<<

. 6
( 22)



>>

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

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
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 — Σ — 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
di¬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 Green™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,
ˆ ˆ ˆ ˆ
ΣR = Σ11 ’ Σ12 = Σ21 ’ Σ22 (5.70)
ˆ ˆ ˆ ˆ
ΣA = Σ11 ’ Σ21 = Σ12 ’ Σ22 (5.71)
ˆ ˆ ˆ ˆ
ΣK = Σ11 + Σ22 = Σ12 + Σ21 (5.72)
ˆ ˆ ˆ ˆ
0 = Σ11 ’ Σ12 + ’ Σ21 + Σ22 . (5.73)
By construction
ˆ
Σ12 (x1 , t1 , x1 , t1 ) t1 > t1
ˆ
Σ11 (x1 , t1 , x1 , t1 ) = (5.74)
ˆ
Σ21 (x1 , t1 , x1 , t1 ) t1 > t1

and
ˆ
Σ21 (x1 , t1 , x1 , t1 ) t1 > t1
ˆ
Σ22 (x1 , t1 , x1 , t1 ) = (5.75)
ˆ
Σ12 (x1 , t1 , x1 , t1 ) t1 > t1
and the matrix self-energy has in the triagonal representation the same triagonal
form as the matrix Green™s function
ΣR ΣK
Σ= . (5.76)
ΣA
0

Exercise 5.6. Introducing
ˆ
Σ< (x1 , t1 , x1 , t1 ) = Σ12 (x1 , t1 , x1 , t1 ) (5.77)

and
ˆ
Σ> (x1 , t1 , x1 , t1 ) = Σ21 (x1 , t1 , x1 , t1 ) (5.78)
show that we have, identically to the relationships for the Green™s functions, the
relation for the retarded self-energy

ΣR (x, t, x , t ) = θ(t ’ t ) Σ> (x, t, x , t ) ’ Σ< (x, t, x , t ) (5.79)

and advanced self-energy

ΣA (x, t, x , t ) = ’θ(t ’ t) Σ> (x, t, x , t ) ’ Σ< (x, t, x , t ) (5.80)
5.5. Non-equilibrium Dyson equations 137


and for the kinetic component

ΣK (x, t, x , t ) = Σ> (x, t, x , t ) + Σ< (x, t, x , t ) . (5.81)

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

ΣA (x, t, x , t ) = ΣR (x , t , x, t) (5.82)

and

ΣK (x, t, x , t ) = ΣK (x , t , x, t) . (5.83)

Exercise 5.7. Show that in the case where the matrix Green™s function is represented
in symmetric form

GA
0
G= (5.84)
GR GK

the matrix self-energy has the form

ΣK ΣR
Σ= . (5.85)
ΣA 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 Green™s function (or ¬elds) we then get,
for the matrix Green™s function, the equations of motion

(i‚t1 ’ h(1))G(1, 1 ) = δ(1 ’ 1 ) + (Σ — G)(1, 1 ) (5.86)

and
(i‚t1 ’ h— (1 ))G(1, 1 ) = δ(1 ’ 1 ) + (G — Σ)(1, 1 ) (5.87)
or introducing the inverse free Green™s function

G’1 (1, 1 ) = (i‚t1 ’ h(1)) δ(1 ’ 1 ) (5.88)
0

the two equations can be expressed through operating with the inverse free matrix
Green™s function from the left

(G’1 ’ Σ) — G = δ(1 ’ 1 ) (5.89)
0

and from the right
G — (G’1 ’ Σ) = δ(1 ’ 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

(G’1 ’ ΣR(A) ) — GR(A) = δ(1 ’ 1 ) (5.91)
0
138 5. Real-time formalism


and
G’1 — GK = ΣR — GK + ΣK — GA . (5.92)
0

Analogously, from Eq. (5.90), we obtain

GR(A) — (G’1 ’ ΣR(A) ) = δ(1 ’ 1 ) (5.93)
0

and
GK — G’1 = GR — ΣK + GK — ΣA . (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
[G’1 — GK ]’ [Σ + ΣA — GK ]’ ’ [ΣK — GR + GA ]’

, , ,
0
2 2

1 1
’ [ΣK — (GR ’ GA )]+ + [(ΣR ’ ΣA ) — 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 Green™s function formalism and diagrammat-
ics is of course ridiculous as ¬nal results follow from Fermi™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 ¬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 di¬erences, and for translational
invariant situations also only on spatial di¬erences, and convolutions are by Fourier
transformation turned into products. In terms of the self-energy we therefore have
for the retarded Green™s function the equilibrium Dyson equation11

GR (p, E) = GR (p, E) + GR (p, E) ΣR (E, p) GR (p, E) (5.96)
0 0
9 In the same vein, if one employs a mean-¬eld approximation, introducing the formalism of
quantum ¬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 Green™s functions

can be expressed in terms of, for example, the (imaginary part of the) retarded Green™s function.
5.6. Equilibrium Dyson equation 139


which we immediately solve to get
1 1
GR (p, E) = = . (5.97)
G’1 (p, E) ’ ΣR (E, p) E’ p ’ Σ (E, p)
R
0

The retarded self-energy determines the analytic structure of the retarded Green™s
function, i.e. the location of the poles of the analytically continued retarded Green™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 Σ(E, p ) and E2 = m Σ(E, p ), and as


dE e’iEt GR (p, E)
R
G (p, t) = (5.98)
’∞

the imaginary part of the self-energy thereby determines the temporal exponential
decay of the Green™s function, i.e. the lifetime of (in the present case) momentum
states. The e¬ect of interactions are clearly to give momentum states a ¬nite lifetime.
For the Fourier transform of Eq. (5.97) we get (in three spatial dimensions for
the prefactor to be correct)

|x’x | 2m(E’ΣR (E, pE p))
i
’m e ˆ
GE (x ’ x ) =
R
(5.99)
|x ’ x |
2π 2

where pE is the solution of the equation pE = 2m(E ’ ΣR (E, pE p)). Interactions
ˆ
will thus provide a ¬nite spatial and temporal range of the Green™s function.
For the case of electrons, say in a metal, the advanced Green™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)

“(E, p)
A(E, p) = (5.100)
2 2
“(E,p)
E ’ ξp ’ eΣR (E, p) + 2

where
1R
eΣ(E, p) ≡ Σ (E, p) + ΣA (E, p) (5.101)
2
and
“(E, p) ≡ i ΣR (E, p) ’ ΣA (E, p) . (5.102)

We note that the sum-rule satis¬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 ¬‚uctuation“dissipation relations
E
ΣR (E, p) ’ ΣA (E, p)
ΣK (E, p) = coth (5.103)
2kT
and for fermions
E
ΣR (E, p) ’ ΣA (E, p)
ΣK (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
Martin“Schwinger“Abrikosov“Gorkov“Dzyaloshinski“Eliashberg“Kadano¬ “Baym“
Langreth analytical continuation procedure. In the classic textbooks of Kadano¬
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
¬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 ρ ∝ e’H/kT and the evolution operator U ∝ e’iHt/ for an isolated
ˆ
system. The imaginary time Green™s function
H ’μ N
G(x, „ ; x , „ ) ≡ ’Tr e’ ˜
T„ (ψ(x, „ ) ψ(x , „ )) (5.105)
kT




is de¬ned in terms of ¬eld operators depending on imaginary time according to (we
suppress all other degrees of freedom than space)

ψ(x) e’
1 1
„ (H’μN ) „ (H’μN )
ψ(x, „ ) = e (5.106)

and
ψ(x, „ ) = e „ (H’μN ) ψ † (x) e’ „ (H’μN ) ,
1 1
˜ (5.107)
where ψ(x) is the ¬eld operator in the Schr¨dinger picture, and T„ provides the
o
imaginary time ordering (with the usual minus sign involved for an odd number
of interchanges of fermi ¬elds). The „ s involved are real variables, the use of the
word imaginary refers to the transformation t ’ ’i„ in which case the time-ordered
real-time Green™s function, Eq. (4.10), transforms into the imaginary-time Green™s
˜
function (more about this shortly). Note that ψ(x, „ ) and ψ(x, „ ) are not each others
5.7. Real-time versus imaginary-time formalism 141


adjoints. Knowledge of the imaginary-time Green™s function allows the calculation
of thermodynamic average values.
The imaginary-time single-particle Green™s function respects the Kubo“Martin“
Schwinger boundary conditions, for example

G(x, „ ; x , 0) = ± G(x, „ ; x , β) , (5.108)

owing to the cyclic invariance property of the trace (the notation β = /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 ¬‚uctuation“
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 Green™s functions are expressed
in terms of a single so-called Matsubara frequency
1
e’iωn („ ’„ ) G(x, x ; ωn ) ,
G(x, „ ; x , „ ) = (5.109)
β ωn

where ωn = 2nπ/β for bosons and ωn = (2n + 1)π/β for fermions, respectively,
n = 0, ±1, ±2, . . . . Equilibrium or thermodynamic properties and linear transport
coe¬cients can therefore be expressed in terms of only one Matsubara frequency,
and the analytical continuation to obtain them from the imaginary-time Green™s
functions is trivial, say the retarded Green™s function is obtained by GR (x, x ; ω) =
G(x, x ; iωn ’ ω + i0+ ) as the two functions coincide according to GR (iωn ) = G(ωn )
for ωn > 0.
The imaginary-time Green™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 Green™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. Let™s iron out unfortunate ¬‚uctuations of the past! Its proliferation also testi¬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 Green™s functions
The imaginary-time Green™s functions are pro¬tably interpreted as contour-ordered
Green™s function, viz. on an imaginary-time contour. First we note, that the times
entering the imaginary-time Green™s function can be interpreted as contour times.
Choosing the times in the time ordered Green™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 ’ iβ, the appendix contour ca in Figure 4.4, turns the expression Eq. (3.61)
into the equation for the imaginary-time Green™s function, Eq. (5.105). This observa-
tion, by the way, gives the standard Feynman diagrammatics for the imaginary-time
Green™s function since Wick™s theorem involving the appendix contour is a trivial
corollary of the general Wick™s theorem of Section 4.3.3. We can thus, for example,
immediately write down the non-equilibrium Dyson equation for the imaginary-time
Green™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 Green™s functions or appendix contour-
ordered Green™s function is

G(1, 1 ) = G0 (1, 1 ) + dx3 d„3 dx2 d„2 G0 (1, 3) Σ(3, 2) G(2, 1 )
ca ca
σ3 σ2



+ dx2 d„2 G0 (1, 2) V (2) G(2, 1 ) . (5.110)
ca
σ2

The appendix contour interpretation of the imaginary-time Green™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 Green™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 Green™s function to real-time Green™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 Green™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 Green™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 Green™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 di¬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 — B)(1, 1 ) ≡ dx2 A(1, 2) B(2, 1 ) (5.111)
σ2

and
(A2B)(1, 1 ) ≡ d„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 in¬nity and back) and ¬nally along the appendix contour to t0 ’ iβ,
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 ’ ’∞, 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 Green™s func-
tion is then given in Eq. (5.110), and for the contour-ordered Green™s function we
144 5. Real-time formalism


analogously have the equation

(0) (0)
GC (1, 1 ) = GC (1, 1 ) + dx3 d„3 dx2 d„2 GC (1, 3) Σ(3, 2) GC (2, 1 )
σ3 σ2
C C



(0)
+ dx2 d„2 GC (1, 2) V (2) GC (2, 1 ) , (5.113)
σ2 C

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

G + G(0) V
G = G0 + G0 Σ G, (5.114)

where Σ 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(„1 , „1 ) = d„ A(„1 , „ ) B(„, „1 ) = (A2B)(„1 , „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 < („1 , „1 ) and C > („1 , „1 ) are analytic functions in the
strips 0 < m(„1 ’ „1 ) < β and ’β < m(„1 ’ „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
„1 appears earlier than the contour time „1 , whatever contour is involved, i.e. we
<
have chosen the relationship „1 c „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 „1 „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 ) = d„ A(„1 , „ ) B(„, „1 ) + d„ A(„1 , „ ) B(„, „1 )
c1 c1




d„ A(„1 , „ ) B < („, „1 ) + d„ A< („1 , „ ) B(„, „1 )
= (5.116)
c1 c1

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


’ : „ < „1
C < (1, 1 ) d„ A> („1 , „ ) B < („, „1 ) c
= c1


c1


← : „ > „1

d„ A< („1 , „ ) B < („, „1 ) c
+ c1


c1


’ : „ < „1

d„ A< („1 , „ ) B < („, „1 ) c
+ c1


c1


c’
← : „ > „1 .
d„ A< („1 , „ ) B > („, „1 ) c
+ (5.117)
1

c’
←1

Parameterizing the contours in terms of the real time variable, as in Eq. (4.35), and
noting that the external contour variables, „1 and „1 , now can be identi¬ed by their
corresponding values on the real time axis, gives (t0 ’ ’∞)
t1
dt (A> (t1 , t) ’ A< (t1 , t)) B < (t, t1 )
<
C (1, 1 ) =
’∞


t1
dt A< (t1 , t)(B < (t, t1 ) ’ B > (t, t1 ))
+ (5.118)
’∞
146 5. Real-time formalism


and thereby


dt θ(t1 ’ t)(A> (t1 , t) ’ A< (t1 , t)) B < (t, t1 )
C < (1, 1 ) =
’∞




dt A< (t1 , t) θ(t1 ’ t)(B < (t, t1 ) ’ B > (t, t1 )) .
+ (5.119)
’∞

Introducing the retarded function
AR (1, 1 ) = θ(t1 ’ t1 ) (A> (1, 1 ) ’ A< (1, 1 )) (5.120)
and the advanced function
AA (1, 1 ) = θ(t1 ’ t1 ) (A> (1, 1 ) ’ A< (1, 1 )) (5.121)
we have the real-time rule for multiplication in series
AR —¦ B < + A< —¦ B A ,
C< = (5.122)
where —¦ symbolizes matrix multiplication in real time, i.e. integration over the inter-
nal real-time variable from minus in¬nity to plus in¬nity of times.
Analogously one shows
AR —¦ B > + A> —¦ 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
θ(t1 ’ t1 ) (AR —¦ B > )(t1 , t1 ) + (A> —¦ B A )(t1 , t1 )
C R (t1 , t1 ) =


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


θ(t1 ’ t1 )((AR —¦ (B > ’ B < ))(t1 , t1 ) + ((A> ’ A< ) —¦ B A )(t1 , t1 )).
=

(5.124)
Expressing retarded and advanced functions according to Eq. (5.120) and Eq. (5.121)
gives
⎛t
1

θ(t1 ’ t1 ) ⎝ dt (A> (t1 , t) ’ A< (t1 , t)) (B > (t, t1 ) ’ B < (t, t1 ))
R
C (t1 , t1 ) =
’∞



t1

dt (A> (t1 , t) ’ A< (t1 , t)) (B > (t, t1 ) ’ B < (t, t1 ))⎠
+
’∞
5.7. Real-time versus imaginary-time formalism 147


t1

θ(t1 ’ t1 ) dt (A> (t1 , t) ’ A< (t1 , t)) (B > (t, t1 ) ’ 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 —¦ B R . (5.126)

Analogously one arrives at

C A = AA —¦ B A . (5.127)

By using Eq. (5.122), Eq. (5.123), Eq. (5.126) and Eq. (5.127) we ¬nd, owing to
the associative property of the composition, that the analytical continuation of the
contour quantity
D = A2B2C (5.128)
becomes
> > > >
D< = AR —¦ B R —¦ C < + AR —¦ B < —¦ C A + A< —¦ B A —¦ 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 Green™s functions.
Employing the analytical continuation procedure, one can thus from the imaginary-
time Green™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 Green™s functions of the form

C(„1 , „1 ) = A(„1 , „1 ) B(„1 , „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 fermion“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 Green™s function or a
contour ordered quantity, say the contour-ordered Green™s functions of Section 4.4,
into products of real-time Green™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 e¬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 Green™s functions are quantum statistically quantities of the same nature,
whereas in the representation introduced in Section 5.3, the Green™s functions carry
distinct information.

5.7.4 Kadano¬“Baym equations
As an example of using the analytical continuation procedure we shall, from the
Dyson equation for the imaginary-time Green™s function in Eq. (5.110) (or the general
contour-ordered Green™s function, Eq. (5.114)), obtain the equations of motion for
the physical correlation functions, the lesser and greater Green™s functions on the
real-time axis. Let us therefore consider the Dyson equation for the imaginary-time
Green™s function or the general contour-ordered Green™s function of the form

G’1 G = δ(1 ’ 1 ) + Σ G, (5.132)
0

where external ¬elds are included in G’1 . Applying the rule for multiplication in
0
series gives
> > >
G’1 —G = Σ —G + Σ — GA
< R < <
(5.133)
0

and similarly for the right-hand Dyson equation
> > >
G< — G’1 = GR — Σ< + G< — ΣA . (5.134)
0

Subtracting the left and right Dyson equations gives
> > > > >
[G’1 — G ]’ = Σ — G ’ G —Σ + Σ —G ’ G —Σ
< R < < A < A R <
, (5.135)
0

which can be rewritten
1R 1>
> >
[G’1 — G< ]’ [Σ + ΣA — G< ]’ ’ [Σ< — GR + GA ]’

, , ,
0
2 2

1 1
’ [Σ< — G> ]+ + [Σ> — G< ]+ .
= , , (5.136)
2 2
These two equations, the Kadano¬“Baym equations, can be used as basis for consid-
ering quantum kinetics.
We recall that the kinetic Green™s function is speci¬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), satis¬ed by GK are satis¬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 Green™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 di¬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 Green™s functions. To get a physically transparent representation, we
introduced the real-time matrix representation of the contour-ordered Green™s func-
tions to describe non-equilibrium states. This allowed us to represent matrix Green™s
function perturbation theory in terms of Feynman diagrams in a standard fashion.
We introduced the physical representation corresponding to the two Green™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 Green™s function components, R, A, K, establishing the real rules. In this
way we were able to express how the di¬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 ¬eld theory from its Feynman diagrams, and show

<<

. 6
( 22)



>>