<<

. 4
( 22)



>>

δ 2 Z[·, · — ]
= ’i T (ψ(x, t) ψ † (x , t )) .
G(x, t; x , t ) = i — (3.99)
δ· (x , t )δ·(x, t)
·=0=· —

The generating functional is a device we shall consider in detail in Chapter 9.
The Green™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 ¬rst two chapters, we
consider brie¬‚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 Green™s functions


3.4 Equilibrium Green™s functions
In this section we shall consider a system in thermal equilibrium. In that case the
state of the system is speci¬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 speci¬ed by the ¬‚uctuation“dissipation
theorem.16 In the canonical ensemble, for example the relation
† †
ψH (x, t) ψH (x , t ) = ψH (x , t ) ψH (x, t + iβ) (3.100)

is valid, where β = /kT , as a consequence of the cyclic invariance of the trace as
the bracket denotes the average

e’H/kT
. . . ≡ Tr ... . (3.101)
Tr(e’H/kT )

The relationship in Eq. (3.100) can, for example, be stated in terms of the Green™s
functions as
G< (x, t + iβ, x , t ) = ± 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
† †
βμ
ψH (x, t ’ iβ) ψH (x , t ) = e ψH (x , t ) ψH (x, t) (3.103)

in which case the average is

e’(H’μN )/kT
. . . ≡ Tr ... (3.104)
Tr(e’(H’μN )/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 Green™s functions we have
βμ
G< (x, t, x , t ) = ± e G> (x, t ’ iβ, x , t ) , (3.105)

where in the grand canonical ensemble for example

“ i ψH (x , t ) ψH (x, t)
G< (x, t, x , t ) =

“i †
Tr(e’(H’μN )/kT ψH (x , t ) ψH (x, t)).
=
Tr(e’(H’μN )/kT )
(3.106)
16 Additional discussion of the ¬‚uctuation“dissipation theorem and its importance in linear re-
sponse theory are continued in Chapter 6. That the operators in Eq. (3.100) are the ¬eld operators
is immaterial; the relationship is valid for arbitrary operators.
3.4. Equilibrium Green™s functions 71


The importance of the canonical ensembles should be stressed for the validity of
these ¬‚uctuation“dissipation relations or so-called Kubo“Martin“Schwinger bound-
ary conditions. They state that the Green™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 di¬erence be-
tween the times, t ’ t , i.e. they are invariant with respect to displacements in time.
If in addition the equilibrium state is translationally invariant, then all Green™s func-
tions are speci¬ed according to

dp dE i (p·(x’x )’E(t’t ))
G(x, t, x , t ) = e G(p, E) (3.107)
(2π )3 2π
’∞

or equivalently

G(p, E, p , E ) = 2π(2π )3 δ(p ’ p ) δ(E ’ E ) G(p, E) . (3.108)

For example,


[ap (t) , a† (t )]± .
i
GK (p, E) = ’i d(t ’ t ) e (t’t )E
(3.109)
p
’∞

The relationship in Eq. (3.105) then takes the form of the detailed balancing
condition E ’μ
G< (p, E) = ± e’ k T G> (p, E) . (3.110)



Exercise 3.16. Show that, for free bosons or fermions speci¬ed by the Hamiltonian
in Eq. (2.21), we have

p2
’ = ’2πi δ(E ’
GR (p, E) GA (p, E) p) , = . (3.111)
p
0 0
2m


Exercise 3.17. Show that, for free longitudinal phonons,, speci¬ed by the Hamilto-
nian in Eq. (1.123), we have

D0 (k, ω) ’ D0 (k, ω) = ’2πi ωk sign(ω) δ(ω 2 ’ ωk ) θ(ωD ’ |ω|)
R A 2 2
(3.112)

where the sign-function, sign(x) = θ(x) ’ θ(’x) = x/|x|, is plus or minus one,
depending on the sign of the argument.
72 3. Quantum dynamics and Green™s functions


Instead of de¬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 ’ μN ,
and we have, according to Eq. (2.36),17

ψμ (x, t) = e’
i i i
μN t μN t μt
ψH (x, t) e =e ψH (x, t) . (3.113)

De¬ning the grand canonical Green™s functions in terms of these ¬elds, we observe
that they are related to those de¬ned according to Eq. (3.106), or those in the canon-
ical ensemble in the thermodynamic limit, according to
i
μ(t’t )
Gμ (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 Green™s
function, formulas have the same appearance in both ensembles.
For the Fourier transformed Green™s function with respect to time, the transition
to the grand canonical ensemble thus corresponds to the substitution E ’ E + μ, 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) = ± e’E/kT G> (E, p) (3.115)

and we have dropped the chemical potential index as these are the Green™s functions
we shall use in the following. The absence of the chemical potential in the exponential
shows that the relationships are speci¬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 Green™s function and the retarded and ad-
vanced Green™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) ’ GA (E, p)) tanh . (3.116)
2kT
In thermal equilibrium, all Green™s functions can thus be speci¬ed once a single of
them is known, say the retarded Green™s functions, and the quantum statistics of the
particles is then re¬‚ected in relations governed by the ¬‚uctuation“dissipation type
relationship such as in Eq. (3.116). Occasionally we keep Boltzmann™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 ¬‚uctuation“
dissipation theorem reads
E
GK (E, p) = (GR (E, p) ’ 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 Green™s functions 73


Exercise 3.19. Show that

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

and
± i G< (E, p) = f“ (E) A(E, p) , (3.119)
where the functions
1
f“ (E) = (3.120)
eE/kT “ 1
denote either the Bose“Einstein distribution or Fermi“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)
a† (t) ap (t) + i a† (t)
H = p p p
2 dt
p
t=0


1 d
’i G< (p, t)
= i + p
2 dt
p
t=0



1 dE
’i G< (p, E)
= (E + p)
2 2π
p ’∞



1 dE
= (E + p) A(p, E) f (E) , (3.121)
2 2π
p ’∞


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

H 1 dp dE
= (E + p) A(p, E) f (E) . (3.122)
(2π )3
V 2 2π
’∞



For a system in thermal equilibrium, the correlation function
† †
ψμ (x, t) ψμ (x , t ) = Tr(e(©’(H’μN ))/kT ψμ (x, t) ψμ (x , t )) (3.123)

can be spectrally decomposed by inserting a complete set of energy states in the
multi-particle space

(H ’ μN )|En , N = (En ’ μN )|En , N (3.124)
74 3. Quantum dynamics and Green™s functions


giving


e(©’En +μN )/kT ei(t’t )(En ’Em +μ) —
ψμ (x, t) ψμ (x , t ) =
N,n,m
N, En |ψ(x, t = 0))|Em , N + 1 —

N + 1, Em |ψ † (x , t = 0)|En , N . (3.125)

From this expression we observe that the G-greater Green™s function G> (x, t, x , t )
considered as a function of imaginary times is an analytic function in the region,
’1/kT < m(t ’ t ) < 0, if the exponential exp{’En (1/kT + i(t ’ 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 ’ t ) < 1/kT .


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


e(©’En +μN )/kT ei(t’t )(En ’Em +μ)
ψμ (x, t) ψμ (x , t ) =
N,n,m

— e’i(x’x )pn m N, En |ψ(0, 0)|Em , N + 1

— N + 1, Em |ψ † (0, 0)|En , N , (3.126)

where pnm = Pn ’ Pm is the di¬erence between the total momentum eigenvalues
for the two states in question. For the Fourier transform we then have

ap (t) a† (t ) = (2π)3 e(©’En +μN )/kT ei(t’t )(En ’Em +μ)
p
N,n,m


— δ(p ’ pmn )| N, En |ψ(0, 0))|Em , N + 1 |2 .
(3.127)

Noting the analyticity in the upper-half ω-plane of the following function, we have
for real values of ω

1
dt θ(t) eiωt = , (3.128)
ω + iδ
’∞

where δ = 0+ , or equivalently

dω e’iωt
θ(t) = . (3.129)
’2πi ω + iδ
’∞
3.4. Equilibrium Green™s functions 75


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

e’(En ’μN )/kT —
GR(A) (p, E) = (2π)3 e©/kT
Nn ,n,m


| Nn , En |ψ(0, 0))|Em , Nm |2
δ(p ’ pmn ) “ (n ” m) , (3.130)
+
E + Emn (’) iδ

where Emn = Em ’En +μ(Nm ’Nn ), and we recall that Nm = Nn ±1. The retarded
(advanced) Green™s function is thus analytic in the upper (lower) half-plane for the
energy variable E. The simple poles for the retarded (advanced) Green™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 speci¬ed in terms of a single Green™s
function, say the retarded or equivalently the spectral function, as by analyticity the
retarded and advanced Green™s functions satisfy the causality or Kramers“Kronig
relations for their real and imaginary parts, or compactly

dE GR (E , p) ’ GA (E , p)
R(A)
G (E, p) =
’2πi E ’ E (’) i0
+
’∞



dE A(E , p)
= . (3.131)
2π E ’ E (’) i0
+
’∞

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

e(©’(En ’μNn ))/kT
’(2π)4
A(p, E) =
Nn ,n,m


— δ(p ’ pmn ) δ(E ’ Emn )| Nn , En |ψ(0, 0))|Em , Nm |2

“ (n ” m)) (3.132)

or equivalently
Emn
e(©’(En ’μNn ))/kT 1 “ e’
(2π)4
A(p, E) = kT

Nn ,n,m


— δ(p ’ pmn ) δ(E ’ Emn )| Nn , En |ψ(0, 0))|Em , Nm |2 , (3.133)

where the upper and lower sign is for bosons and fermions, respectively.
76 3. Quantum dynamics and Green™s functions


The analytic properties of the retarded (or advanced) Green™s function determines
the analytic properties of all the other introduced Green™s functions, and are further
studied in Section 5.6.
The three Green™s functions, GR,A,K , thus carry di¬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 ’ ∞, the retarded and
advanced Green™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 speci¬ed by the
Hamiltonian in Eq. (2.21), one readily obtains for the spectral weight function,
Eq. (3.85),

p2
2π δ(E ’ ξp ) , ’μ= ’ μ,
A0 (p, E) = ξp = (3.135)
p
2m
and according to the ¬‚uctuation“dissipation theorem, all one-particle Green™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

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

’∞




Exercise 3.23. The quantum statistics of particles have, according to the above,
a profound in¬‚uence on the form of the Green™s function. Show that, for the case
of non-interacting fermions at zero temperature, the Fermi surface is manifest in
the time-ordered Green™s function, Eq. (3.61), according to (say) in the canonical
ensemble,
1
G0 (E, p) = , (3.137)
E ’ p + iδ sign(|p| ’ pF )
where δ = 0+ , and the sign-function, sign(x) = θ(x)’θ(’x) = x/|x|, is plus or minus
one depending on the sign of the argument. The grand canonical case corresponds
to the substitution p ’ ξp = p ’ μ.
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 ¬eld
operator, ψ(x) = ξ0 + ψ (x), the creation operator for the lowest energy level is
√ †
singled out, ξ0 = a0 / V , and ξ0 and ξ0 can, for a non-interacting system in the

thermodynamic limit, be regarded as c-numbers, [ξ0 , ξ0 ] = 1/V .
Show that the time ordered Green™s function for non-interacting bosons in the
ground state, |¦N = (N !)’1/2 (a† )N |0 , is given by
0

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

where

G(0) (t ’ t ) = ’i ¦N |T (ξ0 (t) ξ0 (t ))|¦N (3.139)
is speci¬ed at t = t + 0 by iG(0) (0’ ) = n, where n = N/V is the density of the
bosons, and G0 (x, t, x , t ) is speci¬ed by its Fourier transform
1
G0 (E, p) = (3.140)
E’ + iδ
p

corresponding to G-lesser vanishing for the ¬eld ψ , or equivalently, the density of the
bosons is solely provided by the occupation of the lowest energy level. The presence
of a Bose“Einstein condensate at low temperatures thus leads to such modi¬cations
for boson Green™s function expressions.18


As mentioned, perturbation theory and diagrammatic summation schemes are
the main tools in unraveling the e¬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 ¬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 ¬elds
can be treated as external ¬elds. This leads to additional vertices in the Feynman diagrammatics
as encountered in Section 10.6.
78 3. Quantum dynamics and Green™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 ¬eld operators, the Green™s functions of the theory.
In thermal equilibrium, the ¬‚uctuation“dissipation relation leads to simpli¬cation
as all the one-particle or two-point Green™s functions can be expressed in terms of the
spectral weight function. Di¬erent schemes pertaining to equilibrium can be devised
for calculating equilibrium Green™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 Green™s functions of interacting quantum ¬elds in-
volve ever increasing higher-order correlation functions. The rest of the book is
devoted to the study and calculation of Green™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 ¬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 Green™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 ¬eld theory, the Hamiltonian acts on the multi-
particle state space introduced in the ¬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

e’H/kT
ρ(H) = , (4.1)
Tr(e’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 ’ H ’ s μs 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
ρ(t) = U (t, t0 ) ρ(H) U † (t, t0 ) , (4.3)
where (recall Eq. (3.7))
t
U (t, t ) = T e’ dt H(t)
i ¯¯
(4.4)
t


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

A(t) = Tr(ρ(t) A) , (4.5)

where A is the operator representing the physical quantity in question in the Schr¨dinger
o
picture. Transforming to the Heisenberg picture

A(t) = Tr(ρ(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 ¬eld ψ, we have according to Eq. (2.28),3

n(x, t) = ψH (x, t) ψH (x, t) . (4.7)

The average density is seen to be equal to the equal-time and equal-space value of
the G-lesser Green™s function, G< , introduced in Section 3.3

n(x, t) = “iG< (x, t, x, t) , (4.8)

where upper (lower) sign is for bosons and fermions, respectively.
If ¬‚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 ¬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 di¬erent times, one encounters arbitrary long products of Heisenberg
operators.4 Since physical quantities are expressed in terms of the quantum ¬elds of
the particles and interactions in terms of their higher-order correlations, of interest
are the correlation functions, the so-called non-equilibrium Green™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 Green™s function leads to an in¬nite
hierarchy of equations for correlation functions containing ever increasing numbers of
¬eld operators, describing the correlations between the particles set up in the system
by the interactions and external forces. In order to calculate the e¬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 ¬rst chapter. But ¬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 ¬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 ¬elds, H0 , describing the free particles, and a
complicated term, H (i) , describing interactions.
Constructing perturbation theory for zero temperature quantum ¬eld theory, i.e.
the system is in its ground state |G , only the time-ordered Green™s function
† †
G(x, t, x , t ) = ’i T (ψH (x, t) ψH (x , t )) = ’i G|T (ψH (x, t) ψH (x , t ))|G
(4.10)
needs to be considered. Here ψH (x, t) is the ¬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 Green™s function contains more information than seems necessary
for calculating mean or average values, since for times t < t it becomes the G-lesser
Green™s function
G(x, t, x , t ) = G< (x, t, x , t ) (4.11)
5 All transient e¬ects for the above chosen initial condition are of course included. Whether this
choice is appropriate for the study of transient e¬ects depends on the given physical situation.
6 For a reader not familiar with zero temperature quantum ¬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 ¬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 speci¬ed once the time-
ordered Green™s function is known for t < t . However, a perturbation theory involv-
ing only the G-lesser Green™s function can not be constructed.
The time-ordered Green™s function can, instead of being expressed in terms of the
¬eld operator ψH (x, t), i.e. in the Heisenberg picture with respect to H, be expressed
in terms of the ¬eld operators ψH0 (x, t), the Heisenberg picture with respect to H0
or the so-called interaction picture,

ψ(x) e’
i i
H0 (t’tr ) H0 (t’tr )
ψH0 (x, t) = e (4.12)
as they are related according to the unitary transformation
ψH (x, t) = U † (t, tr ) ψH0 (x, t) U (t, tr ) , (4.13)
where
t ¯ (i ) ¯
U (t, tr ) = T e’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{’iH(t ’ tr )}) and

HH0 (t) = eiH0 (t’tr ) H (i) e’iH0 (t’tr ) .
(i)
(4.15)
This is readily seen by noting that the expression on the right-hand side in Eq. (4.13)
satis¬es the ¬rst-order in time di¬erential equation
‚ψ(x, t)
i = [ψ(x, t), H] , (4.16)
‚t
the same equation satis¬ed by the ¬eld ψH (x, t), and at the reference time tr , the two
operators are seen to coincide (coinciding with the ¬eld in the Schr¨dinger picture,
o
ψ(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 † (t, t ) = U (t , t),
the time ordered Green™s function can be expressed in the form

= ’i U † (t, tr )ψH0 (x, t)U (t, t )ψH0 (x , t )U (t , tr ) θ(t ’ t )
G(x, t, x , t )

± i U † (t , tr )ψH0 (x , t )U (t , t)ψH0 (x, t)U (t, tr ) θ(t ’ t) (4.17)

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

G(x, t, x , t ) = ’i U † (tm , tr )T ψH0 (x, t)ψH0 (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

† †
T (ψH (x, t) ψH (x , t )) = U † (tm , tr )T ψH 0 (x, t) ψH 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 = ’∞, and inserting 1 = U (tm , ∞)U (∞, tm ) after
U † gives9

G(x, t, x , t ) = ’i U † (∞, ’∞)T (ψH0 (x, t) ψH0 (x , t ) U (∞, ’∞)) . (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 o¬
adiabatically, say by the substitution HH0 (t) ’ e’ |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, ’∞)|G0 . The on the evolution operator indicates that the
interaction is turned on and o¬ adiabatically. In perturbation theory it can then be
shown, that in the limit of ’ 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 o¬ adiabatically, (the Gell-Mann“Low theorem [4]),10

U (∞, ’∞) |G0 = eiφ |G0 eiφ = G0 |U (∞, ’∞)|G0 .
, (4.20)

As a consequence, the time-ordered Green™s function, Eq. (4.10), can be expressed
in terms of the non-interacting ground state and the ¬elds in the interaction picture
according to

G0 |T (ψH0 (x, t) ψH0 (x , t ) U (∞, ’∞))|G0
G(x, t, x , t ) = ’i . (4.21)
G0 |U (∞, ’∞)|G0

In the next section we will show that the arti¬ce of turning the interaction on
and o¬ 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 ¬nal boundary condition as the particles are then free.11
Since the Gell-Mann“Low theorem fails for states other than the ground state,
and thus even for an equilibrium state at ¬nite temperature, we are in general stuck
9 In fact as an operator identity
† †
T (ψH (x, t) ψH (x , t )) = U † (∞, ’∞)T (ψH 0 (x, t) ψH 0 (x , t ) U (∞, ’∞)).

10 Clearly, it is important that no dissipation or irreversible e¬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
Green™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(∞, ’∞), de¬ned in terms of the full
evolution operator, S(t, t ) = eiH 0 t U (t, t )e’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 † (∞, ’∞) inside the averaging in Eq. (4.19) and Eq. (4.18). At
¬nite temperatures and a fortiori for non-equilibrium states, a perturbation theory
involving only one kind of a real-time Green™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 speci¬ed by the
unitary transformation
t
’i ¯ ¯
OH (t) = V † (t, t0 ) OH (t) V (t, t0 ) ,
dt HH (t)
V (t, t0 ) = T e (4.22)
t0



and

UH (t, t0 ) = e’
i
H(t’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 ¬rst comparing both pictures to the
Schr¨dinger picture obtaining
o
† †
OH (t) = UH (t, t0 ) UH (t, t0 ) OH (t) UH (t, t0 ) UH (t, t0 ) , (4.24)

where
’i dt H(t)
t ¯¯
UH (t, t0 ) = T e (4.25)
t0



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

that V (t, t0 ) and UH (t, t0 ) UH (t, t0 ) satisfy the same ¬rst-order in time di¬erential
equation and the same initial condition. We have thus obtained Dyson™s formula

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

or explicitly
t t
T e’ T e’ dt H(t)
i i i
¯ ¯ ¯¯
dt HH (t) H(t’t0 )
=e . (4.27)
t t


Here Dyson™s formula appeared owing to unitary transformations between Heisen-
berg and interaction pictures, but once conjectured it can of course immediately be
established by direct di¬erentiation. Dyson™s formula is useful in many contexts,
be the time variable real or imaginary, and also for equilibrium states such as when
phase transitions are studied in, for instance, a renormalization group treatment. We
shall in fact apply Dyson™s formula for imaginary times in Section 4.3.2.
We now introduce the contour, the closed time path, which starts at t0 and
proceeds along the real time axis to time t and then back again to t0 , the closed
contour ct as depicted in Figure 4.1.
4.3. Closed time path formalism 85


t
ct

t0

Figure 4.1 The closed time path contour ct .


We then show that the transformation between the two Heisenberg pictures,
Eq. (4.24), can be expressed on closed contour form as (units are chosen to set
equal to one at our convenience)
’i d„ HH („ )
OH (t) = Tct e OH (t) , (4.28)
ct




where „ denotes the contour variable proceeding from t0 along the real-time axis to
t and then back again to t0 , i.e. the variable on ct . The contour ordering symbol Tct
orders products of operators according to the position of their contour time argument
on the closed contour, earlier contour time places an operator to the right.
The crucial equivalence of Eq. (4.24) and Eq. (4.28), which form a convenient
basis for formulating perturbation theory in the closed time path formalism, is based
on the algebra of operators under the contour ordering being equivalent to the algebra
of numbers.12 Expanding the exponential in Eq. (4.28) gives

(’i)n
OH (t) = d„1 . . . d„n Tct (HH („1 ) . . . HH („n ) OH (t)) . (4.29)
n! ct ct
n=0

Let us consider the nth order term. In order to verify Eq. (4.28), we note that the
contour can be split into forward and backward parts

ct = ’ + ← .
’’
c c (4.30)

Splitting the contour into forward and backward contours gives 2n terms. Out of these
there are n!/(m!(n ’ m)!) terms (m = 0, 1, 2, . . . , n), which contain m integrations
over the forward contour, and the rest of the factors, n’ m, have integratons over the
backward contour. Since they di¬er only by a di¬erent dummy integration labeling
they all give the same contribution and
n
n!
d„1 . . . d„n Tct (HH („1 ) . . . HH („n ) OH (t)) =
m!(n ’ m)!
ct ct m=0

— ’
d„m+1 . . . d„n T← (HH („m+1 ) . . . HH („n )) OH (t)

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

’ ’
’ c
c c
12 Even though the Hamiltonian for fermions contains non-commuting objects, the fermi ¬elds,
they appear in pairs and quantum statistical minus signs do not occur.
86 4. Non-equilibrium theory


’ ’
where T’ and T← denotes contour ordering on the forward and backward parts,
c c
respectively. Adding a summation and a compensating Kronecker function the nth-
order term can be rewritten in the form13
∞ ∞
n!
d„1 . . . d„n Tct (HH („1 ) . . . HH („n ) OH (t)) = δn,k+m
m! k!
ct ct m=0
k=0

— ’
d„1 . . . d„k T← (HH („1 ) . . . HH („k )) OH (t)

’ ←
’ c
c c

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

’ ’
’ c
c c
The summation over n in Eq. (4.29) is now trivial, giving
∞ ∞
(’i)k (’i)m
’i d„ HH („ )
T ct e OH (t) =
ct
m! k!
m=0
k=0


— ’
d„1 . . . d„k T← (HH („1 ) . . . HH („k )) OH (t)

’ ←
’ c
c c

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

’ ’
’ c
c c
and thereby
’i ’i
d„ HH („ ) d„ HH („ )

’ ’

’i d„ HH („ )
’ ’
ec ec
T ct e OH (t) = T← OH (t) T’ .
ct
c c

(4.34)

Parameterizing the forward and backward contours according to

„ (t ) = t t [t0 , t] , (4.35)

we get
t
’i d„ HH („ ) ’i

’ dt HH (t )
T’ e c
’ = Te = V (t, t0 ) (4.36)
t0
c

and
’i d„ HH („ )

’ t
= V † (t, t0 )
˜i dt HH (t )

T← e c = Te (4.37)
t0
c

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

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

time ordering, T← = T . The equivalence of Eq. (4.24) and Eq. (4.28) has thus been
c
established. We have shown that the times in V † (t, t0 ) corresponds to contour times
13 Underthe ordering operation, the algebra of non-commuting objects reigning the operators is
not important, and the consideration is essentially the algebra of showing exp(a+b) = exp(a) exp(b).
4.3. Closed time path formalism 87


lying on the backward part, and the times in V (t, t0 ) corresponds to contour times
lying on the forward part.
We shall now use Eq. (4.28) to introduce the contour variable instead of the time
variable. We hereby embark on Schwinger™s closed time path formulation of non-
equilibrium quantum statistical mechanics originally introduced in reference [5].14
We shall thereby develop the diagrammatic perturbative structure of the closed time
path or contour ordered Green™s function.

4.3.1 Closed time path Green™s function
A generalization o¬ers itself, which will lead to a single object in terms of which non-
equilibrium perturbation theory can be formulated. The trick will be to democratize
the status of all times appearing in the time-ordered Green™s function, Eq. (4.18),
i.e. the original real times t and t will be perceived to reside on the closed time path
or contour. The one-particle Green™s function in Eq. (4.18) contains two times; let
us now denote them t1 and t1 . We introduce the contour, which starts at t0 and
proceeds along the real-time axis through t1 and t1 and then back again to t0 , the
closed contour c as depicted in Figure 4.2, c = ’ + ←.15 We have hereby freed the
’’
c c
time variables, which hitherto were tied to the real axis, to lie on either the forward
or return part of the contour, and we introduce the contour variable „ to signify this
two-valued choice of the time variable, examples of which are given in Figure 4.2.16

„1


„1
t0 t1 t1 t

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

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

Tr(e’H/kT Tc (ψH (x1 , sz1 , „1 ) ψH (x1 , sz1 , „1 )))
, „1 ) = ’i
G(x1 , sz1 , „1 , x1 , sz1
Tr(e’H/kT )
(4.38)
14 Reviews of the closed time path formalism stressing various applications are, for example, those
of references [6], [7] and [8].
15 If we discussed a correlation function involving more than two ¬elds, the contour should stretch

all the way to the maximum time value, or in fact we can let the contour extend from t0 to t = ∞
and back again to t0 , since, as we soon realize, beyond max(t1 , t1 , . . .) the forward and backward
evolutions take each other out, producing simply the identity operator.
16 For mathematical rigor, i.e. proper convergence, both the forward and backward contours

should be conceived of as being located in¬nitesimally below the real axis. This will be witnessed
by the analytical continuation procedure discussed in Section 5.7, but in practice this consideration
will not be necessary.
88 4. Non-equilibrium theory


where „1 and „1 can lie on either the forward or backward parts of the closed contour.
We have had a particle with spin in mind, say the electron, but introducing the
condensed notation 1 = (x1 , sz1 , „1 ) we have17
† †
G(1, 1 ) = ’i Tc(ψH (1) ψH (1 )) = ’i Z ’1 Tr(e’H/kT Tc (ψH (1) ψH (1 ))) (4.39)
at which stage any particle could be under discussion as the only relevant thing in
the rest of the section is the contour variable. A contour ordering symbol Tc has been
introduced, which orders operators according to the position of their contour-time
argument on the closed contour, for example for the case of two contour times
c
ψ(x1 , „1 ) ψ † (x1 , „1 ) „1 > „1

Tc (ψ(x1 , „1 ) ψ (x1 , „1 )) = (4.40)
c
“ψ † (x1 , „1 ) ψ(x1 , „1 ) „1 > „1
where the upper (lower) sign is for fermions (bosons) respectively. An obvious nota-
c
tion for ordering along the contour has been introduced, viz. „1 > „1 means that „1
is further along the contour c than „1 irrespective of their corresponding numerical
values on the real axis. The contour ordering thus orders an operator sequence ac-
cording to the contour position; operators with earliest contour times are put to the
right. The algebra of bose ¬elds under the contour ordering is thus like the algebra
of (complex) numbers, whereas the algebra of fermi ¬elds under the contour ordering
is like the Grassmann algebra of anti-commuting numbers.18
We also introduce greater and lesser quantities for the contour ordered Green™s
function, and note according to the contour ordering, Eq. (4.40),
ct
G< (1, 1 ) „1 > „1
G(1, 1 ) = (4.41)
ct
G> (1, 1 ) „1 > „1 .
Here lesser refers to the contour time „1 appearing earlier than contour time „1 , and
vice versa for greater. Note that these relationships are irrespective of the numerical
relationship of their corresponding real time values: if the contour times in G< (1, 1 )
and G> (1, 1 ) are identi¬ed with their corresponding real times we recover their
corresponding real-time Green™s functions discussed in Section 3.3.
Transforming from the Heisenberg picture with respect to the Hamiltonian H(t)
to the Heisenberg picture with respect to the Hamiltonian H, gives, according to
Eq. (4.28),

’i ψH (1) ψH (1 )
G> (1, 1 ) =
’i
’i d„ HH („ ) †
d„ HH („ )
’i Tct 1 e ct
ct
= ψH (1) T ct e ψH (1 )
1 1
1


17 In the following we shall consider the ¬elds as entering the Green™s function, however, for
the following it could be any type of operators and any number of products, G(1, 2, 3, . . .) =
† †
Tc (AH (1) BH (2) CH (3) . . .) . Note that if the operators represent physical quantities, they are
speci¬ed in terms of the ¬elds, and we are back to strings of ¬eld operators modulo the operations
speci¬c to the quantities in question.
18 In Chapter 10 we shall in fact show that in view of this, quantum ¬eld theory can, instead of

being formulated in terms of quantum ¬eld operators, be formulated in terms of scalar or Grassmann
numbers by the use of path integrals.
4.3. Closed time path formalism 89


’i d„ HH („ ) †
’i Tct 1 +ct 1
+c t
ct
= e ψH (1) ψH (1 ) , (4.42)
1 1




where the contours ct1 (ct1 ) starts at t0 and passes through t1 (t1 ), respectively,
and returns to t0 . In the last equality the combined contour, ct1 + ct1 , depicted
in Figure 4.3, has been introduced. It stretches from t0 to min{t1 , t1 } and back to
t0 and then forward to max{t1 , t1 } before ¬nally returning back again to t0 . The
contributions from the hatched parts depicted in Figure 4.3 cancel since for this part
the ¬eld operators at times t1 and t1 are not involved and a closed contour appears
which gives the unit operator, or equivalently U † (t1 , t0 ) U (t1 , t0 ) = 1, and the last
equality in Eq. (4.42) is established. By the same argument, the contour could be
extended from max{t1 , t1 } all the way to plus in¬nity before returning to t0 , and we
encounter the general real-time contour.

t0 t1 t1

c1
c1


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

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

= ’i Tc (ψH (1) ψH (1 ))
G(1, 1 )


= ’i Tc e’i d„ HH („ )
ψH (1) ψH (1 ) , (4.43)
c



where the contour c starts at t0 and stretches through max(t1 , t1 ) (or all the way to
plus in¬nity) and back again to t0 . By introducing the closed contour and contour
ordering we have managed to bring all operators under the ordering operation, which
will prove very useful when it comes to deriving the perturbation theory for the
contour-ordered Green™s function.
Exercise 4.1. From the equation of motion for the ¬eld operator, show that the
equation of motion for the contour-ordered Green™s function is

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

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

where h0 denotes the single-particle Hamiltonian, and we have introduced the contour
delta function
§
⎨ δ(„ ’ „ ) for „ and „ on forward branch
’δ(„ ’ „ ) for „ and „ on return branch
δc („ ’ „ ) = (4.45)
©
0 for „ and „ on di¬erent branches
90 4. Non-equilibrium theory


and Hi („ ) is the interaction part of the Hamiltonian in the Heisenberg picture (recall
Exercise 3.10 on page 66).
The equation of motion for the Green™s function leads, as noted in Section 3.3, to
an in¬nite hierarchy of equations for correlation functions containing an ever increas-
ing number of ¬eld operators describing the correlations between the particles set
up in the system by the interactions and external forces. Needless to say, an exact
solution of a quantum ¬eld theory is a mission impossible in general. At present,
the only general method available for gaining knowledge from the fundamental prin-
ciples about the dynamics of a system is the perturbative study. This goes for
non-equilibrium states a fortiori, and we shall now construct the perturbation the-
ory valid for non-equilibrium states. This consists of dividing the Hamiltonian into
one part representing a simpler well-understood problem and a nontrivial part, the
e¬ect of which is studied order by order.
In the next section we construct the general perturbation theory valid for non-
equilibrium situations. We thus embark on the construction of the diagrammatic
representation starting from the canonical formalism presented in Chapter 1.

4.3.2 Non-equilibrium perturbation theory
We now proceed to obtain the perturbation theory expressions for the contour-
ordered Green™s functions. The Hamiltonian of the system, Eq. (4.9) consists of
a term quadratic in the ¬elds, H0 , describing the free particles, and a complicated
term, H (i) , describing interactions. To get an expression ready-made for a pertur-
bative expansion of the contour-ordered Green™s function, the Hamiltonian in the
weighting factor needs to be quadratic in the ¬elds, i.e. we need to transform the
operators in Eq. (4.42) to the interaction picture with respect to H0 . Quite analogous
to the manipulations in the previous section we have
(i )
’i d„ (HH („ )+HH 0 („ ))
OH (t) = Tct e OH0 (t) , (4.46)
ct 0




where we have further, or directly, transformed from the Heisenberg picture with
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 („ ) 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 Green™s function, Eq. (4.39), that the contour-ordered Green™s
function in the interaction picture is
(i )
Tr e’βH Tc e’i †
d„ (HH („ )+HH 0 („ ))
ψH0 (1) ψH0 (1 )
c 0

G(1, 1 ) = ’i . (4.47)
Tr (e’βH )

We have introduced the notation β = 1/kT for the inverse temperature.
19 We shall later take advantage of the arti¬ce of employing di¬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 Dyson™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 ’i β (i )
e’βH = e’βH0 Tca e’i d„ HH („ )
(4.48)
t0 0



where Tca contour orders along the contour stretching down into the lower complex
time plane from t0 to t0 ’ iβ, the appendix contour ca as depicted in Figure 4.4. We
then get the expression
⎛ ⎛ ⎞ ⎞
t 0 ’i β
(i )
’i (i )
’i d„ (HH („ )+HH 0 („ ))
d„ HH („ )
Tr⎝e’βH0⎝Tca e ⎠Tc e ψH0 (1)ψH0 (1 )⎠

0 0
t0 c



iG(1, 1 ) = t 0 ’i β (i )
Tr e’βH0 Tca e’i t 0 d„ HH („ )
0


(4.49)
ready-made for a perturbative expansion of the contour-ordered Green™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 ’ iβ 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 )
’i †
d„ HH („ ) ’i
Tr e’βH0 Tci d„ HH 0 („ )
e e ψH0 (1) ψH0 (1 )
ci c
0

G(1, 1 ) = ’i .
(i )
’i d„ HH („ ) ’i d„ HH („ )
Tr e’βH0 Tci e e
ci c
0 0


(4.50)
The contour ci stretches from t0 to max{t1 , t1 } (or in¬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 ’ iβ, as depicted in Figure 4.4.


t0 t1 t1

ci


t0 ’ iβ

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 e’i = Tc e’i d„ HH („ ) ’i
d„ (HH („ ) + HH 0 („ )) d„ HH 0 („ )
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 Wick™s theorem
to obtain the perturbative expansion of the contour-ordered Green™s function and
the associated Feynman diagrammatics. Before we show Wick™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 in¬nity) and back again to t0 ,
which since no operators interrupts at intermediate times is just the identity operator
(i )
Tc e’i d„ (HH („ ) + HH 0 („ ))
= 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 arti¬cial relics of the formalisms used in standard zero and
¬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
in¬nity, t0 ’ ’∞, 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
¬eld, retrudes back into the past beyond the microscopic scattering times of the
system, then e¬ectively t0 ’ ’∞, 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 ¬nally have for the contour-
ordered or closed time path Green™s function
(i )
e’i †
d„ (HH („ ) + HH 0 („ ))
G(1, 1 ) = Tr ρ0 TC ψH0 (1) ψH0 (1 )
C 0



(i )
e’i †
d„ HH („ ) ’i d„ HH 0 („ )
= Tr ρ0 TC e ψH0 (1) ψH0 (1 ) , (4.53)
C C
0



where
e’H0 /kT
ρ0 = (4.54)
Tr e’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 o¬ 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 ’ ’∞, 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 e¬ect of intrinsic damping. We note that at ever
increasing temperatures, the appendix contour contribution disappears, since thermal ¬‚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 Schwinger™s closed time path [5], the Schwinger“Keldysh or real-time
contour, which starts at time t = ’∞ and proceeds to time t = ∞ and then back
again to time t = ’∞, as depicted in Figure 4.5.

c1
t

c2


Figure 4.5 The Schwinger“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 e’βH0 Tc e’i d„ HH („ ) ’i d„ HH 0 („ )
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 ¬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 ¬eld V (x, t). In that case the
contour ordered Green™s function has the form ready for a perturbative expansion

’i d„ dx V (x,„ ) ψH (x,„ ) ψH 0 (x,„ )

C 0
GC (1, 1 ) = Tr ρ0 TC e ψH0 (1) ψH0 (1 ) .

(4.56)
Expanding the exponential we get strings of, say, fermi ¬eld operators traced and
weighted with respect to the free statistical operator. The zeroth-order term just
gives the free contour Green™s function

(0)
GC (1, 1 ) = ’iTr ρ0 TC ψH0 (1) ψH0 (1 ) . (4.57)
22 This is an appealing alternative in the quantum ¬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
Green™s function is the trace of products, or strings, of the ¬eld operators of the theory
in the interaction picture weighted by the free part of the Hamiltonian, a quadratic
form in these ¬elds. For example, in the case of electron“phonon interaction a string
of n phonon ¬elds and 2n fermi ¬elds occurs; see Eq. (4.126). The weighted trace over
the fermi and bose ¬elds separates into the two traces over these independent degrees
of freedom. To be explicit, let us ¬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 ¬eld operators, ordered along a contour C.25 We introduce
the representation of the bose ¬eld in terms of its creation and annihilation operators
as in Eq. (2.74) and encounter strings of creation and annihilation operators26
S = tr(ρT TC (c(„n ) c(„n’1 ) . . . c(„2 ) c(„1 )))

≡ TC (c(„n ) c(„n’1 ) . . . c(„2 ) c(„1 )) , (4.60)
where the cs denote either a creation or annihilation operator, a or a† , and
(0)
e’Hb /kT
ρT = (4.61)
(0)
Tr e’Hb /kT

is the statistical operator for the equilibrium state of the non-interacting bosons or
phonons at temperature T , and

a† aq
(0)
Hb = hq = (4.62)
q q
q q

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