’

+

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