)

we obtain

C R = AR ā—¦ B R . (5.126)

Analogously one arrives at

C A = AA ā—¦ B A . (5.127)

By using Eq. (5.122), Eq. (5.123), Eq. (5.126) and Eq. (5.127) we ļ¬nd, owing to

the associative property of the composition, that the analytical continuation of the

contour quantity

D = A2B2C (5.128)

becomes

> > > >

D< = AR ā—¦ B R ā—¦ C < + AR ā—¦ B < ā—¦ C A + A< ā—¦ B A ā—¦ C A (5.129)

and, by induction, Eq. (5.129) generalizes to an arbitrary number of functions mul-

tiplied in series. Note that the retarded and advanced functions appear to the left

and right, respectively, of the greater and lesser Greenā™s functions.

Employing the analytical continuation procedure, one can thus from the imaginary-

time Greenā™s function formalism arrive with equal ease at the real-time non-equilibrium

Dyson equations of Section 5.5.

The only other ingredient encountered in perturbation expansions is the product

of contour-ordered Greenā™s functions of the form

C(Ļ„1 , Ļ„1 ) = A(Ļ„1 , Ļ„1 ) B(Ļ„1 , Ļ„1 ) , (5.130)

multiplication in parallel. This occurs, for example, for pair-creation or electron-

hole excitations, etc., or for a self-energy diagram for example for fermionā“boson

interaction, in which case one might prefer to have the same sequence of the contour

variables in all of the functions as in the self-energy insertion of the diagram in Figure

5.6 (this is, of course, a matter of taste).

Following the above procedure we immediately get for the analytical continuation

of multiplication in parallel

> > >

C < (t1 , t1 ) = A< (t1 , t1 ) B < (t1 , t1 ) . (5.131)

With these tools at hand, we can turn any exact imaginary-time formula, or any

diagram in the perturbative expansion of the imaginary-time Greenā™s function or a

contour ordered quantity, say the contour-ordered Greenā™s functions of Section 4.4,

into products of real-time Greenā™s functions. This automatic mechanical continuation

148 5. Real-time formalism

to real times is much preferable than to do it in the Matsubara frequencies. With

this at hand a very eļ¬ective way of studying non-equilibrium states in the real time

formalism is available, as discussed in the classic text [14], and whether using this or

the other three-fold representation is a matter of taste. However, the G-greater and

G-lesser Greenā™s functions are quantum statistically quantities of the same nature,

whereas in the representation introduced in Section 5.3, the Greenā™s functions carry

distinct information.

5.7.4 Kadanoļ¬ā“Baym equations

As an example of using the analytical continuation procedure we shall, from the

Dyson equation for the imaginary-time Greenā™s function in Eq. (5.110) (or the general

contour-ordered Greenā™s function, Eq. (5.114)), obtain the equations of motion for

the physical correlation functions, the lesser and greater Greenā™s functions on the

real-time axis. Let us therefore consider the Dyson equation for the imaginary-time

Greenā™s function or the general contour-ordered Greenā™s function of the form

Gā’1 G = Ī“(1 ā’ 1 ) + Ī£ G, (5.132)

0

where external ļ¬elds are included in Gā’1 . Applying the rule for multiplication in

0

series gives

> > >

Gā’1 ā—G = Ī£ ā—G + Ī£ ā— GA

< R < <

(5.133)

0

and similarly for the right-hand Dyson equation

> > >

G< ā— Gā’1 = GR ā— Ī£< + G< ā— Ī£A . (5.134)

0

Subtracting the left and right Dyson equations gives

> > > > >

[Gā’1 ā— G ]ā’ = Ī£ ā— G ā’ G ā—Ī£ + Ī£ ā—G ā’ G ā—Ī£

< R < < A < A R <

, (5.135)

0

which can be rewritten

1R 1>

> >

[Gā’1 ā— G< ]ā’ [Ī£ + Ī£A ā— G< ]ā’ ā’ [Ī£< ā— GR + GA ]ā’

ā’

, , ,

0

2 2

1 1

ā’ [Ī£< ā— G> ]+ + [Ī£> ā— G< ]+ .

= , , (5.136)

2 2

These two equations, the Kadanoļ¬ā“Baym equations, can be used as basis for consid-

ering quantum kinetics.

We recall that the kinetic Greenā™s function is speciļ¬ed according to GK = G> +

G< , and note that adding the two equations in Eq. (5.136) we recover Eq. (5.95). We

note that the equations, Eq. (5.129), satisļ¬ed by GK are satisļ¬ed by both G> and

G< , or rather we should appreciate the observation that their equations are identical

with respect to splitting into retarded and advanced Greenā™s functions.

>

Since the equations for G< mixes, through for example self-energies according to

Eq. (5.131), it is economical to work instead solely with GK . However, there can be

5.8. Summary 149

special circumstances where the advantage is reversed, for example when discussing

the dynamics of a tunnel junction. One should note that the quantum statistics of

particle species manifests itself quite diļ¬erently depending on which type of kinetic

propagator one chooses to employ.

5.8 Summary

We have presented the real-time formalism necessary for treating non-equilibrium

situations. For the reader not familiar with equilibrium theory the good news is that

equilibrium theory is just an especially simple case of the presented general theory.

In the real-time formulation of the properties of non-equilibrium states the dynamics

is used to provide a doubling of the degrees of freedom, and one encounters at least

two types of Greenā™s functions. To get a physically transparent representation, we

introduced the real-time matrix representation of the contour-ordered Greenā™s func-

tions to describe non-equilibrium states. This allowed us to represent matrix Greenā™s

function perturbation theory in terms of Feynman diagrams in a standard fashion.

We introduced the physical representation corresponding to the two Greenā™s func-

tions representing the spectral and quantum statistical properties of a system. We

then showed that the matrix notation can be broken down into two simple rules for

the universal vertex structure in the dynamical indices. This allowed us to formulate

the non-equilibrium aspects of the Feynman diagrams directly in terms of the vari-

ous matrix Greenā™s function components, R, A, K, establishing the real rules. In this

way we were able to express how the diļ¬erent features of the spectral and quantum

statistical properties enter into the diagrammatic representation of non-equilibrium

processes. We ended the chapter by showing the equivalence of the imaginary-time

and the closed time path and the real-time formalisms, all formally identical, and

transformed into each other by analytical continuation. In the rest of the book we

shall demonstrate the versatility of the real-time technique. Before constructing the

functional formulation of quantum ļ¬eld theory from its Feynman diagrams, and show

ńņš. 6 |