ńņš. 13 |

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

ńņš. 13 |