Hamiltonian H0 , the relation being equivalent to that in Eq. (4.28). The operator

HH0 („ ) is thus the mechanical external perturbation in the Heisenberg picture with

respect to H0 .19 We have thus analogous to the derivation of the expression Eq. (4.42)

for the contour-ordered Green™s function, Eq. (4.39), that the contour-ordered Green™s

function in the interaction picture is

(i )

Tr e’βH Tc e’i †

d„ (HH („ )+HH 0 („ ))

ψH0 (1) ψH0 (1 )

c 0

G(1, 1 ) = ’i . (4.47)

Tr (e’βH )

We have introduced the notation β = 1/kT for the inverse temperature.

19 We shall later take advantage of the arti¬ce of employing di¬erent dynamics on the forward and

backward paths, making the closed time path formulation a powerful functional tool.

4.3. Closed time path formalism 91

We can now employ Dyson™s formula, Eq. (4.27), for the case of a time-independent

Hamiltonian, H, and imaginary times, to express the Boltzmann weighting factor in

terms of the weighting factor for the free theory

t 0 ’i β (i )

e’βH = e’βH0 Tca e’i d„ HH („ )

(4.48)

t0 0

where Tca contour orders along the contour stretching down into the lower complex

time plane from t0 to t0 ’ iβ, the appendix contour ca as depicted in Figure 4.4. We

then get the expression

⎛ ⎛ ⎞ ⎞

t 0 ’i β

(i )

’i (i )

’i d„ (HH („ )+HH 0 („ ))

d„ HH („ )

Tr⎝e’βH0⎝Tca e ⎠Tc e ψH0 (1)ψH0 (1 )⎠

†

0 0

t0 c

iG(1, 1 ) = t 0 ’i β (i )

Tr e’βH0 Tca e’i t 0 d„ HH („ )

0

(4.49)

ready-made for a perturbative expansion of the contour-ordered Green™s function

valid for the non-equilibrium case. The term involving imaginary times stretching

down into the lower complex time plane from t0 to t0 ’ iβ can be brought under one

contour ordering by adding the appendix contour ca to the contour c giving in total

the contour ci as depicted in Figure 4.4, and we thus have

(i )

’i †

d„ HH („ ) ’i

Tr e’βH0 Tci d„ HH 0 („ )

e e ψH0 (1) ψH0 (1 )

ci c

0

G(1, 1 ) = ’i .

(i )

’i d„ HH („ ) ’i d„ HH („ )

Tr e’βH0 Tci e e

ci c

0 0

(4.50)

The contour ci stretches from t0 to max{t1 , t1 } (or in¬nity) and back again to t0

and has in addition to the contour c the additional appendix ca , i.e. stretches further

down into the lower complex time plane from t0 to t0 ’ iβ, as depicted in Figure 4.4.

t0 t1 t1

ci

t0 ’ iβ

Figure 4.4 The contour ci .

In the numerator we have used the fact that, under contour ordering, operators can

be commuted, leaving operator algebra identical to that of numbers, so that for

example

(i ) (i )

Tc e’i = Tc e’i d„ HH („ ) ’i

d„ (HH („ ) + HH 0 („ )) d„ HH 0 („ )

e . (4.51)

c c c

0 0

92 4. Non-equilibrium theory

The expression in Eq. (4.50) is of a form for which we can use Wick™s theorem

to obtain the perturbative expansion of the contour-ordered Green™s function and

the associated Feynman diagrammatics. Before we show Wick™s theorem in the next

section, some general remarks are in order.

In the denominator in Eq. (4.50), we introduced a closed contour contribution,

that of contour c, stretching from t0 to max{t1 , t1 } (or in¬nity) and back again to t0 ,

which since no operators interrupts at intermediate times is just the identity operator

(i )

Tc e’i d„ (HH („ ) + HH 0 („ ))

= 1. (4.52)

c 0

This was done in order for the expression in Eq. (4.50) to be written on the form where

the usual combinatorial arguments applies to show that unlinked or disconnected

diagrams originating in the numerator are canceled by the vacuum diagrams from

the denominator. However, for the non-equilibrium states of interest here, such

features are actually arti¬cial relics of the formalisms used in standard zero and

¬nite time formalisms. A reader not familiar with these combinatorial arguments

need not bother about these remarks since we shall now specialize to the situation

where this feature is absent.20

We note that only interactions are alive on the appendix contour part, ca , whereas

the external perturbation vanishes on this part of the contour. If we are not interested

in transient phenomena in a system or physics on short time scales of the order

of the collision time scale due to the interactions, we can let t0 approach minus

in¬nity, t0 ’ ’∞, and the contribution from the imaginary part of the contour ci

vanishes. The physical argument is that a propagator with one of its arguments on

the imaginary time appendix is damped on the time scale of the scattering time of the

system. Thus as the initial time, t0 , where the system is perturbed by the external

¬eld, retrudes back into the past beyond the microscopic scattering times of the

system, then e¬ectively t0 ’ ’∞, and contributions due to the imaginary appendix

part ca of the contour vanish.21 The denominator in Eq. (4.49) thus reduces to the

partition function for the non-interacting system and we ¬nally have for the contour-

ordered or closed time path Green™s function

(i )

e’i †

d„ (HH („ ) + HH 0 („ ))

G(1, 1 ) = Tr ρ0 TC ψH0 (1) ψH0 (1 )

C 0

(i )

e’i †

d„ HH („ ) ’i d„ HH 0 („ )

= Tr ρ0 TC e ψH0 (1) ψH0 (1 ) , (4.53)

C C

0

where

e’H0 /kT

ρ0 = (4.54)

Tr e’H0 /kT

20 In Section 9.5, where we start studying physics from scratch in terms of diagrammatics, the

cancellation of the vacuum diagrams is discussed in detail. There, both a diagrammatic proof as

well as the combinatorial proof relevant for the present discussion are given for the cancellation of

the numerator by the separated o¬ vacuum diagrams of the numerator.

21 If the interactions are turned on adiabatically, then as the arbitrary initial time is retruding

back into the past, t0 ’ ’∞, the interaction vanishes in the past, and therefore vanishes on

the imaginary appendix part of the contour. However, there is no need to appeal to adiabatic

coupling since interaction always has the physical e¬ect of intrinsic damping. We note that at ever

increasing temperatures, the appendix contour contribution disappears, since thermal ¬‚uctuations

then immediately wipe out initial correlations.

4.3. Closed time path formalism 93

is the statistical operator for the equilibrium state of the non-interacting system at

the temperature T . The last equality sign follows since the algebra of Hamiltonians

under contour ordering is equivalent to that of numbers. The contour C appearing

in Eq. (4.53) is Schwinger™s closed time path [5], the Schwinger“Keldysh or real-time

contour, which starts at time t = ’∞ and proceeds to time t = ∞ and then back

again to time t = ’∞, as depicted in Figure 4.5.

c1

t

c2

Figure 4.5 The Schwinger“Keldysh closed time path or real-time closed contour.

We note that non-equilibrium perturbation theory in fact has a simpler structure

than the standard equilibrium theory as there is no need for canceling of unlinked or

disconnected diagrams. The contour evolution operator for a closed loop is one: in

the perturbative expansion for the denominator in Eq. (4.50)

(i )

D = Tr e’βH0 Tc e’i d„ HH („ ) ’i d„ HH 0 („ )

e (4.55)

c c

0

only the identity term corresponding to no evolution survives, all other terms comes

in two, one with a minus sign, and the sum cancels. We shall take advantage of the

absence of this so-called denominator-problem in Chapter 12, and this aspect of the

presented non-equilibrium theory is a very important aspect in the many applications

of the closed time path formalism: from the dynamical approach to perform quenched

disorder average to the ¬eld theory of classical statistical dynamics.22

Before turning to obtain the full diagrammatics of non-equilibrium perturbation

theory, let us acquaint ourselves with lowest order terms. The simplest kind of

coupling is that of particles to an external classical ¬eld V (x, t). In that case the

contour ordered Green™s function has the form ready for a perturbative expansion

†

’i d„ dx V (x,„ ) ψH (x,„ ) ψH 0 (x,„ )

†

C 0

GC (1, 1 ) = Tr ρ0 TC e ψH0 (1) ψH0 (1 ) .

(4.56)

Expanding the exponential we get strings of, say, fermi ¬eld operators traced and

weighted with respect to the free statistical operator. The zeroth-order term just

gives the free contour Green™s function

†

(0)

GC (1, 1 ) = ’iTr ρ0 TC ψH0 (1) ψH0 (1 ) . (4.57)

22 This is an appealing alternative in the quantum ¬eld theoretic treatment of quenched disorder,

more physically appealing than the obscure Replica trick or supersymmetry methods, the latter

being limited to systems without interactions.

4.3. Closed time path formalism 95

by expanding the exponentials. When we expand the exponential in Eq. (4.50)

or Eq. (4.53), products of interaction Hamiltonians appear under contour ordering.

The generic case for the perturbative expansion to nth order of the contour-ordered

Green™s function is the trace of products, or strings, of the ¬eld operators of the theory

in the interaction picture weighted by the free part of the Hamiltonian, a quadratic

form in these ¬elds. For example, in the case of electron“phonon interaction a string

of n phonon ¬elds and 2n fermi ¬elds occurs; see Eq. (4.126). The weighted trace over

the fermi and bose ¬elds separates into the two traces over these independent degrees

of freedom. To be explicit, let us ¬rst consider the trace over the bose degrees of

freedom, and of interest is therefore the calculation of the weighted trace of a string

of contour-ordered bose ¬eld operators, ordered along a contour C.25 We introduce

the representation of the bose ¬eld in terms of its creation and annihilation operators

as in Eq. (2.74) and encounter strings of creation and annihilation operators26

S = tr(ρT TC (c(„n ) c(„n’1 ) . . . c(„2 ) c(„1 )))

≡ TC (c(„n ) c(„n’1 ) . . . c(„2 ) c(„1 )) , (4.60)

where the cs denote either a creation or annihilation operator, a or a† , and

(0)

e’Hb /kT

ρT = (4.61)

(0)

Tr e’Hb /kT

is the statistical operator for the equilibrium state of the non-interacting bosons or

phonons at temperature T , and

a† aq

(0)

Hb = hq = (4.62)

q q

q q

(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