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— |x1 § x2 § · · · ( no xn and xm ) · · · § xN , (2.60)
where θ denotes the step function. The second statistics exponent factor is m if
m > n and of the usual form m ’ 1 if m < n, simply adjusting to when operating
with the second annihilation operator the labeling of the state vector di¬ers from the
one used in the de¬nition Eq. (1.72). Then by operating with creation operators we
get

ψ † (x) ψ † (x ) ψ(x ) ψ(x) |x1 § · · · § xN δ(x ’ xn )δ(x ’ xm )
=
m(=n)


— (’1)n’1 (’1)m’θ(n’m) |x § x § x1 § x2 § · · · ( no xn and xm ) · · · § xN


δ(x ’ xn ) δ(x ’ xm ) |x1 § x2 § x3 § · · · § xN
= (2.61)
m(=n)

and multiplying with V (2) (x, x ) and integrating over x and x in Eq. (2.59) there-
fore reproduces Eq. (2.58). Clearly, the operator V on the multi-particle space is
hermitian since the function V (2) (x, x ) is real.
We note that the perhaps more intuitive guess for the two-body interaction in
terms of the density operator
1
dx dx n(x) V (2) (x, x ) n(x )
V = (2.62)
2
di¬ers from the correct expression, Eq. (2.59), by a self-energy term
1
dx V (2) (x, x) n(x) ,
V =V + (2.63)
2
which, for example, for the case of Coulomb interaction would be in¬nite unless no
particles are present, in which case it becomes the other extreme, viz. zero.
The two-particle interaction part of the Hamiltonian, Eq. (2.59), is so-called
normal-ordered, i.e. all annihilation operators appear to the right of any creation op-
erator. We recall that the one-body part of the Hamiltonian is also normal-ordered,
as are those representing physical observables. We note that, as a consequence, the
vacuum state has zero energy and momentum.
44 2. Operators on the multi-particle state space


The derived expression, Eq. (2.59), for two-body interaction of fermions is of
course the same for two-particle interaction of bosons, the derivation being identical,
in fact simpler since no minus sign is involved in the interchange of two bosons.
The Hamiltonian for non-relativistic identical particles interacting through an
instantaneous two-body interaction is thus
2
‚2
1

H = dx ψ (x) ψ(x)
‚x2
2m i

1
dx dx ψ † (x) ψ † (x ) V (2) (x, x ) ψ(x ) ψ(x) .
+ (2.64)
2


Exercise 2.10. Show that the Hamiltonian for an assembly of particles interacting
through two-particle interaction commutes with the number operator.

Exercise 2.11. Show that if the two-particle potential is translational invariant

V (2) (x, x ) = V (x ’ x ) , (2.65)

we have in the momentum representation for the operator on multi-particle space
1 dq
dp dp V (’q) a† a† +q ap ap ,
V = (2.66)
p’q p
(2π )3
2

where V (q) is the Fourier transform of the real potential V (x)

dx e’ x·q
i
V (q) = V (x) . (2.67)

If the potential furthermore is inversion symmetric, V (’x) = V (x), we obtain

1 dq
dp dp V (q) a† a† ’q ap ap .
V = (2.68)
p+q p
(2π )3
2


If the particles possess spin and their two-body interaction is spin dependent, the
interaction in the multi-particle space becomes
1 †

V= dx dx ψ± (x) ψβ (x ) V±± ,ββ (x, x ) ψβ (x ) ψ± (x) , (2.69)
2
±± ,ββ

where, in accordance with custom, the spin degree of freedom appears as an index.
Exercise 2.12. Consider a piece of metal of volume V and describe it in the Som-
merfeld model where the ionic charge is assumed smeared out to form a ¬xed uniform
neutralizing background charge density.
2.4. Interactions 45


Show that, in the momentum representation, the operator on the multi-particle
space representing the interacting electrons is

e2 †
1
a†
V = a a a , (2.70)
2 p+q,σ p ’q,σ p ,σ p,σ
2V 0q
q=0,p,p ,σσ

i.e. the interaction with the background charge eliminates the (q = 0)-term in the
Coulomb interaction.

2.4.2 Fermion“boson interaction
In relativistic quantum theory the creation and annihilation operators, the quan-
tum ¬elds, are necessary to describe dynamics, since particle can be created and
annihilated. Relativistic quantum theory is thus inherently dealing with many-body
systems. In a non-relativistic quantum theory the introduction of the multi-particle
space is never mandatory, but is of convenience since it allows for an automatic way of
respecting the quantum statistics of the particles even when interactions are present.
It is also quite handy, but again not mandatory, when it comes to the description
of symmetry broken states such as the cases of condensed states of fermions in su-
perconductors and super¬‚uid 3 He, and for describing Bose“Einstein condensates of
bosons.
The generic interaction between fermions and bosons is of the form

Hb’f = g dx ψ † (x) φ(x) ψ(x) , (2.71)

where ψ(x) is the fermi ¬eld and φ(x) is the real (hermitian) bose ¬eld, and the
interaction is characterized by some coupling constant g, and possibly dressed up in
some indices characteristic for the ¬elds in question, such as Minkowski and spinor
in the case of QED.5 The fermi and bose ¬elds commute since they operate on
their respective multi-particle spaces making up the total product multi-particle state
space.6 For the fermion“boson interaction which shall be of interest in the following,
viz. the electron“phonon interaction, Eq. ( 2.71) is also a relevant form.

2.4.3 Electron“phonon interaction
Of importance later is the interaction between electrons and the quantized lattice
vibrations in, say, a metal or semiconductor, the electron“phonon interaction. For
illustration it su¬ces to consider the jellium model where the electrons couple only to
longitudinal compressional charge con¬gurations of the lattice ions, the longitudinal
phonons. A deformation of the ionic charge distribution in a piece of matter, will
create an e¬ective potential felt by an electron at point xe , which in the jellium model
5 Even the standard model has only fermionic interactions of this form. The fully indexed theory
will be addressed in Chapter 9.
6 If a theory contains two or more kinematically independent fermion species their corresponding

¬elds are taken to anti-commute.
46 2. Operators on the multi-particle state space


is given by the deformation potential7
n
∇xe · u(xe ) ,
V (xe ) = (2.72)
2N0
where u is the displacement ¬eld of the background ionic charge, N0 is the density of
electron states at the Fermi energy per spin (in three dimensions N0 = mpF /2π 2 3 ),
and n is the electron density. The quantized lattice dynamics leads to the electron“
phonon interaction in the jellium model becoming (recall Eq. (1.131))


n i
ωk [ˆk eik·xe ’ a† e’ik·xe ] (2.73)
ˆ ∇xe · u(xe ) =
ˆ
Ve’ph (xe ) = a ˆk
2N0 2 N0 V
|k|¤kD

where the harmonic oscillator creation and annihilation operators satisfy the commu-
tation relations, [ˆk , a† ] = δk,k , and describe the weakly perturbed collective ionic
a ˆk
oscillations (recall Sections 1.4.1 and 1.4.2). We assume a ¬nite lattice of volume V .
The set of harmonic oscillators is in its multi-particle description thus speci¬ed by
the phonon ¬eld operator


ωk [ak eik·x ’ a† e’ik·x ] , (2.74)
≡ M ni ∇x · u(x) = i
φ(x) c k
2V
|k|¤kD

which is a real scalar bose ¬eld whose quanta, the phonons, are equivalent to bose
particles, the bose ¬eld in the multi-particle space of longitudinal phonons. The
interaction between the lattice of ions and an electron is thus transmitted in discrete
units, the quanta we called phonons. In accordance with custom we leave out hats on
operators on a multi-particle space; the phonon creation and annihilation operators
of course satisfy the above stated canonical commutation relations as well as those of
Eq. (1.113).8 The (longitudinal) phonon ¬eld, Eq. (2.74), is a real or hermitian ¬eld,
φ† (x) = φ(x), and contains a sum of creation and annihilation operators. Except for
the explicit upper (ultraviolet) cut o¬, imposed by the ¬nite lattice constant, it is
thus analogous to the ¬eld describing a spin zero particle.
The electron“phonon interaction in the product of multi-particle spaces for elec-
trons and phonons is according to Eq. (2.72) given in terms of the phonon ¬eld and
the electron density re¬‚ecting that the electrons couple to the (screened) ionic charge
deformations (or equivalently, Eq. (2.72) is a one-body operator for the electrons since
it is a potential-coupling)9

Ve’ph = g dx ne (x) φ(x) = g dx ψ † (x) φ(x) ψ(x) , (2.75)

7 The
electron“phonon interaction is an e¬ective collective description of the underlying screened
electron“ion Coulomb interaction. For the argument leading to the expression of the deformation
potential see, for example, chapter 10 of reference [1].
8 Phonons refer to collective oscillations of the ions and their screening cloud of electrons, similarly

as the e¬ective Coulomb electron“electron interaction describes the interaction between electrons
and their screening clouds. Such objects are referred to as quasi-particles.
9 That the electron“phonon interaction takes this form is the reason for introducing the phonon

¬eld, Eq. (2.74), instead of using the displacement ¬eld.
2.4. Interactions 47


where the electron“phonon interaction coupling constant, g, in the jellium model is
given by
2
1 4
2 F
g= = (2.76)
9 M n i c2
2N0
and for the last rewriting in Eq. (2.75) we have used the fact that fermi and bose
¬elds commute since they are operators on di¬erent parts of the product space con-
sisting of the (tensorial) product of the multi-particle space for fermions and bosons,
respectively. The electron ¬eld operates on its Fock space and the bose ¬eld operates
on its multi-particle space.
We note that in the jellium model, the electron“phonon interaction is local just
as in relativistic interactions,10 but here in the context of solid state physics it is
only an approximation to an in general non-local interaction between the electrons
and the ionic charge deformations. Furthermore, in general the phonon ¬eld is not a
scalar ¬eld as a real crystal supports besides longitudinal also transverse vibrations.
The general form of the electron“phonon interaction is

gkk qb c† σ ckσ aqb + a†
Ve’ph = ’qb , (2.77)
k
k,k ,q,b,σ

where c and a are the electron and phonon ¬elds, respectively, and in addition to
the two transverse phonon branches, optical branches can in general be present if
the unit cell of the crystal contains several atoms. Owing to the presence of the
periodic crystal lattice, the momentum is no longer a good quantum number, and
instead states are labeled by the Bloch or so-called crystal wave vector as de¬ned
by the translations respecting the crystal symmetry. The coupling function, gkk qb ,
vanishes unless the crystal wave vector is conserved modulo a reciprocal lattice vector,
k = k + q + K. The new type of interaction processes, corresponding to K = 0,
so-called Umklapp-processes, are the signature of the periodic crystal structure.
The phonons and electrons have dynamics of their own as described by the Hamil-
tonians of Eq. (1.123) and Eq. (2.64), and we have thus arrived at the Hamiltonian
describing electrons and phonons.

Exercise 2.13. Interaction between photons and electrons is obtained by minimal
coupling, P ’ P ’ eA, where the photon ¬eld in the Schr¨dinger picture is speci¬ed
o
by (recall Exercise 1.10 on page 28)

dk
akp eik·x + a† e’ik·x
A(x) = ep (k) , (2.78)
kp
(2π)3 2c|k| p=1,2

where in the transverse gauge the two perpendicular unit polarization vectors, ep (k),
are also perpendicular to the wave vector, k, of the photon.
10 In relativistic quantum theory the form of the interactions can be inferred from the symmetry
properties of the system. In condensed matter physics the interactions typically originate in the
Coulomb interaction; this is the case for the electron“phonon interaction, which originates in the
Coulomb interaction between the electrons and nuclei constituting a piece of material such as a
metal.
48 2. Operators on the multi-particle state space


The total electron“photon Hamiltonian, for the case of non-relativistic electrons,
then becomes
H = Hph + Hel + Hel’ph (2.79)
where
1
(P ’ eA(x))2
Hel + Hel’ph = (2.80)
2m
and P is the total momentum operator for the electrons, Eq. (2.14).
Show that the electron“photon interaction can be written in the form
e2
Hel’ph = ’ dx jp (x) · A(x) + dx n(x) A2 (x) (2.81)
2m
where the current and density operators for the electrons are speci¬ed in Sections 2.3
and 2.2.


2.5 The statistical operator
Up until now, we have described the physical states of a system in terms of state
vectors in the multi-particle state space. A general state vector, |Ψ , can be expanded
on the basis vectors (using for once the resolution of the identity on the multi-particle
state space)

1
|Ψ = p1 ∨ p2 ∨ · · · ∨ pN |Ψ |p1 ∨ p2 ∨ · · · ∨ pN (2.82)
N! p
1 ,...,pN
N =0

or expressed in terms of the vacuum state with the help of our new, so far only
kinematic gadget, the ¬eld operator

c(p1 , . . . , pN ) a† 1 · · · a† N |0 ,
|Ψ = (2.83)
p p
N =0 p1 ,...,pN

where the cs are the complex amplitude coe¬cients specifying the state. Equivalently
the state vector could be expressed in terms of the ¬eld operator in the position
representation.
The description of a system in terms of its wave function is not the generic one
as systems often can not be considered isolated, and instead will be in a mixture of
states described by a statistical operator or density matrix (at a certain moment in
time)
ρ≡ pk |ψk ψk | (2.84)
k
allowing only the statements for the events of the system that it occurs with prob-
ability pk in the quantum states |ψk in the multi-particle state.11 Certainly this is
11 In general a statistical operator is speci¬ed by a set of (normalized) state vectors, |ψ1 ,
|ψ2 , . . . , |ψn , not necessarily orthogonal, and a set of non-negative numbers adding up to one,
n n
i=1 pi |ψi ψi |. Since the statistical operator is hermitian and
i=1 pi = 1, according to ρ =
non-negative, it is always possible to ¬nd an orthonormal set of states |φ1 , |φ2 , . . . , |φN , so that
ρ = N πn |φn φn |, where πn ≥ 0 and N πn = 1.
n=1 n=1
2.5. The statistical operator 49


the generic situation in condensed matter physics and statistical physics in general.
A diagonal element of the statistical operator, ψ|ρ|ψ , thus gives the probability for
the occurrence of the arbitrary state |ψ .
For the evaluation of the average value of a physical quantity represented by the
operator A, a mixture adds an additional purely statistical averaging, as the quantum
average value in a state is weighted by the probability of occurrence of the state

A≡ pk ψk |A|ψk . (2.85)
k

In view of Eq. (2.84), the average value for a mixture can be expressed in terms of
the statistical operator according to

A = Tr(ρ A) (2.86)

where Tr denotes the trace in the multi-particle state, i.e. the sum of all diagonal
elements evaluated in an arbitrary basis, generalizing the matrix element formula for
the average value in a pure state |ψ , A = Tr(|ψ ψ| A).
The statistical operator is seen to be hermitian and positive, ψ| ρ |ψ ≥ 0, and
has unit trace for a normalized probability distribution. The statistical operator is
only idempotent, i.e. a projector, for the case of a pure state, ρ = |ψ ψ|. For a
mixture we have ρ2 = ρ, and Tr ρ2 < 1.
In practice the most important mixture of states will be the one corresponding to
a system in thermal equilibrium at a temperature T (including as a limiting case the
zero temperature situation where the system de¬nitely is in its ground state). In that
case, applying the zeroth law of thermodynamics (that two systems in equilibrium
at temperature T will upon being brought in thermal contact be in equilibrium at
the same temperature) gives, for the thermal equilibrium statistical operator (Boltz-
mann™s constant, the converter between energy and absolute temperature scale, is
denoted k)
1 ’H/kT
ρT = e , (2.87)
Z
where the normalization factor

Z(T, V, Ns ) = Tr e’H/kT = e’F (T,V,Ns )/kT (2.88)

expressing the normalization of the probability distribution of Boltzmann weights,
is the partition function. Here Tr denotes the trace in the multi-particle state space
of the physical system of interest, but since the number of particles is ¬xed there is
only the contributions from the corresponding N -particle subspace

e’H/kT = e’En /kT (2.89)
n

as we are describing the system in the canonical ensemble. As the temperature ap-
proaches absolute zero, the term with minimum energy, the non-degenerate ground
state energy, dominates the sum, and at zero temperature, the average value of a
physical quantity becomes the N -particle ground state average value. The partition
50 2. Operators on the multi-particle state space


function or equivalently the free energy, F , are functions of the macroscopic param-
eters, the temperature, T , and the volume, V , and the number of particles, Ns , of
di¬erent species in the system, and it contains all thermodynamic information.
When a system consists of a huge number of particles, it is more convenient to
perform calculations in the grand canonical ensemble, where instead of the inconve-
nient constrain of a ¬xed number of particles, their chemical potential and average
number of particles are speci¬ed. In the grand canonical ensemble, the system is thus
described in the multi-particle state space. The system is thus imagined coupled to
particle reservoirs, or a subsystem is considered. The system can exchange particles
with the reservoirs which are described by their chemical potentials (assuming in
general several particle species present). This feature is simply included by introduc-
ing Lagrange multipliers, i.e. tacitly understanding that single-particle energies are
measured relative to their chemical potential, H ’ H ’ s μs Ns . In this case, the
partition function instead of being a function of the number of particles, is a function
of the chemical potentials for the species in question

Zgr (T, μs ) = Tr e’(H’μs Ns )/kT = e’©(T,μs )/kT (2.90)

speci¬ed by the average number of particles according to

‚©
Ns = ’ . (2.91)
‚μs T,V

For systems of particles where the total number of particles is not conserved,
i.e. where the Hamiltonian and the total number operator do not commute, such as
for phonons and photons, the chemical potential vanishes, and the grand canonical
ensemble is employed. For degenerate fermions, such as electrons in a metal, the
chemical potential is a huge energy compared to relevant temperatures, viz. tied to
the Fermi energy as discussed in Exercise 2.15.
The average value in the grand canonical ensemble of a physical quantity repre-
sented by the operator A is thus

e(©’En +μN )/kT N, En |A|En , N .
A = Tr(ρ A) = (2.92)
N,n

As the following exercise shows, if alternatively attempted in the canonical ensem-
ble, calculations run smoothly in the grand canonical ensemble free of the constraint
of a ¬xed number of particles. In the thermodynamic limit, using either of course
gives the same results as the ¬‚uctuations in the particle number in the grand canon-
ical ensemble around the mean value then is negligible.

Exercise 2.14. Show that the grand canonical partition function for non-interacting
non-relativistic fermions or bosons of mass m contained in a volume V is given by

p2
ln (1 “ e’( p ’μ)/kT
) = e’©0 /kT
ln Zgr (T, V, μ) = “
(0)
, = (2.93)
p
2m
p
2.5. The statistical operator 51


and the average number of particles are speci¬ed by the Bose“Einstein or Fermi“Dirac
distributions, respectively,
‚©0 1
=’
N = (2.94)
p ’μ)/kT “1
e(
‚μ p
T,V

from which one readily veri¬es that the thermodynamic potential is speci¬ed by the
pressure, P , and volume of the system according to
©0 (T, μ, V ) = ’P V . (2.95)
Exercise 2.15. Show that for a system of non-interacting degenerate fermions, i.e.
at temperatures where kT μ, the chemical potential is pinned to the Fermi energy,
2 2 2/3
F= (3π n) /2m, as
2
T
1’a
μ= , (2.96)
F
TF
where the fermions of mass m are assumed residing in three spatial dimensions (in
which case the constant of order one is a = π 2 /12) with a density n, and TF = F /k
is the Fermi temperature which for a metal, in view of the large density of conduction
electrons, is seen to be huge, typically of the order 104 K.
Exercise 2.16. At zero temperature, a system of fermions such as a metal contains
high-speed electrons, all states below the Fermi energy are fully occupied, a reservoir
for injecting electrons into other conductors. For bosons the opposite, coming to rest,
can happen. First we observe that for non-interacting bosons the chemical potential
can not be positive, μ ¤ 0, as dictated by the Bose“Einstein distribution function
for occupation of energy levels. As the temperature decreases, the chemical potential
increases and becomes vanishingly small at and below the temperature T0 determined
by the density, n, and mass, m, of the bosons (say, spinless and enclosed in a volume
V ) according to

(m)3/2 1/2
N
=√
n= d . (2.97)
’1
/kT0
V e
2 π2 3
0
At this temperature, the lowest energy level, p=0 = 0, starts to be macroscopically
occupied.
Show that at temperatures below T0 , the number of bosons in the lowest level is
(the population of the other levels are governed by the Bose“Einstein distribution)
3/2
T
1’
N0 = N . (2.98)
T0
In the degenerate region at temperatures below T0 , the bosons comes to rest, the
phenomenon of Bose“Einstein condensation (1925), the bosons become ordered in
momentum space. The total condensation at zero temperature is of course a trivial
feature of the quantum statistics of non-interacting bosons. Using the model of non-
interacting bosons to estimate T0 for the case of 4 He gives T0 3.2 K, quite close to
the temperature 2.2 K of the »-transition where liquid Helium becomes a super¬‚uid
(discovered 1928 and proposed a Bose“Einstein condensate by Fritz London, 1938).
52 2. Operators on the multi-particle state space


2.6 Summary
In this chapter we have constructed the operators of relevance on the multi-particle
space, and shown how they are expressed in terms of the quantum ¬elds, the creation
and annihilation ¬elds. The kinematics of a many-body system, its possible quantum
states and the operators representing its physical quantities, is thus expressed in
terms of these two objects. The Hamiltonians on the multi-particle state space were
constructed for the generic types of many-body interactions. We now turn to consider
the dynamics of many-body systems described by their quantum ¬elds on a multi-
particle state space. In particular the quantum dynamics of a quantum ¬eld theory
describing systems out of equilibrium. This will lead us to the study of correlation
functions for quantum ¬elds, the Green™s functions for non-equilibrium states.
3

Quantum dynamics and
Green™s functions

In the previous chapter we studied the kinematics of many-body systems, and the
form of operators representing the physical properties of a system, all of which were
embodied by the quantum ¬eld. In this chapter we shall study the quantum dynamics
of many-body systems, which can also be embodied by the quantum ¬elds. We shall
employ the fact that the quantum dynamics of a system, instead of being described
in terms of the dynamics of the states or the evolution operator, i.e. as previously
done through the Schr¨dinger equation, can instead be carried by the quantum ¬elds.
o
The quantum dynamics is therefore expressed in terms of the correlation functions
or Green™s functions of the quantum ¬elds evaluated with respect to some state of
the system. In particular we shall consider the general case of quantum dynamics
for arbitrary non-equilibrium states. After introducing various types of Green™s func-
tions and relating them to measurable quantities, we will discuss the simpli¬cations
reigning for the special case of equilibrium states.


3.1 Quantum dynamics
Quantum dynamics can be described in di¬erent ways since quantum mechanics is
a linear theory and the dynamics described by a unitary transformation of states.1
This will come in handy in the next chapter when we study a quantum theory in
terms of its perturbative expansion using the so-called interaction picture. Here we
¬rst discuss the Schr¨dinger and Heisenberg pictures.
o
1 This should be contrasted with classical mechanics where dynamics is speci¬ed in terms of
the physical quantities themselves, the generic case being intractable nonlinear partial di¬erential
equations. We shall study such a classical situation in Chapter 12 with the help of methods borrowed
from quantum ¬eld theory, and where in addition the classical system interacts with an environment
as described by a stochastic force.




53
54 3. Quantum dynamics and Green™s functions


3.1.1 The Schr¨dinger picture
o
Having the Hamiltonian on the multi-particle space at hand we can consider the
dynamics described in the multi-particle space. An arbitrary state in the multi-
particle space has at any time in question the expansion on, say the position basis
states

|Ψ(t) dx1 dx2 . . . dxN ψN (x1 , x2 , . . . , xN , t) |x1 3x2 · · · 3xN ,
= (3.1)
N =0


where 3 stands for ∨ or § for the bose or fermi cases respectively. The coe¬cients

ψN (x1 , x2 , . . . , xN , t) = x1 3x2 · · · 3xN |Ψ(t) (3.2)

are the wave functions describing each N -particle system, and they are symmetric
or antisymmetric due to the symmetry properties of the basis states |x1 3x2 · · · 3xN ,
i.e. no new state is produced by using non-symmetric coe¬cients ψN (x1 , x2 , . . . , xN , t).
The dynamics of a multi-particle particle state is speci¬ed by the Schr¨dinger
o
equation in the multi-particle space
d|Ψ(t)
= H(t) |Ψ(t) ,
i (3.3)
dt
where H(t) is the Hamiltonian on the multi-particle space, which can be explic-
itly time dependent due to external forces as our interest will be to consider non-
equilibrium states. In the multi-particle space, the dynamics of all N -particle sys-
tems are described simultaneously since the above equation contains the in¬nite set
of equations, N = 0, 1, 2, . . ., which in the position representation are

‚ψN (x1 , x2 , . . . , xN , t)
i = dx1 dx2 . . . dxM ψM (x1 , x2 , . . . , xM , t)
‚t
M=0


— x1 3x2 · · · 3xN |H(t)|x1 3x2 · · · 3xM . (3.4)

The even or odd character of a wave function is preserved in time as any Hamil-
ˆ ˆ
tonian for identical particles is symmetric in the pi s and xi s as well as other degrees
of freedom (this is the meaning of identity of particles, no interaction can distinguish
them), so if even- or oddness of a wave function is the state of a¬airs at one moment
in time it will stay this way for all times.2
For the case of two-particle interaction, Eq. (2.59), the Hamiltonian has only
nonzero matrix elements between con¬gurations with the same number of particles,
the total number of particles is a conserved quantity, and the in¬nite set of equa-
tions, Eq. (3.4), splits into independent equations describing systems with the de¬nite
number of particles N = 0, 1, 2, . . . .3 For N = 0, the vacuum state, we have for the
2 Time-invariant subspaces other than the symmetric and anti-symmetric ones do not seem to be
physically relevant.
3 For the case of an N -particle system, the multi-particle space is then not needed, we could

discuss it solely in terms of its N -particle state space.
3.1. Quantum dynamics 55


c-number representing its wave function
dψ0 (t)
i = 0|H(t)|0 ψ0 (t) = 0 , (3.5)
dt
where the last equality sign follows from the fact that since the Hamiltonian for two-
particle interaction, Eq. (2.59), operates ¬rst with the annihilation operator on the
vacuum state, it annihilates it. Since Hamiltonians are normal-ordered, this result is
quite general: the vacuum state is without dynamics.
In the case of electron“phonon interaction, the Hamiltonian has o¬-diagonal ele-
ments with respect to the phonon multi-particle subspace. The number of phonons is
owing to interaction with the electrons not conserved; an electron can emit or absorb
phonons just like an electron in an excited state of an atom can emit a photon in the
decay to a lower energy state. The chemical potential of phonons thus vanishes.
Instead of describing the dynamics in terms of the state vector we can introduce
the time development or time evolution operator4
|ψ(t) = U (t, t ) |ψ(t ) (3.6)

connecting the state vectors at the di¬erent times in question where they provide a
complete description of the system. Solving the Schr¨dinger equation, Eq. (3.3), by
o
iteration gives
U (t, t ) = T e’ t dt H(t) ,
t¯ ¯
i
(3.7)
where T denotes time-ordering.5 The time-ordering operation orders a product of
time-dependent operators into its time-descending sequence (displayed here for the
case of three operators)
§
⎨ A(t1 ) B(t2 ) C(t3 ) for t1 > t2 > t3
B(t2 ) A(t1 ) C(t3 ) for t2 > t1 > t3
T (A(t1 ) B(t2 ) C(t3 )) = (3.8)
©
etc. etc.

In case of fermi ¬elds, the time-ordering (and anti-time-ordering which we shortly en-
counter) shall by de¬nition include a product of minus signs, one for each interchange
of fermi ¬elds. Since the Hamiltonian contains an even number of fermi ¬elds, no sta-
tistical factors are thus involved in interchanging Hamiltonians referring to di¬erent
moments in time under the time-ordering symbol.
The Schr¨dinger equation, Eq. (3.3), then gives the equation of motion for the
o
time evolution operator6
‚U (t, t )
= H(t) U (t, t ) .
i (3.9)
‚t
4 It is amazing how compactly quantum dynamics can be captured, encapsulated in the single
object, the evolution operator.
5 For details see, for example, chapter 2 of reference [1].
6 From a mathematical point of view, convergence properties of limiting processes for operator

sequences At are inherited from the topology of the vector space; i.e. convergence is de¬ned by
convergence of an arbitrary vector At |ψ . The dual space of linear operator on the multi-particle
state space can also be equipped with its own topology by introducing the scalar product for
operators A and B, Tr(B † A). But the result of di¬erentiating is gleaned immediately from simple
algebraic properties.
56 3. Quantum dynamics and Green™s functions


We note the semi-group property of the evolution operator

U (t, t ) U (t , t ) = U (t, t ) (3.10)

and the unitarity of the evolution operator, U † (t, t ) = U ’1 (t, t ), as a state vec-
tor has the scalar product with itself of modulus one enforced by the probability
interpretation of the state vector. As a consequence, U † (t, t ) = U (t , t).

Exercise 3.1. Show that

U † (t, t ) ≡ [U (t, t )]† = T e dt H(t)
t ¯¯
˜i , (3.11)
t


˜
where the anti-time-ordering symbol, T , orders the time sequence oppositely as com-
pared with the time-ordering symbol, T , as the adjoint inverts the order of a sequence
of operators. Use this (or the unitarity of the evolution operator, I = U † (t, t ) U (t, t ))
to verify

‚U † (t, t ) ‚U (t, t )
= U † (t, t ) H(t) ,
’i ’i = U (t, t ) H(t ) . (3.12)
‚t ‚t


The dynamics of a mixture is described by the time dependence of the statistical
operator which, according to Eq. (3.6), is

ρ(t) = U (t, t ) ρ(t ) U † (t, t ) (3.13)

and the statistical operator satis¬es the von Neumann equation
dρ(t)
i = [H(t), ρ(t)] . (3.14)
dt
A diagonal element of the statistical operator, ψ|ρ(t)|ψ , gives the probability for
the occurrence of the arbitrary state |ψ at time t (explaining the use of the word
density matrix).
An important set of mixtures in practice for an isolated system (i.e. the Hamilto-
nian is time independent) is the stationary states in which all physical properties are
time independent. The statistical operator is thus for stationary states a function of
the Hamiltonian of the system, ρ = ρ(H).
For an isolated system, the evolution operator takes the simple form

U (t, t ) = e’
i
(t’t )H
. (3.15)

The generator of time displacements is the only operator in the Heisenberg picture
which, in general, is time independent, and the quantity it represents we call the
energy. For an isolated system, the Hamiltonian thus represents the energy.

3.1.2 The Heisenberg picture
Instead of having the dynamics described by an equation of motion for a state vector
or realisticly by a statistical operator, the Schr¨dinger picture discussed above, it
o
3.1. Quantum dynamics 57


is convenient to transfer the dynamics to the physical quantities, resembling in this
feature the dynamics of classical physics. In this so-called Heisenberg picture, the
state of the system

≡ U † (t, tr ) |ψ(t) = |ψ(tr )
|ψH (3.16)

is time independent, according to Eq. (3.3) and Eq. (3.12), whereas the operators
representing physical quantities are time dependent

AH (t) ≡ U † (t, tr ) A U (t, tr ) . (3.17)

At the arbitrary reference time, tr , the two pictures coincide, the evolution operator
satisfying U (t, t) = 1.
We note that if {|a }a is the set of eigenstates of the operator A then the Heisen-
berg operator has the same spectrum but di¬erent eigenstates

|a, t ≡ U † (t, tr ) |a .
AH (t) |a, t = a |a, t , (3.18)

The operator representing a physical quantity in the Heisenberg picture satis¬es
the equation of motion7
‚AH (t)
= [AH (t), HH (t)] ,
i (3.19)
‚t
where

HH (t) ≡ U † (t, tr ) H(t) U (t, tr ) . (3.20)

Introducing the ¬eld in the Heisenberg picture

ψH (x, t) = U † (t, tr ) ψ(x) U (t, tr ) (3.21)

we obtain its equation of motion
‚ψH (x, t)
= [ψH (x, t), H(t)] .
i (3.22)
‚t
Often context allows us to leave out the subscript, writing ψ(x, t) = ψH (x, t).
In the Heisenberg picture, only the equal time anti-commutator or commutator,
for fermions or bosons, respectively, of the ¬elds is in general a simple quantity, of
course the c-number function:

[ψ(x, t), ψ † (x , t)]s = δ(x ’ x ) . (3.23)

At unequal times, the anti-commutator or commutator of the ¬elds are, owing to
interactions, complicated operators whose unravelling will be done in terms of the
correlation functions of the ¬elds, the Green™s functions we introduce in the next
section.
7 Ifthe Schr¨dinger operator is time dependent, such as can be the case for the current operator
o
in the presence of a time-dependent vector potential representing a classical ¬eld, of course an
additional term appears.
58 3. Quantum dynamics and Green™s functions


Exercise 3.2. Show that the probability density for a particle to be at position x
at time t
n(x, t) = Tr(ρ(t) ψ † (x) ψ(x)) (3.24)
can be rewritten in terms of the ¬elds in the Heisenberg picture
n(x, t) = Tr(ρ ψ † (x, t) ψ(x, t)) , (3.25)
where ρ is an arbitrary statistical operator at the reference time where the two
pictures coincide.


For an isolated system, where the Hamiltonian is time independent, the quantum
¬eld (or any other) operator in the Heisenberg picture is related to the operator in
the Schr¨dinger picture in accordance with Eq. (3.17), which in that case becomes
o
(the coincidence with the Schr¨dinger picture is chosen to be at time t = 0)
o

ψ(x) e’
i i
Ht Ht
ψ(x, t) = e . (3.26)

Exercise 3.3. Show that the time evolution of a free ¬eld in the Heisenberg picture
speci¬ed by the free or kinetic energy Hamiltonian in Eq. (2.21), is given by

ap (t) = ap e’
i
pt
, (3.27)
where p = p2 /2m is the kinetic energy of the free particle with momentum p, and
the coincidence with the Schr¨dinger picture is chosen to be at time t = 0.
o
Show the commutation relations for the free ¬elds is
1

e p·(x’x )’ p (t’t ) .
i i
[ψ0 (x, t), ψ0 (x , t )]s = (3.28)
Vp


For the case of an isolated system of particles interacting through an instanta-
neous two-particle interaction, Eq. (2.59), the Hamiltonian transformed according to
Eq. (3.17) can be expressed in terms of the ¬elds in the Heisenberg picture
2
1 ‚

HH (t) = dx ψH (x, t) ψH (x, t)
2m i ‚x

1 † †
dx dx ψH (x, t) ψH (x , t) V (2) (x, x ) ψH (x , t) ψH (x, t) (3.29)
+
2
and according to Eq. (3.19), H(t) = H, i.e. the Hamiltonian in the Heisenberg picture
is the Hamiltonian, representing the energy of the system.
Our interest shall be the case of non-equilibrium situations where a system is
coupled to external classical ¬elds, for example the coupling of current and density
of charged particles to electromagnetic ¬elds as represented by the Hamiltonian
2
1 ‚

’ eA(x, t)
HA,φ (t) = dx ψH (x, t) + eφ(x, t) ψH (x, t) , (3.30)
2m i ‚x
3.1. Quantum dynamics 59


where the quantum ¬elds are in the Heisenberg picture.
Considering the case of two-particle interaction, and using the operator identities
[A, BC] = [A, B]C ’ B[C, A] = {A, B}C ’ B{C, A} (3.31)
for bose or fermi ¬elds, respectively, and their commutation relations, the equation
of motion for the ¬eld in the Heisenberg picture becomes
‚ψ(x, t)
dx V (2) (x, x ) ψ † (x , t)ψ(x , t) ψ(x, t) ,
i = h(t) ψ(x, t) + (3.32)
‚t
where h = h(’i∇x , x, t) is the free single-particle Hamiltonian, which can be time-
dependent due to external classical ¬elds. For example, for the case of a charged
particle coupled to an electromagnetic ¬eld
2
1 ‚
h(’i ∇x , x, t) = ’ eA(x, t) + e•(x, t) . (3.33)
2m i ‚x
The dynamics of a system, speci¬ed by the time dependence of the quantum ¬eld in
the Heisenberg picture, is thus described in terms of higher-order expressions in the
¬eld operators.

Exercise 3.4. Multiply Eq. (3.32) from the left by ψ † (x, t), and obtain also the
adjoint of this construction. Obtain the continuity or charge conservation equation
in the multi-particle space
‚ n(x, t)
+ ∇x · j(x, t) = 0 , (3.34)
‚t
where
n(x, t) = ψ † (x, t) ψ(x, t) (3.35)
and
ψ † (x, t) ∇x ψ(x, t) ’ (∇x ψ † (x, t)) ψ(x, t)
j(x, t) = (3.36)
2mi
are the probability current and density operators on the multi-particle space in the
Heisenberg picture.
Exercise 3.5. Show that the commutation relation for the displacement ¬eld oper-
ator in the Heisenberg picture at equal times is
‚uβ (x , t)
= i δ±β δ(x ’ x )
u± x, t), ni M (3.37)
‚t ’

re¬‚ecting the canonical commutation relations of non-relativistic quantum mechanics
for the position and momentum operators of the ions in a lattice.
Exercise 3.6. Show that the phonon ¬eld in the Heisenberg picture satis¬es the
equal-time commutation relation (neglecting the ultraviolet or Debye cut-o¬, ωD ’
∞)
c2
‚φ(x , t)
= ’i x δ(x ’ x ) .
φ(x, t), (3.38)
‚t 2

60 3. Quantum dynamics and Green™s functions


Exercise 3.7. Show that, for an isolated system of identical particles interacting
through an instantaneous two-body interaction, V (x ’ x ), the ¬eld operator in the
Heisenberg picture, say in the momentum representation, satis¬es the equation of
motion (recall Exercise 2.10 on page 44)
dap (t) dq
dp V (’q) a† +q (t) ap (t) ap+q (t) .
i = p ap (t) + (3.39)
p
(2π )3
dt
Show that the Hamiltonian in the Heisenberg picture can be expressed in the
form
1 dap (t)
† †
H(t) = p ap (t) ap (t) + i ap (t) . (3.40)
2p dt

Exercise 3.8. Obtain the equation of motion for the electron and phonon ¬elds in
the Heisenberg picture for the case of longitudinal electron“phonon interaction.


Any property of a physical system is expressed in terms of a correlation function
of ¬eld operators taken with respect to the state in question. In Section 3.3, we turn
to introduce these, the Green™s functions. But ¬rst, we will take a short historical
detour.


3.2 Second quantization
Quantum ¬eld theory, as presented in the previous chapters, is simply the quantum
mechanics of an arbitrary number of particles. For the non-relativistic case the
practical task was to lift the N -particle description to the multi-particle state space.
Quantum ¬elds are often referred to as second quantization, which in view of our
general introduction of quantum ¬eld theory for many-body systems is of course a
most unfortunate choice of language. The misnomer has its origin in the following
analogy.
Consider the Schr¨dinger equation for a single particle, say in a potential
o
2
‚ψ(x, t) 1 ‚
i = + V (x, t) ψ(x, t) . (3.41)
‚t 2m i ‚x

Next, interpret the equation as a classical ¬eld equation ` la Maxwell™s equations. A
a
di¬erence is, of course, that the ¬eld is complex, and in the case of the electromagnetic
¬eld there are additional ¬eld components. The Schr¨dinger equation, Eq. (3.41),
o
can be derived from the variational principle

δ dt dx L = 0 , (3.42)

where the Lagrange density is
2
‚ψ(x, t)

∇x ψ — (x, t) · ∇x ψ(x, t) ’ ψ — (x, t) V (x, t) ψ(x, t).
L = ψ (x, t) i ’
‚t 2m
(3.43)
3.2. Second quantization 61


The conjugate ¬eld variable is then
‚L
= i ψ — (x, t)
Π(x, t) = (3.44)
‚ ‚ψ(x,t)
‚t

in analogy with the canonical momenta in classical mechanics
‚L
p= (3.45)

‚x
and the variables of the ¬eld, x, is in the analogy equivalent to the labeling, i, of the
mechanical degrees of freedom, and Π(x) corresponds to pi .
Analogous to Hamilton™s function

pi xi ’ L
H= ™ (3.46)
i

enters the Hamilton function for the classical Schr¨dinger ¬eld
o

‚ψ(x, t)
H= ’L
dx dt Π(x, t) . (3.47)
‚t

In analogy with the canonical commutation relations

[pi , xj ] = δij , [xi , xj ] = 0 , [pi , pj ] = 0 (3.48)
i
the quantum ¬eld theory of the corresponding species is then obtained from the
classical Schr¨dinger ¬eld by imposing the quantization relations for the quantum
o
¬elds (not distinguishing them in notation from their classical counterparts)

δ(x ’ x )
[Π(x, t), ψ(x , t)] = (3.49)
i
and
[ψ(x, t), ψ(x , t)] = 0 , [Π(x, t), Π(x , t)] = 0 . (3.50)
Since according to Eq. (3.45), Π(x, t) = i ψ † (x, t), these are the commutation rela-
tions for a bose ¬eld, Eq. (1.101) and Eq. (1.102). The Hamiltonian, Eq. (3.47), is seen
to be identical to the Hamiltonian operator on the multi-particle space, Eq. (2.19).
In this presentation the bose particles emerge as quanta of the ¬eld in analogy to
the quanta of light in the analogous second quantization of the electromagnetic ¬eld
(recall Section 1.4.2). The quantum ¬eld theory of fermions is similarly constructed
as quanta of a ¬eld, but this time anti-commutation relations are assumed for the
¬eld.8
8A practicing quantum ¬eld theorist need thus not carry much baggage, short-cutting the road
by second quantization.
62 3. Quantum dynamics and Green™s functions


3.3 Green™s functions
An exact solution of a quantum ¬eld theory amounts, according to Eq. (3.32), to
knowing all the correlation functions of the ¬eld variables; needless to say a mis-
sion impossible in general. We shall refer to these correlation functions generally as
Green™s functions.9 We shall also use the word propagator interchangeably for the
various types of Green™s functions.
To get an intuitive feeling for the simplest kind of Green™s function, the single-
particle propagator, consider adding at time t1 a particle in state p1 to the arbitrary
state |Ψ(t1 ) , i.e. we obtain the state a† 1 |Ψ(t1 ) , which at time t has evolved to
p
the state

(t) = e’iH(t’t1 ) a† 1 |Ψ(t1 ) = e’iHt a† 1 (t1 ) |ΨH ,
|Ψp1 (3.51)
p p
t1

where in the last equality we have introduced the creation operator and state vector
in the Heisenberg picture (choosing the time of coincidence with the Schr¨dinger
o
picture at time tr = 0). Similarly, we could consider the state where a particle at
time t1 is added in state p1 . The amplitude for the event that the ¬rst constructed
state is revealed in the other state at the arbitrary (and irrelevant) moment in time
t is then

(t) = ΨH |ap1 (t1 ) a† 1 (t1 ) |ΨH
Ψp1 t1 (t)|Ψp1 (3.52)
p
t1

and the single-particle Green™s function is a measure of the persistence, in time span
|t1 ’ t1 |, of the single-particle character of the excitation consisting of adding a
particle to the system (or determining the persistence of a hole state when removing
a particle upon the interchange a ” a† ).

3.3.1 Physical properties and Green™s functions
Physical quantities for a many-body system such as the average (probability) density
of the particles or the average particle (probability) current density are speci¬ed in
terms of one-particle Green™s functions. For a system in an arbitrary state described
by the statistical operator ρ, the average density at a space-time point for a particle
species of interest is (recall the result of Exercise 3.2 on page 58, which amounted to
employing the cyclical property of the trace)


n(x, t) = Tr(ρ ψH (x, σz , t) ψH (x, σz , t)) , (3.53)
σz

where the quantum ¬eld describing the particles ψH (x, σz , t) is in the Heisenberg
picture with respect to the arbitrary Hamiltonian H(t), and Tr denotes the trace in
the multi-particle state space of the physical system in question. The reference time
where the Schr¨dinger and Heisenberg pictures coincide is chosen as the moment
o
where the state is speci¬ed, i.e. when the arbitrary statistical operator, ρ, repre-
senting the state of the system is speci¬ed. Here σz describes an internal degree of
9 Thus
using the notion in a broader sense than in mathematics, where it denotes the fundamental
solution of a linear partial di¬erential equation as discussed in Appendix C.
3.3. Green™s functions 63


freedom of the identical particles in question. For example, in the case of electrons
this is the spin degree of freedom, and the density is the sum of the density of elec-
trons with spin up and down, respectively.10 The average density is expressed in
terms of the diagonal element of the so-called G-lesser Green™s function

n(x, t) = ±i G< (x, σz , t, x, σz , t) , (3.54)
σz

where11

= “i Tr(ρ ψH (x , σz , t ) ψH (x, σz , t))
G< (x, σz , t, x σz , t )


≡ “i ψH (x , σz , t ) ψH (x, σz , t) , (3.55)
where upper (lower) sign corresponds to bosons (fermions), respectively, and we have
introduced the notation that the bracket means trace of the operators in question
weighted with respect to the state of the system, all quantities in the Heisenberg
picture,
. . . ≡ Tr(ρ . . .) . (3.56)
For the case of a pure state, ρ = |Ψ Ψ|, we see that G< (x, t, x , t ) is the amplitude
for the transition at time t to the state ψH (x , σz , t ) |Ψ , where a particle with
spin σz is removed at position x from state |Ψ , given the system at time t is
in the state ψH (x, σz , t) |Ψ where a particle with spin σz is removed at position
x (assuming t < t , otherwise we are dealing with the complex conjugate of the
opposite transition). Equivalently, it is the amplitude to remain in the state |Ψ after
removing at time t a particle with spin σz at position x and restoring at time t a
particle with spin σz at position x . For the case of a mixture, an additional statistical
averaging over the distribution of initial states takes place. Average quantities, such
as the probability density, can thus be expressed in terms of the one-particle Green™s
function.
The average electric current density for an assembly of identical fermions having
charge e in an electric ¬eld represented by the vector potential A is, according to
Eq. (2.47),
e ‚ ‚
’ G< (x, σz , t, x , σz , t)
j(x, t) =
2m ‚x ‚x
σz
x =x


e2
G< (x, σz , t, x, σz , t) ,
+ i A(x, t) (3.57)
m σz

the particles assumed to have an internal degree of freedom, say spin as is the case
for electrons.
10 One can, of course, also encounter situations where interest is in the density of electrons of a

given spin, in which case one studies n(x, σz , t) = Tr(ρ ψH (x, σz , t) ψH (x, σz , t)) .
11 The annoying presence of the imaginary unit is for later convenience with respect to the Feynman

diagram rules. However, one is entitled to the choice of favorite for de¬ning Green™s functions, and
the corresponding adjustment of the list of Feynman rules.
64 3. Quantum dynamics and Green™s functions


From the equation of motion for the ¬eld operator, Eq. (3.32), the equation
of motion for the Green™s function G-lesser becomes, for the case of two-particle
interaction (assuming spin independent interaction so the spin degree of freedom is
suppressed or using inclusive notation),

’ h(t) G< (x, t, x , t ) = dx V (2) (x, x ) G(2) (x, t, x , t, x , t, x , t ) ,
i
‚t
(3.58)
where

G(2) (x, t, x , t, x , t, x , t ) = ±i ψ(x, t) ψ(x , t) ψ † (x , t) ψ † (x , t ) (3.59)

is a so-called two-particle Green™s function since it involves the propagation of two
particles. The dynamics of a system, speci¬ed by the time dependence of the one-
particle Green™s function, is thus described in terms of higher-order correlation func-
tions in the ¬eld operators. The equation of motion for the one-particle Green™s
function thus leads to an in¬nite hierarchy of equations for correlation functions con-
taining ever increasing numbers of ¬eld operators, describing the correlations set up
in the system by the interactions.12 Since there is no closed set of equations for re-
duced quantities such as Green™s functions, approximations are, in practice, needed
in order to obtain information about the system. On some occasions the system pro-
vides a small parameter that allows controlled approximations; a case to be studied
later is that of electron“phonon interaction in metals. In less controllable situations
one in despair appeals to the tendency of higher-order correlations to average out
for a many-particle system, when it comes to such average properties as densities
and currents, so that the hierarchy of correlations can be broken o¬ self-consistently
at low order. We shall discuss such situations in Section 10.6 and in Chapter 12 in
the context of applying the e¬ective action approach to such di¬ering situations as
a trapped Bose“Einstein condensate and classical statistical dynamics, respectively.

3.3.2 Stable of one-particle Green™s functions
The correlation function G-lesser appeared in the previous section most directly
as related to average properties such as densities and currents. However, we shall
encounter various types of quantum ¬eld correlation functions, i.e. various kinds of
Green™s functions that appear for reasons of their own. For de¬niteness we collect
them all here, though they are not needed until later. The rest of this chapter can
thus be skipped on a ¬rst reading if one shares the view that things should not be
called upon before needed.
We shall also encounter the so-called G-greater Green™s function
† †
G> (x, t, x , t ) = ’i ψH (x, t) ψH (x , t ) = ’iTr(ρ ψH (x, t) ψH (x , t )) , (3.60)

the amplitude for the process of an added particle at position x at time t given a
particle is added at position x at time t , the one-particle propagator in the presence
of interaction with all the other particles.
12 Analogous to the BBGKY-hierachy in classical kinetics or for any description of a system in
terms of a reduced, i.e. partially traced out, quantity.
3.3. Green™s functions 65


We shall later, for reasons of calculation in perturbation theory, encounter the
time-ordered Green™s function

G(x, t, x , t ) = ’i T (ψH (x, t) ψH (x , t )) (3.61)

and we note (valid for both bosons and fermions, recalling the minus sign convention
when two fermi ¬elds are interchanged)

G< (x, t, x , t ) t >t
G(x, t, x , t ) = (3.62)
G> (x, t, x , t ) t>t .

In perturbation theory, the time-ordered Green™s function appears because of the
crucial role of time-ordering in the evolution operator, Eq. (3.7). Quantum dynamics
is ruled by operators, non-commuting objects. However, as shown in Chapter 5, the
necessity of the time-ordered Green™s function is only in one version of perturbation
theory, and then an additional analytic continuation needs to be invoked. Or, if
one is interested only in ground state properties, then perturbation theory can be
formulated in closed form involving only the time-ordered Green™s function. The
general real-time perturbation theory valid for non-equilibrium situations will be
formulated in Chapter 5 in terms of essentially two Green™s functions, and in a way
which displays physical information of systems most transparently.
Finally, in this set-up we shall also later encounter the anti-time-ordered Green™s
function

˜ ˜
G(x, t, x , t ) = ’i T (ψH (x, t) ψH (x , t )) , (3.63)

˜
where T anti-time orders, i.e. orders oppositely to that of T . We note that the
time-ordered and anti-time-ordered Green™s functions can be expressed in terms of
G-greater and G-lesser, for example
˜
G(x, t, x , t ) = θ(t ’ t ) G< (x, t, x , t ) + θ(t ’ t) G> (x, t, x , t ) , (3.64)

where θ is the step or Heaviside function.
Recalling Eq. (3.58), we note for the free Green™s functions the relations

G’1 (x, t) G< (x, t, x , t ) = 0 G’1 (x, t) G> (x, t, x , t ) = 0
, (3.65)
0 0
0 0

and for the time-ordered

G’1 (x, t) G0 (x, t, x , t ) = δ(x ’ x ) δ(t ’ t ) (3.66)
0

and anti-time-ordered

G’1 (x, t) G0 (x, t, x , t ) = ’ δ(x ’ x ) δ(t ’ t ) ,
˜ (3.67)
0

where

G’1 (x, t) = ’h ∇x , x, t
i , (3.68)
0
‚t i
66 3. Quantum dynamics and Green™s functions


which for the case of a charged particle coupled to an electromagnetic ¬eld is
2
‚ 1 ‚
G’1 (x, t) ’ ’ eA(x, t) ’ e•(x, t)
= i . (3.69)
0
‚t 2m i ‚x

Introducing
G’1 (x, t, x , t ) = G’1 (x, t) δ(x ’ x ) δ(t ’ t ) (3.70)
0 0

we obtain a quantity on equal footing with the Green™s function, the inverse free
Green™s function (here in the position representation) as

(G’1 — G0 ) (x, t, x , t ) =
ˆ ˆ δ(x ’ x ) δ(t ’ t ) , (3.71)
0

where — signi¬es matrix multiplication in the spatial and time variables, i.e. internal
integrations over space and for the latter internal integration from minus to plus
in¬nity of times.

Exercise 3.9. The equation of motion for the free phonon ¬eld is (recall Section
2.4.3)
2 φ(x, t) = 0 . (3.72)
Show that the time-ordered free phonon Green™s function

D0 (x, t, x , t ) = ’i T (φ(x, t) φ(x , t )) (3.73)

therefore satis¬es the equation of motion

2 D0 (x, t, x , t ) = ’ x ) δ(t ’ t ) .
x δ(x (3.74)
i
Exercise 3.10. From the equation of motion for the ¬eld operator, show that the
equation of motion for the time-ordered Green™s function is


’ h0 (t) G(x, t, x , t ) δ(x ’ x )δ(t ’ t )
i =
‚t

’ i T ([ψ(x, t), Hi (t)] ψ † (x , t )) , (3.75)

where Hi (t) is the interaction part of the Hamiltonian in the Heisenberg picture.


Other combinations of ¬eld correlations will be of importance in Chapter 5 when
the real-time perturbation theory of general non-equilibrium states are considered,
viz. the retarded Green™s function

’iθ(t ’ t ) [ψ(x, t) , ψ † (x , t )]“
GR (x, t, x , t ) =

θ(t ’ t ) G> (x, t, x , t ) ’ G< (x, t, x , t )
= (3.76)
3.3. Green™s functions 67


and advanced Green™s functions

iθ(t ’ t) [ψ(x, t) , ψ † (x , t )]“
GA (x, t, x , t ) =

’θ(t ’ t) G> (x, t, x , t ) ’ G< (x, t, x , t )
= (3.77)

and the Keldysh or kinetic Green™s function

= ’i [ψ(x, t) , ψ † (x , t )]±
GK (x, t, x , t )

= G> (x, t, x , t ) + G< (x, t, x , t ) , (3.78)

where upper and lower signs, as usual, are for bose and fermi ¬elds, respectively.
Introducing the notation s = ’s, the two kinds of statistics can be combined leaving
¯
the Green™s functions in the forms

’iθ(t ’ t ) [ψ(x, t) , ψ † (x , t )]s
GR (x, t, x , t ) = (3.79)
¯

and

GA (x, t, x , t ) = iθ(t ’ t) [ψ(x, t) , ψ † (x , t )]s (3.80)
¯

and

’i [ψ(x, t) , ψ † (x , t )]s = GS (x, t, x , t )
GK (x, t, x , t ) = (3.81)

where the superscript on the last Green™s function also could remind us of it being
symmetric with respect to the quantum statistics.

Exercise 3.11. Show that the density, up to a state independent constant, can be
expressed in terms of the kinetic Green™s function according to

n(x, t) = ± i GK (x, σz , t, x, σz , t) . (3.82)
σz

Exercise 3.12. Show that the current density can be expressed in terms of the
kinetic Green™s function according to (in the absence of a vector potential)

e ‚ ‚
’ GK (x, t, x , t)
j(x, t) = . (3.83)
2m ‚x ‚x
x =x

The presence of a vector potential just adds the diamagnetic term (recall Eq. (3.57))
in accordance with gauge invariance, ’i ∇ ’ ’i ∇ ’ eA.


We note the relationship

GR (x, t, x , t ) ’ GA (x, t, x , t ) = G> (x, t, x , t ) ’ G< (x, t, x , t ) (3.84)
68 3. Quantum dynamics and Green™s functions


irrespective of the quantum statistics of the particles. The above combination is of
such importance that we introduce the additional notation for the spectral weight
function

= i(GR (x, t, x , t ) ’ GA (x, t, x , t )) = [ψ(x, t) , ψ † (x , t )]“
A(x, t, x , t )

= i G> (x, t, x , t ) ’ G< (x, t, x , t ) . (3.85)

We note as a consequence of the equal time anti-commutation or commutation rela-
tions of the ¬eld operators, that the spectral function at equal times satis¬es

δ(x ’ x )
A(x, t, x , t) = (3.86)

irrespective of the state of the system.

Exercise 3.13. Introduce the mixed or Wigner coordinates13
x+x
r= x’x
R= , (3.87)
2
and
t+t
, t =t’t .
T= (3.88)
2
Show that the spectral weight function expressed in these variables satis¬es the
sum-rule

dE
A(E, p, R, T ) = 1 . (3.89)

’∞

Exercise 3.14. Verify the relations, valid for both bosons and fermions,

GA (x, t, x , t ) = GR (x , t , x, t) (3.90)

and

GK (x, t, x , t ) = ’ GK (x , t , x, t) (3.91)

and

A(x, t, x , t ) = (A(x , t , x, t)) (3.92)

and

G< (x, t, x , t ) = ’ G< (x , t , x, t) (3.93)

and

G> (x, t, x , t ) = ’ G> (x , t , x, t) . (3.94)

Note the relations are valid for arbitrary states.
13 There will be more about Wigner coordinates in Section 7.2.
3.3. Green™s functions 69


For the case of a hermitian bose ¬eld, such as the phonon ¬eld, additional useful
relations exist

DR (x, t, x , t ) = DA (x , t , x, t) (3.95)

and

DK (x, t, x , t ) = DK (x , t , x, t) (3.96)

and

D> (x, t, x , t ) = D< (x , t , x, t) . (3.97)

We thus have that DR(A) (x, t, x , t ) are real functions, whereas DK (x, t, x , t ) is
purely imaginary.
Above, the Green™s function are displayed in terms of the ¬elds in the position
representation. Equally, we can introduce the Green™s function displayed in the
momentum representation, related to the above by Fourier transformation, or for
that matter in any representation speci¬ed by a complete set of states, say the energy
representation speci¬ed in terms of the eigenstates of the Hamiltonian.
Correlation functions of the quantum ¬elds can be obtained by di¬erentiation of
a generating functional. For example, to generate time-ordered Green™s functions we
introduce

+ ψ † (x,t) · — (x,t))
Z[·, · — ] = T ei dx dt (ψ(x,t) ·(x,t)
(3.98)
’∞




generating for example the time-ordered Green™s function for bosons, Eq. (3.61), by
di¬erentiating twice with respect to the complex c-number source function ·,14 to
give

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