Download 6. Second Quantization and Quantum Field Theory

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.

k1 
 kN

k1 ,
, kN

is an orthonormal basis.
Uncertainty principle → identical particles are indistinguishable.
→

, ki ,
  1
,kj,

bosons
fermions
,kj,
, ki ,
2
, ki ,
Bose-Einstein
statistics
Fermi-Dirac
Spin-statistics theorem: this association is due to causality.
,kj,
 i, j
integral
spin
half-integral
Basis with built-in exchange symmetry:
, k N  C     kP1 
k1,
bosons
fermions
 k P N 
P
P
1,
P denotes a permutation
 
P
1

1
if P consists of an
n !
j
C 
With
k1 ,
k ,
1
, N   P 1,
, kN
,k N  , k1 ,
j
N!
k1,
,

k1 ,
even
odd
, kN
, k N '  k1 ,
1

,k N ' 
0
if

, N    P 1 ,
, P  N  
number of transpositions (exchanges)
is orthonormal:
,k N  , k1 , ,k N ' 
k1,
, k N   P k1,
otherwise
2 states with N  N  are always orthogonal.
, k N ' 
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  = | 111 , | 2  = | 110 , | 3  = | 111 , …
Number (n-) representation:
Basis = (symmetrized) eigenstates of number operator
nˆ n0 , n1,
n0 , n1 ,
 n n0 , n1,
, n ,
, n ,
 t  
n0 , n1,

nα= number of particles in | α 
, n ,
, n ,
  n0 n0  n1 n1
n0 , n1 ,
, n ,
n0 , n1 ,
n0 , n1,
, n ,
n0 ,n1 ,
 n n
, n ,
n0 ,n1 , ,n ,


n0 ,n1 , ,n ,
,n ,
t 
orthonormality
 t 
Creation and Annihilation Operators
Conjugate variables in n-rep:
annihilation operators â
 A  n  n0 , n1, , n  1,
aˆ n0 , n1, , n ,
 C  n  n0 , n1, , n  1,
nˆ n0 , n1,
 n n0 , n1,
n  n0 , n1 ,
 A  n 
, n ,
2
n0 , n1 ,

a a n0 , n1,
, n  1,
 C  n 1 A  n  n0 , n1 ,
â
creation operators
aˆ n0 , n1, , n ,
, n ,

nˆ  aˆ aˆ
, n ,
, n ,
n0 , n1,
, n ,
  â 
, n  1,
n0 , n1,
, n ,
 A  n 
2
 C  n 1 A  n 

n  A  n   C  n  1 A  n 
2
A,C real →
For bosons, nα = 0, 1, 2, 3, …
aˆ n0 , n1 ,
, n ,
 A  n  n0 , n1 ,
, n 1,
aˆ n0 , n1 ,
, n ,
 C  n  n0 , n1 ,
, n  1,
 n n0 , n1 ,
, n  1,
 n  1 n0 , n1 ,
, n  1,
→ A(0) = 0, C(1) = 0 and 1 = C(0) A(1).
For fermions, nα = 0, 1
Set: C(0) = A(1) = 1.
aˆ n0 , n1 ,
, n ,
 
S
aˆ n0 , n1 ,
, n ,
 
S
  
S
n n0 , n1 ,
, n  1,
1  n n0 , n1 ,
, n  1,
1  n  n0 , n1 ,
, n  1,
     n n0 , n1 ,
S
, n  1,
S 
 1
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).
0
Commutation Relations
Exchange symmetries of states  Commutation relations between operators
 aˆ 
 aˆ 
 aˆ 
 C m   C m   C m 
 n
n0 , n1 ,
, n ,


n  1
m  0
  0,0,
 n1
1
 n0
0
n1 1

m1  0
n0 1
1
m0  0

Fock
space
0
is the “vacuum”.
n  1
For fermions, nα = 0, 1 →
 C  m   1
m  0
Exchange symmetries are established by requiring
  a, b 
a, b  ab ba  
 a, b
 α
aˆ , â   0
Commutator
Anti-commutator
Boson
Fermion
aˆ , â   0
, n  1,
→
, n  1,
 aˆ aˆ
 aˆ , aˆ 
aˆ aˆ 
 n 1  n 
n 1  n   0

 


1  n 1  n  n  1

→
aˆ , â 
 
0
, n ,
, n ,
 
for
 
Number Representation: Operators
1-P operator :
A  p, x      A  

A   A 
  â 

→
The vacuum projector
i.e.,
 Â   0
  A   

= matrix elements
Aˆ   A  aˆ   aˆ

 
confines A to the 1-particle subspace.
if the number of particles in either  or  is not one.
Many body version :
Aˆ   A  aˆ aˆ

2-Particle Potential
N
1 N
V   V  xi , x j    V  xi , x j 
2 i  j 1
i  j 1
 1 
Basis vector for the 2-P Hilbert space:
 1 
2



2
Completeness condition:
 1   2 
I 
 ,
V

1


2  
1

2 2

1
1
V 

1
2 2

1
1 2 1 2 V  1 2  1 2

2  
1

2 2
2

1

2
 1 2
  1 2
1

   1 2
 ,
1

1  2
1
1 2 1 2 V  1 2  1 2

2 
V
1 2  
 1 2  2 
1

1

2
 aˆ aˆ 

1

2


  aˆ aˆ 


  aˆ aˆ
1
ˆ
V   aˆ aˆ    V    aˆ aˆ
2 
 V    d 3x1  d 3x 2 *  x1  *  x 2 V  x1 , x 2    x 2    x1 
 
confines V to the 2-particle subspace.
Many body version :
1
Vˆ   aˆ aˆ   V   aˆ aˆ
2 
Summary
nˆ  aˆ aˆ
aˆ , aˆ 
aˆ , aˆ    aˆ , aˆ   0
nˆ n0 , n1,
, n ,
 n n0 , n1,
aˆ n0 , n1 ,
, n ,
 
aˆ n0 , n1 ,
, n ,
 
n0 , n1 ,
, n ,
n0 , n1,
1-P operator:
S
S
, n ,
n n0 , n1 ,
, n  1,
1  n n0 , n1 ,
, n ,
  ~     
, n  1,
  n0 n0  n1 n1
 n n
Aˆ    A  aˆ aˆ

2-P operator:
1
ˆ
V     V   aˆ aˆ aˆ aˆ
2 
S 
 1
n


0
6.2.
Field Operators and Observables
Momentum eigenstaes for spinless particles:   k 
Orthonormality:
k k    2    k  k  
3
p
Completeness:
x x    x  x
d 3k
  2 
3
k k 1
3
d
 r x x 1
k  x   x k  e i k x
  x, t   x   t   

where
d 3k
 2 
k  k
3
3
d k
 2 
3
x k k  t 

ei kx 0 aˆ  k    t 


 aˆ  k  0 


 0 aˆ  k 
d 3k
 2 
3
ei k x k   t 
Field Operators
The field operators are defined in the Schrodinger picture by
ˆ  x    x  â    x  â

Momentum basis:
ˆ   x     x â  *  x  â


ˆ  x    d k k  x  aˆ  k 

3

d 3k
 2 
ˆ   x    d 3k k*  x  aˆ   k   
3
ei k x aˆ  k 
d 3k
 2 
3
ei k x aˆ   k 
Commutation relations :
ˆ  x  ,ˆ  x    x  x 

 aˆ , aˆ 
0
ˆ   x  ,ˆ   x    x  x aˆ , aˆ   0

ˆ  x  ,ˆ   x    x   x  aˆ , â    x   x  x x    x  x


ˆ  x   ˆ   x ˆ  x 
 d x ˆ  x    d x ˆ
3
3


 x ˆ  x 
d 3k 
d 3k
3
3
 2    2 

d 3k
 2 
d 3k
d 3k 
3
3
 2    2 
ei k x aˆ   k  ei k x aˆ  k  
aˆ   k  aˆ  k    2    k  k  
aˆ  k  aˆ  k   

3
 d x 
3
3
d 3k
 2 
3
nˆ  k   N = total number of particles
 ρ(x) is the number density operator at x.
Aˆ  
d 3k 
d 3k
 2    2 
3
3
k A  x, p  k  aˆ  k  aˆ  k  



  d 3 x ˆ   x  A  x,  ˆ  x 
 i 
3
3
3
3
d
k
d
k
d
k
d
k4
1


3
1
2

ˆ
ˆ
Vˆ  
k
k
V
x
,
x
k
k
a
k
a




 k 2  aˆ  k 4  aˆ  k 3 
1
2
3
4
1
3
3
3
3
2  2   2   2   2 

1 3
3


ˆ
ˆ

d
x
d
x

x



 x V  x, x ˆ  x ˆ  x 


2
6.3. Equation of Motion & Lagrangian Formalism for Field Operators
Heisenberg picture:
i ˆ ˆ
 i

H t   x  exp   Hˆ t 




ˆ  x, t   exp 
 i ˆ  ˆ
 i

H t   x  exp   Hˆ t 




ˆ   x, t   exp 
i

 i

Hˆ  p, x, t   exp  Hˆ t  Hˆ  p, x  exp   Hˆ t   Hˆ  p, x 




Equal time commutation relations:
ˆ  x, t  , ˆ  x, t   ˆ   x, t  , ˆ   x, t   0
ˆ  x, t  , ˆ   x, t     x  x 
2


Hˆ   d x ˆ  x, t   
 2  U  x  ˆ  x, t 
 2m

1
  d 3 x  d 3 x ˆ   x, t  ˆ   x, t  V  x, x ˆ  x, t ˆ  x, t 
2
3

Equation of Motion
i

ˆ  x, t   ˆ  x, t  , Hˆ 
t
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  x, t  , ˆ   x, t ˆ  x, t  
 ˆ  x, t  , ˆ   x, t  ˆ  x, t   ˆ   x, t  ˆ  x, t  , ˆ  x, t  
   x  x ˆ  x,t 
ˆ  x, t  , ˆ   x, t ˆ   x, t  
 ˆ  x, t  , ˆ   x, t   ˆ   x, t   ˆ   x, t  ˆ  x, t  , ˆ   x, t 
   x  x ˆ   x, t     x  x ˆ   x, t 
i
2



ˆ  x, t    
2  U  x  ˆ  x, t    d 3 x ˆ   x, t V  x, x ˆ  x, t ˆ  x, t 
t
 2m

Lagrangian
2
 

S   dt  d x  *  x, t   i

2  U   x, t    dt  d 3 x L
 t 2m

3
 is complex → it represents 2 degrees of freedom ( Re , Im ) or ( ,  * ).
Variation on  * :
L
0
   t *
E-L eq:
→
L
0
   j *
2

L  
 i

2  U 
 *  t 2m

L
L
L
 t
j
0
 *
   t *
   j *
2

 
i
 
2  U 
t  2m

Schrodinger equation
Variation on  :
L
i  *
  t 
→
2
L

 j *
2
m
   j 
L
 U *

integration
by part
2

 * 
i
 
2  U  *
t
 2m

L
i  *
Generalized momentum conjugate to  =  
   t 
Hamiltonian density
→
2


H    t  L   *  
2  U  
 2m

2


H   d x * 
2  U  
 2m

Quantization rule:
3
~ Classical field
ˆ  x, t    i   x  x
ˆ  x, t  , 


Quantum field theory