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