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δq,q (1 ’ e’ωq /kT )’1 e’iωq (t’t )
=

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)

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