ńņš. 3 |

Ć— |x1 ā§ x2 ā§ Ā· Ā· Ā· ( no xn and xm ) Ā· Ā· Ā· ā§ xN , (2.60)

where Īø denotes the step function. The second statistics exponent factor is m if

m > n and of the usual form m ā’ 1 if m < n, simply adjusting to when operating

with the second annihilation operator the labeling of the state vector diļ¬ers from the

one used in the deļ¬nition Eq. (1.72). Then by operating with creation operators we

get

Ļ ā (x) Ļ ā (x ) Ļ(x ) Ļ(x) |x1 ā§ Ā· Ā· Ā· ā§ xN Ī“(x ā’ xn )Ī“(x ā’ xm )

=

m(=n)

Ć— (ā’1)nā’1 (ā’1)mā’Īø(nā’m) |x ā§ x ā§ x1 ā§ x2 ā§ Ā· Ā· Ā· ( no xn and xm ) Ā· Ā· Ā· ā§ xN

Ī“(x ā’ xn ) Ī“(x ā’ xm ) |x1 ā§ x2 ā§ x3 ā§ Ā· Ā· Ā· ā§ 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

diļ¬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 inļ¬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

ā‚2

1

ā

H = dx Ļ (x) Ļ(x)

ā‚x2

2m i

1

dx dx Ļ ā (x) Ļ ā (x ) V (2) (x, x ) Ļ(x ) Ļ(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 ā’ x ) , (2.65)

we have in the momentum representation for the operator on multi-particle space

1 dq

dp dp V (ā’q) aā aā +q ap ap ,

V = (2.66)

pā’q p

(2Ļ )3

2

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

dx eā’ xĀ·q

i

V (q) = V (x) . (2.67)

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

1 dq

dp dp V (q) aā aā ā’q ap ap .

V = (2.68)

p+q p

(2Ļ )3

2

If the particles possess spin and their two-body interaction is spin dependent, the

interaction in the multi-particle space becomes

1 ā

ā

V= dx dx ĻĪ± (x) ĻĪ² (x ) VĪ±Ī± ,Ī²Ī² (x, x ) ĻĪ² (x ) ĻĪ± (x) , (2.69)

2

Ī±Ī± ,Ī²Ī²

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 ļ¬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 ā

1

aā

V = a a a , (2.70)

2 p+q,Ļ p ā’q,Ļ p ,Ļ p,Ļ

2V 0q

q=0,p,p ,ĻĻ

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

Coulomb interaction.

2.4.2 Fermionā“boson interaction

In relativistic quantum theory the creation and annihilation operators, the quan-

tum ļ¬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 superļ¬‚uid 3 He, and for describing Boseā“Einstein condensates of

bosons.

The generic interaction between fermions and bosons is of the form

Hbā’f = g dx Ļ ā (x) Ļ(x) Ļ(x) , (2.71)

where Ļ(x) is the fermi ļ¬eld and Ļ(x) is the real (hermitian) bose ļ¬eld, and the

interaction is characterized by some coupling constant g, and possibly dressed up in

some indices characteristic for the ļ¬elds in question, such as Minkowski and spinor

in the case of QED.5 The fermi and bose ļ¬elds commute since they operate on

their respective multi-particle spaces making up the total product multi-particle state

space.6 For the fermionā“boson interaction which shall be of interest in the following,

viz. the electronā“phonon interaction, Eq. ( 2.71) is also a relevant form.

2.4.3 Electronā“phonon interaction

Of importance later is the interaction between electrons and the quantized lattice

vibrations in, say, a metal or semiconductor, the electronā“phonon interaction. For

illustration it suļ¬ces to consider the jellium model where the electrons couple only to

longitudinal compressional charge conļ¬gurations of the lattice ions, the longitudinal

phonons. A deformation of the ionic charge distribution in a piece of matter, will

create an eļ¬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

ļ¬elds are taken to anti-commute.

46 2. Operators on the multi-particle state space

is given by the deformation potential7

n

āxe Ā· u(xe ) ,

V (xe ) = (2.72)

2N0

where u is the displacement ļ¬eld of the background ionic charge, N0 is the density of

electron states at the Fermi energy per spin (in three dimensions N0 = mpF /2Ļ 2 3 ),

and n is the electron density. The quantized lattice dynamics leads to the electronā“

phonon interaction in the jellium model becoming (recall Eq. (1.131))

ā

n i

Ļk [Ėk eikĀ·xe ā’ aā eā’ikĀ·xe ] (2.73)

Ė āxe Ā· u(xe ) =

Ė

Veā’ph (xe ) = a Ėk

2N0 2 N0 V

|k|ā¤kD

where the harmonic oscillator creation and annihilation operators satisfy the commu-

tation relations, [Ėk , aā ] = Ī“k,k , and describe the weakly perturbed collective ionic

a Ėk

oscillations (recall Sections 1.4.1 and 1.4.2). We assume a ļ¬nite lattice of volume V .

The set of harmonic oscillators is in its multi-particle description thus speciļ¬ed by

the phonon ļ¬eld operator

ā

Ļk [ak eikĀ·x ā’ aā eā’ikĀ·x ] , (2.74)

ā” M ni āx Ā· u(x) = i

Ļ(x) c k

2V

|k|ā¤kD

which is a real scalar bose ļ¬eld whose quanta, the phonons, are equivalent to bose

particles, the bose ļ¬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 ļ¬eld, Eq. (2.74), is a real or hermitian ļ¬eld,

Ļā (x) = Ļ(x), and contains a sum of creation and annihilation operators. Except for

the explicit upper (ultraviolet) cut oļ¬, imposed by the ļ¬nite lattice constant, it is

thus analogous to the ļ¬eld describing a spin zero particle.

The electronā“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 ļ¬eld and

the electron density reļ¬‚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

Veā’ph = g dx ne (x) Ļ(x) = g dx Ļ ā (x) Ļ(x) Ļ(x) , (2.75)

7 The

electronā“phonon interaction is an eļ¬ective collective description of the underlying screened

electronā“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 eļ¬ective Coulomb electronā“electron interaction describes the interaction between electrons

and their screening clouds. Such objects are referred to as quasi-particles.

9 That the electronā“phonon interaction takes this form is the reason for introducing the phonon

ļ¬eld, Eq. (2.74), instead of using the displacement ļ¬eld.

2.4. Interactions 47

where the electronā“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

ļ¬elds commute since they are operators on diļ¬erent parts of the product space con-

sisting of the (tensorial) product of the multi-particle space for fermions and bosons,

respectively. The electron ļ¬eld operates on its Fock space and the bose ļ¬eld operates

on its multi-particle space.

We note that in the jellium model, the electronā“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 ļ¬eld is not a

scalar ļ¬eld as a real crystal supports besides longitudinal also transverse vibrations.

The general form of the electronā“phonon interaction is

gkk qb cā Ļ ckĻ aqb + aā

Veā’ph = ā’qb , (2.77)

k

k,k ,q,b,Ļ

where c and a are the electron and phonon ļ¬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 deļ¬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 ā’ P ā’ eA, where the photon ļ¬eld in the SchrĀØdinger picture is speciļ¬ed

o

by (recall Exercise 1.10 on page 28)

dk

akp eikĀ·x + aā eā’ikĀ·x

A(x) = ep (k) , (2.78)

kp

(2Ļ)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 electronā“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 electronā“photon Hamiltonian, for the case of non-relativistic electrons,

then becomes

H = Hph + Hel + Helā’ph (2.79)

where

1

(P ā’ eA(x))2

Hel + Helā’ph = (2.80)

2m

and P is the total momentum operator for the electrons, Eq. (2.14).

Show that the electronā“photon interaction can be written in the form

e2

Helā’ph = ā’ dx jp (x) Ā· A(x) + dx n(x) A2 (x) (2.81)

2m

where the current and density operators for the electrons are speciļ¬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, |ĪØ , can be expanded

on the basis vectors (using for once the resolution of the identity on the multi-particle

state space)

ā

1

|ĪØ = p1 āØ p2 āØ Ā· Ā· Ā· āØ pN |ĪØ |p1 āØ p2 āØ Ā· Ā· Ā· āØ 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 ļ¬eld operator

ā

c(p1 , . . . , pN ) aā 1 Ā· Ā· Ā· aā N |0 ,

|ĪØ = (2.83)

p p

N =0 p1 ,...,pN

where the cs are the complex amplitude coeļ¬cients specifying the state. Equivalently

the state vector could be expressed in terms of the ļ¬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)

Ļā” pk |Ļk Ļk | (2.84)

k

allowing only the statements for the events of the system that it occurs with prob-

ability pk in the quantum states |Ļk in the multi-particle state.11 Certainly this is

11 In general a statistical operator is speciļ¬ed by a set of (normalized) state vectors, |Ļ1 ,

|Ļ2 , . . . , |Ļn , not necessarily orthogonal, and a set of non-negative numbers adding up to one,

n n

i=1 pi |Ļi Ļi |. Since the statistical operator is hermitian and

i=1 pi = 1, according to Ļ =

non-negative, it is always possible to ļ¬nd an orthonormal set of states |Ļ1 , |Ļ2 , . . . , |ĻN , so that

Ļ = N Ļn |Ļn Ļn |, where Ļn ā„ 0 and N Ļn = 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, Ļ|Ļ|Ļ , thus gives the probability for

the occurrence of the arbitrary state |Ļ .

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

Aā” pk Ļk |A|Ļk . (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(Ļ 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 |Ļ , A = Tr(|Ļ Ļ| A).

The statistical operator is seen to be hermitian and positive, Ļ| Ļ |Ļ ā„ 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, Ļ = |Ļ Ļ|. For a

mixture we have Ļ2 = Ļ, and Tr Ļ2 < 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 deļ¬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-

mannā™s constant, the converter between energy and absolute temperature scale, is

denoted k)

1 ā’H/kT

ĻT = e , (2.87)

Z

where the normalization factor

Z(T, V, Ns ) = Tr eā’H/kT = eā’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 ļ¬xed there is

only the contributions from the corresponding N -particle subspace

eā’H/kT = eā’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

diļ¬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 ļ¬xed number of particles, their chemical potential and average

number of particles are speciļ¬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 ā’ H ā’ s Ī¼s 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, Ī¼s ) = Tr eā’(Hā’Ī¼s Ns )/kT = eā’Ī©(T,Ī¼s )/kT (2.90)

speciļ¬ed by the average number of particles according to

ā‚Ī©

Ns = ā’ . (2.91)

ā‚Ī¼s 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(Ī©ā’En +Ī¼N )/kT N, En |A|En , N .

A = Tr(Ļ 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 ļ¬xed number of particles. In the thermodynamic limit, using either of course

gives the same results as the ļ¬‚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 ā“ eā’( p ā’Ī¼)/kT

) = eā’Ī©0 /kT

ln Zgr (T, V, Ī¼) = ā“

(0)

, = (2.93)

p

2m

p

2.5. The statistical operator 51

and the average number of particles are speciļ¬ed by the Boseā“Einstein or Fermiā“Dirac

distributions, respectively,

ā‚Ī©0 1

=ā’

N = (2.94)

p ā’Ī¼)/kT ā“1

e(

ā‚Ī¼ p

T,V

from which one readily veriļ¬es that the thermodynamic potential is speciļ¬ed by the

pressure, P , and volume of the system according to

Ī©0 (T, Ī¼, V ) = ā’P V . (2.95)

Exercise 2.15. Show that for a system of non-interacting degenerate fermions, i.e.

at temperatures where kT Ī¼, the chemical potential is pinned to the Fermi energy,

2 2 2/3

F= (3Ļ n) /2m, as

2

T

1ā’a

Ī¼= , (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 = Ļ 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, Ī¼ ā¤ 0, as dictated by the Boseā“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

ā

(m)3/2 1/2

N

=ā

n= d . (2.97)

ā’1

/kT0

V e

2 Ļ2 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 Boseā“Einstein distribution)

3/2

T

1ā’

N0 = N . (2.98)

T0

In the degenerate region at temperatures below T0 , the bosons comes to rest, the

phenomenon of Boseā“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 Ī»-transition where liquid Helium becomes a superļ¬‚uid

(discovered 1928 and proposed a Boseā“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 ļ¬elds, the creation

and annihilation ļ¬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 ļ¬elds on a multi-

particle state space. In particular the quantum dynamics of a quantum ļ¬eld theory

describing systems out of equilibrium. This will lead us to the study of correlation

functions for quantum ļ¬elds, the Greenā™s functions for non-equilibrium states.

3

Quantum dynamics and

Greenā™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 ļ¬eld. In this chapter we shall study the quantum dynamics

of many-body systems, which can also be embodied by the quantum ļ¬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 SchrĀØdinger equation, can instead be carried by the quantum ļ¬elds.

o

The quantum dynamics is therefore expressed in terms of the correlation functions

or Greenā™s functions of the quantum ļ¬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 Greenā™s func-

tions and relating them to measurable quantities, we will discuss the simpliļ¬cations

reigning for the special case of equilibrium states.

3.1 Quantum dynamics

Quantum dynamics can be described in diļ¬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

ļ¬rst discuss the SchrĀØdinger and Heisenberg pictures.

o

1 This should be contrasted with classical mechanics where dynamics is speciļ¬ed in terms of

the physical quantities themselves, the generic case being intractable nonlinear partial diļ¬erential

equations. We shall study such a classical situation in Chapter 12 with the help of methods borrowed

from quantum ļ¬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 Greenā™s functions

3.1.1 The SchrĀØdinger 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

ā

|ĪØ(t) dx1 dx2 . . . dxN ĻN (x1 , x2 , . . . , xN , t) |x1 3x2 Ā· Ā· Ā· 3xN ,

= (3.1)

N =0

where 3 stands for āØ or ā§ for the bose or fermi cases respectively. The coeļ¬cients

ĻN (x1 , x2 , . . . , xN , t) = x1 3x2 Ā· Ā· Ā· 3xN |ĪØ(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 Ā· Ā· Ā· 3xN ,

i.e. no new state is produced by using non-symmetric coeļ¬cients ĻN (x1 , x2 , . . . , xN , t).

The dynamics of a multi-particle particle state is speciļ¬ed by the SchrĀØdinger

o

equation in the multi-particle space

d|ĪØ(t)

= H(t) |ĪØ(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 inļ¬nite set

of equations, N = 0, 1, 2, . . ., which in the position representation are

ā

ā‚ĻN (x1 , x2 , . . . , xN , t)

i = dx1 dx2 . . . dxM ĻM (x1 , x2 , . . . , xM , t)

ā‚t

M=0

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

The even or odd character of a wave function is preserved in time as any Hamil-

Ė Ė

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 aļ¬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 conļ¬gurations with the same number of particles,

the total number of particles is a conserved quantity, and the inļ¬nite set of equa-

tions, Eq. (3.4), splits into independent equations describing systems with the deļ¬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

dĻ0 (t)

i = 0|H(t)|0 Ļ0 (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 ļ¬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 electronā“phonon interaction, the Hamiltonian has oļ¬-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

|Ļ(t) = U (t, t ) |Ļ(t ) (3.6)

connecting the state vectors at the diļ¬erent times in question where they provide a

complete description of the system. Solving the SchrĀØdinger equation, Eq. (3.3), by

o

iteration gives

U (t, t ) = T eā’ t dt H(t) ,

tĀÆ ĀÆ

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)

ā§

āØ 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)

ā©

etc. etc.

In case of fermi ļ¬elds, the time-ordering (and anti-time-ordering which we shortly en-

counter) shall by deļ¬nition include a product of minus signs, one for each interchange

of fermi ļ¬elds. Since the Hamiltonian contains an even number of fermi ļ¬elds, no sta-

tistical factors are thus involved in interchanging Hamiltonians referring to diļ¬erent

moments in time under the time-ordering symbol.

The SchrĀØdinger equation, Eq. (3.3), then gives the equation of motion for the

o

time evolution operator6

ā‚U (t, t )

= H(t) U (t, t ) .

i (3.9)

ā‚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 deļ¬ned by

convergence of an arbitrary vector At |Ļ . 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 ā A). But the result of diļ¬erentiating is gleaned immediately from simple

algebraic properties.

56 3. Quantum dynamics and Greenā™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 ā (t, t ) = U ā’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 ā (t, t ) = U (t , t).

Exercise 3.1. Show that

U ā (t, t ) ā” [U (t, t )]ā = T e dt H(t)

t ĀÆĀÆ

Ėi , (3.11)

t

Ė

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 ā (t, t ) U (t, t ))

to verify

ā‚U ā (t, t ) ā‚U (t, t )

= U ā (t, t ) H(t) ,

ā’i ā’i = U (t, t ) H(t ) . (3.12)

ā‚t ā‚t

The dynamics of a mixture is described by the time dependence of the statistical

operator which, according to Eq. (3.6), is

Ļ(t) = U (t, t ) Ļ(t ) U ā (t, t ) (3.13)

and the statistical operator satisļ¬es the von Neumann equation

dĻ(t)

i = [H(t), Ļ(t)] . (3.14)

dt

A diagonal element of the statistical operator, Ļ|Ļ(t)|Ļ , gives the probability for

the occurrence of the arbitrary state |Ļ 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, Ļ = Ļ(H).

For an isolated system, the evolution operator takes the simple form

U (t, t ) = eā’

i

(tā’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 SchrĀØdinger 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

ā” U ā (t, tr ) |Ļ(t) = |Ļ(tr )

|ĻH (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) ā” U ā (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 diļ¬erent eigenstates

|a, t ā” U ā (t, tr ) |a .

AH (t) |a, t = a |a, t , (3.18)

The operator representing a physical quantity in the Heisenberg picture satisļ¬es

the equation of motion7

ā‚AH (t)

= [AH (t), HH (t)] ,

i (3.19)

ā‚t

where

HH (t) ā” U ā (t, tr ) H(t) U (t, tr ) . (3.20)

Introducing the ļ¬eld in the Heisenberg picture

ĻH (x, t) = U ā (t, tr ) Ļ(x) U (t, tr ) (3.21)

we obtain its equation of motion

ā‚ĻH (x, t)

= [ĻH (x, t), H(t)] .

i (3.22)

ā‚t

Often context allows us to leave out the subscript, writing Ļ(x, t) = ĻH (x, t).

In the Heisenberg picture, only the equal time anti-commutator or commutator,

for fermions or bosons, respectively, of the ļ¬elds is in general a simple quantity, of

course the c-number function:

[Ļ(x, t), Ļ ā (x , t)]s = Ī“(x ā’ x ) . (3.23)

At unequal times, the anti-commutator or commutator of the ļ¬elds are, owing to

interactions, complicated operators whose unravelling will be done in terms of the

correlation functions of the ļ¬elds, the Greenā™s functions we introduce in the next

section.

7 Ifthe SchrĀØdinger 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 ļ¬eld, of course an

additional term appears.

58 3. Quantum dynamics and Greenā™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(Ļ(t) Ļ ā (x) Ļ(x)) (3.24)

can be rewritten in terms of the ļ¬elds in the Heisenberg picture

n(x, t) = Tr(Ļ Ļ ā (x, t) Ļ(x, t)) , (3.25)

where Ļ 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

ļ¬eld (or any other) operator in the Heisenberg picture is related to the operator in

the SchrĀØdinger picture in accordance with Eq. (3.17), which in that case becomes

o

(the coincidence with the SchrĀØdinger picture is chosen to be at time t = 0)

o

Ļ(x) eā’

i i

Ht Ht

Ļ(x, t) = e . (3.26)

Exercise 3.3. Show that the time evolution of a free ļ¬eld in the Heisenberg picture

speciļ¬ed by the free or kinetic energy Hamiltonian in Eq. (2.21), is given by

ap (t) = ap eā’

i

pt

, (3.27)

where p = p2 /2m is the kinetic energy of the free particle with momentum p, and

the coincidence with the SchrĀØdinger picture is chosen to be at time t = 0.

o

Show the commutation relations for the free ļ¬elds is

1

ā

e pĀ·(xā’x )ā’ p (tā’t ) .

i i

[Ļ0 (x, t), Ļ0 (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 ļ¬elds in the Heisenberg picture

2

1 ā‚

ā

HH (t) = dx ĻH (x, t) ĻH (x, t)

2m i ā‚x

1 ā ā

dx dx ĻH (x, t) ĻH (x , t) V (2) (x, x ) ĻH (x , t) ĻH (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 ļ¬elds, for example the coupling of current and density

of charged particles to electromagnetic ļ¬elds as represented by the Hamiltonian

2

1 ā‚

ā

ā’ eA(x, t)

HA,Ļ (t) = dx ĻH (x, t) + eĻ(x, t) ĻH (x, t) , (3.30)

2m i ā‚x

3.1. Quantum dynamics 59

where the quantum ļ¬elds are in the Heisenberg picture.

Considering the case of two-particle interaction, and using the operator identities

[A, BC] = [A, B]C ā’ B[C, A] = {A, B}C ā’ B{C, A} (3.31)

for bose or fermi ļ¬elds, respectively, and their commutation relations, the equation

of motion for the ļ¬eld in the Heisenberg picture becomes

ā‚Ļ(x, t)

dx V (2) (x, x ) Ļ ā (x , t)Ļ(x , t) Ļ(x, t) ,

i = h(t) Ļ(x, t) + (3.32)

ā‚t

where h = h(ā’iāx , x, t) is the free single-particle Hamiltonian, which can be time-

dependent due to external classical ļ¬elds. For example, for the case of a charged

particle coupled to an electromagnetic ļ¬eld

2

1 ā‚

h(ā’i āx , x, t) = ā’ eA(x, t) + eĻ•(x, t) . (3.33)

2m i ā‚x

The dynamics of a system, speciļ¬ed by the time dependence of the quantum ļ¬eld in

the Heisenberg picture, is thus described in terms of higher-order expressions in the

ļ¬eld operators.

Exercise 3.4. Multiply Eq. (3.32) from the left by Ļ ā (x, t), and obtain also the

adjoint of this construction. Obtain the continuity or charge conservation equation

in the multi-particle space

ā‚ n(x, t)

+ āx Ā· j(x, t) = 0 , (3.34)

ā‚t

where

n(x, t) = Ļ ā (x, t) Ļ(x, t) (3.35)

and

Ļ ā (x, t) āx Ļ(x, t) ā’ (āx Ļ ā (x, t)) Ļ(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 ļ¬eld oper-

ator in the Heisenberg picture at equal times is

ā‚uĪ² (x , t)

= i Ī“Ī±Ī² Ī“(x ā’ x )

uĪ± x, t), ni M (3.37)

ā‚t ā’

reļ¬‚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 ļ¬eld in the Heisenberg picture satisļ¬es the

equal-time commutation relation (neglecting the ultraviolet or Debye cut-oļ¬, ĻD ā’

ā)

c2

ā‚Ļ(x , t)

= ā’i x Ī“(x ā’ x ) .

Ļ(x, t), (3.38)

ā‚t 2

ā’

60 3. Quantum dynamics and Greenā™s functions

Exercise 3.7. Show that, for an isolated system of identical particles interacting

through an instantaneous two-body interaction, V (x ā’ x ), the ļ¬eld operator in the

Heisenberg picture, say in the momentum representation, satisļ¬es the equation of

motion (recall Exercise 2.10 on page 44)

dap (t) dq

dp V (ā’q) aā +q (t) ap (t) ap+q (t) .

i = p ap (t) + (3.39)

p

(2Ļ )3

dt

Show that the Hamiltonian in the Heisenberg picture can be expressed in the

form

1 dap (t)

ā ā

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 ļ¬elds in

the Heisenberg picture for the case of longitudinal electronā“phonon interaction.

Any property of a physical system is expressed in terms of a correlation function

of ļ¬eld operators taken with respect to the state in question. In Section 3.3, we turn

to introduce these, the Greenā™s functions. But ļ¬rst, we will take a short historical

detour.

3.2 Second quantization

Quantum ļ¬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 ļ¬elds are often referred to as second quantization, which in view of our

general introduction of quantum ļ¬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 SchrĀØdinger equation for a single particle, say in a potential

o

2

ā‚Ļ(x, t) 1 ā‚

i = + V (x, t) Ļ(x, t) . (3.41)

ā‚t 2m i ā‚x

Next, interpret the equation as a classical ļ¬eld equation ` la Maxwellā™s equations. A

a

diļ¬erence is, of course, that the ļ¬eld is complex, and in the case of the electromagnetic

ļ¬eld there are additional ļ¬eld components. The SchrĀØdinger equation, Eq. (3.41),

o

can be derived from the variational principle

Ī“ dt dx L = 0 , (3.42)

where the Lagrange density is

2

ā‚Ļ(x, t)

ā—

āx Ļ ā— (x, t) Ā· āx Ļ(x, t) ā’ Ļ ā— (x, t) V (x, t) Ļ(x, t).

L = Ļ (x, t) i ā’

ā‚t 2m

(3.43)

3.2. Second quantization 61

The conjugate ļ¬eld variable is then

ā‚L

= i Ļ ā— (x, t)

Ī (x, t) = (3.44)

ā‚ ā‚Ļ(x,t)

ā‚t

in analogy with the canonical momenta in classical mechanics

ā‚L

p= (3.45)

Ė™

ā‚x

and the variables of the ļ¬eld, x, is in the analogy equivalent to the labeling, i, of the

mechanical degrees of freedom, and Ī (x) corresponds to pi .

Analogous to Hamiltonā™s function

pi xi ā’ L

H= Ė™ (3.46)

i

enters the Hamilton function for the classical SchrĀØdinger ļ¬eld

o

ā‚Ļ(x, t)

H= ā’L

dx dt Ī (x, t) . (3.47)

ā‚t

In analogy with the canonical commutation relations

[pi , xj ] = Ī“ij , [xi , xj ] = 0 , [pi , pj ] = 0 (3.48)

i

the quantum ļ¬eld theory of the corresponding species is then obtained from the

classical SchrĀØdinger ļ¬eld by imposing the quantization relations for the quantum

o

ļ¬elds (not distinguishing them in notation from their classical counterparts)

Ī“(x ā’ x )

[Ī (x, t), Ļ(x , t)] = (3.49)

i

and

[Ļ(x, t), Ļ(x , t)] = 0 , [Ī (x, t), Ī (x , t)] = 0 . (3.50)

Since according to Eq. (3.45), Ī (x, t) = i Ļ ā (x, t), these are the commutation rela-

tions for a bose ļ¬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 ļ¬eld in analogy to

the quanta of light in the analogous second quantization of the electromagnetic ļ¬eld

(recall Section 1.4.2). The quantum ļ¬eld theory of fermions is similarly constructed

as quanta of a ļ¬eld, but this time anti-commutation relations are assumed for the

ļ¬eld.8

8A practicing quantum ļ¬eld theorist need thus not carry much baggage, short-cutting the road

by second quantization.

62 3. Quantum dynamics and Greenā™s functions

3.3 Greenā™s functions

An exact solution of a quantum ļ¬eld theory amounts, according to Eq. (3.32), to

knowing all the correlation functions of the ļ¬eld variables; needless to say a mis-

sion impossible in general. We shall refer to these correlation functions generally as

Greenā™s functions.9 We shall also use the word propagator interchangeably for the

various types of Greenā™s functions.

To get an intuitive feeling for the simplest kind of Greenā™s function, the single-

particle propagator, consider adding at time t1 a particle in state p1 to the arbitrary

state |ĪØ(t1 ) , i.e. we obtain the state aā 1 |ĪØ(t1 ) , which at time t has evolved to

p

the state

(t) = eā’iH(tā’t1 ) aā 1 |ĪØ(t1 ) = eā’iHt aā 1 (t1 ) |ĪØH ,

|ĪØp1 (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 SchrĀØdinger

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 ļ¬rst constructed

state is revealed in the other state at the arbitrary (and irrelevant) moment in time

t is then

(t) = ĪØH |ap1 (t1 ) aā 1 (t1 ) |ĪØH

ĪØp1 t1 (t)|ĪØp1 (3.52)

p

t1

and the single-particle Greenā™s function is a measure of the persistence, in time span

|t1 ā’ 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 ā” aā ).

3.3.1 Physical properties and Greenā™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 speciļ¬ed in

terms of one-particle Greenā™s functions. For a system in an arbitrary state described

by the statistical operator Ļ, 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)

ā

n(x, t) = Tr(Ļ ĻH (x, Ļz , t) ĻH (x, Ļz , t)) , (3.53)

Ļz

where the quantum ļ¬eld describing the particles ĻH (x, Ļz , 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 SchrĀØdinger and Heisenberg pictures coincide is chosen as the moment

o

where the state is speciļ¬ed, i.e. when the arbitrary statistical operator, Ļ, repre-

senting the state of the system is speciļ¬ed. Here Ļz 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 diļ¬erential equation as discussed in Appendix C.

3.3. Greenā™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 Greenā™s function

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

Ļz

where11

ā

= ā“i Tr(Ļ ĻH (x , Ļz , t ) ĻH (x, Ļz , t))

G< (x, Ļz , t, x Ļz , t )

ā

ā” ā“i ĻH (x , Ļz , t ) ĻH (x, Ļz , 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,

. . . ā” Tr(Ļ . . .) . (3.56)

For the case of a pure state, Ļ = |ĪØ ĪØ|, we see that G< (x, t, x , t ) is the amplitude

for the transition at time t to the state ĻH (x , Ļz , t ) |ĪØ , where a particle with

spin Ļz is removed at position x from state |ĪØ , given the system at time t is

in the state ĻH (x, Ļz , t) |ĪØ where a particle with spin Ļz 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 |ĪØ after

removing at time t a particle with spin Ļz at position x and restoring at time t a

particle with spin Ļz 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 Greenā™s

function.

The average electric current density for an assembly of identical fermions having

charge e in an electric ļ¬eld represented by the vector potential A is, according to

Eq. (2.47),

e ā‚ ā‚

ā’ G< (x, Ļz , t, x , Ļz , t)

j(x, t) =

2m ā‚x ā‚x

Ļz

x =x

e2

G< (x, Ļz , t, x, Ļz , t) ,

+ i A(x, t) (3.57)

m Ļz

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

ā

given spin, in which case one studies n(x, Ļz , t) = Tr(Ļ ĻH (x, Ļz , t) ĻH (x, Ļz , 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 deļ¬ning Greenā™s functions, and

the corresponding adjustment of the list of Feynman rules.

64 3. Quantum dynamics and Greenā™s functions

From the equation of motion for the ļ¬eld operator, Eq. (3.32), the equation

of motion for the Greenā™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),

ā‚

ā’ h(t) G< (x, t, x , t ) = dx V (2) (x, x ) G(2) (x, t, x , t, x , t, x , t ) ,

i

ā‚t

(3.58)

where

G(2) (x, t, x , t, x , t, x , t ) = Ā±i Ļ(x, t) Ļ(x , t) Ļ ā (x , t) Ļ ā (x , t ) (3.59)

is a so-called two-particle Greenā™s function since it involves the propagation of two

particles. The dynamics of a system, speciļ¬ed by the time dependence of the one-

particle Greenā™s function, is thus described in terms of higher-order correlation func-

tions in the ļ¬eld operators. The equation of motion for the one-particle Greenā™s

function thus leads to an inļ¬nite hierarchy of equations for correlation functions con-

taining ever increasing numbers of ļ¬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 Greenā™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 electronā“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 oļ¬ self-consistently

at low order. We shall discuss such situations in Section 10.6 and in Chapter 12 in

the context of applying the eļ¬ective action approach to such diļ¬ering situations as

a trapped Boseā“Einstein condensate and classical statistical dynamics, respectively.

3.3.2 Stable of one-particle Greenā™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 ļ¬eld correlation functions, i.e. various kinds of

Greenā™s functions that appear for reasons of their own. For deļ¬niteness we collect

them all here, though they are not needed until later. The rest of this chapter can

thus be skipped on a ļ¬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 Greenā™s function

ā ā

G> (x, t, x , t ) = ā’i ĻH (x, t) ĻH (x , t ) = ā’iTr(Ļ ĻH (x, t) ĻH (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. Greenā™s functions 65

We shall later, for reasons of calculation in perturbation theory, encounter the

time-ordered Greenā™s function

ā

G(x, t, x , t ) = ā’i T (ĻH (x, t) ĻH (x , t )) (3.61)

and we note (valid for both bosons and fermions, recalling the minus sign convention

when two fermi ļ¬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 Greenā™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 Greenā™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 Greenā™s function. The

general real-time perturbation theory valid for non-equilibrium situations will be

formulated in Chapter 5 in terms of essentially two Greenā™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 Greenā™s

function

ā

Ė Ė

G(x, t, x , t ) = ā’i T (ĻH (x, t) ĻH (x , t )) , (3.63)

Ė

where T anti-time orders, i.e. orders oppositely to that of T . We note that the

time-ordered and anti-time-ordered Greenā™s functions can be expressed in terms of

G-greater and G-lesser, for example

Ė

G(x, t, x , t ) = Īø(t ā’ t ) G< (x, t, x , t ) + Īø(t ā’ t) G> (x, t, x , t ) , (3.64)

where Īø is the step or Heaviside function.

Recalling Eq. (3.58), we note for the free Greenā™s functions the relations

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

, (3.65)

0 0

0 0

and for the time-ordered

Gā’1 (x, t) G0 (x, t, x , t ) = Ī“(x ā’ x ) Ī“(t ā’ t ) (3.66)

0

and anti-time-ordered

Gā’1 (x, t) G0 (x, t, x , t ) = ā’ Ī“(x ā’ x ) Ī“(t ā’ t ) ,

Ė (3.67)

0

where

ā‚

Gā’1 (x, t) = ā’h āx , x, t

i , (3.68)

0

ā‚t i

66 3. Quantum dynamics and Greenā™s functions

which for the case of a charged particle coupled to an electromagnetic ļ¬eld is

2

ā‚ 1 ā‚

Gā’1 (x, t) ā’ ā’ eA(x, t) ā’ eĻ•(x, t)

= i . (3.69)

0

ā‚t 2m i ā‚x

Introducing

Gā’1 (x, t, x , t ) = Gā’1 (x, t) Ī“(x ā’ x ) Ī“(t ā’ t ) (3.70)

0 0

we obtain a quantity on equal footing with the Greenā™s function, the inverse free

Greenā™s function (here in the position representation) as

(Gā’1 ā— G0 ) (x, t, x , t ) =

Ė Ė Ī“(x ā’ x ) Ī“(t ā’ t ) , (3.71)

0

where ā— signiļ¬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

inļ¬nity of times.

Exercise 3.9. The equation of motion for the free phonon ļ¬eld is (recall Section

2.4.3)

2 Ļ(x, t) = 0 . (3.72)

Show that the time-ordered free phonon Greenā™s function

D0 (x, t, x , t ) = ā’i T (Ļ(x, t) Ļ(x , t )) (3.73)

therefore satisļ¬es the equation of motion

2 D0 (x, t, x , t ) = ā’ x ) Ī“(t ā’ t ) .

x Ī“(x (3.74)

i

Exercise 3.10. From the equation of motion for the ļ¬eld operator, show that the

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

ā‚

ā’ h0 (t) G(x, t, x , t ) Ī“(x ā’ x )Ī“(t ā’ t )

i =

ā‚t

ā’ i T ([Ļ(x, t), Hi (t)] Ļ ā (x , t )) , (3.75)

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

Other combinations of ļ¬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 Greenā™s function

ā’iĪø(t ā’ t ) [Ļ(x, t) , Ļ ā (x , t )]ā“

GR (x, t, x , t ) =

Īø(t ā’ t ) G> (x, t, x , t ) ā’ G< (x, t, x , t )

= (3.76)

3.3. Greenā™s functions 67

and advanced Greenā™s functions

iĪø(t ā’ t) [Ļ(x, t) , Ļ ā (x , t )]ā“

GA (x, t, x , t ) =

ā’Īø(t ā’ t) G> (x, t, x , t ) ā’ G< (x, t, x , t )

= (3.77)

and the Keldysh or kinetic Greenā™s function

= ā’i [Ļ(x, t) , Ļ ā (x , t )]Ā±

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 ļ¬elds, respectively.

Introducing the notation s = ā’s, the two kinds of statistics can be combined leaving

ĀÆ

the Greenā™s functions in the forms

ā’iĪø(t ā’ t ) [Ļ(x, t) , Ļ ā (x , t )]s

GR (x, t, x , t ) = (3.79)

ĀÆ

and

GA (x, t, x , t ) = iĪø(t ā’ t) [Ļ(x, t) , Ļ ā (x , t )]s (3.80)

ĀÆ

and

ā’i [Ļ(x, t) , Ļ ā (x , t )]s = GS (x, t, x , t )

GK (x, t, x , t ) = (3.81)

where the superscript on the last Greenā™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 Greenā™s function according to

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

Ļz

Exercise 3.12. Show that the current density can be expressed in terms of the

kinetic Greenā™s function according to (in the absence of a vector potential)

e ā‚ ā‚

ā’ GK (x, t, x , t)

j(x, t) = . (3.83)

2m ā‚x ā‚x

x =x

The presence of a vector potential just adds the diamagnetic term (recall Eq. (3.57))

in accordance with gauge invariance, ā’i ā ā’ ā’i ā ā’ eA.

We note the relationship

GR (x, t, x , t ) ā’ GA (x, t, x , t ) = G> (x, t, x , t ) ā’ G< (x, t, x , t ) (3.84)

68 3. Quantum dynamics and Greenā™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 ) ā’ GA (x, t, x , t )) = [Ļ(x, t) , Ļ ā (x , t )]ā“

A(x, t, x , t )

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

We note as a consequence of the equal time anti-commutation or commutation rela-

tions of the ļ¬eld operators, that the spectral function at equal times satisļ¬es

Ī“(x ā’ 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= xā’x

R= , (3.87)

2

and

t+t

, t =tā’t .

T= (3.88)

2

Show that the spectral weight function expressed in these variables satisļ¬es the

sum-rule

ā

dE

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

2Ļ

ā’ā

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

ā—

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

and

ā—

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

and

ā—

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

and

ā—

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

and

ā—

G> (x, t, x , t ) = ā’ 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. Greenā™s functions 69

For the case of a hermitian bose ļ¬eld, such as the phonon ļ¬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 Greenā™s function are displayed in terms of the ļ¬elds in the position

representation. Equally, we can introduce the Greenā™s function displayed in the

momentum representation, related to the above by Fourier transformation, or for

that matter in any representation speciļ¬ed by a complete set of states, say the energy

representation speciļ¬ed in terms of the eigenstates of the Hamiltonian.

Correlation functions of the quantum ļ¬elds can be obtained by diļ¬erentiation of

a generating functional. For example, to generate time-ordered Greenā™s functions we

introduce

ā

+ Ļ ā (x,t) Ī· ā— (x,t))

Z[Ī·, Ī· ā— ] = T ei dx dt (Ļ(x,t) Ī·(x,t)

(3.98)

ā’ā

generating for example the time-ordered Greenā™s function for bosons, Eq. (3.61), by

diļ¬erentiating twice with respect to the complex c-number source function Ī·,14 to

give

ńņš. 3 |