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propounded in this chapter.2


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

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