5.1 Real-time matrix representation

To let our physical intuition come into play; we need to get from contour times back

to real times. This is achieved by labeling the forward and return contours of the

closed time path, depicted in Figure 4.5, by numbers, specifying the position of a

contour time by an index. The forward contour we therefore label c1 and the return

contour c2 , i.e. a contour time variable gets tagged by the label 1 or 2 specifying its

belonging to forward or return contour, respectively.3

The contour ordered Green™s function is by this tagging mapped onto a 2 — 2-

1 As the imaginary time Green™s function discussed in Section 5.7.1 does not appeal to intuition.

2 In this chapter we follow the exposition given in references [3] and [9].

3 Instead of labeling the two branches by 1 and 2, one can also label them by ± as in the original

works of Schwinger [5] and Keldysh [10]. However, when stating Feynman rules, numbers are

convenient for labeling.

121

122 5. Real-time formalism

matrix in the dynamical index or Schwinger“Keldysh space

ˆ ˆ

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

ˆ

GC (1, 1 ) ’ G(1, 1 ) ≡ (5.1)

ˆ ˆ

G21 (1, 1 ) G22 (1, 1 )

ˆ

according to the prescription: the ij-component in the matrix Green™s function G is

Gc (1, 1 ) for 1 lying on ci and 1 lying on cj , i, j = 1, 2. The times appearing in the

components of the matrix Green™s function are now standard times, 1 = (x, t1 ), and

we can identify them in terms of our previously introduced Green™s functions, the real-

time Green™s functions introduced in Section 3.3. The matrix structure re¬‚ects the

essence in the real-time formulation of non-equilibrium quantum statistical mechanics

due to Schwinger [5]: letting the quantum dynamics do the doubling of the degrees

of freedom necessary for describing non-equilibrium states.4

The 11-component of the matrix in Eq. (5.1) becomes

G11 (1, 1 ) = ’i T (ψ(x1 , t1 ) ψ † (x1 , t1 )) = ’i T (ψ(1) ψ † (1 )) ,

ˆ (5.2)

the time-ordered Green™s function, where ψ(x, t) = ψH (x, t) is the ¬eld in the

full Heisenberg picture for the species of interest. Contour ordering on the forward

contour is just usual time-ordering.

Analogously, the 21-component becomes

G21 (1, 1 ) = G> (1, 1 ) = ’i ψ(1) ψ † (1 ) ,

ˆ (5.3)

i.e. G-greater, and the 12-component is G-lesser

G12 (1, 1 ) = G< (1, 1 ) = “ i ψ † (1 ) ψ(1) ,

ˆ (5.4)

where upper and lower signs refer to bose and fermi ¬elds, respectively, and the

22-component is the anti-time-ordered Green™s function

G22 (1, 1 ) = G(1, 1 ) = ’i T (ψ(1) ψ † (1 )) .

ˆ ˜ ˜ (5.5)

We note that the time-ordered and anti-time-ordered Green™s functions can be

expressed in terms of G-greater and G-lesser, recall Eq. (3.64), and

ˆ

G11 (1, 1 ) = θ(t1 ’ t1 ) G> (1, 1 ) + θ(t1 ’ t1 ) G< (1, 1 ) . (5.6)

The matrix Green™s function in Eq. (5.1) can therefore be expressed in terms of

the real-time Green™s functions introduced in Section 3.3

G< (1, 1 )

G(1, 1 )

ˆ

G(1, 1 ) = . (5.7)

˜

G> (1, 1 ) G(1, 1 )

The way of representing the information in the contour-ordered Green™s function

as in Eq. (5.1) or equivalently Eq. (5.7) is respectable as, for example, the matrix

4 The thermo-¬eld approach to non-equilibrium theory also employs a doubling of the degrees of

freedom (see, for example, reference [11]), but in our view not in as physically appealing way as

does the real-time version of the closed time path formulation.

5.2. Real-time diagrammatics 123

is anti-hermitian with transposition meaning interchange of all arguments including

that of the dynamical index (note the importance of the sign convention for han-

dling fermi ¬elds under ordering operations). For real bosons the matrix is real and

symmetric. However, when it comes to understanding non-equilibrium contributions

from various processes, as described by Feynman diagrams, the present form of the

matrix Green™s function lacks physical transparency, and o¬ers no basis for develop-

ing intuition. We shall therefore eventually transform to a di¬erent matrix form, and

as a ¬nal act liberate ourselves from the matrix out¬t altogether.

Let us now establish the Feynman rules in the real-time technique for the matrix

Green™s function in the dynamical index or Schwinger“Keldysh space.

5.2 Real-time diagrammatics

Instead of having the diagrammatics represent the perturbative expansion of the

contour Green™s function as in the previous chapter, we shall map the diagrams

to the real-time domain where eventually a proper physical interpretation of the

diagrams can be obtained.

5.2.1 Feynman rules for a scalar potential

We start with the simplest kind of coupling, that of particles interacting with an

external classical ¬eld. For particles interacting with a scalar potential V (x, t), we

have the diagrammatic expansion of the contour ordered Green™s function depicted

in Eq. (4.115) on page 105. The ¬rst-order diagram corresponded to the term

(1) (0) (0)

GC (1, 1 ) = dx2 d„2 GC (1, 2) V (2) GC (2, 1 ) . (5.8)

C

Parameterizing the real-time contour we have

∞ ’∞ ∞ ∞

dt ’

d„2 = dt + dt = dt (5.9)

’∞ ∞ ’∞ ’∞

C

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

∞

ˆ (1) ˆ (0) ˆ (0)

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

’∞

∞

ˆ (0) ˆ (0)

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

’∞

Introducing in Schwinger“Keldysh or dynamical index space the matrix

(3)

ˆ

Vij (1) = V (1) „ij (5.11)

proportional to the third Pauli-matrix

10

„ (3) = (5.12)

0 ’1

124 5. Real-time formalism

we have

∞

ˆ (1) ˆ (0) ˆ (0)

ˆ

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

’∞

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

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

applied and the matrix equation becomes

ˆ ˆ ˆˆ ˆˆ ˆ

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

where — signi¬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 in¬nity of times. For the components of the free equilibrium matrix Green™s

ˆ

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

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

contribution

ˆ

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

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

Schwinger“Keldysh space: each line now represents the free matrix Green™s function,

ˆ

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 ¬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 ¬elds, giving

ˆ ˆ ˆ ˆˆ ˆ ˆ ˆˆ ˆ

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

ˆ

Green™s function, G(0) , satis¬es

G’1 (1) G0 (1, 1 ) = „ (3) δ(1 ’ 1 ) ,

ˆ (5.17)

0

where G’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 Green™s function operating

on the free equilibrium matrix Green™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, „ (3) . For example, introducing the matrix

representation

ˆ ˆ

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

ˇ ˆ

G(1, 1 ) ≡ „ (3) G(1, 1 ) = (5.18)

ˆ ˆ

’G21 (1, 1 ) ’G22 (1, 1 )

5.2. Real-time diagrammatics 125

we then have

G’1 (1) G0 (1, 1 ) = 1 δ(1 ’ 1 ) ,

ˇ (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 Green™s function

G’1 (1, 1 ) = G’1 (1) δ(1 ’ 1 ) 1 (5.20)

0 0

we have

(G’1 — G0 )(1, 1 ) = 1 δ(1 ’ 1 ) = (G0 — G’1 )(1, 1 ) .

ˇ ˇ (5.21)

0 0

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

in the forms

((G’1 ’ V ) — G)(1, 1 ) = 1 δ(1 ’ 1 )

ˇ (5.22)

0

and

(G — (G’1 ’ V ))(1, 1 ) = 1 δ(1 ’ 1 ) .

ˇ (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 Schwinger“Keldysh space

ˇ ˇ ˇˇ

G(1) = G(0) — V G(0) , (5.24)

where

ˇ ˇ

Vij (1) = V (1) δij , 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 fermion“boson interaction or

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

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

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

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

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

jellium model where the electrons couple only to longitudinal compressional charge

con¬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, ¬‚avor, Minkowski, phonon branch, etc.

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

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

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

ˆ (0) ˆ k ˆ (0) ˆ (0) ˆ k

ˆ (1) ˆ (0)

Gij = ig 2 Gii — γi l Gl l Dkk γ lj — Gj j

˜ (5.26)

126 5. Real-time formalism

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

(0) ˆ (0) ˆ k

ˆ (1) ˆk ˆ

Σij = ig 2 γil Gl l Dkk γ lj ,

˜ (5.27)

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

tices are identical

ˆk

(3)

ˆk

γij = δij „jk = γ ij .

˜ (5.28)

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

re¬‚ecting the fact that the ¬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 ’i for our convention of Green™s functions: two factors of ’i

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

Green™s functions. Such features are collected in one™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 Green™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 di¬erent physical properties represented

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

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

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

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

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

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

the third Pauli matrix.

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

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

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

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

Eq. (5.28).

1 3 2 1

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

fermion“boson interaction.

5.3. Triagonal and symmetric representations 127

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

(3) (3)

δij = „ik „kj ),

ˇ (0) ˇ k ˇ (0) ˇ (0) ˇ k

ˇ (1) ˇ (0)

Gij = ig 2 Gii — γi l Gl l Dkk γ lj — Gj j ,

˜ (5.29)

where the absorption vertex is

i

(3) (3) (3)

ˇk ˆk

= γij = γii „i j „k k = δij „jk (5.30)

k

j

and the emission vertex is

i

ˆk (3)

ˇk

= γ ij

˜ = γ ii „i j

˜ = δij δjk . (5.31)

k

j

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

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

ˇ ˇ ˇ ˇ

G(1) = G(0) — Σ(1) — G(0) . (5.32)

5.3 Triagonal and symmetric representations

Since only two components of the matrix Green™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 π/4-

rotation in Schwinger“Keldysh space,

G ’ L G L† ,

ˆ ˆ (5.33)

where the orthogonal matrix, L† = L, is

’1

1 1 1

L = √ (1 ’ i„ (2) ) = √ (5.34)

1 1

2 2

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

’i

0

„ (2) = . (5.35)

i 0

Using the following identities (recall Section 3.3)

ˆ ˆ ˆ ˆ

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

128 5. Real-time formalism

and

ˆ ˆ ˆ ˆ

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

and

ˆ ˆ ˆ ˆ

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

and

ˆ ˆ ˆ ˆ

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

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

amounts to5

ˆ ˆ GA

G11 G12 0

’ (5.40)

ˆ ˆ GR GK

G21 G22

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

introduced in Section 3.3.2.

For real bosons or phonons, the matrix