Now add interactions:
It means anything not quadratic in fields and derivatives.
For example, we can add
 3 ( x),  4 ( x),  ( x) ( x) ( x)
to our Klein-Gordon or Dirac Lagrangian.
These terms will add non-linear terms to the KG equation.
Interaction Hamiltonian:
Now we can study the time evolution.
Schrodinger Picture
States evolve with time, but not the operators.
It is the default choice in wave mechanics.
狀態
波函數
測量
算子
Y(x, t) = x Y(t)
Oˆ
ö
¶Y æ 2 ¶2
ˆ
i
= ç+V (x)÷ Y = HY
2
¶t è 2m ¶x
ø
Evolution Operator:
xˆ  x


ˆp    i 
x 

Only the expectation values are observable.
For the same evolving expectation value, we can instead ask operators to evolve.
Heisenberg Picture
Now the states do not evolve.
We move the time evolution to the operators:
Heisenberg Equation
The rate of change of operators equals their commutators with H.
Heisenberg Picture
Combine the two eq.
KG Equation
Interaction picture
H  H 0  H int
= yI (t) eiH0t/ BS e-iH0t/ yI (t)
We can move just the free H0 to operators.
OI (t) = eiH0t/ OS e-iH0t/
States and Operators both evolve with time in interaction picture:
B = y I (t) BI y I (t) = y I (t) eiH 0t/ BS e-iH 0t/ y I (t)
= yS (t) BS yS (t)
S
Evolution of Operators
OI (t )  eiH0t /  OS eiH0t / 
dOI i
 H 0 , OI 
dt

Operators evolve just like operators in the Heisenberg picture
but with the full Hamiltonian replaced by the free Hamiltonian
For field operators in the interaction picture:
df I i
= [ H 0, fI ]
dt
Field operators are free, as if there is no interaction!
The Fourier expansion of a free field is still valid.
Evolution of States
S
d
 i
 I (t )     H I  I (t )
dt
 
H I (t) = eiH0t/ H int S e-iH0t/
States evolve like in the Schrodinger picture but with
Hamiltonian replaced by HI(t).
HI(t) is just the interaction Hamiltonian Hint in interaction picture!
That means, the field operators in HI(t) are free.
Interaction Picture
Operators evolve just like in the Heisenberg picture but with
the full Hamiltonian replaced by the free Hamiltonian
d i
 H 0 ,  
dt 
States evolve like in the Schrodinger picture but with the full
Hamiltonian replaced by the interaction Hamiltonian.
d
 i
 I (t )     H I  I (t )
dt
 
U operator, the evolution operator of states in the interaction picture
d
y I (t) = -i H I y I (t)
dt
Define time evolution operator U
 I (t )  U (t, t0 )  I (t0 )
d
d
yI (t) = (U(t, t0 ) yI (t0 ) ) = -iH I (U(t, t0 ) yI (t0 )
dt
dt
)
d
U(t, t0 ) = -i H I ×U(t, t0 )
dt
All the problems can be answered if we are able to calculate
this operator. It determines the evolution of states.
U(t, t0 ) y (t0 ) is the state in the future t which evolved from a state ψ in t0
U(¥,-¥) y is the state in long future which evolved from a state ψ from long past.
The amplitude for this state to appear as a state ϕ is their inner product:
f (U(¥,-¥) y ) = f U(¥,-¥) y
The transition amplitude for the decay of A:
can be computed:
A ® B+C
BC U I , A  BC S A
The transition amplitude for the scattering of A:
A+ A ® B + B
can be computed: BB UI (¥,-¥) AA = BB S AA
Perturbation expansion
d
U(t, t0 ) = -i H I ×U(t, t0 )
dt
Solve it by a perturbation expansion in small parameters in HI.
U (t, t0 )  U (0) (t, t0 )  U (1) (t, t0 )  
U (0) (t , t0 )  1
To leading order:
dU (1) (t, t0 )
= -i H I (t)×U (0) (t, t0 ) = -i H I (t)
dt
t
U (1) (t, t0 ) = -i ò dt ' H I (t '')
t0
¥
U (1) (¥,-¥) = -i ò dt ' H I (t '')
-¥
Define S matrix:
¥
¥
¥
S = -i ò dt H(t) = -i ò dt dx H I (x, t) = i ò d 4 x LI (x)
3
-¥
-¥
-¥
It is Lorentz invariant if the interaction Lagrangian is invariant.
Vertex
In ABC model, every particle corresponds to a field:
A  A ( x)  A( x)
Add an interaction term in the Lagrangian:
The transition amplitude for the decay of A:
can be computed:
BC U I , A  BC S A
To leading order:
ˆ B(x)
ˆ
ˆ C(x)
B ( p2 ) C ( p3 ) ig ò d 4 x A(x)
A ( p1 )
ò
d3 p
2w P ( 2p )
3
( aˆ
p ×e
-ip×x
+
p
+ aˆ × e
(
ip×x
) ò
)
» ( aˆ + aˆ ) × bˆ + bˆ × ( cˆ + cˆ
+
+
bˆ
(
( 2p )
d 3 p'
2w P'
+
) ò
3
p'
× e-ip'×x + bˆp'+ × eip'×x
d 3 p''
2w P'' ( 2p )
)
-ip''×x
+
ip''×x
ˆ
ˆ
c
×
e
+
c
×
e
)
p''
3 ( p''
ˆ B(x)
ˆ
ˆ C(x)
B ( p2 ) C ( p3 ) ig ò d 4 x A(x)
A ( p1 )
aˆ p1 bˆ+p2
aˆ p A ( p1 ) = 0 × d 3 ( p - p1 )
cˆ+p3
cˆ+p'' 0 = C ( p'')
bˆ+p' C ( p'') = B ( p') C ( p'')
ò
d3 p
2w P ( 2p )
-ip×x
+
ip×x
ˆ
ˆ
a
×
e
+
a
×
e
(
)
p
p
3
ò
» ( a + a ) × (b + b ) × (c + c
+
d 3 p'
+
2w P' ( 2p )
+
) ò
3
(
bˆp' × e-ip'×x + bˆp'+ × eip'×x
d 3 p''
2w P'' ( 2p )
3
(cˆ
)
-ip''×x
+
ip''×x
ˆ
×
e
+
c
×
e
)
p''
p''
B ( p2 ) C ( p3 ) ig ò d 4 x A(x)B(x)C(x) A ( p1 )
aˆ p1
aˆ p A ( p1 ) = 0 × d 3 ( p - p1 )
bˆ+p2
cˆ+p3
cˆ+p'' 0 = C ( p'')
bˆ+p' C ( p'') = B ( p') C ( p'')
The remaining numerical factor is:
B
C
ig
A
Momentum Conservation
For a toy ABC model
Three scalar particle with masses mA, mB ,mC
pi
1
External Lines
qi
i
q 2j  m 2j  i
Internal Lines
Lines for each kind of particle with appropriate masses.
C
k2
Vertex
A
k1
-ig
k3
2 4 4 k1  k2  k3 
B
The configuration of the vertex determine the interaction of the model.
A
 aa

  a  a
B
C
interaction Lagrangian
  a  a
vertex
Every field operator in the interaction corresponds to one leg in the vertex.
Every field is a linear combination of a and a+
 a  a
Every leg of a vertex can either annihilate or create a particle!
This diagram is actually the combination of 8 diagrams!
A
B
C
Interaction Lagrangian
vertex
The Interaction Lagrangian is integrated over the whole spacetime.
Interaction could happen anywhere anytime.
The amplitudes at various spacetime need to be added up.
The integration yields a momentum conservation.
In momentum space, the factor for a vertex is simply a constant.
LI = lf 4
 a  a
Interaction Lagrangian
Vertex
Every field operator in the interaction corresponds to one leg in the vertex.
Every leg of a vertex can either annihilate or create a particle!
Propagator
Solve the evolution operator to the second order.
d
U(t, t0 ) =- i H I ×U(t, t0 )
dt
t
dU (2) (t, t0 )
(1)
= -i H I (t)×U (t, t0 ) = -H I (t)× ò dt ' [ H I (t ')]
dt
t0
HI is first order.
t ''
é
ù
U (t, t0 ) = - ò dt ''ê H I (t '') ò dt ' [ H I (t ')]ú
êë
úû
t0
t0
t
(2)
The integration of two identical interaction Hamiltonian HI.
The first HI is always later than the second HI
t ''
é
ù
U (t, t0 ) = - ò dt ''ê H I (t '') ò dt ' [ H I (t ')]ú
êë
úû
t0
t0
t
(2)
The integration of two identical interaction Hamiltonian HI.
The first HI is always later than the second HI
t’ and t’’ are just dummy notations and can be exchanged.
t ''
t'
é
ù t é
ùüï
1 ìï t
U (t, t0 ) = - í ò dt ''ê H I (t '') ò dt ' [ H I (t ')]ú + ò dt 'êH I (t ') ò dt '' [ H I (t '')]úý
2 ïî t0
êë
úû t0 êë
úûïþ
t0
t0
(2)
t’’
t ''
t
t'
üï
1 ìï t
= - í ò dt '' ò dt ' [ H I (t '')× H I (t ')] + ò dt ' ò dt '' [ H I (t ')× H I (t '')]ý
2 ïî t0
ïþ
t0
t0
t0
We are integrating over the whole square but always
keep the first H later in time than the second H.
t’
1 t
U (t, t0 ) = - × ò dt ''
2 t0
(2)
t
ò dt ' T [ H (t '') H (t ')]
I
I
t0
T ( A(t1 ) B(t 2 ))   (t1  t 2 ) A(t1 ) B(t 2 )   (t 2  t1 ) B(t 2 ) A(t1 )
t
1
U (2) (t, t0 ) = - × ò dt ''
2 t0
t’’
t
ò dt ' T [ H (t '') H (t ')]
I
I
t0
T ( A(t1 ) B(t 2 ))   (t1  t 2 ) A(t1 ) B(t 2 )   (t 2  t1 ) B(t 2 ) A(t1 )
This definition is Lorentz invariant!
t’
1
S (2) = U (2) (¥,-¥) = - × ò d 4 x1 d 4 x2 T [ LI (x1 ) LI (x2 )]
2
This notation is so powerful, the whole series of
operator U can be explicitly written:
U (t, t0
(n)
-i)
(
)=
n!
n
t
× ò dt1 dt2
dtn T [ H I (t1 ) H I (t2 )
H I (tn )]
t0
The whole series can be summed into an exponential:
ìï é t
ùüï
U(t, t0 ) = T íexp ê-i ò dt ' H I (t ')úý
úûïþ
ïî êë t0
S
(2)
1
= U (¥, -¥) = - × ò d 4 x1 d 4 x2 T [ LI (x1 ) LI (x2 )]
2
Amplitude for scattering A  A  B  B
(2)
B(p3 )B(p4 ) S A(p1 )A(p2 )
= B(p3 )B(p4 )
ò d x d x T [ g A(x ) B(x ) C(x ) × g A(x ) B(x ) C(x )] A(p )A(p )
4
4
1
2
1
1
1
2
2
2
1
2
  d 4 x1 d 4 x2 e i ( p1  p3 ) x2 e i ( p2  p4 ) x1  0 T  C ( x1 ) C ( x2 ) 0
Fourier Transformation
Propagator between x1 and x2
p1-p3 pour into C at x2 p2-p4 pour into C at x1
t1  t2

0 T C( x1 )  C( x2 )  0  0 C( x1 )  C( x2 ) 0  0 a  a
  a  a 

x1
A particle is created at x2 and later annihilated at x1.
B(p4)
B(p3)
B(p3)
B(p4)
x1
x2
A(p1)
C(p1-p3)
C
A(p2)
A(p1)
A(p2)
x2
0
t1  t2

0 T C( x1 )  C( x2 )  0  0 C( x2 )  C( x1 ) 0  0 a  a
  a  a 

x2
A particle is created at x1 and later annihilated at x2.
B(p4)
B(p3)
B(p4)
B(p3)
x2
B(p4)
x1
C
A(p1)
B(p3)
x1
A(p2)
x2
A(p1)
C(p1-p3)
C
A(p2)
A(p1)
A(p2)
x1
0
B(p3 )B(p4 ) S A(p1 )A(p2 ) = -g2 ò d 4 x1 d 4 x2 e-i( p4 -p2 )x1 e
B(p4)
B(p3)
B(p4)
B(p3)
x2
B(p3)
0 T [ C(x1 ) C(x2 )] 0
B(p4)
x1
C
A(p1)
-i( p3 -p1) x2
x1
A(p2)
x2
A(p1)
C(p1-p3)
C
A(p2)
A(p1)
A(p2)
This construction ensures causality of the process. It is actually the
sum of two possible but exclusive processes.
Again every Interaction is integrated over the whole spacetime.
Interaction could happen anywhere anytime and amplitudes
need superposition.
This propagator looks reasonable in coordinate space but
difficult to calculate and the formula is cumbersome.
0 T   ( x)  ( y) 0
0 T   ( x)  ( y) 0
0 T   ( x)  ( y) 0
0 T   ( x)  ( y) 0
This doesn’t look explicitly Lorentz invariant.
But by definition it should be!
So an even more useful form is obtained by extending the
integration to 4-momentum. And in the momentum space,
it becomes extremely simple:
The Fourier Transform of the propagator is simple.
4
4
-i( p4 -p2 )x1
d
x
d
x
e
e
ò 1 2
-i( p3 -p1 ) x2
0 T [ C(x1 ) C(x2 )] 0 =
B(p4)
B(p3)
B(p4)
B(p3)
x2
4
d
(p4 - p2 + p3 - p1 )
2
( p1 - p3 ) - mC
2
B(p3)
B(p4)
x1
C
A(p1)
i
x1
A(p2)
x2
A(p1)
C(p1-p3)
C
A(p2)
A(p1)
A(p2)
0 T   ( x)  ( y) 0
x0  y0
y0 > x0
iw k ( x 0 - y 0 )
-ik ( y - x )
0 T   ( x)  ( y) 0
B(p4)
B(p3)
B(p4)
B(p3)
x2
B(p4)
x1
C
A(p1)
B(p3)
x1
A(p2)
x2
A(p1)
C(p1-p3)
C
A(p2)
A(p1)
A(p2)
For a toy ABC model
qi
Internal Lines
i
q 2j  m 2j  i
Lines for each kind of particle with appropriate masses.
1
S (2) = U (2) (¥, -¥) = - × ò d 4 x1 d 4 x2 T [ LI (x1 ) LI (x2 )]
2
Amplitude for scattering A  A  B  B
B(p3 )B(p4 ) S A(p1 )A(p2 )
= B(p3 )B(p4 )
4
4
d
x
d
ò 1 x2 T [ g A(x1 ) B(x1 ) C(x1 ) × g A(x2 ) B(x2 ) C(x2 )] A(p1 )A(p2 )
  d 4 x1 d 4 x2 e i ( p1  p3 ) x2 e i ( p2  p4 ) x1  0 T  C ( x1 ) C ( x2 ) 0
 a  a
 a  a
 a  a



Every field either couple with another field to form a propagator or
annihilate (create) external particles!
Otherwise the amplitude will vanish when a operators hit vacuum!
For a toy ABC model
Three scalar particle with masses mA, mB ,mC
pi
1
External Lines
qi
i
q 2j  m 2j  i
Internal Lines
Lines for each kind of particle with appropriate masses.
C
k2
Vertex
A
k1
-ig
k3
2 4 4 k1  k2  k3 
B
The configuration of the vertex determine the interaction of the model.
Scalar Antiparticle
Assuming that the field operator is a complex number field.
L0 = ¶m F+¶m F - m2F+F

d3p
( x)  
(2 ) 3
1
2
a e
ipx
p
 b p e ipx

The creation operator b+ in a complex KG field can create a different particle!
H=
P=
Q=
ò
d3 p
(2p )3
ò
d3p
+
+
p
×
a
a
+
b
b
(
p p
p p)
(2p )3
ò
d3p
+
+
a
a
b
b
(
p p
p p)
(2p )3
p + m 2 × ( a p a+p + bp bp+ )
2
The particle b+ create has the same mass but opposite charge.
b+ create an antiparticle.

d3p
( x)  
(2 ) 3
1
2
a e
ipx
p
 b p e ipx

Complex KG field can either annihilate a particle or create an antiparticle!

d3p
 ( x)  
(2 ) 3

1
2
b e
p
ipx
 a p e ipx

Its conjugate either annihilate an antiparticle or create a particle!
The charge difference a field operator generates is always the same!
So we can add an arrow of the charge flow to every leg that corresponds
to a field operator in the vertex.
LI = gF3 + gF+3

d3p
( x)  
(2 ) 3
1
2
a e
p
ipx
 ipx
p
b e

incoming particle or
outgoing antiparticle

d3p
 ( x)  
(2 ) 3

1
2
b e
p
ipx
 a p e ipx
incoming antiparticle or
outgoing particle
charge non-conserving interaction

LI = l × (F F)
+
2
incoming antiparticle or
outgoing particle
incoming particle or
outgoing antiparticle
charge conserving interaction
f » c + c+
LI = g f Y*Y
Y » b + a+
incoming antiparticle or
outgoing particle
Y » a + b+
incoming particle or
outgoing antiparticle
Y » a + b+ can either annihilate a particle or create an antiparticle!
Y » b + a+ can either annihilate an antiparticle or create a particle!
U(1) Abelian Symmetry
L0 = ¶m F+¶m F - m2F+F
The Lagrangian is invariant under the field phase transformation
( x)  e iQ ( x)
    eiQ  eiQ  (x)  
LI = l × (F F)
+
2
LI = gF3 + gF+3
invariant
is not invariant
U(1) symmetric interactions correspond to charge conserving vertices.
If A,B,C become complex, they all carry charges!
C
The interaction is invariant only if QA  QB  QC  0
The vertex is charge conserving.
A
B
Propagator:


0 T  ( x1 ) ( x2 ) 0
t1  t2



0 T  ( x1 )  ( x2 ) 0  0 ( x1 )  ( x2 ) 0  0 b  a
  a  b 
An antiparticle is created at x2 and later annihilated at x1.
B(p4)
B(p3)
B(p3)
B(p4)
x1
x2
A(p1)
C(p1-p3)
C
A(p2)
A(p1)
A(p2)

x1
x2
0
t1  t2



0 T  ( x1 )  ( x2 ) 0  0 ( x2 )   ( x1 ) 0  0 a  b
  b  a 

x2
A particle is created at x1 and later annihilated at x2.
B(p4)
B(p3)
B(p4)
B(p3)
x2
B(p4)
x1
C
A(p1)
B(p3)
x1
A(p2)
x2
A(p1)
C(p1-p3)
C
A(p2)
A(p1)
A(p2)
x1
0


ipx1 iqx2
4
4

d
x
d
x
e
e
0
T

( x1 ) ( x2 ) 0 
 1 2
B(p4)
B(p3)
B(p4)
B(p3)
x2
B(p3)
B(p4)
x1
C
A(p1)
i
4

( p  q)
2
2
q  mC
x1
A(p2)
x2
A(p1)
C(p1-p3)
C
A(p2)
A(p1)
A(p2)
LI = l × B ( A+ A)
B(p4)
B(p3)
B(p4)
B(p3)
x2
x1
C
A(p1)
x1
A(p2)
x2
A(p1)
B(p3)
B(p4)
B ( A+ A)
B ( A+ A)
C(p1-p3)
C
A(p2)
A(p1)
A(p2)
For a toy charged AAB model
Three scalar charged particle with masses mA, mB
pi
1
External Lines
qi
i
q 2j  m 2j  i
Internal Lines
Lines for each kind of particle with appropriate masses.
B
k2
Vertex
-iλ
2 4 4 k1  k2  k3 
A
A
k1
k3
Dirac field and Lagrangian
The Dirac wavefunction is actually a field, though unobservable!
Dirac eq. can be derived from the following Lagrangian.
L   

  

L L

     i     m     i     m  
 
i




   m   0   i     m  0
i




   m   0   i     m  0
Negative energy!
Anti-commutator!
A creation operator!
~
~
b  b, b   b
b annihilate an antiparticle!
a , a  0  a a


p


p
a p a p  0

a p 0


p
Exclusion Principle


p


p


p
a a


p
L I  g 
  a  b
  b  a
External line
When Dirac operators annihilate states, they leave behind a u or v !
 ( x)   a  u  eipx  b  v  eipx 
e  ( p1 )

p

 
3
a p ' p  2 p  2    p  p'  0
Feynman Rules for an incoming particle
 ( x)   b  v  eipx  a   u  eipx 

p
u p1 0
e  ( p1 )
v p1 0
Feynman Rules for an incoming antiparticle
A ( x)  a     a   
LI  gA  

g  
  b  a
u p2
  a  b
u p1