<<

. 5
( 22)



>>

(0) (0)
or for the grand canonical ensemble, substituting in Eq. (4.62) Hb ’ Hb ’ μb Nb
(i.e. we measure energies from the chemical potential, ωq = q ’ μ) and we have
introduced the notation
. . . = tr(ρT . . .) (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 argument™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 „ and „
§
⎨ c(„ ) c(„ ) for „ >C „
TC (c(„ ) c(„ )) = (4.64)
©
c(„ ) c(„ ) for „ >C „
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 ¬elds at equal times, we can in the course of the argument

assume them in¬nitesimally split, and all the contour time variables can thus be considered di¬erent.
96 4. Non-equilibrium theory


where the upper identity is for contour time „ being further along the contour than
„ 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 ¬eld operators as in Eq. (4.60) can now be
decomposed according to Wick™s theorem, which relies only on the simple property

[cq , ρT ] = ρT cq [exp{»c ωq /kB T } ’ 1] (4.65)

valid for a Hamiltonian quadratic in the bose ¬eld (»c = ±1, depending upon whether
cq is a creation or an annihilation operator for state q). We now turn to prove Wick™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(„n ) c(„n’1 ) . . . c(„2 ) c(„1 )) = Tct (cq („ ) cq („ )) (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, Wick™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 Green™s functions, quantities for
which we have explicit expressions.
Before proving Wick™s theorem and the relation Eq. (4.65), we ¬rst observe some
preliminary results. Di¬erent q-labels describe di¬erent momentum degrees of free-
dom, so operators for di¬erent qs commute, and algebraic manipulations with com-
muting operators are just as for usual numbers giving for example

ρT ,
ρT = (4.67)
q
q

where we have introduced the thermal statistical operator for each mode

ρT = zq e’hq /kT
’1
(4.68)
q

and the partition function for the single mode
1
zq = . (4.69)
1 ’ e’ωq /kT
The independence of each mode degree of freedom, as expressed by the commutation
of operators corresponding to di¬erent degrees of freedom, gives
⎛ ⎞

cq ρ T = ⎝ ρ T ⎠ cq ρ T . (4.70)
q q
q (=q)

Now, using the commutation relations for the creation and annihilation operators we
have
cq hq = (hq ’ »c ωq ) cq , (4.71)
4.3. Closed time path formalism 97


where
cq = a†
+1 for q
»c = (4.72)
’1 for cq = a q .
Using Eq. (4.71) repeatedly gives

cq hn = (hq ’ »c ωq )n cq (4.73)
q

and upon expanding the exponential function and re-exponentiating we can commute
through to get
cq ρT = e»c ωq /kT ρT 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 ’ e»c ωq /kT ) cq A
[cq , A] (4.75)

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

= ’tr([cq , ρT ] 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 + e»c ωq /kT ) cq A . (4.77)


Employing Eq. (4.75) with A = 1, 1, a, a† , respectively, we observe that all the
following averages vanish

a† (t) a† (t)a† (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 fermion“boson interaction (and electron“phonon interaction). These
equalities are valid for any state diagonal in the total number of particles, i.e. a state
with a de¬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 e’itHb = ei»c ωq t e’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 e’itHb = cq ei»c ωq 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 speci¬ed by

[aq (t), a† (t )] = δq,q e’iωq (t’t ) . (4.81)
q
98 4. Non-equilibrium theory


According to Eq. (4.75) we thereby have

aq (t) a† (t ) (1 ’ e’ωq /kT )’1 [aq (t), a† (t )]
=
q q


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)
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 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





+
1 5 4 3 2 1 1 5 2 1



Figure 4.9 Fourth-order diagrams in the coupling constant.


The ¬rst three diagrams in Figure 4.9 are solely the result of emission and absorption
of phonons by the electron or bosons by fermions in general. The last diagram is
the signature of the presence of the Fermi sea: a phonon can cause electron“hole
excitations, or in QED a photon can cause electron“positron pair creation. From the
boson point of view, such bubble-diagrams with additional decorations are basic, the
generic boson self-energy diagram, the self-energy being a quantity we introduce in
the next section.
Exercise 4.8. Obtain by brute force application of Wick™s theorem for the fermi and
phonon ¬eld strings the corresponding Feynman diagrams for the fermi propagator
to sixth order in the fermion“boson coupling.
Exercise 4.9. Obtain by brute force application of Wick™s theorem for the fermi and
phonon ¬eld strings the corresponding Feynman diagrams for the phonon propagator
to second order in the coupling.

The feature that the total combinatorial choice factor cancels the factorial factor,
1/n!, originating from the expansion of the exponential function is quite general. For
the N th order term
(’i)N †
Tr e’βH0 TC
(i) (i)
’i d2 . . . d(N + 1) HH0 (2) · · · HH0 (N + 1)ψH0 (1)ψH0 (1 )
N! C C

(4.134)
(i)
all connected combinations that di¬er only by permutations of HH0 give identical
contributions, thus canceling the factor 1/N ! in front, and as a consequence only
topologically di¬erent diagrams appear. This has a very important consequence for
diagrammatics, viz. that it allows separating o¬ sub-parts in a diagram and summing
them separately. We shall shortly return to this in the next section, and in much
more detail in Chapter 9.38
38 We note that the diagrammatic structure of amplitudes for quantum processes can be captured
in the two options: to interact or not to interact! The resulting Feynman diagrams being all
the topologically di¬erent ones constructable by the vertices and propagators of the theory. We
shall take this Shakespearian point of view as the starting point when we construct the general
diagrammatic and formal structure of quantum ¬eld theories in Chapter 9.
4.5. The self-energy 113


The diagrammatic representation of the perturbative expansion of the electron
Green™s function for the case of electron“phonon interaction, or in general the fermion
Green™s function for fermion“boson interaction, thus becomes



= +
x„ x„ x„ x„




+ +



···
+ + . (4.135)



In the perturbative expression for the contour-ordered Green™s function for the
case of electron“phonon interaction, each interaction contains one phonon ¬eld op-
erator, a fermion creation and annihilation ¬eld, all with the same contour time.
The Feynman diagrammatics is thus characterized by a vertex with incoming and
outgoing fermi lines and a phonon line, a three-line vertex.
The totality of diagrams can be captured by the following tool-box and instruc-
tions. With the diagrammatic ingredients, an electron propagator line, a phonon
propagator line and the electron“phonon vertex construct all possible topologically
connected diagrams. This is Wick™s theorem in the language of Feynman diagrams.
We recall that, whenever an odd number of fermi ¬elds are interchanged, a minus
sign appears. Diagrammatically this can be incorporated by the additional sign rule:
for each loop of fermi propagator lines in a diagram a minus sign is attributed. Ac-
companying this are the additional Feynman rules, which for our choices become
the following. In addition to the usual rule of the superposition principle: sum over
all internal labels, our conventions leads for fermion“boson interaction to the addi-
tional Feynman rule: a diagram containing n boson lines is attributed the factor
in g 2n (’1)F , where F is the number of closed loops formed by fermion propagators.


4.5 The self-energy
We have so far only derived diagrammatic formulas from formal expressions. Now
we shall argue directly in the diagrammatic language to generate new diagrammatic
expressions from previous ones, and thereby diagrammatically derive new equations.
In order to get a grasp of the totality of diagrams for the contour-ordered Green™s
function or propagator we shall use their topology for classi¬cation. We introduce
the one-particle irreducible (1PI) propagator, corresponding to all the diagrams that
can not be cut in two by cutting an internal particle line. In the following example
114 4. Non-equilibrium theory




1PI 1PR



(4.136)



the ¬rst diagram is one-particle irreducible, 1PI, whereas the second is one-particle
reducible, 1PR. Here we have used the diagrammatics for the impurity-averaged
propagator in a Gaussian random ¬eld instead of the analogous diagrammatics for
the electron“boson or electron“phonon interaction to illustrate that the arguments
are topological and valid for any type of interaction and its diagrammatics.39
Amputating the external propagator lines of the one-particle irreducible diagrams
(below displayed for the impurity-averaged propagator), we obtain the self-energy:



Σ(1, 1 ) ≡ 1 1
Σ




= +




+ +




···
+ + (4.137)


consisting, by construction, of all amputated diagrams that can not be cut in two by
cutting one bare propagator line.
39 For a detailed discussion of the impurity-averaged propagator, which is of interest in its own
right, we refer to Chapter 3 in reference [1].
4.5. The self-energy 115


We can now go on and uniquely classify all diagrams of the (impurity-averaged)
propagator according to whether they can be cut in two by cutting an internal particle
line at only one place, or at only two, three, etc., places. By construction we uniquely
exhaust all the possible diagrams for the propagator (the subscript is a reminder that
we are considering the contour-ordered Green™s function, but we leave it out from
now on)

GC (1, 1 ) =




= + Σ




+ Σ Σ




+ Σ Σ Σ




+ ··· . (4.138)



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



= + (4.139)
Σ



and we obtain the Dyson equation

G(1, 1 ) = G0 (1, 1 ) + dx3 d„3 dx2 d„2 G0 (1, 3) Σ(3, 2) G(2, 1 ) (4.140)
C C
40 Inthe last term we can interchange the free and full propagator, because iterating from the left
generates the same series as iterating from the right.
116 4. Non-equilibrium theory


which we can write in matrix notation
G = G0 + G0 Σ G (4.141)
where signi¬es matrix multiplication in the spatial variable (and possible internal
degrees of freedom) and contour time variables. Arguing on the topology of the
diagrams has reorganized them and we have obtained a new type of equation.41

4.5.1 Non-equilibrium Dyson equations
The standard topological arguments of the previous section for diagrams organizes
them into irreducible sub-parts and we obtained the Dyson equation, Eq. (4.141).
We could of course have iterated Eq. (4.138) from the other side to obtain
G = G0 + G Σ G0 . (4.142)
For an equilibrium state the two equations are redundant, since time convolutions
by Fourier transformation become simple products for which the order of factors is
irrelevant (as discussed in detail in Section 5.6). However, in a non-equilibrium state,
the two equations contain di¬erent information and subtracting them is a useful way
of expressing the non-equilibrium dynamics as we shall exploit in Chapter 7.
Introduce the inverse of the free contour-ordered Green™s function, Eq. (4.141),
(G’1 G’1 )(1, 1 ) ,
G0 )(1, 1 ) = δ(1 ’ 1 ) = (G0 (4.143)
0 0

where
G’1 (1, 1 ) = G’1 (1) δ(1 ’ 1 ) (4.144)
0 0
and

G’1 (1) = ’ h(1) ,
i (4.145)
0
‚„1
where h denotes the single-particle Hamiltonian for the theory under consideration.
The two non-equilibrium Dyson equations, Eq. (4.141) and Eq. (5.69), can then be
expressed through operating with the inverse free contour-ordered Green™s function
from the left
(G’1 ’ Σ) G = δ(1 ’ 1 ) (4.146)
0
and from the right
(G’1 ’ Σ) = δ(1 ’ 1 ) .
G (4.147)
0
These two non-equilibrium Dyson equations will prove useful in Chapter 7 where
quantum kinetic equations are considered.42
By operating with the inverse of the free propagator, the explicit appearance of
the free propagator (or rather the non-interacting propagator since, possible external
¬elds can be included) has been removed. We can in fact remove its presence com-
pletely by expressing the self-energy in terms of skeleton diagrams where only the
full propagator appears, and we now turn to these.
41 The self-energy is just one out of the in¬nitely many one-particle irreducible vertex functions
which occur in a quantum ¬eld theory. Their signi¬cance will become clear when, in Chapter 10,
we encounter the usefulness of the e¬ective action.
42 If one, by the end of the day, in the Dyson equations uses the lowest-order approximation for the

self-energy, this whole venture into the diagrammatic jungle is hardly worthwhile since a civilized
Golden Rule calculation su¬ces.
4.5. The self-energy 117


4.5.2 Skeleton diagrams
So far we have only a perturbative description of the self-energy; i.e. we have a
representation of the self-energy as a functional of the free contour-ordered Green™s
function and the impurity correlator, Σ[G0 ]. For the case of fermion“boson inter-
action the self-energy is a functional of both types of free contour-ordered Green™s
functions, Σ[G0 , D0 ]. The self-energy, Σ, can in naive perturbation theory be de-
scribed as the sum of diagrams that can not be cut in two by cutting only one
internal free propagator line. In a realistic description of a physical system, we al-
ways need to invoke the speci¬cs of the problem in order to implement a controlled
approximation. To this end we must study the actual correlations in the system,
and it is necessary to have the self-energy expressed in terms of the full propagator.
Coherent quantum processes correspond to an in¬nite repetition of bare processes,
and the diagrammatic approach is precisely useful for capturing this feature, as ir-
reducible re-summations are easily described diagrammatically. In order to achieve
a description of the self-energy in terms of the full propagator, let us consider the
perturbative expansion of the self-energy.
For any given self-energy diagram in the perturbative expansion, Eq. (4.137),
we also encounter self-energy diagrams with all possible self-energy decorations on
internal lines; for example, for the case of particles in a random potential:




’ + ···
+ +




+ ···
= + Σ




’ + Σ
118 4. Non-equilibrium theory




···
+ +
Σ Σ




= . (4.148)


We can uniquely classify all these self-energy decorations in the perturbative expan-
sion according to whether the particle line can be cut into two, three, or more pieces
by cutting particle lines (the step indicated by the second arrow in Eq. (4.148)). We
can therefore partially sum the self-energy diagrams according to the unique pre-
scription: for a given self-energy diagram, remove all internal self-energy insertions,
and substitute for the remaining bare particle propagator lines the full propaga-
tor lines.43 Through this partial summation of the original perturbative expansion
of the self-energy only so-called skeleton diagrams containing the full propagator
will then appear, i.e. Σ[G]. Since in the skeleton expansion we have removed self-
energy insertions (decorations), which allowed a 1PI self-energy diagram to be cut
in two by cutting two lines, we can characterize the skeleton expansion of the self-
energy as the set of skeleton diagrams that can not be cut in two by cutting two
lines (2PI-diagrams). Since propagator and impurity correlator lines, or say phonon
lines, appear topologically equivalently, we can restate quite generally: the skeleton
self-energy expansion consists of all the two-line or two-particle irreducible skeleton
diagrams.
By construction, only self-energy skeleton diagrams that can not be cut in two
by cutting only two full propagator lines appear, and we have the partially summed
diagrammatic expansion for the self-energy:




Σ(1, 1 ) = 1 1




+ 1 1

43 Synonymous names for the full Green™s function or propagator are renormalized or dressed
propagator.
4.6. Summary 119




+ 1 1




···
+ + . (4.149)
1 1




The partial summation of diagrams is unique, since the initial and ¬nal impurity
correlator lines are attached internally in di¬erent ways in each class of summed
diagrams. No double counting of diagrams thus takes place owing to the di¬erent
topology of the skeleton self-energy diagrams, and all diagrams in the perturbative
expansion of the self-energy, Eq. (4.137), are by construction contained in the skeleton
diagrams of Eq. (4.149).
What has been achieved by the partial summation, where each diagram corre-
sponds to an in¬nite sum of terms in perturbation theory, is that the self-energy is
expressed as a functional of the exact propagators or full Green™s functions

Σ(1, 1 ) = Σ(1,1 ) [G, D] . (4.150)
We can continue this topological classi¬cation, and introduce the higher-order
vertex functions; however, we defer this until Chapter 9.

Exercise 4.10. Draw the rest, in Eq. (4.149), of the four skeleton self-energy dia-
grams with three impurity correlators.
Exercise 4.11. Draw the skeleton self-energy diagrams for fermion“boson interac-
tion to fourth order in the coupling.


4.6 Summary
We have presented the formalism needed for treating general non-equilibrium situ-
ations. The closed time path formalism was shown to facilitate a convenient and
compact treatment of non-equilibrium statistical Green™s functions. Perturbation
theory valid for non-equilibrium states turned out in standard fashion, re¬‚ecting a
general Wick theorem for closed time path strings of operators, and the Feynman
diagrams for the contour ordered Green™s functions become of standard form. For
the reader with knowledge of equilibrium theory the good news is thus that the gen-
eral non-equilibrium formalism is formally equivalent to the equilibrium theory if
one elevates time to the contour level. For the reader not familiar with equilibrium
theory the good news is rejoice: knowledge of equilibrium theory is not needed, since
the equilibrium case is just a special simple case of the presented general theory.
However, the apparatus of the closed time path formalism needs a physical inter-
pretation, we need to get back to real time. In the next chapter we shall introduce
120 4. Non-equilibrium theory


the real-time technique and develop the diagrammatic structure of non-equilibrium
theory in a physically appealing language.
5

Real-time formalism

The contour-ordered Green™s function considered in the previous chapter was ideal
for discussing general closed time path properties such as the perturbative diagram-
matic structure for non-equilibrium states. However, the contour-ordered Green™s
function lacks physical transparency and does not appeal to intuition.1 We need a
di¬erent approach, which brings quantities back to real time. To accomplish this we
introduce a representation where forward and return parts of the closed time path
are ordered by numbers, specifying the position of a contour time by two indices,
i = 1, 2. Next is the diagrammatic perturbation theory in the real-time technique
then formulated in a fashion where the aspects of non-equilibrium states emerge in
the physically most appealing way. In particular, we shall construct the representa-
tion where spectral properties and quantum statistics show up on a di¬erent footing
in the diagrams. Lastly, we consider the connection to the imaginary-time treatment
of non-equilibrium states, and establish its equivalence to the real-time approach
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 Green™s function is by this tagging mapped onto a 2 — 2-
1 As the imaginary time Green™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 ± 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 Schwinger“Keldysh space
ˆ ˆ
G11 (1, 1 ) G12 (1, 1 )
ˆ
GC (1, 1 ) ’ G(1, 1 ) ≡ (5.1)
ˆ ˆ
G21 (1, 1 ) G22 (1, 1 )

ˆ
according to the prescription: the ij-component in the matrix Green™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 Green™s function are now standard times, 1 = (x, t1 ), and
we can identify them in terms of our previously introduced Green™s functions, the real-
time Green™s functions introduced in Section 3.3. The matrix structure re¬‚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 ) = ’i T (ψ(x1 , t1 ) ψ † (x1 , t1 )) = ’i T (ψ(1) ψ † (1 )) ,
ˆ (5.2)

the time-ordered Green™s function, where ψ(x, t) = ψH (x, t) is the ¬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 ) = ’i ψ(1) ψ † (1 ) ,
ˆ (5.3)

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

G12 (1, 1 ) = G< (1, 1 ) = “ i ψ † (1 ) ψ(1) ,
ˆ (5.4)

where upper and lower signs refer to bose and fermi ¬elds, respectively, and the
22-component is the anti-time-ordered Green™s function

G22 (1, 1 ) = G(1, 1 ) = ’i T (ψ(1) ψ † (1 )) .
ˆ ˜ ˜ (5.5)

We note that the time-ordered and anti-time-ordered Green™s functions can be
expressed in terms of G-greater and G-lesser, recall Eq. (3.64), and
ˆ
G11 (1, 1 ) = θ(t1 ’ t1 ) G> (1, 1 ) + θ(t1 ’ t1 ) G< (1, 1 ) . (5.6)

The matrix Green™s function in Eq. (5.1) can therefore be expressed in terms of
the real-time Green™s functions introduced in Section 3.3
G< (1, 1 )
G(1, 1 )
ˆ
G(1, 1 ) = . (5.7)
˜
G> (1, 1 ) G(1, 1 )

The way of representing the information in the contour-ordered Green™s function
as in Eq. (5.1) or equivalently Eq. (5.7) is respectable as, for example, the matrix
4 The thermo-¬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 ¬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 Green™s function lacks physical transparency, and o¬ers no basis for develop-
ing intuition. We shall therefore eventually transform to a di¬erent matrix form, and
as a ¬nal act liberate ourselves from the matrix out¬t altogether.
Let us now establish the Feynman rules in the real-time technique for the matrix
Green™s function in the dynamical index or Schwinger“Keldysh space.


5.2 Real-time diagrammatics
Instead of having the diagrammatics represent the perturbative expansion of the
contour Green™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 ¬eld. For particles interacting with a scalar potential V (x, t), we
have the diagrammatic expansion of the contour ordered Green™s function depicted
in Eq. (4.115) on page 105. The ¬rst-order diagram corresponded to the term
(1) (0) (0)
GC (1, 1 ) = dx2 d„2 GC (1, 2) V (2) GC (2, 1 ) . (5.8)
C

Parameterizing the real-time contour we have
∞ ’∞ ∞ ∞
dt ’
d„2 = dt + dt = dt (5.9)
’∞ ∞ ’∞ ’∞

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