=

δq,q (n(ωq ) + 1)e’iωq (t’t )

=

≡ >

i Dqq (t, t ) , (4.82)

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

1 1

n(ωq ) = = . (4.83)

q ’μb )/kT

eωq /kT ’ 1 ’1

e(

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

operators, the correlation function is

a† (t ) aq (t) = tr(ρT a† (t ) aq (t))

≡

<

i Dqq (t, t ) q q

= n(ωq ) δq,q e’iωq (t’t ) . (4.84)

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

are

i a† (t ) aq (t) = itr(ρT a† (t ) aq (t))

G< (t, t ) ≡

qq q q

if ( q ) δq,q e’i q (t’t )

= (4.85)

and

≡ ’i aq (t) a† (t ) = ’itr(ρT aq (t) a† (t ))

G> (t, t )

qq q q

= ’i(1 ’ f ( q )) δq,q e’i q (t’t )

, (4.86)

where f ( q ) is the Fermi function

1

f ( q) = . (4.87)

q ’μ)/kT

e( +1

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

{aq (t), a† (t )} = δq,q e’i q (t’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 di¬erent number of particles or quanta; for example,

√

aq a† aq = Z ’1 e’E({nq }q )/kT ( nq )3 nq |nq ’ 1 = 0 , (4.89)

q

{nq }q

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

As an example of using Wick™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(ρT Tct (a(„1 )a† („2 )a(„3 )a† („4 ))) Tct (a(„1 )a† („2 )) Tct (a(„3 )a† („4 ))

=

Tct (a(„1 )a† („4 )) Tct (a(„3 )a† („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 (a† („ )a† („ )) .

Tct (a(„ )a(„ )) =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(ρT Tct (a(„1 )a† („2 )a(„3 )a† („4 )))

e’E({nq }q )/kT {nq }q |Tct (a(„1 )a† („2 )a(„3 )a† („4 ))|{nq }q .

= (4.92)

{nq }q

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

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

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

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

us therefore consider an N -string with 2N operators

SN = TC (c(„2N ) c(„2N ’1 ) . . . c(„2 ) c(„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 ’ c ’ c is needed. Also in describing a Bose“Einstein condensate it is convenient to work with

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

situation we shall deal with in due time. For the case of electron“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 e¬ectively removed by rede¬ning the equilibrium position of the oscillator.

100 4. Non-equilibrium theory

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

2N ’1

2N

SN = c(„n ) = c(„2N ) c(„n ) . (4.94)

n=1 n=1

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

2N ’1

’1

1’e »c ωq /kT

SN = [c(„2N ), c(„n )] . (4.95)

n=1

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

2N ’1 2N ’2 2N ’2

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

n=1 n=1 n=1

2N ’1

’ c(„n ) c(„2N ) . (4.96)

n=1

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

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

Eq. (4.96), so that

2N ’1 2N ’1 2N ’1

c(„2N ), c(„n ) = [c(„2N ), c(„n )] c(„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 ’ e»c q ωq /kT

[cq („2N ), cq („n )] = δq,q cq („2N ) cq („n ) (4.98)

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

we obtain

2N ’1 2N ’1

SN = c(„2N ) c(„n ) c(„m )

m =1

n=1

m (=n )

⎛ ⎞

2N ’1 2N ’1

⎜ ⎟

T ct ⎝ c(„m )⎠

= Tct (c(„2N ) c(„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 ¬elds involves a minus sign, and an overall sign factor occurs, (’1)ζ P ,

where ζP is the sign of the permutation P bringing the string of ¬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 Wick™s

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

commutator a c-number.29

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

of the bose ¬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 (φ(x2n , „2n ) φ(x2n’1 , „2n’1 ) . . . φ(x2 , „2 ) φ(x1 , „1 ))

TC (φ(xi , „i )φ(xj , „j )) ≡ iN D0 (xi , „i ; xj , „j ) .(4.100)

=

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

The index on the contour-ordered Green™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 Wick™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 e»c ωq /kT

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

where s = “ signi¬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 ’1

2N

SN = c(„n ) = c(„2N ) c(„n )

n=1 n=1

2N ’1

’1

»c („ 2N ) ωq /kT

= 1 + se [c(„2N ), c(„n )]s (4.102)

n=1

and

2N ’1 2N ’2

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

[c(„2N ), c(„n )]s = c(„n )

n=1 n=1

2N ’1

+s c(„n ) c(„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 ¬nite temperature imaginary-time Wick™s theorem,

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

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

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

grand canonical ensemble.

102 4. Non-equilibrium theory

2N ’2

= (’s) c(„2N ’1 ) c(„2N ) c(„n )

n=1

2N ’2

+ [c(„2N ), c(„2N ’1 )]s c(„n )

n=1

2N ’1

+s c(„n ) c(„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(„2N ) through in the ¬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 ’1 2N ’1 2N ’1

n’1

c(„2N ), c(„n ) = (’s) [c(„2N ), c(„n )]s c(„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 e»c q ωq /kT

[cq („2N ), cq („n )]s = δq,q cq („2N ) cq („n ) (4.105)

and taking it outside the thermal average we obtain

2N ’1 2N ’1

SN = c(„2N ) c(„n ) c(„m )

m =1

n=1

m (=n )

⎛ ⎞

2N ’1 2N ’1

⎜ ⎟

T ct ⎝ c(„m )⎠

= Tct (c(„2N ) c(„n )) . (4.106)

m =1

n=1

m (=n )

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

TC (ψ(x2n , „2n ) ψ(x2n’1 , „2n’1 ) . . . ψ(x2 , „2 ) ψ(x1 , „1 ))

(’1)ζP TC (ψ(xi , „i ) ψ(xj , „j ))

=

a.p.p. i=j

≡ (’1)ζP iN G0 (xi , „i ; xj , „j ) , (4.107)

a.p.p. i=j

where the quantum statistical factor (’1)ζP counts the number of transpositions

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

4.4. Non-equilibrium diagrammatics 103

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

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

function is introduced.33

In the perturbative expansion of the Green™s functions, the quantum ¬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

ˆ

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

∞

ft (ˆ (t)+ˆ † (t))

∞ i dt a a

Z[ft ] ≡ T ei dt ft x(t)

ˆ ’∞

= tr ρT T e 2M ω q

’∞

∞ ∞

= e’ 2

1

dt dt ft T (ˆ(t) x(t )) ft

x ˆ

. (4.108)

’∞ ’∞

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

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

4.4 Non-equilibrium diagrammatics

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

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

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

employing Wick™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-de¬nite number of fermions, as useful in the theory of supercon-

ductivity, would lead to the appearance of so-called anomalous Green™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 de¬nition of the

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

of factors at one™s taste.

104 4. Non-equilibrium theory

4.4.1 Particles coupled to a classical ¬eld

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

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

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

or Eq. (4.50), has the form

†

’i d„ dx V (x,„ )ψH (x,„ )ψH 0 (x,„ )

†

GC (1, 1 ) = ’iTr ρ0 Tc

c 0

e ψH0 (1) ψH0 (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 ’ ’∞,

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 ¬eld, which is Gaussian as ρ0 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 ¬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, i.e. decomposed into a product of free thermal

equilibrium contour-ordered Green™s functions.

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

expression

(1) (0) (0)

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