ńņš. 5 |

or for the grand canonical ensemble, substituting in Eq. (4.62) Hb ā’ Hb ā’ Ī¼b Nb

(i.e. we measure energies from the chemical potential, Ļq = q ā’ Ī¼) and we have

introduced the notation

. . . = tr(ĻT . . .) (4.63)

where tr denotes trace with respect to the bose species under consideration. As in

Eq. (4.60) we suppress whenever possible reference to the, for argumentā™s sake, irrele-

vant state labels, here momentum or wave vectors (and possibly spin and longitudinal

and transverse phonon labels).

The contour ordering symbol, TC , orders the operators according to their position

on the contour C (earlier contour positions orders operators to the right) so that, for

example, for two bose operators indexed by contour times Ļ„ and Ļ„

ā§

āØ c(Ļ„ ) c(Ļ„ ) for Ļ„ >C Ļ„

TC (c(Ļ„ ) c(Ļ„ )) = (4.64)

ā©

c(Ļ„ ) c(Ļ„ ) for Ļ„ >C Ļ„

25 Inthe following the contour C can be the real-time contour depicted in Figure 4.5 or the contour

depicted in Figure 4.4, allowing us to include the general case of transient phenomena.

26 Although the Hamiltonian contains ļ¬elds at equal times, we can in the course of the argument

assume them inļ¬nitesimally split, and all the contour time variables can thus be considered diļ¬erent.

96 4. Non-equilibrium theory

where the upper identity is for contour time Ļ„ being further along the contour than

Ļ„ and the lower identity being the ordering for the opposite case (for the fermi case,

we should remember the additional minus sign for interchange of fermi operators).

Such an ordered expression of bose ļ¬eld operators as in Eq. (4.60) can now be

decomposed according to Wickā™s theorem, which relies only on the simple property

[cq , ĻT ] = ĻT cq [exp{Ī»c Ļq /kB T } ā’ 1] (4.65)

valid for a Hamiltonian quadratic in the bose ļ¬eld (Ī»c = Ā±1, depending upon whether

cq is a creation or an annihilation operator for state q). We now turn to prove Wickā™s

theorem, which is the statement that the quadratically weighted trace of a contour-

ordered string of creation and annihilation operators can be decomposed into a sum

over all possible pairwise products

TC (c(Ļ„n ) c(Ļ„nā’1 ) . . . c(Ļ„2 ) c(Ļ„1 )) = Tct (cq (Ļ„ ) cq (Ļ„ )) (4.66)

a.p.p. q,q

where the sum is over all possible ways of picking pairs (a.p.p.) among the n operators,

not distinguishing ordering within pairs. Equivalently, Wickā™s theorem states that

the trace of a contour-ordered string of creation and annihilation operators weighted

with a quadratic Hamiltonian has the Gaussian property. The expressions on the

right are free thermal equilibrium contour-ordered Greenā™s functions, quantities for

which we have explicit expressions.

Before proving Wickā™s theorem and the relation Eq. (4.65), we ļ¬rst observe some

preliminary results. Diļ¬erent q-labels describe diļ¬erent momentum degrees of free-

dom, so operators for diļ¬erent qs commute, and algebraic manipulations with com-

muting operators are just as for usual numbers giving for example

ĻT ,

ĻT = (4.67)

q

q

where we have introduced the thermal statistical operator for each mode

ĻT = zq eā’hq /kT

ā’1

(4.68)

q

and the partition function for the single mode

1

zq = . (4.69)

1 ā’ eā’Ļq /kT

The independence of each mode degree of freedom, as expressed by the commutation

of operators corresponding to diļ¬erent degrees of freedom, gives

ā ā

cq Ļ T = ā Ļ T ā cq Ļ T . (4.70)

q q

q (=q)

Now, using the commutation relations for the creation and annihilation operators we

have

cq hq = (hq ā’ Ī»c Ļq ) cq , (4.71)

4.3. Closed time path formalism 97

where

cq = aā

+1 for q

Ī»c = (4.72)

ā’1 for cq = a q .

Using Eq. (4.71) repeatedly gives

cq hn = (hq ā’ Ī»c Ļq )n cq (4.73)

q

and upon expanding the exponential function and re-exponentiating we can commute

through to get

cq ĻT = eĪ»c Ļq /kT ĻT cq (4.74)

so that for the commutator of interest we have the property stated in Eq. (4.65).

We then prove for an arbitrary operator A that in the bose case

= (1 ā’ eĪ»c Ļq /kT ) cq A

[cq , A] (4.75)

as we ļ¬rst note, by using the cyclic invariance property of the trace, that

= ā’tr([cq , ĻT ] A)

[cq , A] (4.76)

and then by using Eq. (4.65) we get Eq. (4.75).

Exercise 4.2. Show that for the case of fermions

{cq , A} = (1 + eĪ»c Ļq /kT ) cq A . (4.77)

Employing Eq. (4.75) with A = 1, 1, a, aā , respectively, we observe that all the

following averages vanish

aā (t) aā (t)aā (t )

0= a(t) = = a(t)a(t ) = (4.78)

and as a consequence the average value of the interaction energy vanishes, Hi (t) =

0, for the case of fermionā“boson interaction (and electronā“phonon interaction). These

equalities are valid for any state diagonal in the total number of particles, i.e. a state

with a deļ¬nite number of particles.

Repeating the algebraic manipulations leading to Eq. (4.74), or by analytical

continuation of the result, we have

(0) (0)

cq (t) = cq eā’itHb = eiĪ»c Ļq t eā’itHb cq (4.79)

from which we get that the creation and annihilation operators in the interaction

picture have a simple time dependence in terms of a phase factor

(0) (0)

cq (t) = eitHb cq eā’itHb = cq eiĪ»c Ļq t . (4.80)

The commutators formed by creation and annihilation operators in the interaction

picture are thus c-numbers, the only non-vanishing one being speciļ¬ed by

[aq (t), aā (t )] = Ī“q,q eā’iĻq (tā’t ) . (4.81)

q

98 4. Non-equilibrium theory

According to Eq. (4.75) we thereby have

aq (t) aā (t ) (1 ā’ eā’Ļq /kT )ā’1 [aq (t), aā (t )]

=

q q

Ī“q,q (1 ā’ eā’Ļq /kT )ā’1 eā’iĻq (tā’t )

=

Ī“q,q (n(Ļq ) + 1)eā’iĻq (tā’t )

=

ā” >

i Dqq (t, t ) , (4.82)

where the Boseā“Einstein distribution appears as speciļ¬ed by the Bose function

1 1

n(Ļq ) = = . (4.83)

q ā’Ī¼b )/kT

eĻq /kT ā’ 1 ā’1

e(

Exercise 4.3. Show that, for the opposite ordering of the creation and annihilation

operators, the correlation function is

aā (t ) aq (t) = tr(ĻT aā (t ) aq (t))

ā”

<

i Dqq (t, t ) q q

= n(Ļq ) Ī“q,q eā’iĻq (tā’t ) . (4.84)

Exercise 4.4. Show that, for the case of fermi operators, the correlation functions

are

i aā (t ) aq (t) = itr(ĻT aā (t ) aq (t))

G< (t, t ) ā”

qq q q

if ( q ) Ī“q,q eā’i q (tā’t )

= (4.85)

and

ā” ā’i aq (t) aā (t ) = ā’itr(ĻT aq (t) aā (t ))

G> (t, t )

qq q q

= ā’i(1 ā’ f ( q )) Ī“q,q eā’i q (tā’t )

, (4.86)

where f ( q ) is the Fermi function

1

f ( q) = . (4.87)

q ā’Ī¼)/kT

e( +1

Exercise 4.5. Show that, for the case of fermi operators,

{aq (t), aā (t )} = Ī“q,q eā’i q (tā’t )

. (4.88)

q

If the string S, Eq. (4.60), contains an odd number of operators, the expression

equals zero since the expectation value is with respect to the thermal equilibrium

4.3. Closed time path formalism 99

state.27 For an odd number of operators we namely encounter a matrix element

between states with diļ¬erent number of particles or quanta; for example,

ā

aq aā aq = Z ā’1 eā’E({nq }q )/kT ( nq )3 nq |nq ā’ 1 = 0 , (4.89)

q

{nq }q

which is zero by orthogonality of the diļ¬erent energy eigenstates.

As an example of using Wickā™s theorem we write down the term we encounter at

fourth order in the coupling to the bosons (we suppress, for the present consideration,

the immaterial q labels)

tr(ĻT Tct (a(Ļ„1 )aā (Ļ„2 )a(Ļ„3 )aā (Ļ„4 ))) Tct (a(Ļ„1 )aā (Ļ„2 )) Tct (a(Ļ„3 )aā (Ļ„4 ))

=

Tct (a(Ļ„1 )aā (Ļ„4 )) Tct (a(Ļ„3 )aā (Ļ„2 )) .

+

(4.90)

Here we have deleted terms that do not pair creation and annihilation operators,

because such terms, just as above, lead to matrix elements between orthogonal states:

Tct (aā (Ļ„ )aā (Ļ„ )) .

Tct (a(Ļ„ )a(Ļ„ )) =0= (4.91)

At the fourth-order level the ordered Gaussian decomposition can of course be ob-

tained by noting that only by pairing equal numbers of creation and annihilation op-

erators can the number of quanta stay conserved and the matrix element be nonzero

as we have the expression

tr(ĻT Tct (a(Ļ„1 )aā (Ļ„2 )a(Ļ„3 )aā (Ļ„4 )))

eā’E({nq }q )/kT {nq }q |Tct (a(Ļ„1 )aā (Ļ„2 )a(Ļ„3 )aā (Ļ„4 ))|{nq }q .

= (4.92)

{nq }q

Wickā™s theorem is the generalization of this simple observation.

We now turn to the general proof of Wickā™s theorem for the considered case of

bosons. Wickā™s theorem is trivially true for N = 1 (and for N = 2 according to the

above consideration), and we now turn to prove Wickā™s theorem by induction. Let

us therefore consider an N -string with 2N operators

SN = TC (c(Ļ„2N ) c(Ļ„2N ā’1 ) . . . c(Ļ„2 ) c(Ļ„1 )) . (4.93)

We can assume that the contour-time labeling already corresponds to the contour-

ordered one, since the bose operators can be moved freely around under the contour

27 This would not be the case for, say, photons in a coherent state in which case the substitution

c ā’ c ā’ c is needed. Also in describing a Boseā“Einstein condensate it is convenient to work with

a superposition of states containing a diļ¬erent number of particles so that c is non-vanishing, a

situation we shall deal with in due time. For the case of electronā“phonon interaction we thus assume

no linear term in the phonon Hamiltonian, which would correspond to a displaced oscillator, or that

such a term is eļ¬ectively removed by redeļ¬ning the equilibrium position of the oscillator.

100 4. Non-equilibrium theory

ordering, or otherwise we just relabel the indices, and we have28

2N ā’1

2N

SN = c(Ļ„n ) = c(Ļ„2N ) c(Ļ„n ) . (4.94)

n=1 n=1

We then use the above proved relation, Eq. (4.75), to rewrite

2N ā’1

ā’1

1ā’e Ī»c Ļq /kT

SN = [c(Ļ„2N ), c(Ļ„n )] . (4.95)

n=1

In the ļ¬rst term in the commutator we commute c(Ļ„2N ) to the right

2N ā’1 2N ā’2 2N ā’2

c(Ļ„2N ), c(Ļ„n ) = c(Ļ„2N ā’1 ) c(Ļ„2N ) c(Ļ„n ) + [c(Ļ„2N ), c(Ļ„2N ā’1 )] c(Ļ„n )

n=1 n=1 n=1

2N ā’1

ā’ c(Ļ„n ) c(Ļ„2N ) . (4.96)

n=1

We now keep commuting c(Ļ„2N ) through in the ļ¬rst term repeatedly, each time

generating a commutator, and eventually ending up with canceling the last term in

Eq. (4.96), so that

2N ā’1 2N ā’1 2N ā’1

c(Ļ„2N ), c(Ļ„n ) = [c(Ļ„2N ), c(Ļ„n )] c(Ļ„m ) . (4.97)

m =1

n=1 n=1

m (=n )

Then we use that the commutator is a c-number, which according to Eq. (4.75) we

can rewrite as

1 ā’ eĪ»c q Ļq /kT

[cq (Ļ„2N ), cq (Ļ„n )] = Ī“q,q cq (Ļ„2N ) cq (Ļ„n ) (4.98)

and being a c-number it can be taken outside the thermal average in Eq. (4.95), and

we obtain

2N ā’1 2N ā’1

SN = c(Ļ„2N ) c(Ļ„n ) c(Ļ„m )

m =1

n=1

m (=n )

ā ā

2N ā’1 2N ā’1

ā ā

T ct ā c(Ļ„m )ā

= Tct (c(Ļ„2N ) c(Ļ„n )) , (4.99)

m =1

n=1

m (=n )

where we reintroduce the contour ordering. By assumption the second factor can

be written as a sum over all possible pairs (on a.p.p.-form), and by induction the N

28 For

fermions interchange of ļ¬elds involves a minus sign, and an overall sign factor occurs, (ā’1)Ī¶ P ,

where Ī¶P is the sign of the permutation P bringing the string of ļ¬elds to a contour time-ordered

form.

4.3. Closed time path formalism 101

case is then precisely seen to be of that form too. We note, that to prove Wickā™s

theorem we have only exploited that the weight was a quadratic form, leaving the

commutator a c-number.29

The contour label uniquely speciļ¬es from which term in the spatial representation

of the bose ļ¬eld it originates, and since Eq. (4.99) is valid for both creation and

annihilation operators, and therefore for any linear combinations of such, we have

therefore shown30

TC (Ļ(x2n , Ļ„2n ) Ļ(x2nā’1 , Ļ„2nā’1 ) . . . Ļ(x2 , Ļ„2 ) Ļ(x1 , Ļ„1 ))

TC (Ļ(xi , Ļ„i )Ļ(xj , Ļ„j )) ā” iN D0 (xi , Ļ„i ; xj , Ļ„j ) .(4.100)

=

a.p.p. i=j a.p.p. i=j

The index on the contour-ordered Greenā™s functions indicates they are the free ones.

Performing the trace over a string of bose operators weighted by a quadratic form

therefore corresponds to pairing the operators together pairwise in all possible ways.31

For the case of fermi operators, the proof of Wickā™s theorem runs analogous to

the above, in fact the bose and fermi cases can be handled in unison if we unite

Eq. (4.75) and Eq. (4.77) by introducing the notation

1 + s eĪ»c Ļq /kT

[cq , A]s = cq A , (4.101)

where s = ā“ signiļ¬es the case of bose and fermi statistics, respectively. The argu-

ments relating Eq. (4.94) to Eq. (4.106) run identical with commutators replaced by

anti-commutators and a minus sign, or for treating the two cases simultaneous s is

introduced. For the combined case we have

2N ā’1

2N

SN = c(Ļ„n ) = c(Ļ„2N ) c(Ļ„n )

n=1 n=1

2N ā’1

ā’1

Ī»c (Ļ„ 2N ) Ļq /kT

= 1 + se [c(Ļ„2N ), c(Ļ„n )]s (4.102)

n=1

and

2N ā’1 2N ā’2

ā’s(c(Ļ„2N ā’1 ) c(Ļ„2N ) ā’ s[c(Ļ„2N ), c(Ļ„2N ā’1 )]s )

[c(Ļ„2N ), c(Ļ„n )]s = c(Ļ„n )

n=1 n=1

2N ā’1

+s c(Ļ„n ) c(Ļ„2N )

n=1

29 Ifthe weight was not quadratic, we would have encountered correlations that must be handled

additionally.

30 A reader familiar with the standard T = 0 or ļ¬nite temperature imaginary-time Wickā™s theorem,

will recognize that their validity just represents special cases of the above proof.

31 The presented general version of Wickā™s theorem is capable of dealing with many-body systems

of bosons, irrespective of the absence or presence of a Boseā“Einstein condensate, if one employs the

grand canonical ensemble.

102 4. Non-equilibrium theory

2N ā’2

= (ā’s) c(Ļ„2N ā’1 ) c(Ļ„2N ) c(Ļ„n )

n=1

2N ā’2

+ [c(Ļ„2N ), c(Ļ„2N ā’1 )]s c(Ļ„n )

n=1

2N ā’1

+s c(Ļ„n ) c(Ļ„2N ) , (4.103)

n=1

where the (anti- or) commutator, being a c-number, can be taken outside the operator

averaging. We now keep (anti- or ) commuting c(Ļ„2N ) through in the ļ¬rst term

repeatedly, each time generating a (anti- or) commutator and a factor (-s), and

eventually ending up with canceling the last term, so that

2N ā’1 2N ā’1 2N ā’1

nā’1

c(Ļ„2N ), c(Ļ„n ) = (ā’s) [c(Ļ„2N ), c(Ļ„n )]s c(Ļ„m ) . (4.104)

m =1

n=1 n=1

s m (=n )

Then we use the fact that the (anti- or ) commutator is a c-number, which we can

rewrite

1 + s eĪ»c q Ļq /kT

[cq (Ļ„2N ), cq (Ļ„n )]s = Ī“q,q cq (Ļ„2N ) cq (Ļ„n ) (4.105)

and taking it outside the thermal average we obtain

2N ā’1 2N ā’1

SN = c(Ļ„2N ) c(Ļ„n ) c(Ļ„m )

m =1

n=1

m (=n )

ā ā

2N ā’1 2N ā’1

ā ā

T ct ā c(Ļ„m )ā

= Tct (c(Ļ„2N ) c(Ļ„n )) . (4.106)

m =1

n=1

m (=n )

For the case of a fermi ļ¬eld we thus obtain the analogous result to Eq. (4.100)

TC (Ļ(x2n , Ļ„2n ) Ļ(x2nā’1 , Ļ„2nā’1 ) . . . Ļ(x2 , Ļ„2 ) Ļ(x1 , Ļ„1 ))

(ā’1)Ī¶P TC (Ļ(xi , Ļ„i ) Ļ(xj , Ļ„j ))

=

a.p.p. i=j

ā” (ā’1)Ī¶P iN G0 (xi , Ļ„i ; xj , Ļ„j ) , (4.107)

a.p.p. i=j

where the quantum statistical factor (ā’1)Ī¶P counts the number of transpositions

relating the orderings on the two sides. For the case of a state with a deļ¬nite number

4.4. Non-equilibrium diagrammatics 103

of particles, only if fermi creation and annihilation ļ¬elds are paired do we get a

non-vanishing contribution.32 In the last equality, the free contour ordered Greenā™s

function is introduced.33

In the perturbative expansion of the Greenā™s functions, the quantum ļ¬elds, and

their associated multi-particle spaces, have left the stage, absorbed in the expressions

for the free propagators.

The perturbative expansion lends itself to suggestive diagrammatics, the Feynman

diagrammatics for non-equilibrium systems, which we now turn to introduce.

Exercise 4.6. Consider a harmonic oscillator, where x(t) is the position operator in

Ė

the Heisenberg picture, and show that, for the generating functional we have

ā

ft (Ė (t)+Ė ā (t))

ā i dt a a

Z[ft ] ā” T ei dt ft x(t)

Ė ā’ā

= tr ĻT T e 2M Ļ q

ā’ā

ā ā

= eā’ 2

1

dt dt ft T (Ė(t) x(t )) ft

x Ė

. (4.108)

ā’ā ā’ā

In Chapter 9 we will consider the generating functional technique for quantum ļ¬eld

theory. Quantum mechanics is then the case of the zero dimensional ļ¬eld theory.

4.4 Non-equilibrium diagrammatics

Empowered by Wickā™s theorem, we can envisage the whole perturbative expansion

of the contour ordered Greenā™s function. Writing down the nth-order contribution

from the expansion of the exponential in Eq. (4.53) containing the interaction, and

employing Wickā™s theorem, we get expressions involving products of propagators and

vertices. However, the expressions resulting from perturbation theory quickly become

unwieldy. A convenient method of representing perturbative expressions by diagrams

was invented by Feynman. Besides the appealing aspect of representing perturba-

tive expressions by drawings, the diagrammatic method can also be used directly for

reasoning and problem solving. The easily recognizable topology of diagrams makes

the diagrammatic method a powerful tool for constructing approximation schemes as

well as exact equations that may hold true beyond perturbation theory. Furthermore,

by elevating the diagrams to be a representation of possible alternative physical pro-

cesses, the diagrammatic representation becomes a suggestive tool providing physical

intuition into quantum dynamics. In this section we construct the general diagram-

matic perturbation theory valid for non-equilibrium situations. We shall illustrate

the diagrammatics by considering the generic cases.

32 The use of states with a non-deļ¬nite number of fermions, as useful in the theory of supercon-

ductivity, would lead to the appearance of so-called anomalous Greenā™s functions, as we discuss in

Chapter 8.

33 Minus the imaginary unit provided N -fold times from the expansion of the exponential function

containing the interaction, explains why the imaginary unit was introduced in the deļ¬nition of the

contour-ordered Greenā™s function in the ļ¬rst place. However, one is of course entitled to keep track

of factors at oneā™s taste.

104 4. Non-equilibrium theory

4.4.1 Particles coupled to a classical ļ¬eld

The simplest kind of coupling is that of an assembly of identical particle species

coupled to an external classical ļ¬eld, V (x, t). In that case the contour-ordered

Greenā™s function, written in the form ready for a perturbative expansion, Eq. (4.53)

or Eq. (4.50), has the form

ā

ā’i dĻ„ dx V (x,Ļ„ )ĻH (x,Ļ„ )ĻH 0 (x,Ļ„ )

ā

GC (1, 1 ) = ā’iTr Ļ0 Tc

c 0

e ĻH0 (1) ĻH0 (1 ) ,

(4.109)

where c is the contour that starts at t0 and stretches through max(t1 , t1 ) and back

again to t0 , as depicted in Figure 4.4. If t0 is taken to be in the far past, t0 ā’ ā’ā,

we obtain the real-time contour of Figure 4.5. Expanding the exponential we get

strings of, say, fermion operators subdued to the contour-ordering operation and

thermally weighted by the Hamiltonian for the free ļ¬eld, which is Gaussian as Ļ0 is

given by Eq. (4.54). Higher-order terms in the expansion have the same form, they

just contain strings with a larger number of ļ¬elds. In perturbation theory the task

is to evaluate such terms, or rather ļ¬rst break them down into Gaussian products

as accomplished by Wickā™s theorem, i.e. decomposed into a product of free thermal

equilibrium contour-ordered Greenā™s functions.

For the ļ¬rst-order term, Eq. (4.58), we have according to Wickā™s theorem the

expression

(1) (0) (0)

GC (1, 1 ) = dx2 dĻ„2 GC (1, 2) V (2) GC (2, 1 ) (4.110)

c

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 diļ¬ering only by a minus sign.

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

sented by the diagram

(1)

GC (x, Ļ„ ; x , Ļ„ ) = (4.111)

xĻ„ x1 Ļ„1 xĻ„

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

the scalar potential

ā” V (x, Ļ„ ) (4.112)

xĻ„

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

contour-ordered Greenā™s function

R (0)

ā” GC (x, Ļ„ ; x , Ļ„ ) (4.113)

xĻ„ xĻ„

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

4.4. Non-equilibrium diagrammatics 105

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

ā” GC (x, Ļ„ ; x , Ļ„ ) (4.114)

xĻ„ xĻ„

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 ā’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 Greenā™s

function:

GC (x, Ļ„ ; x , Ļ„ ) = = xĻ„ xĻ„

xĻ„ xĻ„

+ +

xĻ„ x1 Ļ„1 xĻ„ xĻ„ x2 Ļ„2 x1 Ļ„1 xĻ„

+ + ... , (4.115)

xĻ„ x3 Ļ„3 x2 Ļ„2 x1 Ļ„1 xĻ„

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

inļ¬nite series

dĻ„2 G0 (1, 2) V (x2 , Ļ„2 ) G0 (2, 1 ) + Ā· Ā· Ā·

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 dĻ„2 G0 (1, 2) V (x2 , Ļ„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 Greenā™s function

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

dx2 dĻ„2 Gā’1 (1, 2) G0 (2, 1 ) = Ī“(1 ā’ 1 ) (4.119)

0

c

we obtain

dx2 dĻ„2 (Gā’1 (1, 2) ā’ V (2)) G(2, 1 ) = Ī“(1 ā’ 1 ) . (4.120)

0

c

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

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

in Ī“-functions in the degrees of freedom, the contour variable Ī“-function speciļ¬ed

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

propagator, in condensed matrix notation

(Gā’1 Gā’1 )(1, 1 )

G)(1, 1 ) = Ī“(1 ā’ 1 ) = (G (4.121)

0 0

where signiļ¬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 ļ¬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 Greenā™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 Greenā™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,Ļ„

= V (x, Ļ„ ) V (x , Ļ„ ) . (4.123)

x ,Ļ„

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 ā’ r) Vimp (x ā’ 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 fermionā“boson interaction, Eq. (2.71), or equiva-

lently, the jellium electronā“phonon interaction, and let the Ļ-ļ¬eld denote the fermi

ļ¬eld in the Greenā™s function we are looking at

(i )

eā’i ā

dĻ„ HH (Ļ„ )

G(1, 1 ) = ā’i Tr Ļ0 TC ĻH0 (1) ĻH0 (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 Greenā™s function, say for fermions, Eq. (4.57). The term linear in the phonon

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

term36

(ā’i)2 ā

(i) (i)

= ā’iTr Ļ0 TC

G(2) (1, 1 ) dĻ„3 HH0 (Ļ„3 ) dĻ„2 HH0 (Ļ„2 ) ĻH0 (1) ĻH0 (1 )

2! C C

i2 ā

g dĻ„3 dĻ„2 dx3 dx2 Tr eā’Ī²H0 TC ĻH0 (x3 , Ļ„3 )ĻH0 (x3 , Ļ„3 )ĻH0 (x3 , Ļ„3 )

=

2! C

ā ā

Ć— ĻH0 (x2 , Ļ„2 ) ĻH0 (x2 , Ļ„2 ) ĻH0 (x2 , Ļ„2 ) ĻH0 (1) ĻH0 (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

ļ¬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-deļ¬nite number of bosons, as useful in the theory of Boseā“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 ļ¬elds weighted

(0) (0)

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

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

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

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

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

ā

ings: the creation fermi ļ¬eld indexed by the external label 1 , ĻH0 (1 ), is paired with

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

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

the ļ¬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 ļ¬elds contains

only two ļ¬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 Greenā™s functions thus becomes

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

GC (1, 1 ) ā’ ig 2 dx3 dĻ„3 dx2 dĻ„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 ā’i for our convention of Greenā™s functions: two factors of ā’i are provided by the

interaction and one provided externally in the deļ¬nition of the Greenā™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 ļ¬rst diagram in Figure 4.6.

,

Figure 4.6 Lowest order fermionā“boson diagrams.

4.4. Non-equilibrium diagrammatics 109

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

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

ordered boson Greenā™s function:

(0)

ā” DC (x, Ļ„ ; x , Ļ„ ) (4.128)

xĻ„ xĻ„

i.e.

DC (1, 1 ) = ā’i trb (Ļb Tc (ĻH0 (1) Ļā 0 (1 ))) = ā’i Tc (ĻH0 (1) Ļā 0 (1 ))

(0) (0)

(4.129)

H H

(0)

as trb denotes the trace with respect to the boson degrees of freedom and Ļb 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 Wickā™s theorem of the second-order expres-

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

ā

1 , ĻH0 (1 ), is paired with the annihilation ļ¬eld associated with an internal point and

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

point 1, thereby ļ¬xing the ļ¬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 ) ā’ ā’ig 2 dx3 dĻ„3 dx2 dĻ„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 ļ¬rst diagram in Figure 4.6

that reļ¬‚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 Greenā™s function taken at equal contour times,

(0)

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

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

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

c

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

lesser Greenā™s function, G< (x, Ļ„ ; x, Ļ„ ), 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 ļ¬nal option for pairings in the Wick decomposition of the second-order ex-

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

ā

external label 1 , ĻH0 (1 ), with the annihilation ļ¬eld indexed by the external label

1, ĻH0 (1). The pairings of the fermi ļ¬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 ) ā’ ā’ig 2 GC (1, 1 ) dx3 dĻ„3 dx2 dĻ„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 electronā“phonon interaction as it contains an overall factor that vanishes.

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

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

0

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

dx D0 (x, Ļ„ ; x , Ļ„ ) = 0 (4.132)

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

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

ordered electron Greenā™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 ļ¬eld. In the Sommerfeld treatment of the Coulomb interaction

in a pure metal, tadpole or Hartree diagrams are also absent, though for a diļ¬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 electronā“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 diļ¬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 Wickā™s theorem for a string of bose

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

(ā’i)4

(2) (i) (i)

ā’i Tr ā’Ļ0 TC

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

4! C C

ā

(i) (i)

Ć— HH0 (4) HH0 (5) ĻH0 (1) ĻH0 (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 ļ¬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

ā

ļ¬eld ĻH0 (1 ) can be paired with any of the fermi annihilation ļ¬elds associated with

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

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

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

three identical contributions, and the creation ļ¬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

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

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

three diļ¬erent contributions as represented by the ļ¬rst three topologically diļ¬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

ā’

+

1 5 4 3 2 1 1 5 2 1

Figure 4.9 Fourth-order diagrams in the coupling constant.

The ļ¬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 electronā“hole

excitations, or in QED a photon can cause electronā“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 Wickā™s theorem for the fermi and

phonon ļ¬eld strings the corresponding Feynman diagrams for the fermi propagator

to sixth order in the fermionā“boson coupling.

Exercise 4.9. Obtain by brute force application of Wickā™s theorem for the fermi and

phonon ļ¬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

(ā’i)N ā

Tr eā’Ī²H0 TC

(i) (i)

ā’i d2 . . . d(N + 1) HH0 (2) Ā· Ā· Ā· HH0 (N + 1)ĻH0 (1)ĻH0 (1 )

N! C C

(4.134)

(i)

all connected combinations that diļ¬er only by permutations of HH0 give identical

contributions, thus canceling the factor 1/N ! in front, and as a consequence only

topologically diļ¬erent diagrams appear. This has a very important consequence for

diagrammatics, viz. that it allows separating oļ¬ 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 diļ¬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 ļ¬eld theories in Chapter 9.

4.5. The self-energy 113

The diagrammatic representation of the perturbative expansion of the electron

Greenā™s function for the case of electronā“phonon interaction, or in general the fermion

Greenā™s function for fermionā“boson interaction, thus becomes

= +

xĻ„ xĻ„ xĻ„ xĻ„

+ +

Ā·Ā·Ā·

+ + . (4.135)

In the perturbative expression for the contour-ordered Greenā™s function for the

case of electronā“phonon interaction, each interaction contains one phonon ļ¬eld op-

erator, a fermion creation and annihilation ļ¬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 electronā“phonon vertex construct all possible topologically

connected diagrams. This is Wickā™s theorem in the language of Feynman diagrams.

We recall that, whenever an odd number of fermi ļ¬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 fermionā“boson interaction to the addi-

tional Feynman rule: a diagram containing n boson lines is attributed the factor

in g 2n (ā’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 Greenā™s

function or propagator we shall use their topology for classiļ¬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 ļ¬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 ļ¬eld instead of the analogous diagrammatics for

the electronā“boson or electronā“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:

Ī£(1, 1 ) ā” 1 1

Ī£

= +

+ +

Ā·Ā·Ā·

+ + (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 Greenā™s function, but we leave it out from

now on)

GC (1, 1 ) =

= + Ī£

+ Ī£ Ī£

+ Ī£ Ī£ Ī£

+ Ā·Ā·Ā· . (4.138)

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

= + (4.139)

Ī£

and we obtain the Dyson equation

G(1, 1 ) = G0 (1, 1 ) + dx3 dĻ„3 dx2 dĻ„2 G0 (1, 3) Ī£(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 Ī£ G (4.141)

where signiļ¬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 Ī£ 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 diļ¬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 Greenā™s function, Eq. (4.141),

(Gā’1 Gā’1 )(1, 1 ) ,

G0 )(1, 1 ) = Ī“(1 ā’ 1 ) = (G0 (4.143)

0 0

where

Gā’1 (1, 1 ) = Gā’1 (1) Ī“(1 ā’ 1 ) (4.144)

0 0

and

ā‚

Gā’1 (1) = ā’ h(1) ,

i (4.145)

0

ā‚Ļ„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 Greenā™s function

from the left

(Gā’1 ā’ Ī£) G = Ī“(1 ā’ 1 ) (4.146)

0

and from the right

(Gā’1 ā’ Ī£) = Ī“(1 ā’ 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

ļ¬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 inļ¬nitely many one-particle irreducible vertex functions

which occur in a quantum ļ¬eld theory. Their signiļ¬cance will become clear when, in Chapter 10,

we encounter the usefulness of the eļ¬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 suļ¬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 Greenā™s

function and the impurity correlator, Ī£[G0 ]. For the case of fermionā“boson inter-

action the self-energy is a functional of both types of free contour-ordered Greenā™s

functions, Ī£[G0 , D0 ]. The self-energy, Ī£, 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 speciļ¬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 inļ¬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:

ā’ + Ā·Ā·Ā·

+ +

+ Ā·Ā·Ā·

= + Ī£

ā’ + Ī£

118 4. Non-equilibrium theory

Ā·Ā·Ā·

+ +

Ī£ Ī£

= . (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. Ī£[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:

Ī£(1, 1 ) = 1 1

+ 1 1

43 Synonymous names for the full Greenā™s function or propagator are renormalized or dressed

propagator.

4.6. Summary 119

+ 1 1

Ā·Ā·Ā·

+ + . (4.149)

1 1

The partial summation of diagrams is unique, since the initial and ļ¬nal impurity

correlator lines are attached internally in diļ¬erent ways in each class of summed

diagrams. No double counting of diagrams thus takes place owing to the diļ¬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 inļ¬nite sum of terms in perturbation theory, is that the self-energy is

expressed as a functional of the exact propagators or full Greenā™s functions

Ī£(1, 1 ) = Ī£(1,1 ) [G, D] . (4.150)

We can continue this topological classiļ¬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 fermionā“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 Greenā™s functions. Perturbation

theory valid for non-equilibrium states turned out in standard fashion, reļ¬‚ecting a

general Wick theorem for closed time path strings of operators, and the Feynman

diagrams for the contour ordered Greenā™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 Greenā™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 Greenā™s

function lacks physical transparency and does not appeal to intuition.1 We need a

diļ¬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 diļ¬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

propounded in this chapter.2

5.1 Real-time matrix representation

To let our physical intuition come into play; we need to get from contour times back

to real times. This is achieved by labeling the forward and return contours of the

closed time path, depicted in Figure 4.5, by numbers, specifying the position of a

contour time by an index. The forward contour we therefore label c1 and the return

contour c2 , i.e. a contour time variable gets tagged by the label 1 or 2 specifying its

belonging to forward or return contour, respectively.3

The contour ordered Greenā™s function is by this tagging mapped onto a 2 Ć— 2-

1 As the imaginary time Greenā™s function discussed in Section 5.7.1 does not appeal to intuition.

2 In this chapter we follow the exposition given in references [3] and [9].

3 Instead of labeling the two branches by 1 and 2, one can also label them by Ā± as in the original

works of Schwinger [5] and Keldysh [10]. However, when stating Feynman rules, numbers are

convenient for labeling.

121

122 5. Real-time formalism

matrix in the dynamical index or Schwingerā“Keldysh space

Ė Ė

G11 (1, 1 ) G12 (1, 1 )

Ė

GC (1, 1 ) ā’ G(1, 1 ) ā” (5.1)

Ė Ė

G21 (1, 1 ) G22 (1, 1 )

Ė

according to the prescription: the ij-component in the matrix Greenā™s function G is

Gc (1, 1 ) for 1 lying on ci and 1 lying on cj , i, j = 1, 2. The times appearing in the

components of the matrix Greenā™s function are now standard times, 1 = (x, t1 ), and

we can identify them in terms of our previously introduced Greenā™s functions, the real-

time Greenā™s functions introduced in Section 3.3. The matrix structure reļ¬‚ects the

essence in the real-time formulation of non-equilibrium quantum statistical mechanics

due to Schwinger [5]: letting the quantum dynamics do the doubling of the degrees

of freedom necessary for describing non-equilibrium states.4

The 11-component of the matrix in Eq. (5.1) becomes

G11 (1, 1 ) = ā’i T (Ļ(x1 , t1 ) Ļ ā (x1 , t1 )) = ā’i T (Ļ(1) Ļ ā (1 )) ,

Ė (5.2)

the time-ordered Greenā™s function, where Ļ(x, t) = ĻH (x, t) is the ļ¬eld in the

full Heisenberg picture for the species of interest. Contour ordering on the forward

contour is just usual time-ordering.

Analogously, the 21-component becomes

G21 (1, 1 ) = G> (1, 1 ) = ā’i Ļ(1) Ļ ā (1 ) ,

Ė (5.3)

i.e. G-greater, and the 12-component is G-lesser

G12 (1, 1 ) = G< (1, 1 ) = ā“ i Ļ ā (1 ) Ļ(1) ,

Ė (5.4)

where upper and lower signs refer to bose and fermi ļ¬elds, respectively, and the

22-component is the anti-time-ordered Greenā™s function

G22 (1, 1 ) = G(1, 1 ) = ā’i T (Ļ(1) Ļ ā (1 )) .

Ė Ė Ė (5.5)

We note that the time-ordered and anti-time-ordered Greenā™s functions can be

expressed in terms of G-greater and G-lesser, recall Eq. (3.64), and

Ė

G11 (1, 1 ) = Īø(t1 ā’ t1 ) G> (1, 1 ) + Īø(t1 ā’ t1 ) G< (1, 1 ) . (5.6)

The matrix Greenā™s function in Eq. (5.1) can therefore be expressed in terms of

the real-time Greenā™s functions introduced in Section 3.3

G< (1, 1 )

G(1, 1 )

Ė

G(1, 1 ) = . (5.7)

Ė

G> (1, 1 ) G(1, 1 )

The way of representing the information in the contour-ordered Greenā™s function

as in Eq. (5.1) or equivalently Eq. (5.7) is respectable as, for example, the matrix

4 The thermo-ļ¬eld approach to non-equilibrium theory also employs a doubling of the degrees of

freedom (see, for example, reference [11]), but in our view not in as physically appealing way as

does the real-time version of the closed time path formulation.

5.2. Real-time diagrammatics 123

is anti-hermitian with transposition meaning interchange of all arguments including

that of the dynamical index (note the importance of the sign convention for han-

dling fermi ļ¬elds under ordering operations). For real bosons the matrix is real and

symmetric. However, when it comes to understanding non-equilibrium contributions

from various processes, as described by Feynman diagrams, the present form of the

matrix Greenā™s function lacks physical transparency, and oļ¬ers no basis for develop-

ing intuition. We shall therefore eventually transform to a diļ¬erent matrix form, and

as a ļ¬nal act liberate ourselves from the matrix outļ¬t altogether.

Let us now establish the Feynman rules in the real-time technique for the matrix

Greenā™s function in the dynamical index or Schwingerā“Keldysh space.

5.2 Real-time diagrammatics

Instead of having the diagrammatics represent the perturbative expansion of the

contour Greenā™s function as in the previous chapter, we shall map the diagrams

to the real-time domain where eventually a proper physical interpretation of the

diagrams can be obtained.

5.2.1 Feynman rules for a scalar potential

We start with the simplest kind of coupling, that of particles interacting with an

external classical ļ¬eld. For particles interacting with a scalar potential V (x, t), we

have the diagrammatic expansion of the contour ordered Greenā™s function depicted

in Eq. (4.115) on page 105. The ļ¬rst-order diagram corresponded to the term

(1) (0) (0)

GC (1, 1 ) = dx2 dĻ„2 GC (1, 2) V (2) GC (2, 1 ) . (5.8)

C

Parameterizing the real-time contour we have

ā ā’ā ā ā

dt ā’

dĻ„2 = dt + dt = dt (5.9)

ā’ā ā ā’ā ā’ā

ńņš. 5 |