2m

2

ˆp (x) · A(x, t) + e

= ’ dx j dx n(x) A2 (x, t) ,

ˆ (2.50)

2m

which becomes the operator on the multi-particle space

e2

’ dx jA(t) (x, t) · A(x, t) ’ dx n(x) A2 (x, t)

HA(t) =

2m

e2

’ dx jp (x) · A(x, t) + dx n(x) A2 (x, t) ,

= (2.51)

2m

where

jp (x) = ψ † (x) ˆ(1) (x, t) ψ(x) . (2.52)

j

The total Hamiltonian on the multi-particle space for an assembly of charged

identical particles interacting with a classical electromagnetic ¬eld is thus4

2

1 ‚

†

’ eA(x, t)

HAφ = dx ψ (x) + eφ(x, t) ψ(x) . (2.53)

2m i ‚x

Physical observables as well as their couplings to classical ¬elds are thus repre-

sented on the multi-particle space by operators quadratic in the ¬elds.

Exercise 2.9. Show that the current density operator for electrons in the momentum

representation takes the form

dp dp p + p ’ i (p ’p)·x †

jp (x) = e e ap σ apσ . (2.54)

(2π )3/2 3/2 2m

(2π )

σ

Having established how to implement one-particle operators on the multi-particle

space, we now turn to implement more complicated operators, viz. those describing

interactions.

4 Here expressed in the so-called Schr¨dinger picture, where the dynamics is carried by state

o

vectors and the quantum ¬eld operators are time independent. The opposite scenario, the Heisenberg

picture, will be discussed in Chapter 3.

42 2. Operators on the multi-particle state space

2.4 Interactions

In this section we shall consider interactions between particles. In relativistic quan-

tum theory, the forms of interaction are determined by Lorentz invariance and ex-

pressed in terms of polynomials of the ¬eld operators describing the creation and

annihilation of particles. Our main interest shall be the typical interactions that are

relevant in condensed matter physics, and there the task is to obtain their form in

the multi-particle space knowing their form on any N -particle system. We therefore

start by considering the case of two-body interaction, and consider fermions as the

case of bosons follows as a simple corollary.

2.4.1 Two-particle interaction

If the identical particles, say fermions, interact through an instantaneous two-body

potential, V (2) (xi , xj ), the interaction between two fermions is represented in the

antisymmetric two-particle state space by the operator

1

ˆ dx1 dx2 |x1 § x2 V (2) (x1 , x2 ) x1 § x2 |

V (2) = (2.55)

2

since

1 (2) 1

ˆ

V (2) |x1 § x2 V (x1 , x2 ) |x1 § x2 ’ V (2) (x2 , x1 ) |x2 § x1

=

2 2

V (2) (x1 , x2 ) |x1 § x2

= (2.56)

and thereby

ˆ

x1 § x2 |V (2) |ψ = V (2) (x1 , x2 ) ψ(x1 , x2 ) , (2.57)

where ψ(x1 , x2 ) is the wave function describing the state of the two fermions, by

construction it is an antisymmetric function of its arguments.

We now show that the two-body interaction which operates on an N -particle basis

state according to

ˆ (2)

VN |x1 § x2 § · · · § xN V (2) (xi , xj ) |x1 § x2 § · · · § xN

=

i<j

1

V (2) (xi , xj ) |x1 § x2 § · · · § xN

= (2.58)

2

i=j

in the Fock space is represented by the operator

1

dx dx ψ † (x) ψ † (x ) V (2) (x, x ) ψ(x ) ψ(x) .

V= (2.59)

2

2.4. Interactions 43

First we note that by twice applying the equation analogous to Eq. (1.72) for the

annihilation operator in the position representation we get

N

ψ(x ) ψ(x) |x1 § · · · § xN (’1)n’1 δ(x ’ xn )|x1 § · · · ( no xn ) · · · § xN

= ψ(x )

n=1

N N

δ(x ’ xn ) (’1)m’θ(n’m) δ(x ’ xm )

n’1

= (’1)

n=1 m=1,(m=n)

— |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