+ ∇x · j(x, t) = 0 , (1.55)

‚t

where the probability current density is10

N N

e 1

dxj δ(xi ’ x) ∇xi ’ ∇xi ψ(x1 , . . . ; t) ψ — (x1 , ..; t)

j(x, t) =

2mi N

i=1 j=1

xi =xi

e

dxi (∇x ’ ∇x ) ψ(x, x2 , . . . ; t) ψ — (x , x2 , . . . ; t)

= , (1.56)

2mi

i=1

x =x

9 For the cases of more than two identical particles there are other time invariant subspaces than

the symmetric and antisymmetric ones. They do not seem to be of physical relevance.

10 In the presence of a vector potential the formula must be amended with the diamagnetic term,

see Exercise 1.4 and Section 2.3.

14 1. Quantum ¬elds

again the last equality follows from the symmetry of the wave function for identical

particles. The Schr¨dinger equation guarantees conservation of probability, i.e. the

o

continuity equation, Eq. (1.55), as a consequence of the Hamiltonian being hermitian.

Exercise 1.4. The Hamiltonian for a charged spinless particle coupled to a vector

potential, A, is11

2

ˆ= 1 ‚

’ eA(x, t)

H . (1.57)

2m i ‚x

Show that the probability current density for the particle in state ψ is

1

∇x ’ 2eA(x, t) ψ(x, t) ψ — (x , t)

∇x ’

j(x, t) = . (1.58)

2m i i

x =x

Rarely can the dynamics of an N -particle system of identical particles be solved

exactly. When it comes to performing actual approximate calculations, the quantum

statistics of the particles will even in the non-relativistic quantum theory of an in-

teracting N -particle system cause havoc, and a more ¬‚exible vehicle for respecting

the quantum statistics of identical particles is convenient. We now turn to introduce

these, the quantum ¬elds. In relativistic quantum theory and conveniently for many-

body systems, the quantum ¬elds instead of the wave function become the carriers

of the dynamics, as we will discuss in Chapter 3.

1.3 Fermi ¬eld

We introduce the fermion creation operator, a† , corresponding to momentum value

p

p, as the linear mapping of F into F

(N ) (N +1)

de¬ned for an arbitrary (not necessarily

12

ordered) basis vector by

a† |p1 § p2 § · · · § pN ≡ |p § p1 § p2 § · · · § pN , (1.59)

p

i.e. it maps an antisymmetrized N -particle state into the antisymmetrized (N + 1)-

particle state where an additional fermion has momentum p. The choice of placing

p at the front is, of course, arbitrary. The other popular choice is to place it at the

end. This re¬‚ects that a creation operator, like a state vector, is de¬ned only modulo

a phase factor.

If in the N -fermion state the momentum state p is already occupied, i.e. exactly

one of the pi s equals p, then owing to the antisymmetric nature of the state

a† |p1 § p2 § · · · § pN = 0N +1 , (1.60)

p

11 The form of the Hamiltonian follows from gauge invariance; i.e. the gauge transformation of

the electromagnetic ¬eld, A(x, t) ’ A(x, t) + ∇Λ(x, t), φ(x, t) ’ φ(x, t) ’ Λ(x, t), and the trans-

™

ie

formation of the wave function ψ(x, t) ’ ψ(x, t) e Λ(x,t) , leaves all physical quantities invariant.

The gauge invariance of quantum mechanics is a consequence of the wave function obtained by the

above phase transformation equally well represents the probability distribution of the particle.

12 As emphasized, the label on the creation operator could refer to any state; usually though, it

refers to a complete set of single-particle states.

1.3. Fermi ¬eld 15

the zero vector of state space H (N +1) . This is the expedience with which the fermion

creation operators respect Pauli™s exclusion principle.

We introduce the sum of state spaces F (N ) and F (N +1) . For example, F (1) + F (2)

consists of states spanned by 2-tuple states, (|p , |p1 § p2 ), and is equipped with the

scalar product, which is the sum of the scalar products in the subspaces, i.e. for the

above vector and (c1 |p , c2 |p1 § p2 ) the scalar product is

( p|, p1 § p2 |)(c1 |p , c2 |p1 § p2 ) = c1 p|p + c2 p1 § p2 |p1 § p2 .

(1.61)

In order for an operator to represent an observable physical quantity it must

map a state space onto itself. In order to facilitate this experience for the fermion

creation operators,13 the multi-particle space or Fock space F (named after the Soviet

physicist Vladimir Fock), is introduced as the sum of the state spaces14

∞

F= F (N ) , (1.62)

N =0

where by de¬nition F (0) is the set of complex numbers.

The inclusion of F (0) is demanded in relativistic quantum theory since relativistic

kinematics predicts the creation and annihilation of particles. The zero vector in Fock

space can not represent the state of absence of any particle.15 Particle species not

present in the initial and ¬nal states must, before and after a reaction, be in a state

in their respective Fock spaces so that their scalar products equal one, thereby not

in¬‚uencing the probabilities for the various possible reactions. Since none of these

particle states is initially and ¬nally occupied, though they may appear virtually

in intermediate states to facilitate the reaction, and since the zero vector does not

respect the above property, the state where particles of a given species are absent,

the vacuum state for these particles is represented by (choosing the simplest phase

choice)

|0 ≡ (1, 01 , 02 , . . .) . (1.63)

Even for a non-relativistic system, the vacuum state is a convenient vehicle for gen-

erating all states of the multi-particle space as we will see shortly.

The set of basis states of the Fock space consists of the vacuum state and all

the basis vectors of each N -particle subspace. In the Fock space, states of the type

(0, |p1 , 02 , |p1 § p2 § p3 , . . . , |p1 § p2 , § · · · § pN , . . .) are thus encountered, su-

perposition of states with di¬erent number of particles. In accordance with the

de¬nition of the scalar product of states in the multi-particle space, it can only

13 Whether a fermi ¬eld is an observable, i.e. a measurable quantity, is doubtful. For example, it

does not have a classical limit as states can at most be singly occupied. A bose ¬eld (introduced

in Section 1.4) on the other hand is an observable, since any number of bosons can occupy a single

state and the average value of a bose ¬eld can thus be nonzero, an example being the classical state

of light, the coherent state, created by a laser.

14 In mathematical terms, the state space is a Hilbert space, and the Fock space is a Hilbert sum

of Hilbert spaces, and itself a Hilbert space.

15 The zero vector in the Fock space is of course (0, 0 , 0 , . . .) ≡ 0, for which the obvious short

12

notation has been introduced.

16 1. Quantum ¬elds

be nonzero if the states have components with the same number of particles. An

N -particle basis state in the multi-particle space is usually shortened according to

(0, 01 , 02 , . . . , |p1 § p2 § · · · § pN , 0N +1 , 0N +2 , . . .) ’ |p1 § p2 § · · · § pN .

For the vacuum state, the creation operator operates also in accordance with its

general prescription of adding a particle

a† |0 = |0, |p , 02 , 03 , . . . . (1.64)

p

The state vector (using the abbreviated notation introduced above)

a† 1 · · · a† N |0 = |p1 § p2 § · · · § pN (1.65)

p p

is an antisymmetric N -particle basis state in the multi-particle state space, provided

that all the momenta are di¬erent of course, otherwise it is the zero-vector.16 The

bracket notation appears a little clumsy in this context, and the notation

= a † 1 · · · a † N ¦0

¦p1 ,...,pN (1.66)

p p

is often used, where ¦0 denotes the vacuum. For a state which is a superposition

of states with di¬erent number of particles we can also express it in terms of the

vacuum state, for example (0, |p1 , |p1 § p2 , 03 , . . . = (a† 1 + a† a† )|0 .

p pp 1 2

By construction, any two fermion creation operators, a† and a† , anti-commute,

p p

i.e.

{a† , a† } ≡ a† a† + a† a† = 0 , (1.67)

pp pp pp

meaning that operating with the anti-commutator {a† , a† } on any vector in Fock

pp

space produces the zero vector in Fock space, 0 ≡ (0, 01 , 02 , . . .), just as multi-

plying any vector in Fock space by the number 0 does. This follows immediately

from the fact that operating with the anti-commutator on any basis vector, say

(0, 01 , 02 , . . . , |p1 § p2 § · · · § pN , 0N +1 , 0N +2 , . . .), gives the sum of two vectors

which di¬er only by a minus sign (or if the momentum labels in the anti-commutator

are equal, the sum of two zero vectors). For the case p = p, the anti-commutation

relation Eq. (1.67) becomes a† a† = ’ a† a† and therefore by itself a† a† = 0. This is

pp pp pp

Pauli™s exclusion principle expressed in terms of the creation operator: two fermions

can not be accommodated in the same state.

We then introduce the fermion annihilation operator, ap , as the adjoint of the

fermion creation operator a† . Since the creation operator maps an N -particle state

p

into an (N + 1)-particle state, the annihilation operator, being the adjoint, will map

an N -particle state into an (N ’ 1)-particle state. To understand its properties we

can restrict attention to the basis vectors of the subspaces F (N ) and F (N ’1) of the

Fock space, and we have

—

p1 § · · · § pN |a† |p2 § · · · § pN

p2 § · · · § pN |ap |p1 § · · · § pN = p

p1 § · · · § pN |p § p2 § · · · § pN ’1

=

det( pi |pj ) ,

= (1.68)

16 With the chosen ordering convention of the previous section it is the ground state for N non-

interacting fermions.

1.3. Fermi ¬eld 17

where, in the last equality, we have introduced the notation p1 = p, and used

Eq. (1.42). Expanding the determinant in terms of its ¬rst column we get

N

—

p2 § · · · § pN |ap |p1 § · · · § pN (’1)n’1 pn |p det( pi |pj (n)

= )

n=1

N

(’1)n’1 pn |p det( pi |pj (n)

= ),

n=1

(1.69)

where the sub-determinant, det( pi |pj (n) ), is the determinant of the matrix Eq. (1.68),

with row n and the ¬rst column removed. Using p|p — = p |p we get

N

p2 § · · · § pN |ap |p1 § · · · § pN (’1)n’1 p|pn det( pj |pi (n)

= )

n=1

(1.70)

and using Eq. (1.42) for the (N ’1)-particle case, the right-hand side can be rewritten

as

N

(’1)n’1 p|pn p2 § · · · § pN |p1 § · · · ( no pn ) .. § pN (1.71)

n=1

and we have

N

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

=

n=1

(1.72)

Thus operating with the fermion annihilation operator labeled by p on an N -particle

basis state produces the zero vector unless exactly one of the momentum values equals

p, and in that case it equals the (N ’ 1)-particle state where none of the fermions

occupies the originally occupied momentum state p. The annihilation operator ap

thus annihilates the particle in state p. In the simplest of situations we have

ap |p |0 .

= (1.73)

Annihilating the single-particle state turns it into the vacuum state.

In particular it follows from Eq. (1.68) that operating with any fermion annihila-

tion operator on the vacuum state produces the zero vector

ap |0 = 0. (1.74)

18 1. Quantum ¬elds

According to Eq. (1.67), the fermion annihilation operators anti-commute

{ap , ap } = 0 . (1.75)

For the case p = p, the anti-commutation relation Eq. (1.75) has the consequence

ap ap = 0, expressing the exclusion principle: no two identical fermions can occupy

the same momentum state.

Next we inquire into the relations obtained by subsequent operations with fermion

creation and annihilation operators, and calculate, according to Eq. (1.72),

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

=

p

p |p |p1 § · · · § pN

=

N

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

+

n=1

(1.76)

and similarly

N

a† a†

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

=

p p

n=1

N

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

=

n=1

(1.77)

and by adding the two equations we realize the relation

{ap , a† } = p |p . (1.78)

p

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 ,

(1.79)

i.e. the operator a† ap counts the number of particles in state p, i.e. the eigenvalue of

p

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

p

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-

formation

a† ’ e’iφ a† .

a» ’ eiφ a» , (1.81)

» »

Note that indeed all the anti-commutation relations remain invariant under the phase

transformation.

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)

p