the real time t0 (recall Exercise 3.21 on page 74). The appendix contour can therefore

matic proofs of conservation laws, i.e. for proving exact relations. The imaginary-time formalism is

seen to be a simple corollary of the closed time path formalism.

5.7. Real-time versus imaginary-time formalism 143

be deformed into the contour depicted in Figure 4.4 on page 91, landing the original

times on the imaginary appendix contour onto the real axis producing the contour-

ordered Green™s function. This provides the analytical continuation from imaginary

times to the real times of interest. We now turn to discuss the procedure in detail.

5.7.3 Analytical continuation procedure

By using the closed time path approach, the analytical continuation procedure is

automated, and the equations of motion for the real-time correlation functions are

obtained without the irrelevant detour into Matsubara frequencies.

The procedure to obtain the real time Dyson equation from either the imaginary-

time formalism, i.e. from Eq. (5.110), or from the general contour formalism Dyson

equation, Eq. (4.141), is in fact the same. In, for example, the perturbative expansion

of the contour-ordered Green™s function or in the Dyson equation, we encounter

objects integrated over the contour depicted in Figure 4.4 on page 91, and we need

to obtain the corresponding formulae in terms of real-time functions, and as we

demonstrate now this is equivalent to analytical continuation.

Since space (and spin) and contour-time variables play di¬erent roles in the fol-

lowing argument, in fact only the contour-time variables play a role, we separate

space and contour-time matrix notations

(A — B)(1, 1 ) ≡ dx2 A(1, 2) B(2, 1 ) (5.111)

σ2

and

(A2B)(1, 1 ) ≡ d„2 A(1, 2) B(2, 1 ) . (5.112)

c

Here the contour-time integration could refer to the imaginary time appendix contour

ca , or the contour ci , stretching from t0 through t1 and t1 and back again to t0 (or

all the way to in¬nity and back) and ¬nally along the appendix contour to t0 ’ iβ,

the contour depicted in Figure 4.4. The latter contour is obtained from the appendix

contour by the allowed analytical continuation procedure as discussed at the end

of the previous section. We shall not be interested in initial correlations, and can

therefore let the initial time protrude to the far past, t0 ’ ’∞, and the contour

in Figure 4.4 becomes the real-time closed contour, C, depicted in Figure 4.5. In

the case of analytical continuation from the imaginary-time appendix contour we

shall also eventually let the real-time t0 protrude to the far past. Everything in the

following, however, would be equally correct if we stick to the general contour, ci ,

depicted in Figure 4.4, allowing treating general initial states and therefore including

the completely general non-equilibrium problem. We would then just in addition

to integrations over the closed time path, have terms with integrations over the

imaginary time appendix contour ca .

Consider the case where the non-equilibrium situation is the result of a time-

dependent potential, V . The Dyson equation for the imaginary-time Green™s func-

tion is then given in Eq. (5.110), and for the contour-ordered Green™s function we

144 5. Real-time formalism

analogously have the equation

(0) (0)

GC (1, 1 ) = GC (1, 1 ) + dx3 d„3 dx2 d„2 GC (1, 3) Σ(3, 2) GC (2, 1 )

σ3 σ2

C C

(0)

+ dx2 d„2 GC (1, 2) V (2) GC (2, 1 ) , (5.113)

σ2 C

which can be written on the form (dropping the contour reminder subscript)

G + G(0) V

G = G0 + G0 Σ G, (5.114)

where Σ denotes the self-energy for the problem of interest.

We thus encounter explicitly contour matrix-multiplication, or multiplication in

series, a term of the form

C(„1 , „1 ) = d„ A(„1 , „ ) B(„, „1 ) = (A2B)(„1 , „1 ) , (5.115)

c

where A and B are functions of the contour variable, and the involved contour could

be any of the three one can encounter as discussed above. Degrees of freedom are

suppressed since they play no role in the following demonstration that contour inte-

grations can be turned into integrations over the real-time axis. To accomplish this

we recall that the functions C < („1 , „1 ) and C > („1 , „1 ) are analytic functions in the

strips 0 < m(„1 ’ „1 ) < β and ’β < m(„1 ’ „1 ) < 0, respectively.

Let us demonstrate the analytical continuation procedure for the case of C < . A

lesser quantity means by the general prescription, Eq. (4.41), that the contour time

„1 appears earlier than the contour time „1 , whatever contour is involved, i.e. we

<

have chosen the relationship „1 c „1 (irrespective of the numerical relationship of

their corresponding real time values in the case of the real-time contour). Exploiting

analyticity, the contour C or ci or the imaginary time contour ca is deformed into

the contour c1 + c1 depicted in Figure 5.7.14

t0 „1 „1

c1

c1

Figure 5.7 Deforming either of the contours C or ci or ca into the contour built by

the contours c1 and c1 .

14 Starting the ascent to real times, we essentially follow Langreth [20].

5.7. Real-time versus imaginary-time formalism 145

The expression in Eq. (5.115), for the chosen contour ordering, therefore becomes

C < (1, 1 ) = d„ A(„1 , „ ) B(„, „1 ) + d„ A(„1 , „ ) B(„, „1 )

c1 c1

d„ A(„1 , „ ) B < („, „1 ) + d„ A< („1 , „ ) B(„, „1 )

= (5.116)

c1 c1

<

and in the last equality, we have used the fact that on contour c1 we have „ c „1 ,

>

and on contour c1 we have „ c „1 . In the event of including initial correlations, or

rather staying with the general exact equation, the additional term with integration

over the appendix contour should be retained in the above equation.

Splitting in forward and return contour parts we have (as a consequence of the

contour positioning of the times on the contour parts in question as indicated to the

right)

’

’ : „ < „1

C < (1, 1 ) d„ A> („1 , „ ) B < („, „1 ) c

= c1

’

’

c1

← : „ > „1

’

d„ A< („1 , „ ) B < („, „1 ) c

+ c1

←

’

c1

’ : „ < „1

’

d„ A< („1 , „ ) B < („, „1 ) c

+ c1

’

’

c1

c’

← : „ > „1 .

d„ A< („1 , „ ) B > („, „1 ) c

+ (5.117)

1

c’

←1

Parameterizing the contours in terms of the real time variable, as in Eq. (4.35), and

noting that the external contour variables, „1 and „1 , now can be identi¬ed by their

corresponding values on the real time axis, gives (t0 ’ ’∞)

t1

dt (A> (t1 , t) ’ A< (t1 , t)) B < (t, t1 )

<

C (1, 1 ) =

’∞

t1

dt A< (t1 , t)(B < (t, t1 ) ’ B > (t, t1 ))

+ (5.118)

’∞

146 5. Real-time formalism

and thereby

∞

dt θ(t1 ’ t)(A> (t1 , t) ’ A< (t1 , t)) B < (t, t1 )

C < (1, 1 ) =

’∞

∞

dt A< (t1 , t) θ(t1 ’ t)(B < (t, t1 ) ’ B > (t, t1 )) .

+ (5.119)

’∞

Introducing the retarded function

AR (1, 1 ) = θ(t1 ’ t1 ) (A> (1, 1 ) ’ A< (1, 1 )) (5.120)

and the advanced function

AA (1, 1 ) = θ(t1 ’ t1 ) (A> (1, 1 ) ’ A< (1, 1 )) (5.121)

we have the real-time rule for multiplication in series

AR —¦ B < + A< —¦ B A ,

C< = (5.122)

where —¦ symbolizes matrix multiplication in real time, i.e. integration over the inter-

nal real-time variable from minus in¬nity to plus in¬nity of times.

Analogously one shows

AR —¦ B > + A> —¦ B A .

C> = (5.123)

We shall also need an expression for C R , and from Eq. (5.122) and Eq. (5.123)

we get

θ(t1 ’ t1 ) (AR —¦ B > )(t1 , t1 ) + (A> —¦ B A )(t1 , t1 )

C R (t1 , t1 ) =

’ (AR —¦ B < )(t1 , t1 ) + (A< —¦ B A )(t1 , t1 )

θ(t1 ’ t1 )((AR —¦ (B > ’ B < ))(t1 , t1 ) + ((A> ’ A< ) —¦ B A )(t1 , t1 )).

=

(5.124)

Expressing retarded and advanced functions according to Eq. (5.120) and Eq. (5.121)

gives

⎛t

1

θ(t1 ’ t1 ) ⎝ dt (A> (t1 , t) ’ A< (t1 , t)) (B > (t, t1 ) ’ B < (t, t1 ))

R

C (t1 , t1 ) =

’∞

⎞

t1

dt (A> (t1 , t) ’ A< (t1 , t)) (B > (t, t1 ) ’ B < (t, t1 ))⎠

+

’∞

5.7. Real-time versus imaginary-time formalism 147

t1

θ(t1 ’ t1 ) dt (A> (t1 , t) ’ A< (t1 , t)) (B > (t, t1 ) ’ B < (t, t1 )) .

=

t1

(5.125)

Using the fact that t1 < t < t1 (as otherwise both left- and right-hand sides vanish)

we obtain

C R = AR —¦ B R . (5.126)

Analogously one arrives at

C A = AA —¦ B A . (5.127)

By using Eq. (5.122), Eq. (5.123), Eq. (5.126) and Eq. (5.127) we ¬nd, owing to

the associative property of the composition, that the analytical continuation of the

contour quantity

D = A2B2C (5.128)

becomes

> > > >

D< = AR —¦ B R —¦ C < + AR —¦ B < —¦ C A + A< —¦ B A —¦ C A (5.129)

and, by induction, Eq. (5.129) generalizes to an arbitrary number of functions mul-

tiplied in series. Note that the retarded and advanced functions appear to the left

and right, respectively, of the greater and lesser Green™s functions.

Employing the analytical continuation procedure, one can thus from the imaginary-

time Green™s function formalism arrive with equal ease at the real-time non-equilibrium

Dyson equations of Section 5.5.

The only other ingredient encountered in perturbation expansions is the product

of contour-ordered Green™s functions of the form

C(„1 , „1 ) = A(„1 , „1 ) B(„1 , „1 ) , (5.130)

multiplication in parallel. This occurs, for example, for pair-creation or electron-

hole excitations, etc., or for a self-energy diagram for example for fermion“boson

interaction, in which case one might prefer to have the same sequence of the contour

variables in all of the functions as in the self-energy insertion of the diagram in Figure

5.6 (this is, of course, a matter of taste).

Following the above procedure we immediately get for the analytical continuation

of multiplication in parallel

> > >

C < (t1 , t1 ) = A< (t1 , t1 ) B < (t1 , t1 ) . (5.131)

With these tools at hand, we can turn any exact imaginary-time formula, or any

diagram in the perturbative expansion of the imaginary-time Green™s function or a

contour ordered quantity, say the contour-ordered Green™s functions of Section 4.4,

into products of real-time Green™s functions. This automatic mechanical continuation

148 5. Real-time formalism

to real times is much preferable than to do it in the Matsubara frequencies. With

this at hand a very e¬ective way of studying non-equilibrium states in the real time

formalism is available, as discussed in the classic text [14], and whether using this or

the other three-fold representation is a matter of taste. However, the G-greater and

G-lesser Green™s functions are quantum statistically quantities of the same nature,

whereas in the representation introduced in Section 5.3, the Green™s functions carry

distinct information.

5.7.4 Kadano¬“Baym equations

As an example of using the analytical continuation procedure we shall, from the

Dyson equation for the imaginary-time Green™s function in Eq. (5.110) (or the general

contour-ordered Green™s function, Eq. (5.114)), obtain the equations of motion for

the physical correlation functions, the lesser and greater Green™s functions on the

real-time axis. Let us therefore consider the Dyson equation for the imaginary-time

Green™s function or the general contour-ordered Green™s function of the form

G’1 G = δ(1 ’ 1 ) + Σ G, (5.132)

0

where external ¬elds are included in G’1 . Applying the rule for multiplication in

0

series gives

> > >

G’1 —G = Σ —G + Σ — GA

< R < <

(5.133)

0

and similarly for the right-hand Dyson equation

> > >

G< — G’1 = GR — Σ< + G< — ΣA . (5.134)

0

Subtracting the left and right Dyson equations gives

> > > > >

[G’1 — G ]’ = Σ — G ’ G —Σ + Σ —G ’ G —Σ

< R < < A < A R <

, (5.135)

0

which can be rewritten

1R 1>

> >

[G’1 — G< ]’ [Σ + ΣA — G< ]’ ’ [Σ< — GR + GA ]’

’

, , ,

0

2 2

1 1

’ [Σ< — G> ]+ + [Σ> — G< ]+ .

= , , (5.136)

2 2

These two equations, the Kadano¬“Baym equations, can be used as basis for consid-

ering quantum kinetics.

We recall that the kinetic Green™s function is speci¬ed according to GK = G> +

G< , and note that adding the two equations in Eq. (5.136) we recover Eq. (5.95). We

note that the equations, Eq. (5.129), satis¬ed by GK are satis¬ed by both G> and

G< , or rather we should appreciate the observation that their equations are identical

with respect to splitting into retarded and advanced Green™s functions.

>

Since the equations for G< mixes, through for example self-energies according to

Eq. (5.131), it is economical to work instead solely with GK . However, there can be

5.8. Summary 149

special circumstances where the advantage is reversed, for example when discussing

the dynamics of a tunnel junction. One should note that the quantum statistics of

particle species manifests itself quite di¬erently depending on which type of kinetic

propagator one chooses to employ.

5.8 Summary

We have presented the real-time formalism necessary for treating non-equilibrium

situations. For the reader not familiar with equilibrium theory the good news is that

equilibrium theory is just an especially simple case of the presented general theory.

In the real-time formulation of the properties of non-equilibrium states the dynamics

is used to provide a doubling of the degrees of freedom, and one encounters at least

two types of Green™s functions. To get a physically transparent representation, we

introduced the real-time matrix representation of the contour-ordered Green™s func-

tions to describe non-equilibrium states. This allowed us to represent matrix Green™s

function perturbation theory in terms of Feynman diagrams in a standard fashion.

We introduced the physical representation corresponding to the two Green™s func-

tions representing the spectral and quantum statistical properties of a system. We

then showed that the matrix notation can be broken down into two simple rules for

the universal vertex structure in the dynamical indices. This allowed us to formulate

the non-equilibrium aspects of the Feynman diagrams directly in terms of the vari-

ous matrix Green™s function components, R, A, K, establishing the real rules. In this

way we were able to express how the di¬erent features of the spectral and quantum

statistical properties enter into the diagrammatic representation of non-equilibrium

processes. We ended the chapter by showing the equivalence of the imaginary-time

and the closed time path and the real-time formalisms, all formally identical, and

transformed into each other by analytical continuation. In the rest of the book we

shall demonstrate the versatility of the real-time technique. Before constructing the

functional formulation of quantum ¬eld theory from its Feynman diagrams, and show

that its classical limit can be used to study classical statistical dynamics, as done in

the last chapter, in the next three chapters we demonstrate various applications of

the real-time formalism to the study of quantum dynamics.

6

Linear response theory

There exists a regime of overlap between the equilibrium and non-equilibrium be-

havior of a system, the non-equilibrium behavior of weakly perturbed states. When

a system is perturbed ever so slightly, its response will be linear in the perturba-

tion, say the current of the conduction electrons in a metal will be proportional to

the strength of the applied electric ¬eld. This regime is called the linear response

regime, and though the system is in a non-equilibrium state all its characteristics

can be inferred from the properties of its equilibrium state. In the next chapter we

shall go beyond the linear regime by showing how to obtain quantum kinetic equa-

tions. The kinetic-equation approach to transport is a general method, and allows

in principle nonlinear e¬ects to be considered. However, in many practical situa-

tions one is interested only in the linear response of a system to an external force.

The linear response limit is a tremendous simpli¬cation in comparison with general

non-equilibrium conditions, and is the subject matter of this chapter. In particular