ńņš. 21 |

136 5. Real-time formalism

be cut in two be cutting only two full propagator lines. It is the latter representation

that is useful in the Dyson equation.

Equivalently, by iterating from the left gives the matrix Dyson equation

G = G0 + G ā— Ī£ ā— G0 . (5.69)

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. However, in a non-equilibrium state, the two matrix equations contain

diļ¬erent information and subtracting them is a useful way of expressing the non-

equilibrium dynamics, and we shall exploit this in Chapters 7 and 8.

Since the transformation of the real-time matrix self-energy is identical to the one

for the matrix Greenā™s function we get, analogously to the equations from Eq. (5.36)

to Eq. (5.39), and therefore for the components of the self-energy matrix,

Ė Ė Ė Ė

Ī£R = Ī£11 ā’ Ī£12 = Ī£21 ā’ Ī£22 (5.70)

Ė Ė Ė Ė

Ī£A = Ī£11 ā’ Ī£21 = Ī£12 ā’ Ī£22 (5.71)

Ė Ė Ė Ė

Ī£K = Ī£11 + Ī£22 = Ī£12 + Ī£21 (5.72)

Ė Ė Ė Ė

0 = Ī£11 ā’ Ī£12 + ā’ Ī£21 + Ī£22 . (5.73)

By construction

Ė

Ī£12 (x1 , t1 , x1 , t1 ) t1 > t1

Ė

Ī£11 (x1 , t1 , x1 , t1 ) = (5.74)

Ė

Ī£21 (x1 , t1 , x1 , t1 ) t1 > t1

and

Ė

Ī£21 (x1 , t1 , x1 , t1 ) t1 > t1

Ė

Ī£22 (x1 , t1 , x1 , t1 ) = (5.75)

Ė

Ī£12 (x1 , t1 , x1 , t1 ) t1 > t1

and the matrix self-energy has in the triagonal representation the same triagonal

form as the matrix Greenā™s function

Ī£R Ī£K

Ī£= . (5.76)

Ī£A

0

Exercise 5.6. Introducing

Ė

Ī£< (x1 , t1 , x1 , t1 ) = Ī£12 (x1 , t1 , x1 , t1 ) (5.77)

and

Ė

Ī£> (x1 , t1 , x1 , t1 ) = Ī£21 (x1 , t1 , x1 , t1 ) (5.78)

show that we have, identically to the relationships for the Greenā™s functions, the

relation for the retarded self-energy

Ī£R (x, t, x , t ) = Īø(t ā’ t ) Ī£> (x, t, x , t ) ā’ Ī£< (x, t, x , t ) (5.79)

and advanced self-energy

Ī£A (x, t, x , t ) = ā’Īø(t ā’ t) Ī£> (x, t, x , t ) ā’ Ī£< (x, t, x , t ) (5.80)

5.5. Non-equilibrium Dyson equations 137

and for the kinetic component

Ī£K (x, t, x , t ) = Ī£> (x, t, x , t ) + Ī£< (x, t, x , t ) . (5.81)

Show that the components of the self-energy matrix satisļ¬es

ā—

Ī£A (x, t, x , t ) = Ī£R (x , t , x, t) (5.82)

and

ā—

Ī£K (x, t, x , t ) = Ī£K (x , t , x, t) . (5.83)

Exercise 5.7. Show that in the case where the matrix Greenā™s function is represented

in symmetric form

GA

0

G= (5.84)

GR GK

the matrix self-energy has the form

Ī£K Ī£R

Ī£= . (5.85)

Ī£A 0

We shall not at present take the diagrammatics beyond the self-energy to higher-

order vertices, since in the following chapters only the Dyson equation is needed. In

Chapter 9 we shall study diagrammatics in their full glory.

From the equation of motion for the free Greenā™s function (or ļ¬elds) we then get,

for the matrix Greenā™s function, the equations of motion

(iā‚t1 ā’ h(1))G(1, 1 ) = Ī“(1 ā’ 1 ) + (Ī£ ā— G)(1, 1 ) (5.86)

and

(iā‚t1 ā’ hā— (1 ))G(1, 1 ) = Ī“(1 ā’ 1 ) + (G ā— Ī£)(1, 1 ) (5.87)

or introducing the inverse free Greenā™s function

Gā’1 (1, 1 ) = (iā‚t1 ā’ h(1)) Ī“(1 ā’ 1 ) (5.88)

0

the two equations can be expressed through operating with the inverse free matrix

Greenā™s function from the left

(Gā’1 ā’ Ī£) ā— G = Ī“(1 ā’ 1 ) (5.89)

0

and from the right

G ā— (Gā’1 ā’ Ī£) = Ī“(1 ā’ 1 ) . (5.90)

0

These two non-equilibrium Dyson equations will prove useful in Chapter 7 where

quantum kinetic equations are considered.

The matrix equation, Eq. (5.89), comprises the three coupled equations for GR,A,K

(Gā’1 ā’ Ī£R(A) ) ā— GR(A) = Ī“(1 ā’ 1 ) (5.91)

0

138 5. Real-time formalism

and

Gā’1 ā— GK = Ī£R ā— GK + Ī£K ā— GA . (5.92)

0

Analogously, from Eq. (5.90), we obtain

GR(A) ā— (Gā’1 ā’ Ī£R(A) ) = Ī“(1 ā’ 1 ) (5.93)

0

and

GK ā— Gā’1 = GR ā— Ī£K + GK ā— Ī£A . (5.94)

0

Exercise 5.8. Show that, subtracting the left and right Dyson equations for GK ,

the resulting equation can be written in the form

1R 1

[Gā’1 ā— GK ]ā’ [Ī£ + Ī£A ā— GK ]ā’ ā’ [Ī£K ā— GR + GA ]ā’

ā’

, , ,

0

2 2

1 1

ā’ [Ī£K ā— (GR ā’ GA )]+ + [(Ī£R ā’ Ī£A ) ā— GK ]+ . (5.95)

= , ,

2 2

If at the end of the day, one makes the lowest-order approximation for the self-

energy (as often done!), introducing the Greenā™s function formalism and diagrammat-

ics is of course ridiculous as ļ¬nal results follow from Fermiā™s Golden Rule.9 A virtue

of the real-time formalism and its associated Feynman diagrams is that nontrivial

approximations can be established using the diagrammatic estimation technique, and

higher-order correlations studied systematically, as we shall consider in the following

chapters, not least in chapter 10.10

Before studying applications of the real-time technique we shall make obsolete

one version of the imaginary-time formalism, viz. the too pervasive Matsubara tech-

nique. The general imaginary-time formalism has virtues for special Euclidean ļ¬eld

theory purposes as well as for expedient proofs establishing conserving approxima-

tions. After the discussion of the equilibrium Dyson equation in the next section,

we demonstrate the equivalence of the imaginary-time formalism to the closed time

path formulation and the real-time technique introduced in this chapter.

5.6 Equilibrium Dyson equation

In equilibrium all quantities depend only on time diļ¬erences, and for translational

invariant situations also only on spatial diļ¬erences, and convolutions are by Fourier

transformation turned into products. In terms of the self-energy we therefore have

for the retarded Greenā™s function the equilibrium Dyson equation11

GR (p, E) = GR (p, E) + GR (p, E) Ī£R (E, p) GR (p, E) (5.96)

0 0

9 In the same vein, if one employs a mean-ļ¬eld approximation, introducing the formalism of

quantum ļ¬eld theory seems excessive. This point of view was taken in references [1] and [13].

10 For a discussion of the diagrammatic estimation technique see chapter 3 of reference [1].

11 We recall the result of Section 3.4, that in thermal equilibrium all the various Greenā™s functions

can be expressed in terms of, for example, the (imaginary part of the) retarded Greenā™s function.

5.6. Equilibrium Dyson equation 139

which we immediately solve to get

1 1

GR (p, E) = = . (5.97)

Gā’1 (p, E) ā’ Ī£R (E, p) Eā’ p ā’ Ī£ (E, p)

R

0

The retarded self-energy determines the analytic structure of the retarded Greenā™s

function, i.e. the location of the poles of the analytically continued retarded Greenā™s

function onto the second Riemann sheet through the branch cut along the real axis

(recall Section 3.4), the generic situation being that of a simple pole. For given

momentum value the simple pole is located at E = E1 + iE2 , determined by E1 =

p + e Ī£(E, p ) and E2 = m Ī£(E, p ), and as

ā

dE eā’iEt GR (p, E)

R

G (p, t) = (5.98)

ā’ā

the imaginary part of the self-energy thereby determines the temporal exponential

decay of the Greenā™s function, i.e. the lifetime of (in the present case) momentum

states. The eļ¬ect of interactions are clearly to give momentum states a ļ¬nite lifetime.

For the Fourier transform of Eq. (5.97) we get (in three spatial dimensions for

the prefactor to be correct)

ā

|xā’x | 2m(Eā’Ī£R (E, pE p))

i

ā’m e Ė

GE (x ā’ x ) =

R

(5.99)

|x ā’ x |

2Ļ 2

where pE is the solution of the equation pE = 2m(E ā’ Ī£R (E, pE p)). Interactions

Ė

will thus provide a ļ¬nite spatial and temporal range of the Greenā™s function.

For the case of electrons, say in a metal, the advanced Greenā™s function likewise

describes the attenuation of the holes.

Exercise 5.9. Show that the spectral function in equilibrium is given by (using now

the grand canonical ensemble)

Ī“(E, p)

A(E, p) = (5.100)

2 2

Ī“(E,p)

E ā’ Ī¾p ā’ eĪ£R (E, p) + 2

where

1R

eĪ£(E, p) ā” Ī£ (E, p) + Ī£A (E, p) (5.101)

2

and

Ī“(E, p) ā” i Ī£R (E, p) ā’ Ī£A (E, p) . (5.102)

We note that the sum-rule satisļ¬ed by the spectral weight function, Eq. (3.89), sets

limitation on the dependence of the self-energy on the energy variable. The general

features of interaction is to broaden the peak in the spectral weight function and to

shift, renormalize, energies.

140 5. Real-time formalism

Exercise 5.10. Show that for bosons in equilibrium at temperature T , their self-

energy components satisfy the ļ¬‚uctuationā“dissipation relations

E

Ī£R (E, p) ā’ Ī£A (E, p)

Ī£K (E, p) = coth (5.103)

2kT

and for fermions

E

Ī£R (E, p) ā’ Ī£A (E, p)

Ī£K (E, p) = tanh . (5.104)

2kT

5.7 Real-time versus imaginary-time formalism

Although we shall mainly use the real-time technique presented in this chapter

throughout, it is useful to be familiar with the equivalent imaginary-time formalism

in view of the vast amount of literature where this method has been employed. Or

more importantly to realize the link between the imaginary-time formalism and the

Martinā“Schwingerā“Abrikosovā“Gorkovā“Dzyaloshinskiā“Eliashbergā“Kadanoļ¬ ā“Baymā“

Langreth analytical continuation procedure. In the classic textbooks of Kadanoļ¬

and Baym [14] and Abrikosov, Gorkov and Dzyaloshinski [15] on non-equilibrium

statistical mechanics, the imaginary-time formalism introduced by Matsubara [16]

and Fradkin [17] and Martin and Schwinger [18] was used. Being then a Euclidean

ļ¬eld theory it possesses nice convergence properties. However, it lacks appeal to

intuition.

5.7.1 Imaginary-time formalism

The workings of the imaginary-time formalism are based on the mathematical formal

resemblance of the Boltzmann statistical weighting factor in the equilibrium statis-

tical operator Ļ ā eā’H/kT and the evolution operator U ā eā’iHt/ for an isolated

Ė

system. The imaginary time Greenā™s function

H ā’Ī¼ N

G(x, Ļ„ ; x , Ļ„ ) ā” ā’Tr eā’ Ė

TĻ„ (Ļ(x, Ļ„ ) Ļ(x , Ļ„ )) (5.105)

kT

is deļ¬ned in terms of ļ¬eld operators depending on imaginary time according to (we

suppress all other degrees of freedom than space)

Ļ(x) eā’

1 1

Ļ„ (Hā’Ī¼N ) Ļ„ (Hā’Ī¼N )

Ļ(x, Ļ„ ) = e (5.106)

and

Ļ(x, Ļ„ ) = e Ļ„ (Hā’Ī¼N ) Ļ ā (x) eā’ Ļ„ (Hā’Ī¼N ) ,

1 1

Ė (5.107)

where Ļ(x) is the ļ¬eld operator in the SchrĀØdinger picture, and TĻ„ provides the

o

imaginary time ordering (with the usual minus sign involved for an odd number

of interchanges of fermi ļ¬elds). The Ļ„ s involved are real variables, the use of the

word imaginary refers to the transformation t ā’ ā’iĻ„ in which case the time-ordered

real-time Greenā™s function, Eq. (4.10), transforms into the imaginary-time Greenā™s

Ė

function (more about this shortly). Note that Ļ(x, Ļ„ ) and Ļ(x, Ļ„ ) are not each others

5.7. Real-time versus imaginary-time formalism 141

adjoints. Knowledge of the imaginary-time Greenā™s function allows the calculation

of thermodynamic average values.

The imaginary-time single-particle Greenā™s function respects the Kuboā“Martinā“

Schwinger boundary conditions, for example

G(x, Ļ„ ; x , 0) = Ā± G(x, Ļ„ ; x , Ī²) , (5.108)

owing to the cyclic invariance property of the trace (the notation Ī² = /kT is used).

The periodic boundary condition is for bosons, and the anti-periodic boundary con-

dition is for fermions (the identical consideration in connection with the ļ¬‚uctuationā“

dissipation theorem was discussed in Section 3.4, and is further discussed in Section

6.5). We note the crucial role of the (grand) canonical ensemble as elaborated in

Section 3.4.

In its simple equilibrium applications in statistical mechanics, thermodynamics, or

in linear response theory, the involved imaginary-time Greenā™s functions are expressed

in terms of a single so-called Matsubara frequency

1

eā’iĻn (Ļ„ ā’Ļ„ ) G(x, x ; Ļn ) ,

G(x, Ļ„ ; x , Ļ„ ) = (5.109)

Ī² Ļn

where Ļn = 2nĻ/Ī² for bosons and Ļn = (2n + 1)Ļ/Ī² for fermions, respectively,

n = 0, Ā±1, Ā±2, . . . . Equilibrium or thermodynamic properties and linear transport

coeļ¬cients can therefore be expressed in terms of only one Matsubara frequency,

and the analytical continuation to obtain them from the imaginary-time Greenā™s

functions is trivial, say the retarded Greenā™s function is obtained by GR (x, x ; Ļ) =

G(x, x ; iĻn ā’ Ļ + i0+ ) as the two functions coincide according to GR (iĻn ) = G(Ļn )

for Ļn > 0.

The imaginary-time Greenā™s functions can also be used to study non-equilibrium

states by letting the external potential depend on the imaginary time. The Matsub-

ara technique is then a bit cumbersome, but can be used to derive exact equations,

say, the Dyson equation for real-time Greenā™s functions. In fact this was the method

used originally to study non-equilibrium superconductivity in the quasi-classical ap-

proximation [19].12 However, for general non-equilibrium situations, the necessary

analytical continuation in arbitrarily many Matsubara frequencies becomes nontrivial

(and are usually left out of textbooks), and are more involved than using the real-

time technique. Furthermore, when approximations are made, the real-time results

obtained upon analytical continuation can be spurious. However, the main disad-

vantage of the imaginary-time formalism is that it lacks physical transparency. We

shall therefore not discuss it further in the way it is usually done in textbooks, but

use a contour formulation to show its equivalence to the real-time formalism.13

12 Amazingly, the non-equilibrium theory of superconductivity was originally obtained using the

Matsubara technique [19], as, I guess, the imaginary-time formalism was in rule at the Landau

Institute. A plethora of papers and textbooks have perpetrated the use of the imaginary-time

formalism. It is the contestant to be the most important frozen accident in the evolution of non-

equilibrium theory. Letā™s iron out unfortunate ļ¬‚uctuations of the past! Its proliferation also testiļ¬es

to the fact that idiosyncratically written papers, such as the seminal paper of Schwinger [5], can be

a long time in germination.

13 The imaginary-time formalism can be useful for special purpose applications such as diagram-

142 5. Real-time formalism

5.7.2 Imaginary-time Greenā™s functions

The imaginary-time Greenā™s functions are proļ¬tably interpreted as contour-ordered

Greenā™s function, viz. on an imaginary-time contour. First we note, that the times

entering the imaginary-time Greenā™s function can be interpreted as contour times.

Choosing the times in the time ordered Greenā™s function in Eq. (3.61), instead to lie

on the contour starting at, say, t0 and ending down in the lower complex time plane

at t0 ā’ iĪ², the appendix contour ca in Figure 4.4, turns the expression Eq. (3.61)

into the equation for the imaginary-time Greenā™s function, Eq. (5.105). This observa-

tion, by the way, gives the standard Feynman diagrammatics for the imaginary-time

Greenā™s function since Wickā™s theorem involving the appendix contour is a trivial

corollary of the general Wickā™s theorem of Section 4.3.3. We can thus, for example,

immediately write down the non-equilibrium Dyson equation for the imaginary-time

Greenā™s function, t1 and t1 lying on the appendix contour ca . Considering the case

where the non-equilibrium situation is the result of a time-dependent potential, V ,

the Dyson equation for the imaginary-time Greenā™s functions or appendix contour-

ordered Greenā™s function is

G(1, 1 ) = G0 (1, 1 ) + dx3 dĻ„3 dx2 dĻ„2 G0 (1, 3) Ī£(3, 2) G(2, 1 )

ca ca

Ļ3 Ļ2

+ dx2 dĻ„2 G0 (1, 2) V (2) G(2, 1 ) . (5.110)

ca

Ļ2

The appendix contour interpretation of the imaginary-time Greenā™s function is in

certain situations more expedient than the real-time formulation when it for example

comes to diagrammatically proving exact relationships, since it has fewer diagrams

than the real-time approach if unfolding its matrix structure is needed. It should

thus be used in such situations, but then it is preferable not to use the Matsubara

frequency technique, but instead stick to the appendix contour formalism.

In practice we need to know how to analytically continue imaginary-time quanti-

ties to real-time functions, say for the imaginary-time Dyson equation, or for terms

appearing in perturbative expansions of imaginary-time quantities. Instead of turn-

ing to standard textbook imaginary-time formalism, the Matsubara technique, it is in

view of the above preferable to go to the appendix contour ordered Greenā™s functions,

and perform the analytical continuation from there. In fact, this analytical contin-

uation procedure becomes equivalent to the analytical continuation of the contour

ordered functions in the general contour formalism, for example the transition from

the contour-ordered Greenā™s function to real-time Greenā™s functions, which we now

turn to discuss. We illustrate this in the next section by performing the proceduce

for the Dyson equation.

The Boltzmann factor can guarantee analyticity of the Greenā™s function for times

ńņš. 21 |