of the signs of the two permutations, sign(Q) = sign(P ) · sign(P ), and a permutation

and its inverse have the same sign (owing to their equal number of transpositions),

ζP ’1 = ζP . Antisymmetrization of a permuted state gives the same antisymmetric

state multiplied by the sign of the permutation permuting the original N -particle

state since

1

ˆ

A |pP 1 , pP 2 , . . . pP N (’1)ζP |pQ 1 |pQ 2 · · · |pQ N

= (1.32)

N!

P

and as P runs through all the permutations so does Q = P P , and therefore

1

ˆ

A |pP 1 , pP 2 , . . . pP N (’1)ζQ |pQ 1 |pQ 2 · · · |pQ N

= (’1)ζP

N!

Q

ˆ

= (’1)ζP A |p1 , p2 , . . . , pN . (1.33)

Therefore, if any two single-particle states are identical, the antisymmetrized state

vector equals the zero vector, since the two states obtained by permuting the two

identical labels are identical and yet upon antisymmatrization they di¬er by a minus

sign, i.e. Pauli™s exclusion principle for fermions: no two fermions can occupy the

same state.

8 1. Quantum ¬elds

Further, according to Eq. (1.33), applying the antisymmetrization operator twice

ˆ1

ˆ

A2 |p1 , p2 , . . . , pN =A (’1)ζP |pP 1 |pP 2 · · · |pP N

N!

P

1 ˆ

(’1)ζP (’1)ζP A |p1 , p2 , . . . , pN

=

N!

P

ˆ

= A |p1 , p2 , . . . , pN (1.34)

gives the same state as applying it only once, i.e. the symmetrization operators are

ˆ ˆˆ ˆ

projectors, A2 = A, S 2 = S. The presence of the factor 1/N ! in the de¬nitions,

Eq. (1.29) and Eq. (1.30), is thus there to ensure the operators are normalized pro-

jectors. Representing mutually exclusive symmetry properties, they are orthogonal

projectors, their product is the operator that maps any vector onto the zero vector

ˆˆ 0 ˆˆ

AS =ˆ=SA (1.35)

since symmetrizing an antisymmetric state, or vice versa, gives the zero vector.

The symmetrization operators are hermitian, A† = A, S † = S, as veri¬ed for

ˆ ˆˆ ˆ

ˆ

example for A by ¬rst noting that according to the de¬nition of the adjoint operator

p1 , . . . , pN |A† |p1 , p2 , . . . , pN —

ˆ ˆ

p1 , . . . , pN |A|p1 , p2 , . . . , pN

=

1 — —

(’1)ζP p1 |pP 1 · · · pN |pP N

=

N!

P

(’1)ζS

pS 1 |p1 · · · pS N |pN

= (1.36)

N!

the matrix element being nonzero only if the set {pi }i=1,...,N is a permutation of the

set {pi }i=1,...,N , S being the permutation that brings the set {pi }i=1,...,N into the set

{pi }i=1,...,N , pS i = pi . Permuting both sets of indices by the inverse permutation

S ’1 of S, and using that ζS ’1 = ζS , we get

1

p1 , . . . , pN |A† |p1 , p2 , . . . , pN

ˆ (’1)ζS ’1 p1 |pS ’1 · · · pN |pS ’1

=

N! 1 N

1

(’1)ζP p1 , . . . , pN |pP1 , . . . , pPN

=

N!

P

ˆ

p1 , . . . , pN |A|p1 , . . . , pN .

= (1.37)

Exercise 1.1. Show that the adjoint of a product of linear operators A and B is the

product of their adjoint operators in opposite sequence

(A B)† = B † A† (1.38)

and generalize to the case of an arbitrary number of operators.

1.2. N-particle system 9

Exercise 1.2. The vector space of state vectors, the kets, and the dual space of

linear functionals on the state space, the bras, are isomorphic vector spaces, which

we express by the adjoint operation, |ψ † = ψ| and ψ|† = |ψ . This mapping

is anti-linear and isomorphic, and we use the same symbol as for the adjoint of an

operator.

Show that for arbitrary state vectors and operators on the state space the rela-

tionship (X|ψ )† = ψ|X † . An operator being its own adjoint, X † = X, is said to

ˆ ˆ ˆ ˆ

be a hermitian operator and its eigenvalues are real, such operators being of primary

importance in quantum mechanics.

ˆ

Exercise 1.3. Show that the symmetrization operator, S, is hermitian.

ˆ ˆ

The linear operators S and A project any state onto either of the two orthogonal

subspaces of symmetric or antisymmetric states.8 The state space for a physical

system consisting of N identical particles is thus not H (N ) , the N -fold product of

the one-particle state space, but either the symmetric subspace, B (N ) , for bosons,

or antisymmetric subspace, F (N ) , for fermions, obtained by projecting the states of

H (N ) by either type of symmetrization operator depending on the statistics of the

particles in question.

1.2.2 Kinematics of fermions

Let us introduce the orthogonal, normalized up to a phase factor, antisymmetric

basis states in the antisymmetric N -particle state space F (N )

√

ˆ

|p1 § p2 § · · · § pN ≡ N ! A |p1 , p2 , . . . , pN

1

√ (’1)ζP |pP 1 — |pP 2 — · · · — |pP N

=

N! P

1

√ (’1)ζP |pP 1 |pP 2 · · · |pP N

=

N! P

1

√ (’1)ζP |pP 1 , pP 2 , . . . , pP N .

= (1.39)

N! P

We demonstrate that they are orthogonal by using the properties of the antisym-

metrization operator (we ¬rst for simplicity of the Kronecker function assume box

normalization, i.e, the momentum values are discrete)

N ! p1 , . . . , pN |A† A|p1 , . . . , pN

ˆˆ

p1 § · · · § pN |p1 § · · · § pN =

ˆ

N ! p1 , . . . , pN |A|p1 , . . . , pN

=

8 Only for the case of two particles do the two subspaces of symmetric and antisymmetric states

span the original state space, H (2) = H — H. In general, the other subspaces for the case of more

than two particles do not seem to be state spaces for systems of identical particles.

10 1. Quantum ¬elds

p1 , . . . , pN | (’1)ζP |pP 1 , . . . , pP

=

N

P

§

{p }i ≡ {p}i

⎨ (’1)ζS

= (1.40)

©

0 otherwise

where {pi }i=1,...,N ≡ {pi }i=1,...,N is short for the labels {pi }i=1,...,N being a permuta-

tion of the labels {pi }i=1,...,N , and S the permutation that takes the set {pi }i=1,...,N

into {pi }i=1,...,N , pS i = pi . Or simply in words, only if the primed set of momenta

is a permutation of the unprimed set is the scalar product of the states nonzero (we

have of course assumed that all momenta are di¬erent since otherwise for fermions

the vector is the zero-vector).

Incidentally we have

§1

⎨ √N ! (’1) S {p }i ≡ {p}i

ζ

p1 § p2 , § · · · § pN | p1 , p2 , .., pN = (1.41)

©

0 otherwise

expressing that additional permutations are redundant, for example an additional an-

tisymmetrization is redundant as expressed by the second equality sign in Eq. (1.40),

or equivalently that the symmetrization operators are hermitian projectors.

The scalar product of antisymmetric states is the determinant of the N — N

matrix with entries pi |pj

p1 § · · · § pN |p1 § · · · § pN det( pi |pj )

=

(’1)ζP p1 |pP 1 · · · pN |pP

= , (1.42)

N

P

the Slater determinant.

In the operator calculus perturbation theory for a single particle, the resolution

of the identity plays a crucial e¬cient role. For an assembly of identical particles

this role will be taken over by the commutation rules for the quantum ¬elds we shall

shortly introduce. The resolutions of the identity on the symmetrized subspaces

re¬‚ect the redundancy of antisymmetrized or symmetrized states. Though not of

much practical use, we include them for completeness (the resolution of the identity

makes a short appearance in Section 3.1.1). The resolution of the identity on the

antisymmetric state space can be written in terms of the N -state identity operator

since the identity operator commutes with any operator

A†

ˆˆ ˆ ˆˆ ˆ ˆ ˆ ˆ

A I (N ) A = A I1 — I2 — I3 — · · · — IN

1=

|p1 p1 | — |p2 p2 | — · · · — |pN pN | A†

ˆ ˆ

A

=

p1 ,...,pN

|p1 , p2 , . . . , pN p1 , p2 , . . . , pN | A†

ˆ ˆ

A

=

p1 ,...,pN

1.2. N-particle system 11

1

|p1 § p2 § · · · § pN p1 § p2 § · · · § pN |

=

N! p

1 ,...,pN

|p1 § p2 § · · · § pN p1 § p2 § · · · § pN | .

= (1.43)

|p1 |<|p2 |< ··· <|pN |

In obtaining the last equality we have used the fact that if the momenta of the

particles are interchanged in the N -particle state to be antisymmetrized, the same

antisymmetric state vector is obtained modulo a phase factor ±1, for example

ˆ ˆ

A |p1 , p2 , . . . , pN = ’ A |p2 , p1 , . . . , pN . (1.44)

In the sum in the second last expression in Eq. (1.43), there are thus N ! identical

terms.

The symmetrization phase factor in Eq. (1.40) can always be chosen to equal 1 by

considering proper orderings in the de¬nition of the basis states, thereby removing

the redundancy in the general de¬nition, Eq. (1.39), of the basis states. For example,

if we choose to use only basis vectors where the momenta appear ordered according

to the ordering |p1 | < |p2 | < · · · < |pN |, this restriction on de¬ning the set of

basis states |p1 § p2 § · · · § pN results in them forming an orthonormal basis in the

antisymmetric state space FN , as also expressed by the last equality in Eq. (1.43).

1.2.3 Kinematics of bosons

We now turn to a discussion of the state space relevant for N identical bosons. In

the symmetric state space, B (N ) , we introduce the symmetric orthogonal basis states

√

ˆ

|p1 ∨ p2 ∨ · · · ∨ pN ≡ N ! S |p1 , p2 , . . . , pN

1

√ |pP 1 — |pP 2 — · · · — |pP N

=

N! P

1

√ |pP 1 |pP 2 · · · |pP N

=

N! P

1

√ |pP 1 , pP 2 , . . . , pP N .

= (1.45)

N! P

All derivations of formulas to be obtained for symmetric basis states runs equivalent

to those for antisymmetric basis states. For example,

p1 ∨ · · · ∨ pN |p1 ∨ · · · ∨ pN p1 |pP 1 · · · pN |pP

=

N

P

per( pi |pj ) ,

= (1.46)

where the last equality de¬nes the permanent of the N — N matrix which has the

entries pi |pj .

12 1. Quantum ¬elds

In fact, the bose and fermi cases, i.e. the symmetric and antisymmetric basis

states, can be treated simultaneously if we introduce the factor ( )ζP inside the

summation sign

1

√

|p1 3 · · · 3pN ≡ ( )ζP |pP 1 , pP 2 , . . . , pP N (1.47)

N! P

since then the fermi case corresponds to = ’1 and the bose case to = +1, and 3

stands for ∨ or § for the bose and fermi cases, respectively.

The states introduced in Eq. (1.45) provide a resolution of the identity in the

symmetric state space, B (N ) , speci¬ed by

S†

ˆˆ ˆ ˆˆ ˆ ˆ ˆ ˆ

S I (N ) S = S I1 — I2 — I3 — · · · — IN

1=

|p1 p1 | — |p2 p2 | — · · · — |pN pN | S †

ˆ ˆ

S

=

p1 ,...,pN

|p1 , p2 , . . . , pN p1 , p2 , . . . , pN | S †

ˆ ˆ

S

=

p1 ,...,pN

1

|p1 ∨ p2 ∨ · · · ∨ pN p1 ∨ p2 ∨ · · · ∨ pN | .

= (1.48)

N! p

1 ,···,pN

The symmetric states introduced in Eq. (1.45) are not normalized in general, since

for bosons the momenta need not di¬er. Of course, if the momentum values are all

di¬erent, the state |p1 ∨ p2 ∨ · · · ∨ pN is a sum of N ! normalized N -particle states

which are all orthogonal to each other, and the state is therefore normalized in view

of the overall prefactor. However, if say n1 of the momentum values equals p1 and all

the rest are di¬erent, the state |p1 ∨ p2 ∨ · · · ∨ pN will be a sum of N !/n1 ! N -particle

states each orthogonal to each other but now appearing with the prefactor n1 ! ,

since permutations among the identical labels produce the same N -particle state. In

general, if ni is the number of times pi occurs among the vectors p1 , p2 , . . . , pN , nj

being equal to 0 if the momentum value pj does not appear, then the set of ordered

vectors, choosing for example the ordering according to |p1 | ¤ |p2 | ¤ · · · ¤ |pN |,

1 N!

√ ˆ

|p1 ∨ p2 ∨ · · · ∨ pN S |p1 , p2 , . . . , pN (1.49)

=

n1 !n2 ! · · · nN !

n1 !n2 ! · · · nN !

constitute an orthonormal basis for the symmetric state space.

Equivalently we can state for the scalar product in Eq. (1.46)

§

⎨ n1 ! n2 ! · · · {p }i ≡ {p}i

p1 ∨ · · · ∨ pN |p1 ∨ · · · ∨ pN = (1.50)

©

0 otherwise.

The resolution of the identity in the symmetric N -particle state space can there-

fore also be expressed in terms of orthonormal states according to

1

ˆ(N ) |p1 ∨ p2 ∨ · · · ∨ pN p1 ∨ p2 ∨ · · · ∨ pN |. (1.51)

IS =

n1 !n2 !n3 ! · · ·

|p1 |¤|p2 |¤···¤|pN |

1.2. N-particle system 13

1.2.4 Dynamics and probability current and density

The quantum dynamics of an N -particle system of identical particles is given by the

Schr¨dinger equation

o

‚ψ(x1 , x2 , . . . ; t)

i = H ψ(x1 , x2 , . . . ; t) , (1.52)

‚t

where H is the Hamiltonian in the position representation for the N -particle sys-

tem. For example, for the case of N non-relativistic electrons interacting through

instantaneous two-particle interaction the Hamiltonian is

N 2

1 ‚ 1

V (xi ’ xj ) .

H= + (1.53)

2m i ‚xi 2

i=1 i=j

In non-relativistic quantum mechanics the even or odd character of a wave function

is preserved in time as any Hamiltonian for identical particles is symmetric in the

degrees of freedom, here in the momenta and positions, but as well as other degrees

of freedom in general (this is the meaning of identity of particles, no interaction can

distinguish them). So if even- or oddness of a wave function is the state of a¬airs at

one moment in time it will stay this way for all times.9

All physical properties are expressible in terms of the wave function, for example

the average density of the particles, or rather the probability for the event that one

of the particles is at position x, is

N N

1

dxj δ(xi ’ x) |ψ(x1 , x2 , . . . ; t)|2 = dxi |ψ(x, x2 , . . . ; t)|2

n(x, t) =

N

i=1 j=1 i=1

(1.54)

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

particles.

Taking the time derivative of the probability density, and using the Schr¨dinger

o

equation, gives the continuity equation