### Document

```The essence of Particle Physics
Particles are actually not like balls but essentially more fields!
Well, not quite.
Field 場的觀念原來是來自一個粒子系統的連續極限！
y

, i=1,2⋯

→  ,

,  ≡

=

1
=  −  =  2 +
2

,  ,   ,
,

L= ℒ   3

ℒ  = ℒ   ,

=
∙  =
ℒ   ,
∙  3  =
ℒ ,   ∙  4
4  ≡  3

=
ℒ ,   ∙  4

ℒ ,   = ℒ ′,  ′

（一次微分項違背羅倫茲不變性！）
1
1 2 2

ℒ =     −
2
2
=
1 2 1
−
2
2
−
2
1
− 2  2
2
≡

=
= 0 0 −  2

= 0

=

Euler Equation 即是純量場  必須滿足的運動方程式

1
1 2 2

ℒ =     −
2
2
2  2
= 22

=

+ 2   = 0
=    +    = 2
Klein-Gordon Equation

+ 2   = 0
+ 2   = 0
2 + 2   = 0

=    = 0 0 −  2
2
−  2 + 2  = 0
2

2
22 = 0
−

2
2 + 2   = 0

3  ∙

,
2 3
,  =

2
−  2 + 2  = 0
2

2
+
2
2
+ 2

∙  ,  = 0
2
+ 2 ∙  ,  = 0
2

=

2
+ 2
KG Field is just a collection of SHO’s.
Each SHO is characterized by its k or p: “momentum”.

KG Field is just a collection of SHO’s.
Each SHO is characterized by its k or p: “momentum”.
=  =

2
+ 2

…….

4
4q
4
3q
3
3
2q
2
2
q

0
13

=  =

2
+ 2
KG場的每一個傅立葉分析量子化後，就是一個多粒子系統。

2
+ 2 。這正是一顆相對論粒子。

pi =
¶L
¶qi
H º å pi qi - L
i

= d 3x
Conjugate Momentum

ℒ   ,
For Klein-Gordon Fields:
ℒ=
1
1
1
1
− 2  2 =  2 −
2
2
2
2
Conjugate Momentum
=
ℋ∙
3
=
=
3
ℒ

2
1
− 2  2 ~ −
2
=
1 2 1
∙  +
2
2
2
1 2 2
+   ~ +
2
Now! Quantum Field Theory
Just copy the procedure of Quantization from classical mechanics to quantum mechanics:
Upgrade all observable to operators.
Impose a commutation relation between position and momentum:
Quantum Field Theory

For a quantum theory of field, the field is promoted to operators!

Particle Quantum Mechanics
x (t )
ˆ
x(t)

That is why Particle QM 一般來說 is not Lorentz invariant.
Upgrade fields to operators.
→
Impose a commutation relation between fields and their momenta:
x are actually indices!
i x
3
,  ,  ,
=
−
,  ,  ,
=  ,  ,  ,
=0
Canonical Commutation Relation or Canonical Quantization Condition
Quantum Field Theory is done!

=
†
+  −

2
2
=
p 
3
1
−∙  † +   ∙

=  ,

2 3 2
Conjugate Momentum 共軛動量:
,  =
3
−
2 3
∙
−  −∙ †
2

3  ∙

2 3
3
1
∙  +  − ∙  †

2 3 2
,  =
aa
i 2

†

2
33 11
∙
∙
−
∙
∙ ††

+
+

−

2
2 33 2
2
,
, ==
=
=
†
− −

q 
aa

,  ,  ,
=
3
−
†

†
, ′
= 2 3
3
− ′
†
, ′ = † , ′
=0
†
, ′
= 2 3
3
− ′
†
, ′ = † , ′
=0

,  ,  ,
=
3   3 ′ 1
−
2 3 2 3 2
=
3   3 ′ − ′
∙−′∙  ,  † +  −∙+′∙  † ,
−

′

2 3 2 3 2
=
=
3  ∙  3 ′
3  ∙  ∙
′
2
†
∙  +  −∙ † ,  ′∙ ′ −  −′∙ ′
− ′
−  ∙−′∙
2
−
=
3
−
3
− ′ +  −∙+′∙
3
− ′

=
3
1
†
+
2 3    2

† 可以增加能量一個量子 。  則降低能量一個量子 。
=
3
1
†

+
2 3    2
† 可以增加能量一個量子 。  則降低能量一個量子 。
There is a conserved momentum.
† 可以增加動量一個量子 。  則降低動量一個量子。
0 = 0

†
0 =
†

0 ∝ ,

,  =  ∙ ,
,  =  ∙ ,
† :Creation operator,  :Annihilation operator
of a particle with momentum p and energy Ep

†1 †2 †3 0 = 1 , 2 , 3

Every Fourier Component  behaves like a SHO with ω
=  =

2
+ 2

…….

4
4q
4
3q
3
3
2q
2
2
q

0
27

†1 †2 = †2 †1

1 , 2 = †1 †2 0 = †2 †1 0 = 2 , 1

Bose統計在量子力學是外加的假設，在量子場論卻是推導的結果。

Ψ 1 , 2 = Ψ 2 , 1

ℒ =  Φ†  Φ − 2 Φ† Φ

Φ=
ℒ=
1 + 2
2
1
1
1
1
1  1 − 2 12 +  2  2 − 2 22
2
2
2
2

  m   0
2
2
 m  0
2
2

ℒ =  Φ†  Φ − 2 Φ† Φ → − Φ†   Φ − 2 Φ† Φ
= − Φ† ∙  2 Φ − 2 Φ

2 Φ − 2 Φ = 0
ℒ
Φ†
Complex KG field不需要是實數，自由度是 Real KG Field的兩倍。
3
1
∙  +  −∙ †
3
2
2
,  =

3
1
∙  +  − ∙  †

2 3 2
Φ ,  =

3
−
2 3
†
Π ,  = Φ =
∙

−  −∙ †
2

Φ ,  , Π ,
Φ ,  , Φ ,
=
3
= Π ,  , Π ,
=
†
†
, ′
=  , ′
= 2 3
−
3
∙
2 3
=0

2
3
− ′
, ′ =  , ′ =  , ′ = 0
+ 2 ∙  † +  †

The creation operator b+ can create another particle!
=
3
∙
2 3
=
3
†
†
∙

∙

+

2 3
=
3
†
†
∙

−

2 3

2
+ 2
∙
†
+
†
=
3
∙  ∙  † +  †
3
2
The particle b+ create has the same mass but opposite charge.
b+ create an antiparticle.
Φ ,  =
3
1
∙  +  −∙  †

2 3 2
The terms in Φ can either annihilate a particle or create an antiparticle!
Φ†
,  =
3
1
∙  +  − ∙  †

2 3 2
The terms in Φ† its conjugate either annihilate an antiparticle or create a particle!
The charge difference a field operator generates is always the same!

ℒ =  Φ†  Φ − 2 Φ† Φ

Φ  →  − Φ

Φ† Φ → Φ†   ∙  − Φ = Φ† Φ
Φ†  Φ →  Φ†   ∙  −  Φ =  Φ†  Φ

U(1) Abelian Symmetry

ℒ=
1†  1 − 2 1† 1 +

ℒ 可以寫成矩陣形式：
ℒ =   †   − 2 †
2†  2 − 2 2† 2
1
2

→ ′ =
 H 1 x a  b   H 1  x  
     U  

U






H
x
H
x
 2  c d   2

3

U U 1
= exp −
det U  1
=1

∙
2
ℒ =   †   − 2 †
ℒ 在此SU(2)變換下是不變的！因此系統有此對稱性。
ℒ →  †  † ∙    − 2 †  † ∙  =   †   − 2 † = ℒ

SU(2) Non-Abelian Symmetry

1
2
→
′1
′2
=
1
2

2
1
′1
1
→
=
2
′2
2

 H 1( x) 


H
(
x
)
 2

 H 1 x  
 H 1 x  

  U  





H
x
H
x
 2

 2

 u x  
 u x  

  U  





d
x
d
x




SU(2)是兩個性質相同的量子場的unitary線性變換

Assume there are N fields with identical masses:
 1

 2
  3

 

 n








Φ → Φ′ = Φ

Φ†  Φ − 2 Φ∗ Φ ≡  Φ†  Φ − 2 Φ† Φ
ℒ=
=1

ℒ →  Φ†  †   Φ − 2 Φ†  † Φ =  Φ†  Φ − 2 Φ† Φ= ℒ
If the  particles have identical masses, the free theory has a SU(N) symmetry!
SU(N) Non-Abelian Symmetry

```