Flavor symmetry
p,n 在原子核中、在核力作用時性質相近
p
n
假如我們將電磁及弱力關掉，自然界在 p-n 互
換下是對稱。這樣的對稱就稱為近似對稱。
或稱部分對稱，只有核力遵守這個對稱！
實用主義者！
這個世界是部分美麗的！
Flavor symmetry
p,n 在原子核中性質相近
u
d
p-n 互換對稱其實是u-d 互換對稱
u
u-d 互換對稱
d
Z2
這個變換群只有兩個變換，互
換一次，互換兩次即回到原狀。
量子力學下互換群卻變得更大！
量子力學容許量子態的疊加
u
u
a
u
d
c
u
+ b
d
d
古典
+ d
d
量子
u-d 互換對稱
a
c
2
2
 b
2
 d
2
ac  bd
*
UU

1
1
*
0
a
 
c
u 
u 
   U   ,
d 
d 
b   a*
   *
d   b
*
c 
 1
* 
d 
a
U  
c
b

d
This is quite general.
2
1
3

N
古典
1’
2
2’

N’
N
i ,
i  1 N
is a set of orthonormal bases.
i' ,
i  1 N
must be a set of orthonormal bases.
There is a unitary operator U connecting the two bases
i'  U i
1
量子
量子力學下互換群卻變得更大！
量子力學容許量子態的疊加
u
u
a
u
d
c
u
+ b
d
d
u-d 互換對稱 UU


U U 1
det U  1
+ d
d
SU(2)
這個變換群包含無限多個變換，由連續實數來標訂，
Groups that can be parameterized by continuous variables are
called Lie groups.
U ( k )
It is natural to assign the variable to zero when there is no
transformation.
U ( )  
 1  i k Tk  
0
Tk   i
U
 k
generators
 0
These generators form a linear space.
Group elements can be expressed as the exponent
of their linear combinations.
U ( )  exp i  k T k 
For Lie groups, communicators of generators are
linear combinations of generator.
T , T   iC
i
j
k
ij
Tk
This communicator is almost like a multiplication.
A linear space with a multiplication structure is called an algebra.
Communicators form a Lie Algebra!
The most important theorem:
The property of a Lie group is totally determined by its Lie algebra.
Lie Groups with identical Lie algebras are equivalent!
U ( )  exp i  k T k 
U
UU



U U 1
 exp  i  k T k

T


T
det U  exp i  k tr T k 
det U  1
tr T  0
For SU(N), generators are N by N traceless hermitian matrices.
For SU(2)
T

tr T  0
T
There are 3 independent generators.
We can choose the Pauli Matrices:
0
 1  
1
1

0
0
 2  
i
 i

0 
1
 3  
0
0 

 1
For SU(N), generators are N by N traceless hermitian matrices.
There are N2-1 independent generators.
For generators:
Ji 
i
2
SU(2) algebra structure:
J
i
,J
j
  i
ijk
Jk
This is just the commutation relation for angular momenta.
旋轉的大小是由三個連續的角度來表示：
旋轉對稱的世界要求所有物理量，必定
是純量、向量或張量！
SU(2)的結構與三度空間旋轉群O(3)一模一樣！
We can change the base of the Lie algebra:
J   J 1  iJ 2
J  , J 3    J 
J
J  , J    2 J 3
could raise (lower) the eigenvalues of J 3
J3 m  m m
J 3 J  m
  J J
J m
is a eigenstate of J3 with a eigenvalue m  1

3
m
  J

, J 3   m J  m
  J

m
  m  1   J

m

SU(2) Representations
We can organize a representation by eigenstates of J3.
J3 m  m m
In every rep, there must be a eigenstate with the highest J3 eigenvalue j ,
J j  0
From this state, we can continue lower the eigenvalue by J-:
J j 
j j 1
in general
J j  k 
k  12 j  k 
j  k 1
until the lowest eigenvalue j - l
J j  l  0
j
l
2
The coefficient must vanish:
l  1  2 j  l   0
A representation can be denoted by j.
For every l and therefore every j, there is one and only one representation.
j
l
2
m
are the basis of the rep. m  j , j  1,   j  1,  j
The rep is dim 2 j  1
From J  j  k 
k  12 j  k 
j  k 1
we can derive the actions of J- J+ and hence Ji on the basis vectors.
Then the actions of Ji on the whole representation follow.
Doublet
j
1
2
2D rep
m 
1
m 
i
2
0
 1  
1
1

0
Two basis vectors
2
2
A state in the rep:
Ji 
1
2
a
 
b
0
 
i
 i

0 
3
1
 
0
0 

 1
j 1
3D rep
m 1
Triplet

m 0


0


m  1
3 basis vectors

在同一個Representation中的態，是可以由對稱群的變換互相聯結，
因此對稱性要求其性質必需相同！
Triplets of SU(2) is actually equivalent to vectors in O(3).
There is only one 3D rep.
m 1 
x i y
m  1 
,
2
W


W 1  iW 2
2
x i y
2
,W


W 1  iW 2
2
,
m 0  z
除了不帶電的Pion，還有兩種帶電的Pion，質量非常接近：




0


帶電Pion的存在正是”為了”維護這個p,n互換的對稱
p

n

n
p
n

p

p
n
當p,n互換，Pion也要互換：
Isospin SU(2)
p
n



0



Cartan proved there are only finite numbers of forms of Lie algebras
SU ( N ), SO ( N ), Sp ( 2 N ), E 6 , 7 , 8 , G 2 , F 4
Élie Joseph Cartan
1869-1951
想像u,d,s三個物體性質相近，彼此可以互換：
量子力學中這個互換對稱可以擴大為由3 × 3矩陣所代表的變換：
u
d
三個風味的夸克的互換對稱：
s
a

U  d
g


U U 1
b
e
h
c

f 
i 
SU(3)
Generators are divided into two groups.
commute
We can use their eigenstates to organize a representation.
The remaining 6 generators form 3 sets of lowering and raising generators
U
could raise (lower) the eigenvalues (t3,y) of T 3 , Y
by   1, 1 
如果核力遵守SU(3)對稱，此對稱會要求所有參與核力的
的粒子，必須可分類為質量相近的群組：
就像牛頓力學的旋轉對稱，要求所有力學量必能分類為
純量、向量或張量！
將性質質量相近的粒子列表
Fortune telling diagrams?
這是核力遵守SU(3)對稱的證據
(8) Octet
自旋3/2的重子，(10) decuplet