Lagrange ¬eld equations following from Hamilton™s principle of least action, can be

expressed in terms of the displacement ¬eld

2u(x, t) = 0 uk (t) + c2 k 2 uk (t) = 0

¨

, (1.129)

speci¬ed by the d™Alembertian

1 ‚2

2= x’ . (1.130)

c2 ‚t2

The solution is, for example, the running normal mode expansion with periodic

boundary conditions

1

[ck (t) eik·x + c— (t) e’ik·x ] , ck (t) = ck e’iωk t

u(x, t) = , ωk = c|k|

k

V

k=0

(1.131)

or equivalently for the Fourier components

uk (t) = ck (t) + c— (t) , (1.132)

’k

as the vector ¬eld u(x, t) is real.

Introducing the momentum density of the medium19

™

‚u(x, t) δL[u, u]

Π(x, t) ≡ ρ = (1.133)

™

‚t δ u(x, t)

and recalling that the Hamilton and Lagrange functions are related through a Legen-

dre transformation (see, for example, Eq. (3.46) or Eq. (A.10)), we have the Hamilton

functional for the dynamics of the elastic medium

ρ c2

1 2

(∇ · u(x, t))2

Hb = dx (Π(x, t)) + . (1.134)

ρ 2

Introducing

k · ck (t)

ρ ωk 1/2

ak (t) = (1.135)

V k

we obtain

1

ωk (ak (t) a— (t) + a— (t) ak (t)) .

Hb = (1.136)

k k

2

k

The classical theory is now quantized by letting the normal mode expansion co-

e¬cients become operators, a— (t) ’ a† (t), which satisfy the harmonic oscillator

k k

creation and annihilation equal time commutation relations20

[ak (t), a† (t)] = δk,k [a† (t), a† (t)] = 0

, , [ak (t), ak (t)] = 0 . (1.137)

k k k

19 Functional di¬erentiation is discussed in section 9.2.1.

20 This so-called second quantization procedure is discussed further in Section 3.2.

28 1. Quantum ¬elds

There is for the present purpose nothing conspicuous about this quantization proce-

dure, as it gives the same Hamiltonian as the one derived quantum mechanically in

the previous section, where these commutation relations are directly inherited from

the canonical commutation relations for the position and momentum operators of

the individual ions of the material. The continuum description was appropriate since

only long wave length oscillations, long compared with the inter-ionic distance, were

of relevance.

In classical physics, kinematics and dynamics of physical quantities are expressed

in terms of the same quantities. Kinematics, i.e. the description of the physical state

of an object is, in classical physics, intuitive, described in terms of the position and

velocity of an object, (x(t), v(t)): we can point to the position of an object and from

its motion construct its velocity. The classical dynamics is expressed in terms of

the time dependence of the positions and velocities (or momenta) of the concerned

objects, say in terms of Hamilton™s equations. In quantum mechanics, dynamics and

kinematics can be separated, as is the case in the Schr¨dinger picture, where the

o

dynamics is carried by a state vector and the physical properties of a system by

operators. When quantizing a classical theory, the Hamiltonian is thus obtained in

the so-called Heisenberg picture, where the operators representing physical quantities

are time dependent and also carrying the dynamics of the system. The quantized

elastic medium Hamiltonian is therefore expressed in the Heisenberg picture, and the

Hamiltonian in the Schr¨dinger picture is here obtained simply by removing the time

o

variable, recalling ak (t) = ak e’iωk t , i.e. we implement the commutation relations

for the ak quantities, and we recover the expression in Eq. (1.123) for the phonon

Hamiltonian. The Schr¨dinger and Heisenberg pictures are discussed in detail in

o

Section 3.1.2.

A similar prescription in fact works for quantizing the free Maxwell equations

of classical electrodynamics, producing the quantum theory of electromagnetism,

quantum electrodynamics or QED, as discussed in Exercise 1.10, where the quanta

of the ¬eld are Einstein™s photons. In the case of phonons, the quanta describe

the quantum states of small oscillations of an assembly of atoms as described by the

Schr¨dinger equation. However, for the case of electromagnetism, the photons do not

o

refer to any dynamics of a medium. The non-relativistic ¬eld theory of interacting

electrons, described by the Hamiltonian for Coulomb interaction, Eq. (1.53), is only a

limiting case of QED, but the one relevant for the dynamics of, say, electrons in solids.

In the next chapter we shall therefore take the approach to non-relativistic quantum

¬eld theory which starts from the known interactions of an N -particle system and

then construct their forms on the multi-particle state spaces. However, when we

eventually consider the dynamics of a quantum ¬eld theory in terms of its Feynman

diagrammatics in Chapter 9, all theories appear on an equal footing, particulars are

just embedded in the various indices possibilities for propagators and vertices.

Exercise 1.10. Maxwell™s equations, the classical equations of motion for the elec-

tromagnetic ¬eld, can for vacuum, the free theory, be obtained from Hamilton™s

principle of least action with the Lagrange density (SI units are employed)

4π

L= E2 + μ0 B2 . (1.138)

0

2

1.5. Occupation number representation 29

Representing the electric ¬eld solely in terms of a vector potential, • = 0, and

choosing the Coulomb or radiation gauge, ∇ · A = 0, show that the Lagrange density

becomes

4π ™2

L= 0 A + μ0 (∇ — A)

2

(1.139)

2

and the Euler“Lagrange equation becomes

1 ‚2

∇’2 2

2

A(x, t) = 0 , (1.140)

c ‚t

√

where c = 1/ μ0 0 denotes the velocity of light. Note that manifest Lorentz and

gauge invariance have been sacri¬ced in the Coulomb gauge. Expressing the solution

in terms of running normal modes, obtain that the Hamiltonian for free photons has

the form

dk

c|k| a† akp ,

Hph = (1.141)

kp

3

(2π) p=1,2

where since we are in the transverse gauge two perpendicular polarizations occur,

and the creation and annihilation operators for photons with wave vectors k and

polarizations p satisfy the commutation relations

[akp , a† p ] = (2π)3 δ(k ’ k ) δpp . (1.142)

k

1.5 Occupation number representation

In this section we make a side remark which is not necessary for understanding any

of the further undertakings; we just include it for its historical relevance, since this is

how quantum ¬eld theory traditionally was presented, originating in the treatment

of the electromagnetic ¬eld and emulated for fermions in many textbooks.

The operator

N » = a† a» (1.143)

»

counts, as noted in the previous sections, the number of particles in state » in any

N -particle basis state expressed in terms of these states

¦»1 »2 ...»N = a† 1 a† 2 · · · a† N |0 . (1.144)

» » »

The set of numbers, {n»i }i , counted in the basis states by the set of number operators

{N»i }i therefore uniquely characterizes the basis states, and the set of these oper-

ators therefore forms a complete set of commuting operators in the corresponding

multi-particle space, symmetric or antisymmetric. They therefore give rise to a repre-

sentation, the occupation number representation. As a basis set in the multi-particle

space, we can therefore equally well use the occupation number representation, where

the orthonormal basis states are de¬ned by this complete set of commuting opera-

tors, and simply are labeled by stating how many particles are present in any of the

30 1. Quantum ¬elds

single particle states », |n»1 , n»2 , n»3 , . . . . These states are related to our previous

basis states according to

1

|n»1 , n»2 , n»3 , . . . ≡ √ |»1 3 · · · 3»1 3»2 3 · · · 3»2 3»3 3 · · · 3»3 3 · · · ,

n1 !n2 !n3 ! · · ·

(1.145)

where ni is the number of times state »i occurs, and 3 stands for ∨ or § for the bose

or fermi case, respectively. In the fermion case, each »i can of course at most occur

once, i.e. ni = 0 or ni = 1.

We note that if, as in the following, the »-label refers to the single-particle energy

states, the sum of single-particle energies

E0 ({n» }» ) = n» (1.146)

»

»

of an assembly of identical particle is the energy eigenvalue of the free Hamiltonian

H0 = N» (1.147)

»

»

in state |n»1 , n»2 , n»3 , . . . , i.e. the single-particle or free Hamiltonian can be ex-

pressed in terms of the number operators.

The occupation number representation is not necessary, since the introduction

of the workings of the creation and annihilation operators as done in the previous

sections is easier. However, one notices that, for the case of bosons, the creation

operator operates on an occupation number eigenstate according to21

√

a† i |n»1 , n»2 , n»3 , . . . = ni + 1 |n»1 , n»2 , n»3 , . . . , n»i + 1, . . . (1.148)

»

and the annihilation operator according to

√

a»i |n»1 , n»2 , n»3 , . . . ni |n»1 , n»2 , n»3 , . . . , n»i ’ 1, . . .

= (1.149)

and we realize that, in the bose case, creation and annihilation operators act analo-

gously to creation and annihilation operators for a harmonic oscillator. The quanta

in a harmonic oscillator thus have an equivalent interpretation in terms of particles

occupying the energy states of a harmonic oscillator. Here emerges the reason for

the success of Einstein™s revolutionary interpretation of Planck™s lumps of energy in

the electromagnetic ¬eld as particles. This is how the quanta of the electromagnetic

¬eld oscillators are interpreted as particles, viz. photons. Vice versa, the collec-

tive small oscillations of lattice ions performed by the atoms or ions in a solid can

be represented in terms of harmonic oscillators, the so-called phonons as described

in Section 1.4.1, or equivalently has identical properties to particles obeying bose

statistics. For bosons such as photons, i.e. in quantum optics, the occupation, or

just number representation, is of course of fundamental relevance.

21 For fermions additional sign factors appear as discussed in Exercise 1.11.

1.6. Summary 31

Exercise 1.11. Consider the case of fermions and de¬ne the basis states in the

number representation in terms of the vacuum state economically according to

(a† )n1 (a† )n2 (a† )n3 . . . (a† )n∞ |0

|n1 , n2 , n3 , . . . , n∞ = (1.150)

∞

1 2 3

where the ni s can take on the values 0 or 1.

Show that, for ns = 1,

(’1)Ss (a† )n1 . . . (as a† ) . . . (a† )n∞ |0 ,

as |n1 , n2 , n3 , . . . , n∞ = (1.151)

∞

s

1

where Ss = n1 + n2 + · · ·+ ns’1 counts how many anti-commutations it takes to move

as to its place displayed on the right-hand side. If ns = 0, the annihilation operator

as could be moved all the way to act on the vacuum, producing the zero-vector.

Use the above observations to show that

√

a s | . . . , ns , . . . (’1)Ss ns | . . . , ns ’ 1, . . .

= (1.152)

and √

a † | . . . , ns , . . . (’1)Ss ns + 1 | . . . , ns + 1, . . .

= (1.153)

s

and thereby

N s | . . . , ns , . . . n s | . . . , ns , . . . .

= (1.154)

Here we have used modulo one-notation in the ns -state labeling: 1 + 1 = 0 and

0 ’ 1 = 0. These relations are therefore similar to those for bosons except that

obnoxious sign factors occur owing to the Fermi statistics. The occupation number

representation for fermions is therefore not attractive as the wedge is not explicit.

1.6 Summary

In this chapter we have considered the quantum mechanical description of systems

which can be in superposition of states with an arbitrary content of particles. To

deal with such situations, endemic to relativistic quantum theory, quantum ¬elds were

introduced, describing the creation and annihilation of particles. The states in the

multi-particle state space could be simply expressed by operating with the creation

¬eld on the vacuum state, the state corresponding to absence of particles. The

whole kinematics of a many-body system is thus expressed in terms of just these two

operators. Our ¬rst encounter with a quantum ¬eld theory was the case of quantized

lattice vibrations, phonons, and equivalent to the quantum mechanics of a set of

harmonic oscillators, and the archetype resulting from the scheme of quantizing a

classical ¬eld theory. The scheme was then exploited to quantize the electromagnetic

¬eld where the quanta of the ¬eld, the photons, were particles with two internal spin

or polarization or helicity states. In the case of phonons, the continuum quantum

¬eld description was only an appropriate long wave length description, whereas in

the case of photons the quantum ¬eld theory is truly a description of a system with

an in¬nite number of degrees of freedom. In the next chapter we shall consider

non-relativistic many-body systems, and the task is therefore not to assess the form

32 1. Quantum ¬elds

of the Hamiltonian, but the more mundane task of elevating a known N -particle

Hamiltonian to its form on the multi-particle state space.

As we develop the various topics of the book the following conclusion will emerge:

quantum ¬elds are the universal vehicle for describing ¬‚uctuations whatever their

nature, being quantum or thermal or purely classical stochastic.

2

Operators on the

multi-particle state space

A physical property A is characterized by the total set of possible values {a}a it can

exhibit. In quantum mechanics, the same information is expressed by the operator

representing the physical quantity in question, expressed by the weighted sum of

projection operators

ˆ a |a a|

A= (2.1)

a

weighted by the eigenvalues of the operator in question.1 We now want to ¬nd the

expression for the operator on the multi-particle space whose restriction to any N -

particle subspace reduces to the operator in question for the system consisting of N

identical bosons or fermions. We show that all operators for an N -particle system are

lifted very simply to the multi-particle space through an expression in terms of the

creation and annihilation operators in a way analogous to the bra and ket expression

in Eq. (2.1).

2.1 Physical observables

In quantum mechanics, physical properties are represented by operators, say momen-

ˆ

tum by an operator denoted p, and for an N -particle system their total momentum

ˆ

is represented by the operator in Eq. (1.25), denoted PN . We now want to ¬nd

the expression for the operator on the multi-particle space whose restriction to any

N -particle subspace reduces to the total momentum operator for the N identical par-

ticles. In the following we consider the case of fermions; as usual for kinematics the

case of bosons is a trivial corollary. We have for the operation of the total momentum

1 For details on the construction of operators from values of physical outcomes, we refer to chapter

1 of reference [1].

33

34 2. Operators on the multi-particle state space

operator on a general antisymmetric N -particle basis state

N

1

√

ˆ

PN |»1 § »2 § · · · § »N (’1)ζP |»P 1 |»P 2 · · · |»P N

ˆ

= pi

N!

i=1 P

(ˆ |»P 1 )|»P 2 · · · |»P N

(’1)ζP

= p

P

|»P 1 (ˆ |»P 2 ) · · · |»P N + ···

+ p

|»P 1 |»P 2 · · · (ˆ |»P N )

+ . (2.2)

p

Presently we are discussing the one-body momentum operator

dp p |p p|

ˆ

p= (2.3)

but it is in fact appropriate ¬rst to access how the general one-body transition oper-

ator |p p| is implemented, and thereby the whole operator algebra.2

ˆ

For a general one-body operator, f (1) , the corresponding operator for the N -

particle system

N

ˆ (1) ˆ(1)

FN = fi (2.4)

i=1

operates according

ˆ (1)

FN |»1 § »2 § · · · § »N |f (1) »1 § »2 § · · · § »N

=

|»1 § f (1) »2 § · · · § »N + ···

+

|»1 § »2 § · · · § f (1) »N ,

+ (2.5)

ˆ

where f (1) » labels the state which f (1) maps the state labeled by » into

ˆ

|f (1) » = f (1) |» . (2.6)

ˆ (pp )

The operator, FN , on the N -particle space corresponding to the one-body

ˆ

operator f (1) = |p p| thus operates according to

ˆ (pp )

FN |»1 § »2 § · · · § »N p|»1 |p § »2 § · · · § »N

=