### Power Point File

The essence of Particle Physics
The essence of Particle Physics
Particles are actually not like balls but essentially more fields!
Well, not quite.
They are quantized fields.
Fields when quantized are not like fields any longer, but more like particles.
Quantum Field Theory
Classical Field Theory
f ( x, t )
f ( xm )
Space and time are treated equally as parameters.
It is manifestly Lorentz Invariant.
But they are quantized fields.
Dirac Notation of Quantum Mechanics

Ket


Bra


Oˆ
  Oˆ 

(
) (
)
Oˆ = Y Oˆ Y = Y Oˆ + Y = Y Oˆ Y
算子夾在 Ψ 及 Ψ 間的 matrix element

 (x )

Oˆ

 


i



x




Oˆ   * ( x)  Oˆ ( x)dx


 ( x)  x 
 * ( x)   x
Classical Field Theory
f ( x, t )
f ( xm )
Space and time are treated equally as parameters.
It is manifestly Lorentz Invariant.
Quantum Field Theory
f ( xm )
fˆ ( x m )
For a quantum theory of field, the field is promoted to operators!
It is still manifestly Lorentz Invariant.
Particle Quantum Mechanics
x(t )
ˆ
x(t)
Space is operator while time remains a number parameter.
That is why Particle QM can not be Lorentz invariant.

y
ri (t) ® yi (t) i =1 N

ix
yi (t )  y( x, t ) 所以位置基本上是足標，但與時間一樣都是數字！

Classical Field Theory
j ( x, t ) ® j ( x m )
It is just like electric field but simpler.

ϕ is a scalar not a vector as electric field: E
It is easiest to describe fields using Lagrangian and Hamiltonian.
 x  
Action is defined as the integral over time of the Lagrange.
S=

The equation of motion is given by the principle of Least Action.
For the fields spreading over space, the Lagrangian would be the
integral over space of the Lagrange Density (Lagrangian) ℒ.
L= ℒ   ,
Hence the action is an integral over spacetime.
3
Hence the action is a integral over spacetime.
The integration is Lorenz Invariant.
4  =  4 ′
The Lorenz invariance of the Lagrangian density will guarantee
the Lorenz invariance of Action.
If ℒ   ,    =ℒ ′  ′ ,  ′ ′ ′
then  =
ℒ   ,
4  ==
ℒ ′  ′ ,  ′ ′ ′
The invariance of Action under Lorentz transformation will
guarantee the invariance of equation of motion!
4 ′ = ′
The equation of motion is given by the principle of Least Action.
Under arbitrary change of fields f ( x ) ® f ( x ) + df ( x) the change of action is zero
=
Euler Equation

Euler Equation

ℒ=
1
1
− 2  2
2
2



     m 2  0
Klein-Gordon Equation

¶2
(¶m ) (¶ ) = ¶t 2 - Ñ × Ñ
m


     m 2  0
¶2
(¶m ) (¶ ) = ¶t 2 - Ñ × Ñ
m

2

 m2   0
Klein-Gordon Equation


     m 2  0
Klein-Gordon Equation

2 y  2 y

2
x
 t 2

2

 m 2   0 Klein-Gordon Equation

1
1 2 2

ℒ =     −
2
2

Invariant 。

Solving KG Equation:
2

 m2   0
Expand the KG field in terms of Fourier Series

Plug into KG Eq.:
2
2
 t   E    t   0

p
p
p
t 2
Every Fourier Component  behaves like a SHO with ω
p  Ep 
2
p  m2
KG Field is just a collection of SHO’s.
Each SHO is characterized by its k or p “momentum”.
The frequency ω or “energy” of the SHO is just that of a
relativistic particle with mass m.

e

E
1 2 1 2
kx 
p
2
2m

1 2 1 2
ˆ
E  kxˆ 
pˆ
2
2m

E  hf  
E  hf  

2
2
 t   E    t   0

p
p
p
t 2
KG Field is just a collection of SHO’s.
Every Fourier Component  behaves like a SHO with ω
p  Ep 
2
p  m2

ℒ =  Φ∗  Φ − 2 Φ∗ Φ

Φ=
1 + 2
2
1
1 2 2 1
1 2 2

ℒ =  1  1 −  1 +  2  2 −  2
2
2
2
2

 2  m 2  0
 2  m2  0

ℒ =  Φ∗  Φ − 2 Φ∗ Φ → − Φ∗   Φ − 2 Φ∗ Φ
= − Φ∗ ∙  2 Φ − 2 Φ

Φ∗
2 Φ − 2 Φ = 0
Non-Relativistic Field Theory
1
ℒ =  0  −
∗ ∙  − 2 ∗
2
∗
2

ℒ

ℒ
∗

+∙
ℒ
∗
+∙
−
ℒ

ℒ
−
=0

ℒ
1
2 ∗   = 0
→
−

∙

−

−
2
0
∗
2
1
∙  + 22 ∗   = 0
2
Schrodinger Equation
ℒ

1
−
→  ∗ −
∙ ∗ − 22 ∗  ∗ = 0

2
0  +

ℒ

+∙
ℒ

The symmetry of Complex KG Field
ℒ =  Φ∗  Φ − 2 Φ∗ Φ
The Lagrangian is invariant under the phase transformation of the field operator:
( x)  e iQ ( x)

    eiQ  eiQ  (x)  
             eiQ  e  iQ     (x)        
U(1) Abelian Symmetry

SU(N) Non-Abelian Symmetry
Assume there are N fields:
 1 


2 
  3 


  
 
 n

Φ∗  Φ − 2 Φ∗ Φ ≡  Φ†  Φ − 2 Φ† Φ
ℒ=
=1
This Lagrangian is invariant under SU(N)!
( x)  U  ( x)  e i T  ( x)
i i
Φ† Φ → Φ†  † Φ = Φ† Φ
If the particles have identical masses, the free theory has a SU(N) symmetry!
Isospin SU(2)變換
u
u 
   U   
d 
d 

2 × 2 矩陣 U 如同旋轉一般可以以三個角度 (like Euler Angles)來標定
Every particle corresponds to a field.
u Particle
u (x) Field
 u ( x) 
 u ( x) 

  U 1 ,  2 ,  3  

 d ( x) 
 d ( x) 
ℒ=
†   − 2 †  +
†   − 2  †
The Lagrangian is invariant under SU(2)
U (1 ,2 ,3 )

2

 m2   0


 


d p
 ip  x
( x, t )  
  p, t   e
3
2 
3






2
2
  p, t   E p    p, t   0
2
t
2
For every p, the frequency of the SHO has two solutions:  E p E p  p  m2

 ( p, t )  a p  e  iEt  c p  eiEt


3


 
d p
d p
iEt  ip x
iEt  ip x
 ( x)  
a p  e

c p  e
3
3
2 
2 
3




These SHO’s correspond to the plane wave solutions of KG Eq.
e
-iEt+ip×x
iEt+ip×x
or e

1
0  +
∙  = 0
2

 


d p
 ip  x
 ( x, t )  
  p, t   e
3
2 
3



 

 
p p
i   p, t  
   p, t   0
t
2m

2
=
2

 iE p t

 ( p, t )  a p  e

 
d3p
iE p t  ip x
 ( x)  
a p  e
3
2 



A general solution is a linear superposition of all plane waves.

3


 
d p
d p
iEt  ip x
iEt  ip x
 ( x)  
a p  e

c p  e
3
3
2 
2 
3
3


 
d p
d p'
iEt  ip x
iEt ip ' x

a p  e

c p '  e
3
3
2 
2 
3
d3p
d
p' + iEt-ip'×x
-iEt+ip×x
=ò
ap × e
+ò
bp' × e
3
3
( 2p )
( 2p )
3
=
ò
d3p
( 2p )






(
)


(
-ip×x
+
ip×x
a
×
e
+
b
×
e
)
p
3( p


p'   p
)
∙  =  −  ∙
Explicitly Lorentz Invariant
The solution of complex KG Equation:

d p
ip x

ip x
 e
 e
x   
a

b
p
p
3
2 
3


For real field:    
The real solution of KG Equation:
f ( x) =
ò
d3 p
( 2p )
-ip×x
+
ip×x
a
×
e
+
a
×
e
)
p
3( p
To quantize the field theory, it’s easier to use Hamiltonian Formalism
pi =
¶L
¶qi
H º å pi qi - L
i
For field system, remember the space coordinates are just indices
= d 3x
Conjugate Momentum
For Klein-Gordon Fields:
 x  
L 

  x  
δ
t



pˆ    i 
x 

xˆ  x

 x 
Oˆ
ˆ x 
O


Oˆ   * ( x)  Oˆ ( x)dx


Oˆ   o

Oˆ   o

Eigenfunction

Eigenvalue


Oˆ  Oˆ

2
 Oˆ 2  Oˆ
 Oˆ 2    Oˆ 2 
 o 2    o  
2

2

2
 o2  o2  0

p  0 ei ( kxt )
p  k


pˆ p ( x, t )  pi
 p ( x,t )0 ei kxt   kp  pp ( x, t )
x
pˆ  p

Oˆ   o

pˆ
p

xˆ
pˆ p  p p
x

x

xˆ x  x x

E  E

2
2
2


ˆ
p


 2   V ( xˆ )
Hˆ 
 V ( xˆ )  
2m
2m  x 
Hˆ   E
2  2 
 2  ( x)  V ( xˆ ) ( x)  E ( x)

2m  x 
d 2 2m
 2 V ( x)  E  
dx2


xˆ  pˆ  pˆ  xˆ

 

 x
ˆ
ˆ
 p  x     i x   i    x   i  xˆ  pˆ 
x 
x 

 x
xˆ  pˆ  pˆ  xˆ  xˆ, pˆ   i  0
xˆ, pˆ   i
Canonical Commutation Relation

xˆ  pˆ  pˆ  xˆ  xˆ, pˆ   i  0

Oˆ , Oˆ  Oˆ Oˆ  Oˆ Oˆ  0
Oˆ , Oˆ  Oˆ Oˆ  Oˆ Oˆ  0

1
2
1
2
2
1
Lˆ , Lˆ  Lˆ Lˆ  Lˆ Lˆ  0
x
z
x
z
z
x
1
2
1
2
2
1
Lˆ , Lˆ  Lˆ Lˆ  Lˆ Lˆ  0
2
2
z
2
z
z
Canonical Commutation Relation
xˆ, pˆ   i

Now! Quantum Field Theory
We use Canonical Quantization to go
from mechanics to quantum mechanics:
Upgrade all observable to operators and impose a
commutation relation between position and momentum:
Fields grow out of systems of particles
y

ix
yi (t )  y( x, t )
Space coordinates x are actually indices!
We know how to quantize particle system and
hence we know how to quantize fields!
Upgrade all observables to operators and impose a
commutation relation between fields and their momenta:
qi  qˆi
 x  ˆx
ix



( 3) 
 ( x, t ),  y, t   i ( x  y),




 ( x, t ),  y, t    ( x, t ),  y, t   0
Quantum Field Theory is done!
 x  ˆx
a, b  aˆ, bˆ

d p
ip x

ip x
ˆ
 e
 e
ˆ
ˆ
 x   
a

a
p
p
3
2 
3


What is the commutation relation of the a operators?
KG Field is just a collection of SHO’s.
Hints from Quantum
SHO:
=

=

a  a
q
2

a  a
p
i 2

fˆ ( x ) =
ò
d3 p
( 2p )
3
( aˆ
p
× e-ip×x + aˆ +p × eip×x )

SHO of different p are decoupled and hence their operators commute.
fˆ ( x ) =
ò
d3p
2w P ( 2p )
3
( aˆ
 ( x, t ),  y, t   i (3) ( x  y),
 ( x, t ),  y, t    ( x, t ),  y, t   0
p
×e
-ip×x
+
p
+ aˆ × e
ip×x
)
fˆ ( y) =
ò
d3p
2w P ( 2p )
3
( aˆ
-ip×y
+
ip×y
ˆ
×
e
+
a
×
e
)
p
p

a  a
q
2
a  a
p
i 2
AB, C  ABC  CAB  AB, C A, CB
H E =E E
The operator a+ can be used to raise the energy by one quantum while
the operator a can be used to lower the energy by one quantum
a+ is called Raising Operator while a Lowering Operator.

Quantum Field Theory is just a series of quantum SHO.
The operator ap+ can be used to raise the energy by one quantum ωp
while the operator ap can be used to lower the energy by ωp.
There is a conserved momentum.
The operator ap+ can be used to raise the momentum by one quantum p
while the operator ap can be used to lower the energy by p.

fˆ ( x ) =
ò
d3p
2w P ( 2p )
a 
 n

p
3
( aˆ
p
× e-ip×x + aˆ +p × eip×x )

0  p, n


H p, n  nEp p, n
 
 
P p, n  np p, n
Particle space are built.
ap+ Creation operator and ap Annihilation operator
of a particle with momentum p and energy Ep
Scalar Antiparticle
Assuming that the field operator is a complex number field.
ℒ =  Φ∗  Φ − 2 Φ∗ Φ
x   

d p
3
2 
3
2
a

p
 e ip x  b p  eip x

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!
3
d
p
  ( 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!
Non-Relativistic Field Theory
1
ℒ =  0  −
∗ ∙
2

∗
1
0  +
∙  = 0
2


 
d3p
iE p t  ip x
ˆ ( x)  
aˆ p  e
3
2 


ap+ Creation operator and ap Annihilation operator of a particle
with momentum  and energy  =
2
2