ńņš. 4 |

= ā’i T (Ļ(x, t) Ļ ā (x , t )) .

G(x, t; x , t ) = i ā— (3.99)

Ī“Ī· (x , t )Ī“Ī·(x, t)

Ī·=0=Ī· ā—

The generating functional is a device we shall consider in detail in Chapter 9.

The Greenā™s functions introduced in this section are the correlation functions for

the case of an arbitrary state. Before we embark on the construction of the general

non-equilibrium perturbation theory and its diagrammatic representation starting

from the canonical formalism as presented here and in the ļ¬rst two chapters, we

consider brieļ¬‚y equilibrium theory, in particular the general property characterizing

equilibrium.15

14 For the case of fermions, the sources must be anti-commuting c-numbers, so-called Grassmann

variables. We elaborate on this point in Chapter 9.

15 In Chapter 9 we proceed the other way around, and the reader inclined to take diagrammatics

as a starting point of a physical theory can thus start from there.

70 3. Quantum dynamics and Greenā™s functions

3.4 Equilibrium Greenā™s functions

In this section we shall consider a system in thermal equilibrium. In that case the

state of the system is speciļ¬ed by the Boltzmann statistical operator, Eq. (2.87),

characterized by its macroscopic parameter, the temperature T .

In thermal equilibrium, the correlation functions of a system are subdued to a

boundary condition in imaginary time as speciļ¬ed by the ļ¬‚uctuationā“dissipation

theorem.16 In the canonical ensemble, for example the relation

ā ā

ĻH (x, t) ĻH (x , t ) = ĻH (x , t ) ĻH (x, t + iĪ²) (3.100)

is valid, where Ī² = /kT , as a consequence of the cyclic invariance of the trace as

the bracket denotes the average

eā’H/kT

. . . ā” Tr ... . (3.101)

Tr(eā’H/kT )

The relationship in Eq. (3.100) can, for example, be stated in terms of the Greenā™s

functions as

G< (x, t + iĪ², x , t ) = Ā± G> (x, t, x , t ) (3.102)

valid for arbitrary interactions among the particles in the system.

Exercise 3.15. Show that in the grand canonical ensemble, for example the following

relation, is valid

ā ā

Ī²Ī¼

ĻH (x, t ā’ iĪ²) ĻH (x , t ) = e ĻH (x , t ) ĻH (x, t) (3.103)

in which case the average is

eā’(Hā’Ī¼N )/kT

. . . ā” Tr ... (3.104)

Tr(eā’(Hā’Ī¼N )/kT

and the Hamiltonian and the total number operator commute if the chemical poten-

tial is nonzero (recall Eq. (2.36)). Stated in terms of Greenā™s functions we have

Ī²Ī¼

G< (x, t, x , t ) = Ā± e G> (x, t ā’ iĪ², x , t ) , (3.105)

where in the grand canonical ensemble for example

ā

ā“ i ĻH (x , t ) ĻH (x, t)

G< (x, t, x , t ) =

ā“i ā

Tr(eā’(Hā’Ī¼N )/kT ĻH (x , t ) ĻH (x, t)).

=

Tr(eā’(Hā’Ī¼N )/kT )

(3.106)

16 Additional discussion of the ļ¬‚uctuationā“dissipation theorem and its importance in linear re-

sponse theory are continued in Chapter 6. That the operators in Eq. (3.100) are the ļ¬eld operators

is immaterial; the relationship is valid for arbitrary operators.

3.4. Equilibrium Greenā™s functions 71

The importance of the canonical ensembles should be stressed for the validity of

these ļ¬‚uctuationā“dissipation relations or so-called Kuboā“Martinā“Schwinger bound-

ary conditions. They state that the Greenā™s functions are anti-periodic or periodic

in imaginary time depending on the particles being fermions or bosons, the interval

of periodicity being set by the inverse temperature. This is the crucial observation

for the Euclidean or imaginary-time formulation of quantum statistical mechanics,

as further discussed in Section 5.7.

In thermal equilibrium, correlation functions only depend on the diļ¬erence be-

tween the times, t ā’ t , i.e. they are invariant with respect to displacements in time.

If in addition the equilibrium state is translationally invariant, then all Greenā™s func-

tions are speciļ¬ed according to

ā

dp dE i (pĀ·(xā’x )ā’E(tā’t ))

G(x, t, x , t ) = e G(p, E) (3.107)

(2Ļ )3 2Ļ

ā’ā

or equivalently

G(p, E, p , E ) = 2Ļ(2Ļ )3 Ī“(p ā’ p ) Ī“(E ā’ E ) G(p, E) . (3.108)

For example,

ā

[ap (t) , aā (t )]Ā± .

i

GK (p, E) = ā’i d(t ā’ t ) e (tā’t )E

(3.109)

p

ā’ā

The relationship in Eq. (3.105) then takes the form of the detailed balancing

condition E ā’Ī¼

G< (p, E) = Ā± eā’ k T G> (p, E) . (3.110)

Exercise 3.16. Show that, for free bosons or fermions speciļ¬ed by the Hamiltonian

in Eq. (2.21), we have

p2

ā’ = ā’2Ļi Ī“(E ā’

GR (p, E) GA (p, E) p) , = . (3.111)

p

0 0

2m

Exercise 3.17. Show that, for free longitudinal phonons,, speciļ¬ed by the Hamilto-

nian in Eq. (1.123), we have

D0 (k, Ļ) ā’ D0 (k, Ļ) = ā’2Ļi Ļk sign(Ļ) Ī“(Ļ 2 ā’ Ļk ) Īø(ĻD ā’ |Ļ|)

R A 2 2

(3.112)

where the sign-function, sign(x) = Īø(x) ā’ Īø(ā’x) = x/|x|, is plus or minus one,

depending on the sign of the argument.

72 3. Quantum dynamics and Greenā™s functions

Instead of deļ¬ning the unitary transformation to the Heisenberg picture according

to Eq. (3.26), we can let it be governed by the grand canonical Hamiltonian, H ā’ Ī¼N ,

and we have, according to Eq. (2.36),17

ĻĪ¼ (x, t) = eā’

i i i

Ī¼N t Ī¼N t Ī¼t

ĻH (x, t) e =e ĻH (x, t) . (3.113)

Deļ¬ning the grand canonical Greenā™s functions in terms of these ļ¬elds, we observe

that they are related to those deļ¬ned according to Eq. (3.106), or those in the canon-

ical ensemble in the thermodynamic limit, according to

i

Ī¼(tā’t )

GĪ¼ (x, t, x , t ) = G(x, t, x , t ) e . (3.114)

Since average densities and currents are expressed in terms of the equal time Greenā™s

function, formulas have the same appearance in both ensembles.

For the Fourier transformed Greenā™s function with respect to time, the transition

to the grand canonical ensemble thus corresponds to the substitution E ā’ E + Ī¼, as

energies will appear measured from the chemical potential. The detailed balancing

condition, Eq. (3.110), can therefore, for a translationally invariant state, equivalently

be stated in the form

G< (E, p) = Ā± eā’E/kT G> (E, p) (3.115)

and we have dropped the chemical potential index as these are the Greenā™s functions

we shall use in the following. The absence of the chemical potential in the exponential

shows that the relationships are speciļ¬ed in the grand canonical ensemble, where

energies are measured relative to the chemical potential (upper and lower signs refer

as usual to bosons and fermions, respectively).

In thermal equilibrium, the kinetic Greenā™s function and the retarded and ad-

vanced Greenā™s functions, or rather the spectral weight function, are thus related for

the case of fermions according to

E

GK (E, p) = (GR (E, p) ā’ GA (E, p)) tanh . (3.116)

2kT

In thermal equilibrium, all Greenā™s functions can thus be speciļ¬ed once a single of

them is known, say the retarded Greenā™s functions, and the quantum statistics of the

particles is then reļ¬‚ected in relations governed by the ļ¬‚uctuationā“dissipation type

relationship such as in Eq. (3.116). Occasionally we keep Boltzmannā™s constant, k,

explicitly, the non-essential converter between energy and temperature scales.

Exercise 3.18. Show that, for bosons in equilibrium at temperature T , the ļ¬‚uctuationā“

dissipation theorem reads

E

GK (E, p) = (GR (E, p) ā’ GA (E, p)) coth . (3.117)

2kT

17 The number operator is assumed to commute with the Hamiltonian. If the number operator

does not commute with the Hamiltonian, such as for phonons, the description is of course in the

grand canonical ensemble and the chemical potential vanishes.

3.4. Equilibrium Greenā™s functions 73

Exercise 3.19. Show that

i G> (E, p) = (1 Ā± fā“ (E)) A(E, p) (3.118)

and

Ā± i G< (E, p) = fā“ (E) A(E, p) , (3.119)

where the functions

1

fā“ (E) = (3.120)

eE/kT ā“ 1

denote either the Boseā“Einstein distribution or Fermiā“Dirac distribution, for bosons

and fermions respectively.

Exercise 3.20. Show that the average energy in the thermal equilibrium state for

the case of two-body interaction between fermions (recall Exercise 3.7 on page 60),

for example, can be expressed as

1 dap (t)

aā (t) ap (t) + i aā (t)

H = p p p

2 dt

p

t=0

1 d

ā’i G< (p, t)

= i + p

2 dt

p

t=0

ā

1 dE

ā’i G< (p, E)

= (E + p)

2 2Ļ

p ā’ā

ā

1 dE

= (E + p) A(p, E) f (E) , (3.121)

2 2Ļ

p ā’ā

where f is the Fermi function, and thereby for the energy density

ā

H 1 dp dE

= (E + p) A(p, E) f (E) . (3.122)

(2Ļ )3

V 2 2Ļ

ā’ā

For a system in thermal equilibrium, the correlation function

ā ā

ĻĪ¼ (x, t) ĻĪ¼ (x , t ) = Tr(e(Ī©ā’(Hā’Ī¼N ))/kT ĻĪ¼ (x, t) ĻĪ¼ (x , t )) (3.123)

can be spectrally decomposed by inserting a complete set of energy states in the

multi-particle space

(H ā’ Ī¼N )|En , N = (En ā’ Ī¼N )|En , N (3.124)

74 3. Quantum dynamics and Greenā™s functions

giving

ā

e(Ī©ā’En +Ī¼N )/kT ei(tā’t )(En ā’Em +Ī¼) Ć—

ĻĪ¼ (x, t) ĻĪ¼ (x , t ) =

N,n,m

N, En |Ļ(x, t = 0))|Em , N + 1 Ć—

N + 1, Em |Ļ ā (x , t = 0)|En , N . (3.125)

From this expression we observe that the G-greater Greenā™s function G> (x, t, x , t )

considered as a function of imaginary times is an analytic function in the region,

ā’1/kT < m(t ā’ t ) < 0, if the exponential exp{ā’En (1/kT + i(t ā’ t ))} dominates

the convergence of the sum.

Exercise 3.21. Show similarly that G< (x, t, x , t ) is an analytic function in the

region of imaginary times, 0 < m(t ā’ t ) < 1/kT .

Assuming a translational invariant system and using Eq. (2.16) we have

ā

e(Ī©ā’En +Ī¼N )/kT ei(tā’t )(En ā’Em +Ī¼)

ĻĪ¼ (x, t) ĻĪ¼ (x , t ) =

N,n,m

Ć— eā’i(xā’x )pn m N, En |Ļ(0, 0)|Em , N + 1

Ć— N + 1, Em |Ļ ā (0, 0)|En , N , (3.126)

where pnm = Pn ā’ Pm is the diļ¬erence between the total momentum eigenvalues

for the two states in question. For the Fourier transform we then have

ap (t) aā (t ) = (2Ļ)3 e(Ī©ā’En +Ī¼N )/kT ei(tā’t )(En ā’Em +Ī¼)

p

N,n,m

Ć— Ī“(p ā’ pmn )| N, En |Ļ(0, 0))|Em , N + 1 |2 .

(3.127)

Noting the analyticity in the upper-half Ļ-plane of the following function, we have

for real values of Ļ

ā

1

dt Īø(t) eiĻt = , (3.128)

Ļ + iĪ“

ā’ā

where Ī“ = 0+ , or equivalently

ā

dĻ eā’iĻt

Īø(t) = . (3.129)

ā’2Ļi Ļ + iĪ“

ā’ā

3.4. Equilibrium Greenā™s functions 75

Therefore for the retarded and advanced Greenā™s functions we have the spectral

representations

eā’(En ā’Ī¼N )/kT Ć—

GR(A) (p, E) = (2Ļ)3 eĪ©/kT

Nn ,n,m

| Nn , En |Ļ(0, 0))|Em , Nm |2

Ī“(p ā’ pmn ) ā“ (n ā” m) , (3.130)

+

E + Emn (ā’) iĪ“

where Emn = Em ā’En +Ī¼(Nm ā’Nn ), and we recall that Nm = Nn Ā±1. The retarded

(advanced) Greenā™s function is thus analytic in the upper (lower) half-plane for the

energy variable E. The simple poles for the retarded (advanced) Greenā™s function

are thus spread densely just below (above) the real axis, and in the thermodynamic

limit this spectrum of simple poles coalesces into a continuum creating a branch cut

for the functions along the real axis.

In equilibrium all propagators are thus speciļ¬ed in terms of a single Greenā™s

function, say the retarded or equivalently the spectral function, as by analyticity the

retarded and advanced Greenā™s functions satisfy the causality or Kramersā“Kronig

relations for their real and imaginary parts, or compactly

ā

dE GR (E , p) ā’ GA (E , p)

R(A)

G (E, p) =

ā’2Ļi E ā’ E (ā’) i0

+

ā’ā

ā

dE A(E , p)

= . (3.131)

2Ļ E ā’ E (ā’) i0

+

ā’ā

The spectral weight function, has according to Eq. (3.130), the spectral decom-

position

e(Ī©ā’(En ā’Ī¼Nn ))/kT

ā’(2Ļ)4

A(p, E) =

Nn ,n,m

Ć— Ī“(p ā’ pmn ) Ī“(E ā’ Emn )| Nn , En |Ļ(0, 0))|Em , Nm |2

ā“ (n ā” m)) (3.132)

or equivalently

Emn

e(Ī©ā’(En ā’Ī¼Nn ))/kT 1 ā“ eā’

(2Ļ)4

A(p, E) = kT

Nn ,n,m

Ć— Ī“(p ā’ pmn ) Ī“(E ā’ Emn )| Nn , En |Ļ(0, 0))|Em , Nm |2 , (3.133)

where the upper and lower sign is for bosons and fermions, respectively.

76 3. Quantum dynamics and Greenā™s functions

The analytic properties of the retarded (or advanced) Greenā™s function determines

the analytic properties of all the other introduced Greenā™s functions, and are further

studied in Section 5.6.

The three Greenā™s functions, GR,A,K , thus carry diļ¬erent information about the

many-body system: GR,A the spectral properties and GK in addition the quantum

statistics of the concerned particles. In Chapter 5 we will construct the diagrammatic

perturbation theory that, even for non-equilibrium states, keeps these important

features explicit.

Exercise 3.22. Show that for large energy variable, E ā’ ā, the retarded and

advanced Greenā™s functions always have the asymptotic behavior

1

GR(A) (E, p) . (3.134)

E

In the absence of interactions, i.e. for free bosons or fermions speciļ¬ed by the

Hamiltonian in Eq. (2.21), one readily obtains for the spectral weight function,

Eq. (3.85),

p2

2Ļ Ī“(E ā’ Ī¾p ) , ā’Ī¼= ā’ Ī¼,

A0 (p, E) = Ī¾p = (3.135)

p

2m

and according to the ļ¬‚uctuationā“dissipation theorem, all one-particle Greenā™s func-

tion are then immediately obtained.

In the presence of interactions, the delta-spike in the spectral weight function

will be broadened and a tail appears, however, subject to the general sum-rule of

Eq. (3.89) which for the equilibrium state reads

ā

dE

A(E, p) = 1 . (3.136)

2Ļ

ā’ā

Exercise 3.23. The quantum statistics of particles have, according to the above,

a profound inļ¬‚uence on the form of the Greenā™s function. Show that, for the case

of non-interacting fermions at zero temperature, the Fermi surface is manifest in

the time-ordered Greenā™s function, Eq. (3.61), according to (say) in the canonical

ensemble,

1

G0 (E, p) = , (3.137)

E ā’ p + iĪ“ sign(|p| ā’ pF )

where Ī“ = 0+ , and the sign-function, sign(x) = Īø(x)ā’Īø(ā’x) = x/|x|, is plus or minus

one depending on the sign of the argument. The grand canonical case corresponds

to the substitution p ā’ Ī¾p = p ā’ Ī¼.

3.5. Summary 77

Exercise 3.24. For N non-interacting bosons in a volume V at zero temperature,

they all occupy the lowest energy level corresponding to the label p = 0. In the ļ¬eld

operator, Ļ(x) = Ī¾0 + Ļ (x), the creation operator for the lowest energy level is

ā ā

singled out, Ī¾0 = a0 / V , and Ī¾0 and Ī¾0 can, for a non-interacting system in the

ā

thermodynamic limit, be regarded as c-numbers, [Ī¾0 , Ī¾0 ] = 1/V .

Show that the time ordered Greenā™s function for non-interacting bosons in the

ground state, |Ī¦N = (N !)ā’1/2 (aā )N |0 , is given by

0

G0 (x, t, x , t ) = G(0) (t ā’ t ) + G0 (x, t, x , t ) , (3.138)

where

ā

G(0) (t ā’ t ) = ā’i Ī¦N |T (Ī¾0 (t) Ī¾0 (t ))|Ī¦N (3.139)

is speciļ¬ed at t = t + 0 by iG(0) (0ā’ ) = n, where n = N/V is the density of the

bosons, and G0 (x, t, x , t ) is speciļ¬ed by its Fourier transform

1

G0 (E, p) = (3.140)

Eā’ + iĪ“

p

corresponding to G-lesser vanishing for the ļ¬eld Ļ , or equivalently, the density of the

bosons is solely provided by the occupation of the lowest energy level. The presence

of a Boseā“Einstein condensate at low temperatures thus leads to such modiļ¬cations

for boson Greenā™s function expressions.18

As mentioned, perturbation theory and diagrammatic summation schemes are

the main tools in unraveling the eļ¬ects of interactions on the equilibrium properties

of a system. This has been dealt with in textbooks mainly using the imaginary-

time formalism (which we will discuss in Section 5.7.1), and unfortunately most

numerously in the so-called Matsubara technique. This technique, which is based on

a purely mathematical feature, lacks physical transparency. A main purpose of this

book is to show that the real-time technique, which is based on the basic feature of

quantum dynamics, has superior properties in terms of physical insight. Furthermore,

there is no need to delve into equilibrium theory Feynman diagrammatics since it will

be a simple corollary of the general real-time non-equilibrium theory we now turn to

develop.

3.5 Summary

In this chapter we have shown that by transforming to the Heisenberg picture, the

quantum dynamics of a many-body system can be described by the time development

of the ļ¬eld operator in the Heisenberg picture. The measurable physical quantities

of a system were thus expressed in terms of strings of Heisenberg operators weighted

18 Ifthe ground state of a system of interacting bosons has no condensed phase, standard zero-

temperature perturbation theory can not be applied. In the opposite case, the zero-momentum ļ¬elds

can be treated as external ļ¬elds. This leads to additional vertices in the Feynman diagrammatics

as encountered in Section 10.6.

78 3. Quantum dynamics and Greenā™s functions

with respect to a state, generally a mixture described by a statistical operator. The

dynamics of such systems are therefore described in terms of the correlation functions

of ļ¬eld operators, the Greenā™s functions of the theory.

In thermal equilibrium, the ļ¬‚uctuationā“dissipation relation leads to simpliļ¬cation

as all the one-particle or two-point Greenā™s functions can be expressed in terms of the

spectral weight function. Diļ¬erent schemes pertaining to equilibrium can be devised

for calculating equilibrium Greenā™s functions but we shall not entertain them here as

they will be simple corollaries of the general non-equilibrium theory presented in the

next chapter.

The equations of motion for Greenā™s functions of interacting quantum ļ¬elds in-

volve ever increasing higher-order correlation functions. The rest of the book is

devoted to the study and calculation of Greenā™s functions for non-equilibrium states

using diagrammatic and functional methods. We therefore turn to develop the for-

malism necessary for obtaining information about the properties of systems when

they are out of equilibrium.

4

Non-equilibrium theory

In this chapter we will develop the general formalism necessary for dealing with

non-equilibrium situations. We ļ¬rst formulate the non-equilibrium problem, and

discuss why the standard method applicable for the study of ground state properties

fails for arbitrary states. We then introduce the closed time path formulation, and

construct the perturbation theory for the closed time path or contour ordered Greenā™s

function valid for non-equilibrium states. The diagrammatic perturbation theory in

the closed time path formulation is then formulated, and generic types of interaction

are considered.1

4.1 The non-equilibrium problem

Let us consider an arbitrary physical system described by the Hamiltonian H. Since

we consider non-equilibrium quantum ļ¬eld theory, the Hamiltonian acts on the multi-

particle state space introduced in the ļ¬rst chapter, consisting of products of multi-

particle spaces for the species involved. The generic non-equilibrium problem can be

construed as follows: far in the past, prior to time t0 , the system can be thought

of as having been brought to the equilibrium state characterized by temperature T .

The state of the system is thus at time t0 described by the statistical operator2

eā’H/kT

Ļ(H) = , (4.1)

Tr(eā’H/kT )

where Tr denotes the trace in the multi-particle state space of the physical system

in question. At times larger than t = t0 , a possible time-dependent mechanical

perturbation, described by the Hamiltonian H (t), is applied to the system. The

1 In this chapter we follow the exposition given in reference [3].

2 We can also imagine and treat the case where the particles in the system are coupled to particle

reservoirs described by their chemical potentials as this is simply included by tacitly understanding

that single-particle energies are measured relative to their chemical potentials, H ā’ H ā’ s Ī¼s Ns ,

i.e. shifting from the canonical to the grand canonical ensemble. In fact, in actual calculations it is

the more convenient choice, as discussed in Sections 2.5 and 3.4.

79

80 4. Non-equilibrium theory

total Hamiltonian is thus

H(t) = H + H (t) . (4.2)

The simplest non-equilibrium problem is concerned with the calculation of some

average value of a physical quantity A at times t > t0 . The state of the system is

evolved to

Ļ(t) = U (t, t0 ) Ļ(H) U ā (t, t0 ) , (4.3)

where (recall Eq. (3.7))

t

U (t, t ) = T eā’ dt H(t)

i ĀÆĀÆ

(4.4)

t

is the evolution operator, and the average value of the quantity of interest is thus

A(t) = Tr(Ļ(t) A) , (4.5)

where A is the operator representing the physical quantity in question in the SchrĀØdinger

o

picture. Transforming to the Heisenberg picture

A(t) = Tr(Ļ(H) AH (t)) = AH (t) , (4.6)

where, as discussed in Section 3.1.2, AH (t) denotes the operator representing the

physical quantity in question in the Heisenberg picture with respect to H(t), and we

have chosen the reference time in Eq. (3.17) to be t0 . The average value is typically

a type of quantity of interest for a macroscopic system, i.e. a system consisting of a

huge number of particles. For example, for the average (probability) density of the

particle species described by the quantum ļ¬eld Ļ, we have according to Eq. (2.28),3

ā

n(x, t) = ĻH (x, t) ĻH (x, t) . (4.7)

The average density is seen to be equal to the equal-time and equal-space value of

the G-lesser Greenā™s function, G< , introduced in Section 3.3

n(x, t) = ā“iG< (x, t, x, t) , (4.8)

where upper (lower) sign is for bosons and fermions, respectively.

If ļ¬‚uctuations are of interest or importance we encounter higher order correlation

functions, generically according to Section 2.1, then appears the trace over products

of pairs of Heisenberg ļ¬eld operators for particle species weighted by the initial

state. If one is interested in the probability that a certain sequence of properties are

measured at diļ¬erent times, one encounters arbitrary long products of Heisenberg

operators.4 Since physical quantities are expressed in terms of the quantum ļ¬elds of

the particles and interactions in terms of their higher-order correlations, of interest

are the correlation functions, the so-called non-equilibrium Greenā™s functions.

Owing to interactions, memory of the initial state of a subsystem is usually rapidly

lost. We shall not in practice be interested in transient properties but rather steady

3 Possible

spin degrees of freedom are suppressed, or imagined absorbed in the spatial variable.

4 See

chapter 1 of reference [1] for a discussion of such probability connections or histories with

a modern term.

4.2. Ground state formalism 81

states, where the dependence on the initial state is lost, and the time dependence is

governed by external forces. Initial correlations can be of interest in their own right,

even in many-body systems.5 In fact, for all of the following, the statistical operator

in the previous formulae, say Eq. (4.6), could have been chosen as arbitrary. This

would lead to additional features which we point out as we go along, and in practice

each case then has to be dealt with on an individual basis.

The equation of motion for the one-particle Greenā™s function leads to an inļ¬nite

hierarchy of equations for correlation functions containing ever increasing numbers of

ļ¬eld operators, describing the correlations between the particles set up in the system

by the interactions and external forces. In order to calculate the eļ¬ects of interac-

tions, we now embark on the construction of perturbation theory and the diagram-

matic representation of non-equilibrium theory starting from the canonical formalism

presented in the ļ¬rst chapter. But ļ¬rst we describe why the zero temperature, i.e.

ground state, formalism is not capable of dealing with general non-equilibrium situ-

ations, before embarking on ļ¬nding the necessary remedy, and eventually construct

non-equilibrium perturbation theory and its corresponding diagrammatic represen-

tation.

4.2 Ground state formalism

To see the need for the closed time path description consider the problem of pertur-

bation theory. The Hamiltonian of a system

H = H0 + H (i) (4.9)

consists of a term quadratic in the ļ¬elds, H0 , describing the free particles, and a

complicated term, H (i) , describing interactions.

Constructing perturbation theory for zero temperature quantum ļ¬eld theory, i.e.

the system is in its ground state |G , only the time-ordered Greenā™s function

ā ā

G(x, t, x , t ) = ā’i T (ĻH (x, t) ĻH (x , t )) = ā’i G|T (ĻH (x, t) ĻH (x , t ))|G

(4.10)

needs to be considered. Here ĻH (x, t) is the ļ¬eld operator in the Heisenberg picture

with respect to H for one of the species of particles described by the Hamiltonian.6

The time-ordered Greenā™s function contains more information than seems necessary

for calculating mean or average values, since for times t < t it becomes the G-lesser

Greenā™s function

G(x, t, x , t ) = G< (x, t, x , t ) (4.11)

5 All transient eļ¬ects for the above chosen initial condition are of course included. Whether this

choice is appropriate for the study of transient eļ¬ects depends on the given physical situation.

6 For a reader not familiar with zero temperature quantum ļ¬eld theory, no such thing is required.

It will be a simple corollary of the more powerful formalism presented in Section 4.3.2, and devel-

oped to its ļ¬nal real-time formalism presented in Chapter 5. The reason for the usefulness of the

time ordering operation is to be expected remembering the crucial appearance of time-ordering in

the evolution operator. Also, under the governing of the time-ordering symbol, operators can be

commuted without paying a price except for the possible quantum statistical minus signs in the

case of fermions.

82 4. Non-equilibrium theory

and thereby are all average values of physical quantities speciļ¬ed once the time-

ordered Greenā™s function is known for t < t . However, a perturbation theory involv-

ing only the G-lesser Greenā™s function can not be constructed.

The time-ordered Greenā™s function can, instead of being expressed in terms of the

ļ¬eld operator ĻH (x, t), i.e. in the Heisenberg picture with respect to H, be expressed

in terms of the ļ¬eld operators ĻH0 (x, t), the Heisenberg picture with respect to H0

or the so-called interaction picture,

Ļ(x) eā’

i i

H0 (tā’tr ) H0 (tā’tr )

ĻH0 (x, t) = e (4.12)

as they are related according to the unitary transformation

ĻH (x, t) = U ā (t, tr ) ĻH0 (x, t) U (t, tr ) , (4.13)

where

t ĀÆ (i ) ĀÆ

U (t, tr ) = T eā’i dt HH (t)

(4.14)

tr 0

is the evolution operator in the interaction picture (leaving out for brevity an index

to distinguish it from the full evolution operator exp{ā’iH(t ā’ tr )}) and

HH0 (t) = eiH0 (tā’tr ) H (i) eā’iH0 (tā’tr ) .

(i)

(4.15)

This is readily seen by noting that the expression on the right-hand side in Eq. (4.13)

satisļ¬es the ļ¬rst-order in time diļ¬erential equation

ā‚Ļ(x, t)

i = [Ļ(x, t), H] , (4.16)

ā‚t

the same equation satisļ¬ed by the ļ¬eld ĻH (x, t), and at the reference time tr , the two

operators are seen to coincide (coinciding with the ļ¬eld in the SchrĀØdinger picture,

o

Ļ(x)).

Transforming to the interaction picture, and using the semi-group property of the

evolution operator, U (t, t ) U (t , t ) = U (t, t ),7 and the relation U ā (t, t ) = U (t , t),

the time ordered Greenā™s function can be expressed in the form

ā

= ā’i U ā (t, tr )ĻH0 (x, t)U (t, t )ĻH0 (x , t )U (t , tr ) Īø(t ā’ t )

G(x, t, x , t )

ā

Ā± i U ā (t , tr )ĻH0 (x , t )U (t , t)ĻH0 (x, t)U (t, tr ) Īø(t ā’ t) (4.17)

which can also be expressed on the form (tm denotes max{t, t })

ā

G(x, t, x , t ) = ā’i U ā (tm , tr )T ĻH0 (x, t)ĻH0 (x , t )U (tm , tr ) (4.18)

since the time-ordering symbol places the operators in the original order.8

7 For

a detailed discussion of the evolution operator and the Heisenberg and interaction pictures

we refer to chapter 2 of reference [1].

8 In fact the operator identity

ā ā

T (ĻH (x, t) ĻH (x , t )) = U ā (tm , tr )T ĻH 0 (x, t) ĻH 0 (x , t ) U (tm , tr )

is valid since only transformation of operators was involved, and nowhere is advantage taken of the

averaging with respect to the state in question.

4.2. Ground state formalism 83

Usually, say in a scattering experiment realized in a particle accelerator, only

transitions from an initial state in the far past are of interest so that the reference

time is chosen in the far past, tr = ā’ā, and inserting 1 = U (tm , ā)U (ā, tm ) after

U ā gives9

ā

G(x, t, x , t ) = ā’i U ā (ā, ā’ā)T (ĻH0 (x, t) ĻH0 (x , t ) U (ā, ā’ā)) . (4.19)

If the average is with respect to the ground state of the system, one can make use

of the trick of adiabatic switching, i.e. the interaction is assumed turned on and oļ¬

adiabatically, say by the substitution HH0 (t) ā’ eā’ |t| HH0 (t). The non-interacting

(i) (i)

(non-degenerate) ground state |G0 , H0 |G0 = E0 |G0 , is evolved by the full adiabatic

evolution operator U into the normal ground state of the interacting system at time

t = 0, |G = U (0, ā’ā)|G0 . The on the evolution operator indicates that the

interaction is turned on and oļ¬ adiabatically. In perturbation theory it can then be

shown, that in the limit of ā’ 0, the true interacting ground state at time t = 0

is obtained modulo a phase factor that is obtained from the limiting expression of

turning the interaction on and oļ¬ adiabatically, (the Gell-Mannā“Low theorem [4]),10

U (ā, ā’ā) |G0 = eiĻ |G0 eiĻ = G0 |U (ā, ā’ā)|G0 .

, (4.20)

As a consequence, the time-ordered Greenā™s function, Eq. (4.10), can be expressed

in terms of the non-interacting ground state and the ļ¬elds in the interaction picture

according to

ā

G0 |T (ĻH0 (x, t) ĻH0 (x , t ) U (ā, ā’ā))|G0

G(x, t, x , t ) = ā’i . (4.21)

G0 |U (ā, ā’ā)|G0

In the next section we will show that the artiļ¬ce of turning the interaction on

and oļ¬ adiabatically is not needed when using the closed time path formulation

and generalizing time-ordering to contour-ordering, and it can also be avoided by

using functional methods as in Chapter 9, and plays no role in the non-equilibrium

formalism. In describing a scattering experiment, adiabatic switching is of course an

innocent initial and ļ¬nal boundary condition as the particles are then free.11

Since the Gell-Mannā“Low theorem fails for states other than the ground state,

and thus even for an equilibrium state at ļ¬nite temperature, we are in general stuck

9 In fact as an operator identity

ā ā

T (ĻH (x, t) ĻH (x , t )) = U ā (ā, ā’ā)T (ĻH 0 (x, t) ĻH 0 (x , t ) U (ā, ā’ā)).

10 Clearly, it is important that no dissipation or irreversible eļ¬ects takes place. Contrarily, in

statistical physics, reduced dynamics is the main interest, i.e. certain degrees of freedom are left

unobserved and emission and absorption takes place, technically partial traces occurs.

11 As will become clear from the following sections, the denominator in Eq. (4.21) is diagrammati-

cally the sum of all the vacuum diagrams that therefore cancel all the disconnected diagrams in the

numerator, and one obtains the standard connected Feynman diagrammatics for the time-ordered

Greenā™s function for a system at zero temperature such as is relevant in, say, QED. In QED one

works with the so-called scattering matrix or S-matrix, S(ā, ā’ā), deļ¬ned in terms of the full

evolution operator, S(t, t ) = eiH 0 t U (t, t )eā’iH 0 t , so that the matrix elements of the S-matrix are

expressed in terms of the free-particle states.

84 4. Non-equilibrium theory

with the operator U ā (ā, ā’ā) inside the averaging in Eq. (4.19) and Eq. (4.18). At

ļ¬nite temperatures and a fortiori for non-equilibrium states, a perturbation theory

involving only one kind of a real-time Greenā™s functions can not be obtained. In

order to construct a single object which contains all the dynamical information we

shall follow Schwinger and introduce the closed time path formulation [5].

4.3 Closed time path formalism

Let us return to the non-equilibrium situation of Section 4.1 where the dynamics is

determined by a time dependent Hamiltonian H(t) = H + H (t), where H is the

Hamiltonian for the isolated system of interest and H (t) is a time-dependent pertur-

bation acting on it. The unitary transformation relating operators in the Heisenberg

pictures governed by the Hamiltonians H(t) and H, respectively, is speciļ¬ed by the

unitary transformation

t

ā’i ĀÆ ĀÆ

OH (t) = V ā (t, t0 ) OH (t) V (t, t0 ) ,

dt HH (t)

V (t, t0 ) = T e (4.22)

t0

and

ā

UH (t, t0 ) = eā’

i

H(tā’t0 )

HH (t) = UH (t, t0 ) H (t) UH (t, t0 ) , (4.23)

and we have chosen t0 as reference time where the two pictures coincide. This

relation between the two pictures is obtained by ļ¬rst comparing both pictures to the

SchrĀØdinger picture obtaining

o

ā ā

OH (t) = UH (t, t0 ) UH (t, t0 ) OH (t) UH (t, t0 ) UH (t, t0 ) , (4.24)

where

ā’i dt H(t)

t ĀÆĀÆ

UH (t, t0 ) = T e (4.25)

t0

is the evolution operator corresponding to the Hamiltonian H(t). Then one notes

ā

that V (t, t0 ) and UH (t, t0 ) UH (t, t0 ) satisfy the same ļ¬rst-order in time diļ¬erential

equation and the same initial condition. We have thus obtained Dysonā™s formula

ā

V (t, t0 ) = UH (t, t0 ) UH (t, t0 ) (4.26)

or explicitly

t t

T eā’ T eā’ dt H(t)

i i i

ĀÆ ĀÆ ĀÆĀÆ

dt HH (t) H(tā’t0 )

=e . (4.27)

t t

Here Dysonā™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 diļ¬erentiation. Dysonā™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 Dysonā™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)

ā’i dĻ„ HH (Ļ„ )

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

ct

where Ļ„ 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

ā

(ā’i)n

OH (t) = dĻ„1 . . . dĻ„n Tct (HH (Ļ„1 ) . . . HH (Ļ„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 = ā’ + ā .

ā’ā’

c c (4.30)

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

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

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

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

they all give the same contribution and

n

n!

dĻ„1 . . . dĻ„n Tct (HH (Ļ„1 ) . . . HH (Ļ„n ) OH (t)) =

m!(n ā’ m)!

ct ct m=0

Ć— ā’

dĻ„m+1 . . . dĻ„n Tā (HH (Ļ„m+1 ) . . . HH (Ļ„n )) OH (t)

ā

ā’ ā

ā’ c

c c

Ć— ā’

dĻ„1 . . . dĻ„m Tā’ (HH (Ļ„1 ) . . . HH (Ļ„m )) , (4.31)

ā’

ā’ ā’

ā’ c

c c

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

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

86 4. Non-equilibrium theory

ā’ ā’

where Tā’ and Tā 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

ā ā

n!

dĻ„1 . . . dĻ„n Tct (HH (Ļ„1 ) . . . HH (Ļ„n ) OH (t)) = Ī“n,k+m

m! k!

ct ct m=0

k=0

Ć— ā’

dĻ„1 . . . dĻ„k Tā (HH (Ļ„1 ) . . . HH (Ļ„k )) OH (t)

ā

ā’ ā

ā’ c

c c

Ć— ā’

dĻ„1 . . . dĻ„m Tā’ (HH (Ļ„1 ) . . . HH (Ļ„m )) . (4.32)

ā’

ā’ ā’

ā’ c

c c

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

ā ā

(ā’i)k (ā’i)m

ā’i dĻ„ HH (Ļ„ )

T ct e OH (t) =

ct

m! k!

m=0

k=0

Ć— ā’

dĻ„1 . . . dĻ„k Tā (HH (Ļ„1 ) . . . HH (Ļ„k )) OH (t)

ā

ā’ ā

ā’ c

c c

Ć— ā’

dĻ„1 . . . dĻ„m Tā’ (HH (Ļ„1 ) . . . HH (Ļ„m )) (4.33)

ā’

ā’ ā’

ā’ c

c c

and thereby

ā’i ā’i

dĻ„ HH (Ļ„ ) dĻ„ HH (Ļ„ )

ā

ā’ ā’

ā’

ā’i dĻ„ HH (Ļ„ )

ā’ ā’

ec ec

T ct e OH (t) = Tā OH (t) Tā’ .

ct

c c

(4.34)

Parameterizing the forward and backward contours according to

Ļ„ (t ) = t t [t0 , t] , (4.35)

we get

t

ā’i dĻ„ HH (Ļ„ ) ā’i

ā’

ā’ dt HH (t )

Tā’ e c

ā’ = Te = V (t, t0 ) (4.36)

t0

c

and

ā’i dĻ„ HH (Ļ„ )

ā

ā’ t

= V ā (t, t0 )

Ėi dt HH (t )

ā’

Tā e c = Te (4.37)

t0

c

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

ā’

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

c

ā’

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

c

established. We have shown that the times in V ā (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 Schwingerā™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 Greenā™s function.

4.3.1 Closed time path Greenā™s function

A generalization oļ¬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 Greenā™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 Greenā™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 = ā’ + ā.15 We have hereby freed the

ā’ā’

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 Ļ„ to signify this

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

Ļ„1

Ļ„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 Greenā™s function or the contour-

ordered Greenā™s function

ā

Tr(eā’H/kT Tc (ĻH (x1 , sz1 , Ļ„1 ) ĻH (x1 , sz1 , Ļ„1 )))

, Ļ„1 ) = ā’i

G(x1 , sz1 , Ļ„1 , x1 , sz1

Tr(eā’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 ļ¬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 = ā

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 inļ¬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 Ļ„1 and Ļ„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 , Ļ„1 ) we have17

ā ā

G(1, 1 ) = ā’i Tc(ĻH (1) ĻH (1 )) = ā’i Z ā’1 Tr(eā’H/kT Tc (ĻH (1) ĻH (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

Ļ(x1 , Ļ„1 ) Ļ ā (x1 , Ļ„1 ) Ļ„1 > Ļ„1

ā

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

c

ā“Ļ ā (x1 , Ļ„1 ) Ļ(x1 , Ļ„1 ) Ļ„1 > Ļ„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. Ļ„1 > Ļ„1 means that Ļ„1

is further along the contour c than Ļ„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 ļ¬elds under the contour ordering is thus like the algebra

of (complex) numbers, whereas the algebra of fermi ļ¬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 Greenā™s

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

ct

G< (1, 1 ) Ļ„1 > Ļ„1

G(1, 1 ) = (4.41)

ct

G> (1, 1 ) Ļ„1 > Ļ„1 .

Here lesser refers to the contour time Ļ„1 appearing earlier than contour time Ļ„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 identiļ¬ed with their corresponding real times we recover their

corresponding real-time Greenā™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),

ā

ā’i ĻH (1) ĻH (1 )

G> (1, 1 ) =

ā’i

ā’i dĻ„ HH (Ļ„ ) ā

dĻ„ HH (Ļ„ )

ā’i Tct 1 e ct

ct

= ĻH (1) T ct e ĻH (1 )

1 1

1

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

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

ā ā

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

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

speciļ¬c to the quantities in question.

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

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

numbers by the use of path integrals.

4.3. Closed time path formalism 89

ā’i dĻ„ HH (Ļ„ ) ā

ā’i Tct 1 +ct 1

+c t

ct

= e ĻH (1) ĻH (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 ļ¬nally returning back again to t0 . The

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

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

which gives the unit operator, or equivalently U ā (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 inļ¬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

ā

= ā’i Tc (ĻH (1) ĻH (1 ))

G(1, 1 )

ā

= ā’i Tc eā’i dĻ„ HH (Ļ„ )

ĻH (1) ĻH (1 ) , (4.43)

c

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

plus inļ¬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 Greenā™s function.

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

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

ā‚

ā’ h0 (Ļ„ ) + Ī¼ G(x, Ļ„, x , Ļ„ ) = Ī“(x ā’ x ) Ī“c (Ļ„ ā’ Ļ„ )

i

ā‚Ļ„

i Tc ([Ļ ā (x, Ļ„ ), Hi (Ļ„ )]Ļ ā (x , Ļ„ )) , (4.44)

ā’

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

delta function

ā§

āØ Ī“(Ļ„ ā’ Ļ„ ) for Ļ„ and Ļ„ on forward branch

ā’Ī“(Ļ„ ā’ Ļ„ ) for Ļ„ and Ļ„ on return branch

Ī“c (Ļ„ ā’ Ļ„ ) = (4.45)

ā©

0 for Ļ„ and Ļ„ on diļ¬erent branches

90 4. Non-equilibrium theory

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

Exercise 3.10 on page 66).

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

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

ing number of ļ¬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 ļ¬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

eļ¬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 Greenā™s functions. The Hamiltonian of the system, Eq. (4.9) consists of

a term quadratic in the ļ¬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 Greenā™s function, the Hamiltonian in the

weighting factor needs to be quadratic in the ļ¬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 )

ā’i dĻ„ (HH (Ļ„ )+HH 0 (Ļ„ ))

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

ct 0

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

respect to the Hamiltonian H to the Heisenberg picture with respect to the free

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

HH0 (Ļ„ ) 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 Greenā™s function, Eq. (4.39), that the contour-ordered Greenā™s

function in the interaction picture is

(i )

Tr eā’Ī²H Tc eā’i ā

dĻ„ (HH (Ļ„ )+HH 0 (Ļ„ ))

ĻH0 (1) ĻH0 (1 )

c 0

G(1, 1 ) = ā’i . (4.47)

Tr (eā’Ī²H )

We have introduced the notation Ī² = 1/kT for the inverse temperature.

19 We shall later take advantage of the artiļ¬ce of employing diļ¬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 Dysonā™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 ā’i Ī² (i )

eā’Ī²H = eā’Ī²H0 Tca eā’i dĻ„ HH (Ļ„ )

(4.48)

t0 0

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

time plane from t0 to t0 ā’ iĪ², the appendix contour ca as depicted in Figure 4.4. We

then get the expression

ā ā ā ā

t 0 ā’i Ī²

(i )

ā’i (i )

ā’i dĻ„ (HH (Ļ„ )+HH 0 (Ļ„ ))

dĻ„ HH (Ļ„ )

Trāeā’Ī²H0āTca e ā Tc e ĻH0 (1)ĻH0 (1 )ā

ā

0 0

t0 c

iG(1, 1 ) = t 0 ā’i Ī² (i )

Tr eā’Ī²H0 Tca eā’i t 0 dĻ„ HH (Ļ„ )

0

(4.49)

ready-made for a perturbative expansion of the contour-ordered Greenā™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 ā’ iĪ² 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 )

ā’i ā

dĻ„ HH (Ļ„ ) ā’i

Tr eā’Ī²H0 Tci dĻ„ HH 0 (Ļ„ )

e e ĻH0 (1) ĻH0 (1 )

ci c

0

G(1, 1 ) = ā’i .

(i )

ā’i dĻ„ HH (Ļ„ ) ā’i dĻ„ HH (Ļ„ )

Tr eā’Ī²H0 Tci e e

ci c

0 0

(4.50)

The contour ci stretches from t0 to max{t1 , t1 } (or inļ¬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 ā’ iĪ², as depicted in Figure 4.4.

t0 t1 t1

ci

t0 ā’ iĪ²

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 eā’i = Tc eā’i dĻ„ HH (Ļ„ ) ā’i

dĻ„ (HH (Ļ„ ) + HH 0 (Ļ„ )) dĻ„ HH 0 (Ļ„ )

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 Wickā™s theorem

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

the associated Feynman diagrammatics. Before we show Wickā™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 inļ¬nity) and back again to t0 ,

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

(i )

Tc eā’i dĻ„ (HH (Ļ„ ) + HH 0 (Ļ„ ))

= 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 artiļ¬cial relics of the formalisms used in standard zero and

ļ¬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

inļ¬nity, t0 ā’ ā’ā, 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

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

system, then eļ¬ectively t0 ā’ ā’ā, 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 ļ¬nally have for the contour-

ordered or closed time path Greenā™s function

(i )

eā’i ā

dĻ„ (HH (Ļ„ ) + HH 0 (Ļ„ ))

G(1, 1 ) = Tr Ļ0 TC ĻH0 (1) ĻH0 (1 )

C 0

(i )

eā’i ā

dĻ„ HH (Ļ„ ) ā’i dĻ„ HH 0 (Ļ„ )

= Tr Ļ0 TC e ĻH0 (1) ĻH0 (1 ) , (4.53)

C C

0

where

eā’H0 /kT

Ļ0 = (4.54)

Tr eā’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 oļ¬ 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 ā’ ā’ā, 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 eļ¬ect of intrinsic damping. We note that at ever

increasing temperatures, the appendix contour contribution disappears, since thermal ļ¬‚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 Schwingerā™s closed time path [5], the Schwingerā“Keldysh or real-time

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

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

c1

t

c2

Figure 4.5 The Schwingerā“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 eā’Ī²H0 Tc eā’i dĻ„ HH (Ļ„ ) ā’i dĻ„ HH 0 (Ļ„ )

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 ļ¬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 ļ¬eld V (x, t). In that case the

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

ā

ā’i dĻ„ dx V (x,Ļ„ ) ĻH (x,Ļ„ ) ĻH 0 (x,Ļ„ )

ā

C 0

GC (1, 1 ) = Tr Ļ0 TC e ĻH0 (1) ĻH0 (1 ) .

(4.56)

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

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

gives the free contour Greenā™s function

ā

(0)

GC (1, 1 ) = ā’iTr Ļ0 TC ĻH0 (1) ĻH0 (1 ) . (4.57)

22 This is an appealing alternative in the quantum ļ¬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

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

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

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

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

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

of freedom. To be explicit, let us ļ¬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 ļ¬eld operators, ordered along a contour C.25 We introduce

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

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

S = tr(ĻT TC (c(Ļ„n ) c(Ļ„nā’1 ) . . . c(Ļ„2 ) c(Ļ„1 )))

ā” TC (c(Ļ„n ) c(Ļ„nā’1 ) . . . c(Ļ„2 ) c(Ļ„1 )) , (4.60)

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

(0)

eā’Hb /kT

ĻT = (4.61)

(0)

Tr eā’Hb /kT

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

phonons at temperature T , and

aā aq

(0)

Hb = hq = (4.62)

q q

q q

ńņš. 4 |