and equivalently for higher order terms. The term where the external points are

paired, giving rise to a disconnected or unlinked diagram with a vacuum diagram

contribution, clearly vanishes owing to the integration along both the forward and

return parts of the contour, giving two terms di¬ering only by a minus sign.

The generic component in a diagram, the ¬rst order term, is graphically repre-

sented by the diagram

(1)

GC (x, „ ; x , „ ) = (4.111)

x„ x1 „1 x„

where a cross has been introduced to symbolize the interaction of the particles with

the scalar potential

≡ V (x, „ ) (4.112)

x„

and a thin line is used to represent the zeroth-order or free thermal equilibrium

contour-ordered Green™s function

R (0)

≡ GC (x, „ ; x , „ ) (4.113)

x„ x„

in order to distinguish it from the contour-ordered Green™s function in the presence

4.4. Non-equilibrium diagrammatics 105

of the potential V , the full contour-ordered Green™s function

≡ GC (x, „ ; x , „ ) (4.114)

x„ x„

depicted as a thick line.

With this dictionary or stenographic rules, the analytical form, Eq. (4.110), is ob-

tained from the diagram, Eq. (4.111), since integration is implied over the variables

of the internal points where interaction with the potential takes place. The only dif-

ference to equilibrium standard Feynman diagrammatics is that internal integrations

are not over time or imaginary time, but over the contour variable.

The second-order expression in perturbation theory leads to two terms giving

identical contributions, since interchange of pairs of fermion operators introduces no

factor of ’1. The resulting factor of two exactly cancels the factor of two originat-

ing from the expansion of the exponential in Eq. (4.109). This feature repeats for

the higher-order terms, and for particles interacting with a scalar potential V (x, t),

we have the following diagrammatic representation of the contour-ordered Green™s

function:

GC (x, „ ; x , „ ) = = x„ x„

x„ x„

+ +

x„ x1 „1 x„ x„ x2 „2 x1 „1 x„

+ + ... , (4.115)

x„ x3 „3 x2 „2 x1 „1 x„

where all ingredients now represent contour quantities according to the above dictio-

nary.

Exercise 4.7. Show that for a particle coupled to a scalar potential V (x, t), the

in¬nite series

d„2 G0 (1, 2) V (x2 , „2 ) G0 (2, 1 ) + · · ·

G(1, 1 ) = G0 (1, 1 ) + dx2 (4.116)

c

by iteration can be captured in the Dyson equation

G(1, 1 ) = G0 (1, 1 ) = dx2 d„2 G0 (1, 2) V (x2 , „2 ) G(2, 1 ) , (4.117)

c

which has the diagrammatic representation

= + . (4.118)

x1 t1 x1 t1 x1 t1 x1 t1 x1 t1 x1 t1

x2 t2

106 4. Non-equilibrium theory

If in Eq. (4.117) we operate with the inverse free contour ordered Green™s function

which satis¬es (recall Exercise 4.1 on page 89)

dx2 d„2 G’1 (1, 2) G0 (2, 1 ) = δ(1 ’ 1 ) (4.119)

0

c

we obtain

dx2 d„2 (G’1 (1, 2) ’ V (2)) G(2, 1 ) = δ(1 ’ 1 ) . (4.120)

0

c

As expected, the coupling to a classical ¬eld can be accounted for by adding the

potential term to the free Hamiltonian. The δ-function contains, besides products

in δ-functions in the degrees of freedom, the contour variable δ-function speci¬ed

in Eq. (4.45). We shall write the equation, absorbing the potential in the inverse

propagator, in condensed matrix notation

(G’1 G’1 )(1, 1 )

G)(1, 1 ) = δ(1 ’ 1 ) = (G (4.121)

0 0

where signi¬es matrix multiplication in the spatial variable (and possible inter-

nal degrees of freedom) and contour time variables. The latter, adjoint, equation

corresponds to the choice of iterating from the left instead of the right.

4.4.2 Particles coupled to a stochastic ¬eld

If the potential V (x, t) of the previous section is treated as a stochastic Gaussian

random variable (with zero mean), the diagrams in perturbation theory, Eq. (4.115),

will be turned into the diagrams for the averaged Green™s function according to

the prescription: pair together pairwise potential crosses in all possible ways and

substitute for the paired crosses the Gaussian correlator of the stochastic variable.

For the lowest order contribution to the averaged contour ordered Green™s function

we thus have the diagram

(2)

GC (1, 1 ) =

(4.122)

where the outermost labels 1 and 1 as well as the internal labels 2 and 3 are sup-

pressed, and the following notation has been introduced for the correlator:

x,„

= V (x, „ ) V (x , „ ) . (4.123)

x ,„

4.4. Non-equilibrium diagrammatics 107

If the stochastic variable is taken as time independent, V (x), we cover the case of

particles in a random impurity potential (treated in the Born approximation), and

the correlator, the impurity correlator, is given by

V (x)V (x ) = ni dr Vimp (x ’ r) Vimp (x ’ r) . (4.124)

where Vimp (x) is the potential created at position x by a single impurity at the origin,

and ni is their concentration.34

4.4.3 Interacting fermions and bosons

The next level of complication is the important case of interacting fermions and

bosons. Let us look at the generic fermion“boson interaction, Eq. (2.71), or equiva-

lently, the jellium electron“phonon interaction, and let the ψ-¬eld denote the fermi

¬eld in the Green™s function we are looking at

(i )

e’i †

d„ HH („ )

G(1, 1 ) = ’i Tr ρ0 TC ψH0 (1) ψH0 (1 ) . (4.125)

C 0

Here the contour C is either the real-time contour of Figure 4.5, or the general contour

of Figure 4.4.35

Expanding the exponential we get terms in increasing order of the coupling con-

stant. The zeroth-order term just gives the free or thermal equilibrium contour-

ordered Green™s function, say for fermions, Eq. (4.57). The term linear in the phonon

or boson ¬eld vanishes as discussed in Section 4.3.3, and we consider the second-order

term36

(’i)2 †

(i) (i)

= ’iTr ρ0 TC

G(2) (1, 1 ) d„3 HH0 („3 ) d„2 HH0 („2 ) ψH0 (1) ψH0 (1 )

2! C C

i2 †

g d„3 d„2 dx3 dx2 Tr e’βH0 TC ψH0 (x3 , „3 )ψH0 (x3 , „3 )φH0 (x3 , „3 )

=

2! C

† †

— ψH0 (x2 , „2 ) ψH0 (x2 , „2 ) φH0 (x2 , „2 ) ψH0 (1) ψH0 (1 ) . (4.126)

The expression has the form of a string of fermi and bose operators subdued to the

contour-ordering operation and thermally weighted by the Hamiltonian for the free

¬elds which is Gaussian. The trace over these independent degrees of freedom splits

34 Fordetails on quenched disorder and impurity averaging see Chapter 3 of reference [1].

35 For the general contour of Figure 4.4, we should recall the cancelation of the disconnected

diagrams against the vacuum diagrams of the denominator. However, the uninitiated reader need

not worry about this by adopting the closed real-time contour. For the general case, the proof of

cancelation can be consulted in Section 9.5.2.

36 The use of states with a non-de¬nite number of bosons, as useful in the theory of Bose“Einstein

condensation, will be discussed in Section 10.6.

108 4. Non-equilibrium theory

into a product of two separate traces containing only fermi or bose ¬elds weighted

(0) (0)

by their respective free ¬eld Hamiltonians, H0 = Hf + Hb . Higher-order terms in

the expansion have the same form, they just contain strings with a larger number of

¬elds. In perturbation theory the task is to evaluate such terms, or rather ¬rst break

them down into Gaussian products as accomplished by Wick™s theorem.

Consider the expression in Eq. (4.126), and perform the following choice of pair-

†

ings: the creation fermi ¬eld indexed by the external label 1 , ψH0 (1 ), is paired with

the annihilation ¬eld associated with an internal point whose creation ¬eld is paired

with the annihilation ¬eld associated with the other internal point, thereby ¬xing

the ¬nal fermion pairing. Since the internal points represents dummy integrations

this kind of choice gives rise to two identical expressions, an observation that can

be used to cancel the factorial factor, 1/2!, originating from the expansion of the

exponential function in Eq. (4.125). The string of boson or phonon ¬elds contains

only two ¬elds simply leading to the appearance of their contour-ordered thermal

average. The considered second-order expression from the Wick decomposition for

the contour-ordered fermion Green™s functions thus becomes

(2) (0) (0) (0) (0)

GC (1, 1 ) ’ ig 2 dx3 d„3 dx2 d„2 GC (1, 3) GC (3, 2) DC (3, 2) GC (2, 1 ) .

C C

(4.127)

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 function.

The next step is then to visualize these unwieldy expressions arising in perturba-

tion theory in terms of diagrams and a few stenographic rules, the Feynman rules.

The considered second-order term in the coupling constant, Eq. (4.127), can be rep-

resented by the ¬rst diagram in Figure 4.6.

,

Figure 4.6 Lowest order fermion“boson diagrams.

4.4. Non-equilibrium diagrammatics 109

Here the straight line represents the free or thermal equilibrium contour ordered

fermion Green™s function and the wavy line represents the thermal equilibrium contour-

ordered boson Green™s function:

(0)

≡ DC (x, „ ; x , „ ) (4.128)

x„ x„

i.e.

DC (1, 1 ) = ’i trb (ρb Tc (φH0 (1) φ† 0 (1 ))) = ’i Tc (φH0 (1) φ† 0 (1 ))

(0) (0)

(4.129)

H H

(0)

as trb denotes the trace with respect to the boson degrees of freedom and ρb is the

thermal equilibrium statistical operator for the free bosons. As a Feynman rule, each

vertex carries a factor of the coupling constant.

Another decomposition according to Wick™s theorem of the second-order expres-

sion in Eq. (4.126) corresponds to when the fermi ¬eld indexed by the external label

†

1 , ψH0 (1 ), is paired with the annihilation ¬eld associated with an internal point and

the creation ¬eld of that vertex is paired with the ¬eld corresponding to the external

point 1, thereby ¬xing the ¬nal fermion pairing, and again giving rise to two identical

expressions, which in this case are the expression

(2) (0) (0) (0) (0)

GC (1, 1 ) ’ ’ig 2 dx3 d„3 dx2 d„2 GC (1, 2) GC (2, 1 ) DC (3, 2) GC (3, 3) .

C C

(4.130)

The corresponding expression can, according to the above dictionary for Feynman

diagrams, be represented by the second diagram in Figure 4.6. We note the relative

minus sign compared with the term represented by the ¬rst diagram in Figure 4.6

that re¬‚ects a general feature, which in diagrammatic terms can be stated as the

Feynman rule: associated with a closed loop of fermion propagators is a factor of

minus one.

The considered expressison corresponding to the second diagram in Figure 4.6

contains the fermion contour-ordered Green™s function taken at equal contour times,

(0)

GC (x, „ ; x, „ ), and therefore needs interpretation. Recalling that the annihilation

¬eld occurs to the right of the creation ¬eld originally in the interaction Hamiltonian,

and labeling the contour variable of the latter by „ , we then have for the contour

c

variables of these ¬elds „ < „ , and the propagator closing on itself represents the G-

lesser Green™s function, G< (x, „ ; x, „ ), corresponding to the density of the fermions.

0

This is indicated by the direction of the arrow on the propagator closing on itself in

the second diagram in Figure 4.6.

110 4. Non-equilibrium theory

The ¬nal option for pairings in the Wick decomposition of the second-order ex-

pression in Eq. (4.126) corresponds to pairing the fermi creation ¬eld indexed by the

†

external label 1 , ψH0 (1 ), with the annihilation ¬eld indexed by the external label

1, ψH0 (1). The pairings of the fermi ¬elds labeled by the internal points can again

be done in a two-fold way, and the corresponding expression arises

(2) (0) (0) (0) (0)

GC (1, 1 ) ’ ’ig 2 GC (1, 1 ) dx3 d„3 dx2 d„2 GC (3, 2) DC (3, 2) GC (2, 3) ,

C C

(4.131)

which can be represented by the diagram depicted in Figure 4.7.

1 1

Figure 4.7 Unlinked or second-order vacuum diagram contribution to GC .

The vacuum bubble gives a vanishing overall factor owing to forward and return

contour integrations canceling each other for the case of the real-time closed contour.

The expression corresponding to the second diagram in Figure 4.6 vanishes for

the case of electron“phonon interaction as it contains an overall factor that vanishes.

Letting x, „ represent the internal point where the fermi propagator closes on itself

(representing the quantity G< (x, „ ; x, „ ), the free fermionic density which is inde-

0

pendent of the variables), the term involving the phonon propagator then becomes

dx D0 (x, „ ; x , „ ) = 0 (4.132)

since the integrand is the divergence of a function with a vanishing boundary term.37

The second-order contribution in the electron“phonon coupling to the contour-

ordered electron Green™s function is thus represented by the diagram depicted in

Figure 4.8.

37 Thus the theory does not contain any so-called tadpole diagrams, which is equivalent to the

vanishing of the average of the phonon ¬eld. In the Sommerfeld treatment of the Coulomb interaction

in a pure metal, tadpole or Hartree diagrams are also absent, though for a di¬erent reason. They

are canceled by the interaction with the homogeneous background charge (recall Exercise 2.12 on

page 44).

4.4. Non-equilibrium diagrammatics 111

1 3 2 1

Figure 4.8 Second-order contribution to GC for the electron“phonon interaction.

We observe the usual Feynman rule expressing the superposition principle: inte-

grate over all internal space points (and sum over all internal spin degrees of freedom)

and integrate over the internal contour time variable associated with each vertex. In

addition we have the Feynman rule: only topologically di¬erent diagrams appear; in-

terchange of internal dummy integration variables has been traded with the factorial

from the exponential function.

The next non-vanishing term will, according to Wick™s theorem for a string of bose

¬elds, be the fourth order term for the fermion“boson coupling, and the expression

(’i)4

(2) (i) (i)

’i Tr ’ρ0 TC

GC (1, 1 ) = d2 . . . d5 HH0 (2) HH0 (3)

4! C C

†

(i) (i)

— HH0 (4) HH0 (5) ψH0 (1) ψH0 (1 ) (4.133)

needs to be Wick de-constructed. To get the diagrammatic expression for this term

plot down four dots on a piece of paper representing the four internal points in

the fourth-order perturbative expression; label them 2, 3, 4 and 5, and the two

external states, 1 and 1 . Attach at each internal dot, or vertex, a wiggly stub

and incoming and outgoing stubs representing the three ¬eld operators for each

interaction Hamiltonian. To get connected diagrams (the unlinked diagrams again

vanish owing to the vanishing of vacuum bubbles) we proceed as follows. The external

†

¬eld ψH0 (1 ) can be paired with any of the fermi annihilation ¬elds associated with

the internal points, giving rise to four identical contributions since the internal points

represent dummy integration variables. The creation ¬eld emerging from this point

can be paired with annihilation ¬elds at the remaining three vertices, giving rise to

three identical contributions, and the creation ¬eld emerging from this vertex has

two options: either connecting to one of the two remaining internal vertices or to the

external point. In both cases, two identical terms arise, thereby canceling the overall

factor from the expansion of the exponential function, 1/4!, in Eq. (4.133). In the

latter case, the factor of two occurs because of the two-fold way of pairing the boson

¬elds, and this latter term is thus, according to the above dictionary, represented

by the last diagram in Figure 4.9. Pairing the boson ¬elds for the former case gives

three di¬erent contributions as represented by the ¬rst three topologically di¬erent

diagrams in Figure 4.9.

112 4. Non-equilibrium theory

+

1 5 4 3 2 1 1 5 4 3 2 1

4 3