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+
1 5 4 3 2 1 1 5 2 1



Figure 4.9 Fourth-order diagrams in the coupling constant.


The ¬rst three diagrams in Figure 4.9 are solely the result of emission and absorption
of phonons by the electron or bosons by fermions in general. The last diagram is
the signature of the presence of the Fermi sea: a phonon can cause electron“hole
excitations, or in QED a photon can cause electron“positron pair creation. From the
boson point of view, such bubble-diagrams with additional decorations are basic, the
generic boson self-energy diagram, the self-energy being a quantity we introduce in
the next section.
Exercise 4.8. Obtain by brute force application of Wick™s theorem for the fermi and
phonon ¬eld strings the corresponding Feynman diagrams for the fermi propagator
to sixth order in the fermion“boson coupling.
Exercise 4.9. Obtain by brute force application of Wick™s theorem for the fermi and
phonon ¬eld strings the corresponding Feynman diagrams for the phonon propagator
to second order in the coupling.

The feature that the total combinatorial choice factor cancels the factorial factor,
1/n!, originating from the expansion of the exponential function is quite general. For
the N th order term
(’i)N †
Tr e’βH0 TC
(i) (i)
’i d2 . . . d(N + 1) HH0 (2) · · · HH0 (N + 1)ψH0 (1)ψH0 (1 )
N! C C

(4.134)
(i)
all connected combinations that di¬er only by permutations of HH0 give identical
contributions, thus canceling the factor 1/N ! in front, and as a consequence only
topologically di¬erent diagrams appear. This has a very important consequence for
diagrammatics, viz. that it allows separating o¬ sub-parts in a diagram and summing
them separately. We shall shortly return to this in the next section, and in much
more detail in Chapter 9.38
38 We note that the diagrammatic structure of amplitudes for quantum processes can be captured
in the two options: to interact or not to interact! The resulting Feynman diagrams being all
the topologically di¬erent ones constructable by the vertices and propagators of the theory. We
shall take this Shakespearian point of view as the starting point when we construct the general
diagrammatic and formal structure of quantum ¬eld theories in Chapter 9.
4.5. The self-energy 113


The diagrammatic representation of the perturbative expansion of the electron
Green™s function for the case of electron“phonon interaction, or in general the fermion
Green™s function for fermion“boson interaction, thus becomes



= +
x„ x„ x„ x„




+ +



···
+ + . (4.135)



In the perturbative expression for the contour-ordered Green™s function for the
case of electron“phonon interaction, each interaction contains one phonon ¬eld op-
erator, a fermion creation and annihilation ¬eld, all with the same contour time.
The Feynman diagrammatics is thus characterized by a vertex with incoming and
outgoing fermi lines and a phonon line, a three-line vertex.
The totality of diagrams can be captured by the following tool-box and instruc-
tions. With the diagrammatic ingredients, an electron propagator line, a phonon
propagator line and the electron“phonon vertex construct all possible topologically
connected diagrams. This is Wick™s theorem in the language of Feynman diagrams.
We recall that, whenever an odd number of fermi ¬elds are interchanged, a minus
sign appears. Diagrammatically this can be incorporated by the additional sign rule:
for each loop of fermi propagator lines in a diagram a minus sign is attributed. Ac-
companying this are the additional Feynman rules, which for our choices become
the following. In addition to the usual rule of the superposition principle: sum over
all internal labels, our conventions leads for fermion“boson interaction to the addi-
tional Feynman rule: a diagram containing n boson lines is attributed the factor
in g 2n (’1)F , where F is the number of closed loops formed by fermion propagators.


4.5 The self-energy
We have so far only derived diagrammatic formulas from formal expressions. Now
we shall argue directly in the diagrammatic language to generate new diagrammatic
expressions from previous ones, and thereby diagrammatically derive new equations.
In order to get a grasp of the totality of diagrams for the contour-ordered Green™s
function or propagator we shall use their topology for classi¬cation. We introduce
the one-particle irreducible (1PI) propagator, corresponding to all the diagrams that
can not be cut in two by cutting an internal particle line. In the following example
114 4. Non-equilibrium theory




1PI 1PR



(4.136)



the ¬rst diagram is one-particle irreducible, 1PI, whereas the second is one-particle
reducible, 1PR. Here we have used the diagrammatics for the impurity-averaged
propagator in a Gaussian random ¬eld instead of the analogous diagrammatics for
the electron“boson or electron“phonon interaction to illustrate that the arguments
are topological and valid for any type of interaction and its diagrammatics.39
Amputating the external propagator lines of the one-particle irreducible diagrams
(below displayed for the impurity-averaged propagator), we obtain the self-energy:



Σ(1, 1 ) ≡ 1 1
Σ




= +




+ +




···
+ + (4.137)


consisting, by construction, of all amputated diagrams that can not be cut in two by
cutting one bare propagator line.
39 For a detailed discussion of the impurity-averaged propagator, which is of interest in its own
right, we refer to Chapter 3 in reference [1].
4.5. The self-energy 115


We can now go on and uniquely classify all diagrams of the (impurity-averaged)
propagator according to whether they can be cut in two by cutting an internal particle
line at only one place, or at only two, three, etc., places. By construction we uniquely
exhaust all the possible diagrams for the propagator (the subscript is a reminder that
we are considering the contour-ordered Green™s function, but we leave it out from
now on)

GC (1, 1 ) =




= + Σ




+ Σ Σ




+ Σ Σ Σ




+ ··· . (4.138)



By iteration, this equation is seen to be equivalent to the equation40



= + (4.139)
Σ



and we obtain the Dyson equation

G(1, 1 ) = G0 (1, 1 ) + dx3 d„3 dx2 d„2 G0 (1, 3) Σ(3, 2) G(2, 1 ) (4.140)
C C
40 Inthe last term we can interchange the free and full propagator, because iterating from the left
generates the same series as iterating from the right.
116 4. Non-equilibrium theory


which we can write in matrix notation
G = G0 + G0 Σ G (4.141)
where signi¬es matrix multiplication in the spatial variable (and possible internal
degrees of freedom) and contour time variables. Arguing on the topology of the
diagrams has reorganized them and we have obtained a new type of equation.41

4.5.1 Non-equilibrium Dyson equations
The standard topological arguments of the previous section for diagrams organizes
them into irreducible sub-parts and we obtained the Dyson equation, Eq. (4.141).
We could of course have iterated Eq. (4.138) from the other side to obtain
G = G0 + G Σ G0 . (4.142)
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 (as discussed in detail in Section 5.6). However, in a non-equilibrium state,
the two equations contain di¬erent information and subtracting them is a useful way
of expressing the non-equilibrium dynamics as we shall exploit in Chapter 7.
Introduce the inverse of the free contour-ordered Green™s function, Eq. (4.141),
(G’1 G’1 )(1, 1 ) ,
G0 )(1, 1 ) = δ(1 ’ 1 ) = (G0 (4.143)
0 0

where
G’1 (1, 1 ) = G’1 (1) δ(1 ’ 1 ) (4.144)
0 0
and

G’1 (1) = ’ h(1) ,
i (4.145)
0
‚„1
where h denotes the single-particle Hamiltonian for the theory under consideration.
The two non-equilibrium Dyson equations, Eq. (4.141) and Eq. (5.69), can then be
expressed through operating with the inverse free contour-ordered Green™s function
from the left
(G’1 ’ Σ) G = δ(1 ’ 1 ) (4.146)
0
and from the right
(G’1 ’ Σ) = δ(1 ’ 1 ) .
G (4.147)
0
These two non-equilibrium Dyson equations will prove useful in Chapter 7 where
quantum kinetic equations are considered.42
By operating with the inverse of the free propagator, the explicit appearance of
the free propagator (or rather the non-interacting propagator since, possible external
¬elds can be included) has been removed. We can in fact remove its presence com-
pletely by expressing the self-energy in terms of skeleton diagrams where only the
full propagator appears, and we now turn to these.
41 The self-energy is just one out of the in¬nitely many one-particle irreducible vertex functions
which occur in a quantum ¬eld theory. Their signi¬cance will become clear when, in Chapter 10,
we encounter the usefulness of the e¬ective action.
42 If one, by the end of the day, in the Dyson equations uses the lowest-order approximation for the

self-energy, this whole venture into the diagrammatic jungle is hardly worthwhile since a civilized
Golden Rule calculation su¬ces.
4.5. The self-energy 117


4.5.2 Skeleton diagrams
So far we have only a perturbative description of the self-energy; i.e. we have a
representation of the self-energy as a functional of the free contour-ordered Green™s
function and the impurity correlator, Σ[G0 ]. For the case of fermion“boson inter-
action the self-energy is a functional of both types of free contour-ordered Green™s
functions, Σ[G0 , D0 ]. The self-energy, Σ, can in naive perturbation theory be de-
scribed as the sum of diagrams that can not be cut in two by cutting only one
internal free propagator line. In a realistic description of a physical system, we al-
ways need to invoke the speci¬cs of the problem in order to implement a controlled
approximation. To this end we must study the actual correlations in the system,
and it is necessary to have the self-energy expressed in terms of the full propagator.
Coherent quantum processes correspond to an in¬nite repetition of bare processes,
and the diagrammatic approach is precisely useful for capturing this feature, as ir-
reducible re-summations are easily described diagrammatically. In order to achieve
a description of the self-energy in terms of the full propagator, let us consider the
perturbative expansion of the self-energy.
For any given self-energy diagram in the perturbative expansion, Eq. (4.137),
we also encounter self-energy diagrams with all possible self-energy decorations on
internal lines; for example, for the case of particles in a random potential:




’ + ···
+ +




+ ···
= + Σ




’ + Σ
118 4. Non-equilibrium theory




···
+ +
Σ Σ




= . (4.148)


We can uniquely classify all these self-energy decorations in the perturbative expan-
sion according to whether the particle line can be cut into two, three, or more pieces
by cutting particle lines (the step indicated by the second arrow in Eq. (4.148)). We
can therefore partially sum the self-energy diagrams according to the unique pre-
scription: for a given self-energy diagram, remove all internal self-energy insertions,
and substitute for the remaining bare particle propagator lines the full propaga-
tor lines.43 Through this partial summation of the original perturbative expansion
of the self-energy only so-called skeleton diagrams containing the full propagator
will then appear, i.e. Σ[G]. Since in the skeleton expansion we have removed self-
energy insertions (decorations), which allowed a 1PI self-energy diagram to be cut
in two by cutting two lines, we can characterize the skeleton expansion of the self-
energy as the set of skeleton diagrams that can not be cut in two by cutting two
lines (2PI-diagrams). Since propagator and impurity correlator lines, or say phonon
lines, appear topologically equivalently, we can restate quite generally: the skeleton
self-energy expansion consists of all the two-line or two-particle irreducible skeleton
diagrams.
By construction, only self-energy skeleton diagrams that can not be cut in two
by cutting only two full propagator lines appear, and we have the partially summed
diagrammatic expansion for the self-energy:




Σ(1, 1 ) = 1 1




+ 1 1

43 Synonymous names for the full Green™s function or propagator are renormalized or dressed
propagator.
4.6. Summary 119




+ 1 1




···
+ + . (4.149)
1 1




The partial summation of diagrams is unique, since the initial and ¬nal impurity
correlator lines are attached internally in di¬erent ways in each class of summed
diagrams. No double counting of diagrams thus takes place owing to the di¬erent
topology of the skeleton self-energy diagrams, and all diagrams in the perturbative
expansion of the self-energy, Eq. (4.137), are by construction contained in the skeleton
diagrams of Eq. (4.149).
What has been achieved by the partial summation, where each diagram corre-
sponds to an in¬nite sum of terms in perturbation theory, is that the self-energy is
expressed as a functional of the exact propagators or full Green™s functions

Σ(1, 1 ) = Σ(1,1 ) [G, D] . (4.150)
We can continue this topological classi¬cation, and introduce the higher-order
vertex functions; however, we defer this until Chapter 9.

Exercise 4.10. Draw the rest, in Eq. (4.149), of the four skeleton self-energy dia-
grams with three impurity correlators.
Exercise 4.11. Draw the skeleton self-energy diagrams for fermion“boson interac-
tion to fourth order in the coupling.


4.6 Summary
We have presented the formalism needed for treating general non-equilibrium situ-
ations. The closed time path formalism was shown to facilitate a convenient and
compact treatment of non-equilibrium statistical Green™s functions. Perturbation
theory valid for non-equilibrium states turned out in standard fashion, re¬‚ecting a
general Wick theorem for closed time path strings of operators, and the Feynman
diagrams for the contour ordered Green™s functions become of standard form. For
the reader with knowledge of equilibrium theory the good news is thus that the gen-
eral non-equilibrium formalism is formally equivalent to the equilibrium theory if
one elevates time to the contour level. For the reader not familiar with equilibrium
theory the good news is rejoice: knowledge of equilibrium theory is not needed, since
the equilibrium case is just a special simple case of the presented general theory.
However, the apparatus of the closed time path formalism needs a physical inter-
pretation, we need to get back to real time. In the next chapter we shall introduce
120 4. Non-equilibrium theory


the real-time technique and develop the diagrammatic structure of non-equilibrium
theory in a physically appealing language.
5

Real-time formalism

The contour-ordered Green™s function considered in the previous chapter was ideal
for discussing general closed time path properties such as the perturbative diagram-
matic structure for non-equilibrium states. However, the contour-ordered Green™s
function lacks physical transparency and does not appeal to intuition.1 We need a
di¬erent approach, which brings quantities back to real time. To accomplish this we
introduce a representation where forward and return parts of the closed time path
are ordered by numbers, specifying the position of a contour time by two indices,
i = 1, 2. Next is the diagrammatic perturbation theory in the real-time technique
then formulated in a fashion where the aspects of non-equilibrium states emerge in
the physically most appealing way. In particular, we shall construct the representa-
tion where spectral properties and quantum statistics show up on a di¬erent footing
in the diagrams. Lastly, we consider the connection to the imaginary-time treatment
of non-equilibrium states, and establish its equivalence to the real-time approach

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