Transcript Document
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 12 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 12 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