T e’ T e’ dt H(t)

i i i

¯ ¯ ¯¯

dt HH (t) H(t’t0 )

=e . (4.27)

t t

Here Dyson™s formula appeared owing to unitary transformations between Heisen-

berg and interaction pictures, but once conjectured it can of course immediately be

established by direct di¬erentiation. Dyson™s formula is useful in many contexts,

be the time variable real or imaginary, and also for equilibrium states such as when

phase transitions are studied in, for instance, a renormalization group treatment. We

shall in fact apply Dyson™s formula for imaginary times in Section 4.3.2.

We now introduce the contour, the closed time path, which starts at t0 and

proceeds along the real time axis to time t and then back again to t0 , the closed

contour ct as depicted in Figure 4.1.

4.3. Closed time path formalism 85

t

ct

t0

Figure 4.1 The closed time path contour ct .

We then show that the transformation between the two Heisenberg pictures,

Eq. (4.24), can be expressed on closed contour form as (units are chosen to set

equal to one at our convenience)

’i d„ HH („ )

OH (t) = Tct e OH (t) , (4.28)

ct

where „ denotes the contour variable proceeding from t0 along the real-time axis to

t and then back again to t0 , i.e. the variable on ct . The contour ordering symbol Tct

orders products of operators according to the position of their contour time argument

on the closed contour, earlier contour time places an operator to the right.

The crucial equivalence of Eq. (4.24) and Eq. (4.28), which form a convenient

basis for formulating perturbation theory in the closed time path formalism, is based

on the algebra of operators under the contour ordering being equivalent to the algebra

of numbers.12 Expanding the exponential in Eq. (4.28) gives

∞

(’i)n

OH (t) = d„1 . . . d„n Tct (HH („1 ) . . . HH („n ) OH (t)) . (4.29)

n! ct ct

n=0

Let us consider the nth order term. In order to verify Eq. (4.28), we note that the

contour can be split into forward and backward parts

ct = ’ + ← .

’’

c c (4.30)

Splitting the contour into forward and backward contours gives 2n terms. Out of these

there are n!/(m!(n ’ m)!) terms (m = 0, 1, 2, . . . , n), which contain m integrations

over the forward contour, and the rest of the factors, n’ m, have integratons over the

backward contour. Since they di¬er only by a di¬erent dummy integration labeling

they all give the same contribution and

n

n!

d„1 . . . d„n Tct (HH („1 ) . . . HH („n ) OH (t)) =

m!(n ’ m)!

ct ct m=0

— ’

d„m+1 . . . d„n T← (HH („m+1 ) . . . HH („n )) OH (t)

←

’ ←

’ c

c c

— ’

d„1 . . . d„m T’ (HH („1 ) . . . HH („m )) , (4.31)

’

’ ’

’ c

c c

12 Even though the Hamiltonian for fermions contains non-commuting objects, the fermi ¬elds,

they appear in pairs and quantum statistical minus signs do not occur.

86 4. Non-equilibrium theory

’ ’

where T’ and T← denotes contour ordering on the forward and backward parts,

c c

respectively. Adding a summation and a compensating Kronecker function the nth-

order term can be rewritten in the form13

∞ ∞

n!

d„1 . . . d„n Tct (HH („1 ) . . . HH („n ) OH (t)) = δn,k+m

m! k!

ct ct m=0

k=0

— ’

d„1 . . . d„k T← (HH („1 ) . . . HH („k )) OH (t)

←

’ ←

’ c

c c

— ’

d„1 . . . d„m T’ (HH („1 ) . . . HH („m )) . (4.32)

’

’ ’

’ c

c c

The summation over n in Eq. (4.29) is now trivial, giving

∞ ∞

(’i)k (’i)m

’i d„ HH („ )

T ct e OH (t) =

ct

m! k!

m=0

k=0

— ’

d„1 . . . d„k T← (HH („1 ) . . . HH („k )) OH (t)

←

’ ←

’ c

c c

— ’

d„1 . . . d„m T’ (HH („1 ) . . . HH („m )) (4.33)

’

’ ’

’ c

c c

and thereby

’i ’i

d„ HH („ ) d„ HH („ )

←

’ ’

’

’i d„ HH („ )

’ ’

ec ec

T ct e OH (t) = T← OH (t) T’ .

ct

c c

(4.34)

Parameterizing the forward and backward contours according to

„ (t ) = t t [t0 , t] , (4.35)

we get

t

’i d„ HH („ ) ’i

’

’ dt HH (t )

T’ e c

’ = Te = V (t, t0 ) (4.36)

t0

c

and

’i d„ HH („ )

←

’ t

= V † (t, t0 )

˜i dt HH (t )

’

T← e c = Te (4.37)

t0

c

i.e. contour ordering along the forward contour is identical to ordinary time ordering,

’

T’ = T , whereas contour ordering along the backward contour corresponds to anti-

c

’

time ordering, T← = T . The equivalence of Eq. (4.24) and Eq. (4.28) has thus been

c

established. We have shown that the times in V † (t, t0 ) corresponds to contour times

13 Underthe ordering operation, the algebra of non-commuting objects reigning the operators is

not important, and the consideration is essentially the algebra of showing exp(a+b) = exp(a) exp(b).

4.3. Closed time path formalism 87

lying on the backward part, and the times in V (t, t0 ) corresponds to contour times

lying on the forward part.

We shall now use Eq. (4.28) to introduce the contour variable instead of the time

variable. We hereby embark on Schwinger™s closed time path formulation of non-

equilibrium quantum statistical mechanics originally introduced in reference [5].14

We shall thereby develop the diagrammatic perturbative structure of the closed time

path or contour ordered Green™s function.

4.3.1 Closed time path Green™s function

A generalization o¬ers itself, which will lead to a single object in terms of which non-

equilibrium perturbation theory can be formulated. The trick will be to democratize

the status of all times appearing in the time-ordered Green™s function, Eq. (4.18),

i.e. the original real times t and t will be perceived to reside on the closed time path

or contour. The one-particle Green™s function in Eq. (4.18) contains two times; let

us now denote them t1 and t1 . We introduce the contour, which starts at t0 and

proceeds along the real-time axis through t1 and t1 and then back again to t0 , the

closed contour c as depicted in Figure 4.2, c = ’ + ←.15 We have hereby freed the

’’

c c

time variables, which hitherto were tied to the real axis, to lie on either the forward

or return part of the contour, and we introduce the contour variable „ to signify this

two-valued choice of the time variable, examples of which are given in Figure 4.2.16

„1

„1

t0 t1 t1 t

Figure 4.2 Examples of real times being elevated to contour times.

We are thus led to study the closed time path Green™s function or the contour-

ordered Green™s function

†

Tr(e’H/kT Tc (ψH (x1 , sz1 , „1 ) ψH (x1 , sz1 , „1 )))

, „1 ) = ’i

G(x1 , sz1 , „1 , x1 , sz1

Tr(e’H/kT )

(4.38)

14 Reviews of the closed time path formalism stressing various applications are, for example, those

of references [6], [7] and [8].

15 If we discussed a correlation function involving more than two ¬elds, the contour should stretch

all the way to the maximum time value, or in fact we can let the contour extend from t0 to t = ∞

and back again to t0 , since, as we soon realize, beyond max(t1 , t1 , . . .) the forward and backward

evolutions take each other out, producing simply the identity operator.

16 For mathematical rigor, i.e. proper convergence, both the forward and backward contours

should be conceived of as being located in¬nitesimally below the real axis. This will be witnessed

by the analytical continuation procedure discussed in Section 5.7, but in practice this consideration

will not be necessary.

88 4. Non-equilibrium theory

where „1 and „1 can lie on either the forward or backward parts of the closed contour.

We have had a particle with spin in mind, say the electron, but introducing the

condensed notation 1 = (x1 , sz1 , „1 ) we have17

† †

G(1, 1 ) = ’i Tc(ψH (1) ψH (1 )) = ’i Z ’1 Tr(e’H/kT Tc (ψH (1) ψH (1 ))) (4.39)

at which stage any particle could be under discussion as the only relevant thing in

the rest of the section is the contour variable. A contour ordering symbol Tc has been

introduced, which orders operators according to the position of their contour-time

argument on the closed contour, for example for the case of two contour times

c

ψ(x1 , „1 ) ψ † (x1 , „1 ) „1 > „1

†

Tc (ψ(x1 , „1 ) ψ (x1 , „1 )) = (4.40)

c

“ψ † (x1 , „1 ) ψ(x1 , „1 ) „1 > „1

where the upper (lower) sign is for fermions (bosons) respectively. An obvious nota-

c

tion for ordering along the contour has been introduced, viz. „1 > „1 means that „1

is further along the contour c than „1 irrespective of their corresponding numerical

values on the real axis. The contour ordering thus orders an operator sequence ac-

cording to the contour position; operators with earliest contour times are put to the

right. The algebra of bose ¬elds under the contour ordering is thus like the algebra

of (complex) numbers, whereas the algebra of fermi ¬elds under the contour ordering

is like the Grassmann algebra of anti-commuting numbers.18

We also introduce greater and lesser quantities for the contour ordered Green™s

function, and note according to the contour ordering, Eq. (4.40),

ct

G< (1, 1 ) „1 > „1

G(1, 1 ) = (4.41)

ct

G> (1, 1 ) „1 > „1 .

Here lesser refers to the contour time „1 appearing earlier than contour time „1 , and

vice versa for greater. Note that these relationships are irrespective of the numerical

relationship of their corresponding real time values: if the contour times in G< (1, 1 )

and G> (1, 1 ) are identi¬ed with their corresponding real times we recover their

corresponding real-time Green™s functions discussed in Section 3.3.

Transforming from the Heisenberg picture with respect to the Hamiltonian H(t)

to the Heisenberg picture with respect to the Hamiltonian H, gives, according to

Eq. (4.28),

†

’i ψH (1) ψH (1 )

G> (1, 1 ) =

’i

’i d„ HH („ ) †

d„ HH („ )

’i Tct 1 e ct

ct

= ψH (1) T ct e ψH (1 )

1 1

1

17 In the following we shall consider the ¬elds as entering the Green™s function, however, for

the following it could be any type of operators and any number of products, G(1, 2, 3, . . .) =

† †

Tc (AH (1) BH (2) CH (3) . . .) . Note that if the operators represent physical quantities, they are

speci¬ed in terms of the ¬elds, and we are back to strings of ¬eld operators modulo the operations

speci¬c to the quantities in question.

18 In Chapter 10 we shall in fact show that in view of this, quantum ¬eld theory can, instead of

being formulated in terms of quantum ¬eld operators, be formulated in terms of scalar or Grassmann

numbers by the use of path integrals.

4.3. Closed time path formalism 89

’i d„ HH („ ) †

’i Tct 1 +ct 1

+c t

ct

= e ψH (1) ψH (1 ) , (4.42)

1 1

where the contours ct1 (ct1 ) starts at t0 and passes through t1 (t1 ), respectively,

and returns to t0 . In the last equality the combined contour, ct1 + ct1 , depicted

in Figure 4.3, has been introduced. It stretches from t0 to min{t1 , t1 } and back to

t0 and then forward to max{t1 , t1 } before ¬nally returning back again to t0 . The

contributions from the hatched parts depicted in Figure 4.3 cancel since for this part

the ¬eld operators at times t1 and t1 are not involved and a closed contour appears

which gives the unit operator, or equivalently U † (t1 , t0 ) U (t1 , t0 ) = 1, and the last

equality in Eq. (4.42) is established. By the same argument, the contour could be

extended from max{t1 , t1 } all the way to plus in¬nity before returning to t0 , and we

encounter the general real-time contour.

t0 t1 t1

c1

c1

Figure 4.3 Parts of contour evolution operators canceling in Eq. (4.42).

We have an analogous situation for G< (1, 1 ), and we have shown that

†

= ’i Tc (ψH (1) ψH (1 ))

G(1, 1 )

†

= ’i Tc e’i d„ HH („ )

ψH (1) ψH (1 ) , (4.43)

c

where the contour c starts at t0 and stretches through max(t1 , t1 ) (or all the way to

plus in¬nity) and back again to t0 . By introducing the closed contour and contour

ordering we have managed to bring all operators under the ordering operation, which

will prove very useful when it comes to deriving the perturbation theory for the

contour-ordered Green™s function.

Exercise 4.1. From the equation of motion for the ¬eld operator, show that the

equation of motion for the contour-ordered Green™s function is

‚

’ h0 („ ) + μ G(x, „, x , „ ) = δ(x ’ x ) δc („ ’ „ )

i

‚„

i Tc ([ψ † (x, „ ), Hi („ )]ψ † (x , „ )) , (4.44)

’

where h0 denotes the single-particle Hamiltonian, and we have introduced the contour

delta function

§

⎨ δ(„ ’ „ ) for „ and „ on forward branch

’δ(„ ’ „ ) for „ and „ on return branch

δc („ ’ „ ) = (4.45)

©

0 for „ and „ on di¬erent branches

90 4. Non-equilibrium theory

and Hi („ ) is the interaction part of the Hamiltonian in the Heisenberg picture (recall

Exercise 3.10 on page 66).

The equation of motion for the Green™s function leads, as noted in Section 3.3, to

an in¬nite hierarchy of equations for correlation functions containing an ever increas-

ing number of ¬eld operators describing the correlations between the particles set

up in the system by the interactions and external forces. Needless to say, an exact

solution of a quantum ¬eld theory is a mission impossible in general. At present,

the only general method available for gaining knowledge from the fundamental prin-

ciples about the dynamics of a system is the perturbative study. This goes for

non-equilibrium states a fortiori, and we shall now construct the perturbation the-

ory valid for non-equilibrium states. This consists of dividing the Hamiltonian into

one part representing a simpler well-understood problem and a nontrivial part, the

e¬ect of which is studied order by order.

In the next section we construct the general perturbation theory valid for non-

equilibrium situations. We thus embark on the construction of the diagrammatic

representation starting from the canonical formalism presented in Chapter 1.

4.3.2 Non-equilibrium perturbation theory

We now proceed to obtain the perturbation theory expressions for the contour-

ordered Green™s functions. The Hamiltonian of the system, Eq. (4.9) consists of

a term quadratic in the ¬elds, H0 , describing the free particles, and a complicated

term, H (i) , describing interactions. To get an expression ready-made for a pertur-

bative expansion of the contour-ordered Green™s function, the Hamiltonian in the

weighting factor needs to be quadratic in the ¬elds, i.e. we need to transform the

operators in Eq. (4.42) to the interaction picture with respect to H0 . Quite analogous

to the manipulations in the previous section we have

(i )

’i d„ (HH („ )+HH 0 („ ))

OH (t) = Tct e OH0 (t) , (4.46)

ct 0

where we have further, or directly, transformed from the Heisenberg picture with