. 2
( 22)


ap a† |p1 § · · · § pN ap |p § p1 § · · · § pN

p |p |p1 § · · · § pN

(’1)n p |pn |p § p1 § · · · ( no pn ) · · · § pN


and similarly
a† a†
ap |p1 § · · · § pN (’1)n’1 p |pn |p1 § · · · ( no pn ) · · · § pN
p p

(’1)n’1 p |pn |p § p1 § · · · ( no pn ) · · · § pN ,

and by adding the two equations we realize the relation

{ap , a† } = p |p . (1.78)

The anti-commutator of fermion creation and annihilation operators is not an oper-
ator but a c-number, i.e. proportional to the identity operator. This is the funda-
mental relation obeyed by the fermion creation and annihilation operators, and its
virtue is that it makes respecting the quantum statistics a trivial matter. When do-
ing calculations for fermion processes, we can in fact, as we show later, forget all the
previous index-nightmare Fock state vector formalism, and we need only remember
the fundamental anti-commutation relation.
We note that, according to Eq. (1.77) and Eq. (1.72),

|p1 § · · · § pN if exactly one of the pi s equals p
a† ap |p1 § · · · § pN =
p 0N ’1 otherwise ,

i.e. the operator a† ap counts the number of particles in state p, i.e. the eigenvalue of
the operator is either 1 or 0, depending on the state in question being occupied or not.
1.3. Fermi ¬eld 19

The operator np = a† ap is therefore referred to as the number operator for state
or mode p. The number of particles counted in the vacuum state is correctly zero.
One readily veri¬es (see Exercise 1.6 below), that all the mode number operators
commute and each number operator has only two eigenvalues, 0 or 1. The total
set of momentum state number operators, {np }p , thus constitutes a complete set of
commuting operators as specifying the eigenvalues for each number operator uniquely
speci¬es a basis vector. They can therefore be used to de¬ne a representation, as
discussed in Section 1.5.
Had we used any other complete set of single particle states, say labeled by
index », we would analogously have obtained for the commutation relations for the
operators creating and annihilating particles in states »1 and »2

{a»1 , a† 2 } = »1 |»2 = δ»1 ,»2 , (1.80)

where the set of chosen single-particle states here is assumed orthonormal and discrete
unless we use compact notation to include a continuum as well, Kronecker including
delta. An example could be that of the energy eigenstates. In the case of momentum
states, we encounter in Eq. (1.78) either a Kronecker function or a delta function
depending on whether the particles are con¬ned or not.
Since the creation and annihilation operators are de¬ned in terms of operations
on state vectors, they inherit their invariance with respect to a global phase trans-

a† ’ e’iφ a† .
a» ’ eiφ a» , (1.81)
» »

Note that indeed all the anti-commutation relations remain invariant under the phase
Exercise 1.5. Show for arbitrary operators A, B and C the relations

[A, BC] = B [A, C] + [A, B] C = [A, B] C ’ B [C, A] (1.82)

and analogously for [AB, C], and in terms of anti-commutators

[A, BC] = {A, B} C ’ B {C, A} . (1.83)

Exercise 1.6. Let us familiarize ourselves with the consequences of the algebra of
creation and annihilation operators

{a† 1 , a† 2 } = 0 = {a»1 , a»2 } (1.84)
» »

and, according to Eq. (1.80) for di¬erent state labels, creation and annihilation op-
erators also anti-commute as

{a»1 , a† 2 } = 0 . (1.85)

It therefore su¬ces to consider a single pair of creation and annihilation operators,
denoted a = a» , and we have {a, a† } = 1. As a consequence of the anti-commutation
relations, Eq. (1.84),

a2 = 0 = (a† )2 (1.86)
20 1. Quantum ¬elds

and verify therefore

(a† a)2 = a† a . (1.87)

Show that for any c-number
a† a
= aa† + e a† a .
e (1.88)

Show that for the number operator, n = a† a, we have its characteristic equation

n(n ’ 1) = 0 (1.89)

demonstrating that its eigenvalues can be either zero or one.
Show that for di¬erent state labels, the number operators commute as

[n» , n» ] = 0 (1.90)

even though the creation and annihilation operators all anti-commute, and the num-
ber operators behave as bose operators. Or, in general, polynomials containing an
even number of anti-commuting operators behave algebraically as numbers.
Exercise 1.7. Show that the in¬nite product state

up + vp a† a†
|BCS p‘ ’p“ |0
= (1.91)

is normalized provided that |up |2 + |vp |2 = 1 for all p.

If, instead of momentum states, position states had been used we would analo-
gously have encountered creation operators which create fermions at de¬nite posi-
tions, for example

a† 1 a† 2 . . . a† N |0 = |x1 § x2 § · · · § xN . (1.92)
x x x

Creation operators of di¬erent representations are related through their transforma-
tion functions. For position and momentum we have, according to Eq. (1.5),
e’ p·x |p
|x dp |p p|x
= = (1.93)
(2π )3/2
and therefore the relationship
a† = e’ p·x a† .
x p
(2π )

In accordance with tradition, instead of using the notation a† we shall introduce

ψ † (x) = a† . (1.95)

To obtain the inverse relation we use
e x·p |x
|p dx |x x|p
= = (1.96)
(2π )3/2
1.3. Fermi ¬eld 21

and we have the relationships between the operators creating particles with de¬nite
position and momentum
dp dx
ψ † (x) = e’ p·x a† a† = e p·x ψ † (x) .
i i
, (1.97)
p p
(2π )3/2 (2π )3/2

Taking the adjoint we obtain analogously for the annihilation operators or quantum
dp dx
e p·x ap e’ p·x ψ(x) .
i i
ψ(x) = , ap = (1.98)
(2π )3/2 (2π )3/2

How one prefers to keep track of factors of 2π in the above Fourier transfor-
mations, is, of course, a matter of taste. With the above convention determined by
the fundamental choice of Eq. (1.5), no such factors appear in the fundamental anti-
commutation relation, Eq. (1.78). If the fermions are con¬ned to a box of volume V
we shall use (guided by our preference for ¬elds to have the same dimensions as wave
1 1
dx e’ p·x
i i
ψ(x) = √ ap = √
e ap , ψ(x) , (1.99)

leaving the fundamental anti-commutation relation Eq. (1.78)

{ap , a† } = δp,p , (1.100)

where the discrete allowed momentum values are speci¬ed by the boundary condition
for the states, say periodic boundary conditions.
One readily veri¬es, as a consequence of the analogous Eq. (1.78), or by using
Eq. (1.97) and Eq. (1.98), the fundamental anti-commutation relations for Fermi
¬elds in the position representation17

{ψ(x), ψ † (x )} = δ(x ’ x ) (1.101)


{ψ(x), ψ(x )} = 0 = {ψ † (x), ψ † (x )} . (1.102)

For equal position the latter two equations have, as a consequence, ψ(x) ψ(x) = 0
and ψ † (x) ψ † (x) = 0, expressing the exclusion principle: no two identical fermions
can occupy the same position.
For the N -particle basis state with particles at the indicated locations we shall
also use the notation
† † †
¦x1 x2 ...xN = ψx1 ψx2 · · · ψxN |0 (1.103)
17 If the particles represented by the ¬elds have internal degrees of freedom, say spin, we have
{ψ(x, σz ) , ψ† (x , σz )} = δσ z ,σ z δ(x ’ x ) .

Often the notation ψσ z (x) = ψ(x, σz ) is used.
22 1. Quantum ¬elds

where |0 denotes the vacuum state for the fermions.
As stressed, any complete set of single-particle states, not just the position or
momentum states, could have been employed. For example in the absence of transla-
tion invariance and using the single-particle energy eigenstates we have analogously
for the quantum ¬elds

ψ(x) = x|» a» = ψ» (x) a» , a» = dx ψ» (x) ψ(x) , (1.104)
» »

where ψ» (x) are the orthonormal eigenstates of a single-particle Hamiltonian.
Instead of characterizing the quantum statistics of a collection of fermions in
terms of the antisymmetry of their state vectors, which as we have seen is a bit
messy or at least requires a substantial amount of indices-writing, it is now taken
care of by the simple anti-commutation relations for the creation and annihilation
operators. The price paid for this enormous simpli¬cation is of course that the
operators now are operators on a super-space, the multi-particle space. As shown in
the next section, the implementation for bosons is identical to the above except that
the quantum statistics is taken care of by the commutation relations of the creation
and annihilation operators.
Exercise 1.8. For N non-interacting spin one-half fermions, an ideal Fermi gas, the
ground state is obtained from the vacuum state according to
⎛ ⎞

|G0 = ⎝ a† ⎠ |0 , (1.105)

i.e all the states below the Fermi energy, F = p2 /2m, are occupied in accordance
with Pauli™s exclusion principle, and all states above are empty for the case of the
ground state. Pictorially, the ground state is that of a ¬lled sphere in momentum
space, the Fermi sea, with the Fermi surface separating occupied and unoccupied
Show that the one-particle Green™s function or density matrix becomes

Gσ (x ’ x ) ≡ G0 |ψσ (x)ψσ (x )|G0

3n sin kF |x ’ x | ’ kF |x ’ x | cos kF |x ’ x |
= , (1.106)
(kF |x ’ x |)3

where n is the density of the fermions, and kF = pF / , and in the considered three
dimensions kF = 3π 2 n. The considered amplitude speci¬es the overlap between the

state where a particle with spin σ at position x has been removed from the ground
state and the state where a particle with spin σ at position x has been removed
from the ground state. Or equivalently, it speci¬es the amplitude for transition to
the ground state of the state where a particle with spin σ at position x has been
removed from the ground state and subsequently a particle with spin σ has been
added at position x.
1.4. Bose ¬eld 23

At small spatial separation
(kF |x ’ x |)2
† †
G0 |ψσ (x)ψσ (x 1’
)|G0 (1.107)
2 10

and at x = x it counts the density of fermions per spin at the position in question.
Show that the pair correlation function is related to the one-particle density ma-
trix according to
⎪ n2
⎨2 σ =σ

G0 |ψσ (x)ψσ (x )ψσ (x )ψσ (x)|G0 = (1.108)
⎪ n2
© ’ Gσ (x ’ x ) σ = σ.

Interpret the result and note in particular the anti-bunching of non-interacting fermions:
the avoidance of identical fermions to be at the same position in space, a repulsion
solely due to the exchange symmetry, the exclusion principle at work in real space.

So far the creation operators are just a kinematic gadget giving an equivalent way
of describing the N -particle state space for arbitrary N , since for example

a† 1 a† 2 · · · a† N |0 = |p1 § p2 § · · · § pN (1.109)
p p p

speci¬es the basis states in terms of the creation operators and the vacuum state.
In Chapter 2, we shall show how operators representing physical quantities can be
expressed in terms of the creation and annihilation operators, and thereby realize in
Chapter 3 their usefulness in describing quantum dynamics in the most general case
where the number of particles is not conserved. But ¬rst we consider the kinematics
for the case where the identical particles are bosons.

1.4 Bose ¬eld
The bose particle creation operator, a† , is introduced according to its action on the
basis states of Eq. (1.45)

a† |p1 ∨ p2 ∨ · · · ∨ pN ≡ |p ∨ p1 ∨ p2 ∨ · · · ∨ pN (1.110)

and the adjoint operates according to
ap |p1 § · · · § pN p|pn |p1 § · · · ( no pn ) · · · § pN , (1.111)

i.e. it annihilates a particle in state p, and is referred to as the bose annihilation
operator. As previously noted, the derivation is equivalent to the antisymmetric case.
Since no minus signs ever occur, the bose creation and annihilation operators sat-
isfy the commutation relations (the analogous equations to Eq. (1.76) and Eq. (1.77)
are now subtracted to give the following result)

[ap , a† ] = p |p (1.112)
24 1. Quantum ¬elds


[a† , a† ] = 0 = [ap , ap ] . (1.113)

We note that, according to the equation for bosons analogous to Eq. (1.77), the
operator a† ap counts the number of particles in state p

a† ap |p1 ∨ · · · ∨ pN = np |p1 ∨ · · · ∨ pN (1.114)

where np denotes the number of particles in momentum state p in the basis state
|p1 ∨ · · · ∨ pN , i.e. the number of pi s which are equal to p, and the operator np =
a† ap is referred to as the number operator for state or mode p. In contrast to the
case of fermions, the boson number operators have besides the eigenvalue 0 all natural
numbers as eigenvalues. As in the case of fermions, the total set of momentum state
number operators, {np }p , thus constitute a complete set of commuting operators
giving rise to a representation as discussed in Section 1.5.
Quite analogous to the case of fermions, creation and annihilation operators with
respect to position can be introduced. For the N -particle basis state with particles
at the indicated locations we have
† † †
¦x1 x2 ... xN = ψx1 ψx2 . . . ψxN |0 , (1.115)

where |0 denotes the vacuum state for the bosons.
Kinematically, independent boson ¬elds are assumed to commute, and bose ¬elds
commute with fermi ¬elds (at equal times).
Though already stated, the expression for the resolution of the identity is not
of much practical use; the job has been taken over by the creation and annihilation
operators, we include it for completeness. The resolution of the identity in the multi-
particle space takes the form (and identically for fermions by using the antisymmetric

|p1 ∨ p2 ∨ · · · ∨ pN p1 ∨ p2 ∨ · · · ∨ pN |
1 =
N =0 p1 ,p2 ,...,pN

|p1 ∨ p2 ∨ · · · ∨ pN p1 ∨ p2 ∨ · · · ∨ pN | ,
n1 ! · · · nN !
p1 ¤p2 ¤···
N =0

where the term N = 0 denotes the projection operator onto the vacuum, |0 0|.
Exercise 1.9. Compact notation encompassing both bosons and fermions can some-
times be convenient. Writing an anti-commutator {A, B} ≡ [A, B]+ , the double val-
ued variable, s = ±, comprises both anti-commutators and commutators, [A, B]s ,
and distinguishes the two types of quantum statistics. Show that

[np , a† ]s = a† , [np , ap ]s = s ap . (1.117)
p p
1.4. Bose ¬eld 25

1.4.1 Phonons
The bose ¬eld does not occur only in connection with the elementary bosonic par-
ticles of the standard model, but can be useful in describing collective phenomena
such as the long wave length oscillations of the ions in say a metal or a semicon-
ductor, and we turn to see how this comes about. The Hamiltonian describing the
ions of mass M and density ni in a crystal lattice is given by the kinetic energy
term for the ions and an e¬ective ion“ion interaction determined by the screened
Coulomb interaction. Expanding the e¬ective ion“ion interaction potential to low-
est, quadratic, order, neglecting anharmonic e¬ects and thus only accounting for
small oscillations of the ions, the Hamiltonian can be diagonalized by an orthogo-
nal transformation rendering it equivalent to that of a set of independent harmonic
oscillators. In this long wave length description, the background dynamics can be
described by a continuum limit quantum ¬eld, the quantum displacement ¬eld, u(x),
a coarse-grained description of the ionic displacements at position x. For longer than
interatomic distance, the screened Coulomb interaction is e¬ectively a delta function,
Ve¬ (x ’ x ) = Z 2 /2N0 δ(x ’ x ), and together with the kinetic energy of the back-
ground ions, the background Hamiltonian functional valid for small displacements
then becomes18

M n i c2
1 2
(∇x · u(x))2
Hb = dx (Π(x)) + , (1.118)
2M ni 2

where the components of the momentum density and the displacement ¬eld inherit
the canonical commutation relations of the ions

δ±β δ(x ’ x )
[Π± (x) , uβ (x)] = (1.119)
and the sound velocity is given by
Zn Zm 2
c2 = = v (1.120)
3M F
2N0 M
where n = Zni is the equilibrium electron density and m the electron mass. We
note that the longitudinal sound velocity is typically smaller by a factor of 100 than
the Fermi velocity, vF . The continuum description of the oscillations of the in fact
discretely located ions appeared because the ions were assumed to exhibit only small
The Hamiltonian describing the dynamics of the background is in fact just a set
of harmonic oscillators, as obtained by diagonalizing the Hamiltonian. Introducing
the normal mode operators
k · ck
2M ni ωk
where ck + c† = dx e’ix·k u(x)
ak = , (1.121)
V k
18 For details of these arguments, starting from the quantum mechanics of the individual ions and
then taking the continuum limit, we refer the reader to, for example, chapter 10 of reference [1].
26 1. Quantum ¬elds

(ck eik·x + c† e’ik·x )
u(x) = (1.122)

the background Hamiltonian becomes the free longitudinal phonon Hamiltonian
a† ak +
Hph = Hb = ωk (1.123)

with linear dispersion ωk = c |k|, and the operators satisfy the harmonic oscillator
normal mode commutation relations
[ak , a† ] = δk,k [a† , a† ] = 0 ,
, [ak , ak ] = 0 (1.124)
k kk

inherited from the canonical commutation relations for the position and momentum
operators of the individual ions. A quantum of an oscillator, a quantized sound mode,
is referred to as a phonon. In the Debye model, the lattice vibrations are assumed
to have linear dispersion all the way to the cut-o¬ wave vector kD .
However, instead of the above quantum mechanical argument, we can also here
take the opportunity to discuss the classical ¬eld theory of oscillations in an isotropic
elastic medium, and then obtain the corresponding quantum ¬eld theory by quan-
tizing the dynamics of the normal modes. This trick can then be elevated to give us
the quantum theory of the electromagnetic ¬eld.

1.4.2 Quantizing a classical ¬eld theory
As an example of quantizing a classical ¬eld theory we consider an elastic isotropic
medium of volume V speci¬ed by its longitudinal sound velocity c and mass density
ρ. In terms of the displacement ¬eld, u(x, t), describing the displacement of the
background matter at position x at time t, we have for small displacements
δnb (x, t)
= ’ ∇ · u(x, t) , (1.125)
where δnb (x, t) is the deviation of the medium density from the average density
ni . Newton™s equation and the continuity equation leads for small δnb (x, t) to this
density disturbance satisfying the wave equation
1 ‚2
x’ 2 δnb (x, t) = 0 (1.126)
c ‚t2
or the dynamics of the elastic medium is equivalently, through the principle of least
action, described by the Lagrange functional
ρ ‚u(x, t)
’ c2 (∇ · u(x, t))2

L[u, u] = dx . (1.127)
2 ‚t

In accordance with the assumed isotropy of the elastic medium, it exhibits no
shear or vorticity, sustaining only longitudinal waves
∇x — u(x, t) = 0 , k — uk (t) = 0 , uk (t) . (1.128)
1.4. Bose ¬eld 27

The classical equations of motion for the displacement ¬eld of the medium, the
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
[ck (t) eik·x + c— (t) e’ik·x ] , ck (t) = ck e’iωk t
u(x, t) = , ωk = c|k|
or equivalently for the Fourier components

uk (t) = ck (t) + c— (t) , (1.132)

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

k · ck (t)
ρ ωk 1/2
ak (t) = (1.135)
V k
we obtain
ωk (ak (t) a— (t) + a— (t) ak (t)) .
Hb = (1.136)
k 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
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
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
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
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)

L= E2 + μ0 B2 . (1.138)
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
4π ™2
L= 0 A + μ0 (∇ — A)
and the Euler“Lagrange equation becomes

1 ‚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
c|k| a† akp ,
Hph = (1.141)
(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)

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
|n»1 , n»2 , n»3 , . . . ≡ √ |»1 3 · · · 3»1 3»2 3 · · · 3»2 3»3 3 · · · 3»3 3 · · · ,
n1 !n2 !n3 ! · · ·
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)


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)

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.

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)

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

34 2. Operators on the multi-particle state space

operator on a general antisymmetric N -particle basis state

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

(ˆ |»P 1 )|»P 2 · · · |»P N
= p

|»P 1 (ˆ |»P 2 ) · · · |»P N + ···
+ p

|»P 1 |»P 2 · · · (ˆ |»P N )
+ . (2.2)

Presently we are discussing the one-body momentum operator

dp p |p p|
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
For a general one-body operator, f (1) , the corresponding operator for the N -
particle system
ˆ (1) ˆ(1)
FN = fi (2.4)
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)
where f (1) » labels the state which f (1) maps the state labeled by » into
|f (1) » = f (1) |» . (2.6)
ˆ (pp )
The operator, FN , on the N -particle space corresponding to the one-body
operator f (1) = |p p| thus operates according to
ˆ (pp )
FN |»1 § »2 § · · · § »N p|»1 |p § »2 § · · · § »N

p|»2 |»1 § p § »3 § · · · § »N + ···

p|»N |»1 § »2 § · · · § »N ’1 § p .
+ (2.7)
2 Weare here guided by the knowledge that a bra has the feature of an annihilation operator
and a ket has the feature of a creation operator, and the transition operators constitute the basis
of the measurement algebra of a quantum system, i.e. completeness of a basis in the state space
» |» »| = 1, and the set {|» » |}»,» is a
is expressed, in the »-representation, by the identity
basis in the dual space, the space of linear operators on the state space. For details see chapter 1
in reference [1].
2.1. Physical observables 35

Since by antisymmetrization

(’1)n’1 |p § »1 § · · · § ( no »n ) § · · · § »N

= |»1 § · · · § »n’1 § p § »n+1 § · · · § »N (2.8)

we have
ˆ (pp )
FN |»1 § · · · § »N p|»n (’1)n’1 |»1 § · · · (no »n instead p ) · · · § »N ,

but according to Eq. (1.77) this is the same state which is obtained when operating
with the operator a† ap so that

a† ap |»1 § · · · § »N .
ˆ (pp )
FN |»1 § · · · § »N = (2.10)

We have thus established how to implement a one-body operator onto the multi-
particle space so that its restriction to any N -particle subspace is the corresponding
N -particle operator. The implementation for bosons is identical to the above, as
usual the derivation is completely analogous, in fact simpler since no minus signs
There is of course nothing special about momentum labels; the formal machinery,
i.e. the combinatorics, works for any set of one-particle states, say labeled by μ,
so that corresponding to the one-particle operator |μ2 μ1 | corresponds the operator
F (1) in the multi-particle space

F (1) = a† 2 aμ1 . (2.11)

An arbitrary one-particle operator has, in an arbitrary basis, the form

ˆ ˆ
|» »|f (1) |» »|
f (1) = (2.12)

and by linearity the corresponding operator F (1) in the multi-particle space is thus

»|f (1) |» a† a» .
F (1) = (2.13)

We note that if f (1) is hermitian in the one-particle state space, as is F (1) in the
multi-particle state space.
The total momentum operator P in the multi-particle space is thus

dp p a† ap = dx ψ † (x) ∇x ψ(x)
P= (2.14)
expressed in either the momentum or position representation of the ¬eld.
36 2. Operators on the multi-particle state space

Exercise 2.1. Show that the commutator of the total momentum operator and the
¬eld is
[ψ(x), P] = ∇x ψ(x) (2.15)
or equivalently
ψ(x) = e’ x·P ψ(0) e x·P .
i i

We now have the prescription for mapping any one-particle operator into the
corresponding operator on the multi-particle space. For a non-relativistic particle of
mass m in a potential V the Hamiltonian is

p2 p2
ˆ ˆ
H= + V (ˆ , t) = + dx n(x) V (x, t) ,
ˆ (2.17)
2m 2m
where in the second equality we have introduced the probability density operator for
a particle, n(x) = δ(ˆ ’ x) (recall Section 1.2.4). In the position representation the
ˆ x
Hamiltonian has the matrix elements
p2 ‚2
ˆ 1
+ V (ˆ , t) |x δ(x ’ x )
= + V (x, t) (2.18)
x| x
2m 2m i

and according to Eq. (2.13) the corresponding Hamiltonian on the multi-particle
space becomes

H= dx ψ (x) + V (x, t) ψ(x) . (2.19)
2m i

We note that the single particle properties can be expressed in terms of the
occupation number operators. For the case of energy, the energy eigenstates should be
used, recall Eq. (1.147) and see Exercise 2.2, and of course for the case of momentum,
Eq. (2.14), the reference states should be the momentum states.

Exercise 2.2. Show that the kinetic energy operator for an assembly of non-relativistic
free identical particles of mass m

H= dx ψ (x) ψ(x) (2.20)
2m i

in the momentum and energy representation has the form

a† ap =
H= np , (2.21)
p p
p p

where p = p2 /2m is the kinetic energy of the free particle with momentum p.
The sum over momenta occurs, one momentum state per momentum volume ”p =
(2π )3 /V in three dimensions, as the particles are assumed enclosed in a box of
volume V .
2.2. Probability density and number operators 37

Exercise 2.3. Show that the average value of the kinetic energy operator for an
electron gas consisting of N electrons in the ground state, i.e. the energy of N free
electrons in the ground state, is
3 p2
a† F
ap = N, (2.22)
p p
5 2m

where pF is the Fermi momentum, the radius of the sphere of occupied momentum
states (in three dimensions pF = (3π 2 n)1/3 , where n = N/V is the density of the
Exercise 2.4. Show that the vacuum state is non-degenerate and uniquely charac-
terized by all the eigenvalues of the state number operators np being zero.
Exercise 2.5. For the quantities discussed so far, a possible (say) spin degree of free-
dom of the particles did not have its two spin states discriminated, and its presence
was left implicit in the notation. To consider a situation where spin states needs to be
speci¬ed explicitly, consider (say) electrons interacting with the magnetic moments
of impurities. The interaction of an electron interacting with the magnetic moments
of impurities is
u(x ’ xa ) Sa · σ ±,±
V±,± (x) = (2.23)
where xa is the location of a magnetic impurity with spin Sa and σ represents the
electron spin. In the multi-particle space, the interaction of the impurity spins and
the electrons thus becomes

dx u(x ’ xa ) ψ± (x) Sa · σ ±,± ψ± (x) .
Vsf = (2.24)

Show it can be rewritten in the form
† †
S ’ ψ‘ (x)ψ“ (x) + S + ψ“ (x)ψ‘ (x)
dx u(x ’ xa )
Vsf =

† †
S z ψ‘ (x)ψ‘ (x) ’ ψ“ (x)ψ“ (x)
+ , (2.25)

where S ± = S x ± iS y .

Density and current density operators are important as well as their coupling
to external ¬elds, and we now turn to their construction in the multi-particle state

2.2 Probability density and number operators
The one-particle probability density operator (recall Section 1.2.4)
n(x) = δ(ˆ ’ x) = |x x|
ˆ (2.26)
38 2. Operators on the multi-particle state space

maps according to the general prescription, Eq. (2.13), to the operator on the multi-
particle space
dx1 dx2 x1 |ˆ (x)|x2 ψ † (x1 ) ψ(x2 )
n(x) = n (2.27)

and therefore the probability density operator in the multi-particle space is
n(x) = ψ † (x) ψ(x) . (2.28)
By construction this operator reduces in each N -particle subspace to the N -particle
density operator
δ(ˆ i ’ x) .
n(x) =
ˆ (2.29)

The identity operator in the one-particle state space3

ˆ dx |x x| =
I= dx n(x)
ˆ (2.30)

becomes in the multi-particle space according to the general prescription, Eq. (2.13),
the operator
dx ψ † (x) ψ(x) = dp a† ap ,
N= (2.31)

which is the operator that counts the total number of particles in each N -particle
state, the total number operator. For example, according to the equation analogous
to Eq. (1.77) but in the position representation

dx ψ † (x) ψ(x) |x1 § x2
N |x1 § x2 =

dx ( x|x1 |x § x2 ’ x|x2 |x § x1 )

2 |x1 § x2 .
= (2.32)
Or more e¬ciently by just using the basic anti-commutation or commutation relation
for the ¬elds, we obtain by consecutively anti-commuting or commuting, depending
on the particles being fermions or bosons, the ψ(x)-operator in the number operator
to the right and eventually killing the vacuum, that for the basis state Eq. (1.115)
N ¦x1 x2 ... xn = n ¦x1 x2 ... xn . (2.33)
For the case of the vacuum state the eigenvalue is zero, there are no particles
in the vacuum. In non-relativistic quantum mechanics, the total number operator
for each set of species is always conserved; however, this is of course not the case
in relativistic quantum theory. We note that the vacuum state has zero energy and
momentum (and of course, as noted, zero number of particles).
3 The
physical interpretation of the identity operator in the one-particle state space is the number
operator, counting the particle number in any one-particle state, I |ψ = 1 |ψ .
2.2. Probability density and number operators 39

Exercise 2.6. Show that if |ψn represents a state with n particles, N |ψn = n |ψn ,
N ψ † (x) |ψn = (n + 1) ψ † (x) |ψn (2.34)
i.e. ψ † (x) |ψn is a state with (n + 1) particles.

Since ψ(x) removes a particle from any state, the relationship

ψ(x) f (N ) = f (N + 1) ψ(x) (2.35)

is valid for an arbitrary function, f , of the number operator. In particular we have

e’±N ψ(x) e±N = e± ψ(x) . (2.36)

Exercise 2.7. Show that the number operator for electrons in the momentum rep-
resentation takes the form

dp a† apσ .
N= (2.37)


Exercise 2.8. The state considered in Exercise 1.7 on page 20 is the famous BCS-
state, which describes remarkably well the ground state properties of many s-wave
superconductors as realized by J. Bardeen, L. N. Cooper and J. R. Schrie¬er (in
1957). Note its total disrespect for the sacred conservation law of non-relativistic
Fermi systems, the conservation of the number of particles or, equivalently, we can
say that the state corresponds to a situation with broken global gauge invariance.
For the reader interested in BCS-ology (which is further investigated in Section
8.1), verify for the average of the number operator

≡ BCS|N |BCS |vp |2
N =2 (2.38)

and for the variance

(N ’ N )2 = N2 ’ N 2
u2 vp .
=4 (2.39)

Show that a† |BCS and a’p“ |BCS represent the same state and that they are
orthogonal to the state |BCS . The BCS-pairing state consists of linear superpositions
of particle and hole states. Show that, as a consequence, anomalous moments are
non-vanishing in the state |BCS , for example

BCS|ap‘ a’p“ |BCS = ’u— vp . (2.40)
40 2. Operators on the multi-particle state space

2.3 Probability current density operator
For a single particle, the probability current density operator is, according to Eq.
j(x) = {ˆ , n(x)} .
pˆ (2.41)
For particles carrying electric charge, e, the electric current density operator or charge
current density operator in the presence of a vector potential, A(x, t), is speci¬ed in
terms of the kinematic momentum operator (ˆ can being the operator satisfying the
canonical commutation relation, Eq. (1.8))

pkin = pcan ’ eA(ˆ , t)
ˆt ˆ (2.42)

and the charge current density operator is
ˆt (x) = {ˆ kin , n(x)} .
ˆ (2.43)
j p
2m t
The current density operator then has two distinct parts
ˆt (x) = ˆp (x) + ˆd (x) (2.44)
j j jt

consisting of the so-called paramagnetic current density operator (or simply the cur-
rent density operator in the absence of a vector potential)

ˆp (x) = e {ˆ can , n(x)}
ˆ (2.45)
j p
and in the present case a time-dependent so-called diamagnetic current density op-
2 2
ˆd (x) = ’ e {A(ˆ , t), n(x)} = ’ e n(x) A(ˆ , t) ,
ˆ ˆ (2.46)
jt x x
2m m
the last equality sign following from the fact that the two operators commute.
For particles carrying electric charge e, the electric current density operator on
the multi-particle space is therefore, according to Eq. (2.12),

jA(t) (x, t) = ψ † (x) ˆA(t) (x, t) ψ(x) , (2.47)

ˆA(t) (x, t) = ˆ(1) (x, t) ’ e A(x, t) (2.48)
j j
is the one-particle current density operator in the position representation in the
presence of an external vector potential A(x, t) and
’ ←
e ‚ ‚
ˆ(1) ’
(x, t) = , (2.49)
2mi ‚x ‚x

the arrows indicating on which ¬eld, to the left or right, the di¬erential operator
2.3. Probability current density operator 41

The interaction between an electron and an electromagnetic ¬eld represented by
a vector potential, A(x, t), can be written in terms of the current density operator
and the density operator

= ’ dx ˆt (x) · A(x, t) ’
ˆ dx n(x) A2 (x, t)
HA(t) ˆ

ˆp (x) · A(x, t) + e
= ’ dx j dx n(x) A2 (x, t) ,
ˆ (2.50)

which becomes the operator on the multi-particle space

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

’ dx jp (x) · A(x, t) + dx n(x) A2 (x, t) ,
= (2.51)


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

The total Hamiltonian on the multi-particle space for an assembly of charged
identical particles interacting with a classical electromagnetic ¬eld is thus4
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 )

Having established how to implement one-particle operators on the multi-particle
space, we now turn to implement more complicated operators, viz. those describing
4 Here expressed in the so-called Schr¨dinger picture, where the dynamics is carried by state
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
ˆ dx1 dx2 |x1 § x2 V (2) (x1 , x2 ) x1 § x2 |
V (2) = (2.55)
1 (2) 1
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
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

V (2) (xi , xj ) |x1 § x2 § · · · § xN
= (2.58)

in the Fock space is represented by the operator

dx dx ψ † (x) ψ † (x ) V (2) (x, x ) ψ(x ) ψ(x) .
V= (2.59)
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
ψ(x ) ψ(x) |x1 § · · · § xN (’1)n’1 δ(x ’ xn )|x1 § · · · ( no xn ) · · · § xN
= ψ(x )

δ(x ’ xn ) (’1)m’θ(n’m) δ(x ’ xm )
= (’1)
n=1 m=1,(m=n)


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( 22)