6. Second Quantization and Quantum Field Theory

6.0. Preliminary

6.1. The Occupation Number Representation

6.2. Field Operators and Observables

6.3. Equation of Motion and Lagrangian Formalism for Field Operators

6.0. Preliminary

Systems with variable numbers of particles ~ Second quantization• High energy scattering and decay processes.

• Relativistic systems.

• Many body systems (not necessarily relativistic).

1st quantization:

• Dynamical variables become operators;

• E, L, … take on only discrete values.

2nd quantization:

• Wave functions become field operators.

• Properties described by counting numbers of 1-particle states being occupied.

• Processes described in terms of exchange of real or virtual particles.

For system near ground state:

→ Quasi-particles (fermions) or elementary excitations (bosons).

→ Perturbative approach.

6.1. The Occupation Number Representation

Many body problem ~ System of N identical particles.

{ | k } = set of complete, orthonormal, 1-particle states that satisfy the BCs.

1 1, ,N Nk k k k is an orthonormal basis.

Uncertainty principle → identical particles are indistinguishable.

→ , , , , , , , ,i j j ik k k k ,i j2 , , , ,i jk k

1 bosons fermions

Bose-Einstein Fermi-Dirac

statisticsintegral half-integral

spin

Spin-statistics theorem: this association is due to causality.

Basis with built-in exchange symmetry:

1 1, ,P

N P P NP

k k C k k bosons fermions

P denotes a permutation 1, , 1, , 1 , ,N P N P P N

1

1P

if P consists of an

even odd

number of transpositions (exchanges)

!

!

jj

n

CN

is orthonormal:

1, , Nk k

1 1 '1 1 ' , , , , ,, , , ,N NN N k k k kk k k k

1 1 ', , , , ,

1

0N Nk k k k

1 1 ', , , ,N Nk k P k k

otherwise

if

With ,

2 states with N N are always orthogonal.

Number Representation: States

Let { | α } be a set of complete, orthonormal 1-P basis.

α = 0,1,2,3,… denotes a set of quantum numbers with increasing E.

E.g., one electron spinless states of H atom: | α = | nlm

| 0 = | 100 , | 1 = | 111 , | 2 = | 110 , | 3 = | 111 , …

Number (n-) representation:

Basis = (symmetrized) eigenstates of number operator

0 1 0 1ˆ , , , , , , , ,n n n n n n n n nα= number of particles in | α

0 0 1 10 1 0 1, , , , , , , , n n n n n nn n n n n n orthonormality

0 1

0 1 0 1, , , ,

, , , , , , , ,n n n

t n n n n n n t

0 1

0 1

0 1 , , , ,, , , ,

, , , , n n nn n n

n tn n

Creation and Annihilation Operators

Conjugate variables in n-rep:

a aannihilation operators creation operators

ˆ ˆ ˆn a a

a

0 1 0 1ˆ , , , , , , , ,n n n n n n n n

0 1 0 1ˆ , , , , , , , 1,a n n n A n n n n

0 1 0 1ˆ , , , , , , , 1,a n n n C n n n n

0 1 0 1, , , , , , , ,n n n n a a n n n

2

0 1 0 1, , , 1, , , , 1,A n n n n n n n 2A n

1C n A n 0 1 0 11 , , , , , , , ,C n A n n n n n n n

For bosons, nα = 0, 1, 2, 3, …

0 1 0 1ˆ , , , , , , , 1,a n n n A n n n n

0 1 0 1ˆ , , , , , , , 1,a n n n C n n n n

For fermions, nα = 0, 1 → A(0) = 0, C(1) = 0 and 1 = C(0) A(1).

Set: C(0) = A(1) = 1.

0 1 0 1ˆ , , , , , , , 1,S

a n n n n n n n

0 1 0 1ˆ , , , , 1 , , , 1,S

a n n n n n n n

0 11 , , , 1,S

n n n n

0 1, , , 1,S

n n n n

21n A n C n A n A,C real →

0 1, , , 1,n n n n

0 11 , , , 1,n n n n

Completeness of this basis is with respect to the Fock space.

There exists many particle states that cannot be constructed in this manner. E.g., BCS states (Cooper pairs).

1

0

S n

Commutation Relations Exchange symmetries of states Commutation relations between operators

1 0

1 0

1 0

1 0

0 1 1 1 1

1 00 0 0

ˆ ˆ ˆ, , , ,

n n n

n n n

m m m

a a an n n

C m C m C m

0,0, is the “vacuum”.

For fermions, nα = 0, 1 → 1

0

1n

m

C m

α

Fock space

Exchange symmetries are established by requiring , 0ˆ aa

Boson Fermion

,a b ab ba

Commutator

Anti-commutator

,

,

a b

a b

, 0ˆ aa → ˆ ˆ 0,a a

ˆ ˆ, 1, , 1, , , , ,

1 1 0for

1 1 1

ˆ ˆn n a a n n

n n n n

n n n

a a

ˆ , aa →

Number Representation: Operators

1-P operator : ,A A p x A

A A = matrix elements

a ˆ ˆ ˆA A a a

→

The vacuum projector confines A to the 1-particle subspace.

i.e., 0A if the number of particles in either or is not one.

Many body version : ˆ ˆ ˆA A a a

2-Particle Potential

1

1,

2

N

i ji j

V V

x x 1

,N

i ji j

V

x x

Basis vector for the 2-P Hilbert space: 1 21 2 1 2

1 2 2 1

1 22 1

Completeness condition:

1 2 2 1,

I

,

1 2 1 2

1 2 2 1 1 2 2 1

1

2V V

1 2 1 2 1 2 1 2

1

2V

1 2 1 2 1 2 1 2

1

2V V

1 2 1 2 ˆ ˆa a

1 2 2 1 1 2

ˆ ˆa a

ˆ ˆa a

ˆ ˆ ˆ1

2ˆˆ a a VV a a

*1

3 3 *1 2 2 1 2 12,V d d V x x x x x xx x

confines V to the 2-particle subspace.

Many body version : ˆ ˆˆ1

2ˆˆ a Va aV a

Summary

ˆ ˆ ˆn a a

ˆ ˆ ˆ 0ˆ, ,a aa a

ˆ ~ˆ ,a a

0 1 0 1ˆ , , , , , , , ,n n n n n n n n

0 1 0 1ˆ , , , , , , , 1,S

a n n n n n n n

0 1 0 1ˆ , , , , 1 , , , 1,S

a n n n n n n n

0 0 1 10 1 0 1, , , , , , , , n n n n n nn n n n n n

ˆ ˆ ˆA A a a 1-P operator:

2-P operator: ˆˆ ˆ1

2ˆˆ V a aa aV

1

0

S n

6.2. Field Operators and Observables

Momentum eigenstaes for spinless particles: p

k

32 k k k k

3

3 12

d k

k k

Orthonormality: Completeness:

x x x x 3 1d r x x

k x x k ie k x

, t t x x

3

32

d kt

x k k

3

32

id ke t

k x k

3

3ˆ0

2ik

e tad

k x k

k k ˆ 0a k ˆ0 a kwhere

The field operators are defined in the Schrodinger picture by

ˆ a

xx a

x ˆ a xx * a

x

Momentum basis: 3ˆ ˆd k a kx x k

3

3ˆ

2id k

e a

k x k

3 *ˆ ˆd k a kx x k

3

3ˆ

2id k

e a

k x k

Commutation relations :

ˆ ˆ 0ˆ ˆ, , aa

x x x x

ˆ ˆ ˆ 0ˆ, ,a a

x x x x

ˆ ˆ ˆ, , aa

x x x x

x x x x x x

Field Operators

ˆ ˆ ˆ x x x

3 3ˆ ˆ ˆd x d x x x x

3 3

33 3

ˆ ˆ2 2

i id k d kd x e a e a

k x k xk k

3 33

3 3ˆ 2

2ˆ

2a

d k d ka

k k k k

3

3ˆ ˆ

2a a

d k

k k

3

3ˆ

2

d kn N

k = total number of particles

ρ(x) is the number density operator at x.

3 3

3 3ˆ,

2ˆ

2ˆA A

d ka a

d k

k x p k k k 3 ˆ ˆ,d x A

i

x x x

33 3 331 2 4

3 3 3 1 2 3 4 1 2 4 332 2 2

1ˆ ˆ ˆ ˆ ˆ,2 2

V Vd kd k d k d k

a a a a

k k x x k k k k k k

3 31ˆ ˆ ˆ ˆ,

2d x d x V x x x x x x

6.3. Equation of Motion & Lagrangian Formalism for Field Operators

Heisenberg picture:

ˆ ˆˆ ˆ, exp expi i

t H t H t

x x

ˆ ˆˆ ˆ, exp expi i

t H t H t

x x

ˆ ˆ ˆ ˆ, , exp , expi i

H t H t H H t

p x p x ˆ ,H p x

Equal time commutation relations:

ˆ ˆ ˆ ˆ, , , , , , 0t t t t x x x x

ˆ ˆ, , ,t t x x x x

23

3 3

2ˆ ˆ ˆ, ,2

1ˆ ˆ ˆ ˆ, , , , ,

2

H d x t tm

d x d x t

U

t V t t

x x

x x x x

x

x x

Equation of Motion ˆˆ ˆ, , ,i t t Ht

x x

,a bc abc bca abc bac bac bca , ,a b c b a c

,ab c abc cab abc acb acb cab , ,a b c a c b

ˆ ˆ ˆ, , , ,

ˆ ˆ ˆ ˆ ˆ ˆ, , , , , , , ,

t t t

t t t t t t

x x x

x x x x x x

ˆ , t x x x

ˆ ˆ ˆ, , , ,

ˆ ˆ ˆ ˆ ˆ ˆ, , , , , , , ,

t t t

t t t t t t

x x x

x x x x x x

ˆ ˆ, ,t t x x x x x x

2

32ˆ ˆ ˆ ˆ ˆ, , , , , ,2

i t t d x t tt m

U t V x x x x xx x x

Lagrangian

32

2* , ,2

S dt d x t Ui tt m

x x 3dt d x L

is complex → it represents 2 degrees of freedom ( Re , Im ) or ( , * ).

Variation on * :

0

*t

L

0*j

L 2

2

* 2U

t mi

L

0* * *t j

t j

L L L

E-L eq:

→2

2

2U

t mi

Schrodinger equation

Variation on :

*

t

i

L

2

*2 j

j m

L*U

L

→2

2 **

2U

t mi

integration by part

Generalized momentum conjugate to = *

t

i

L

Hamiltonian density t H L

22*

2U

m

32

2*2

Um

H d x

→

Quantization rule: ˆˆ , , ,t t i x x x x

Quantum field theory

~ Classical field