flavor symmetry
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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
u-d 互換對稱
u
d
p,n 在原子核中性質相近
p-n 互換對稱其實是 u-d 互換對稱
u
d
2Z
這個變換群只有兩個變換,互換一次,互換兩次即回到原狀。
u-d 互換對稱
量子力學容許量子態的疊加
a + b
c + d
1**
**
db
ca
dc
baUU
u
u
u
d
d
dd
u
量子力學下互換群卻變得更大!
0
1
1
**
22
22
bdac
dc
ba
dc
baU
d
uU
d
u,
古典量子
1 2
1’1
古典
This is quite general.
3 N2’2
N’N
量子Nii 1,
Nii 1,'
is a set of orthonormal bases.must be a set of orthonormal bases.
There is a unitary operator U connecting the two bases
iUi '
u-d 互換對稱
量子力學容許量子態的疊加
a + b
c + d
1 UUUU SU(2)
u
u
u
d
d
dd
u
量子力學下互換群卻變得更大!
這個變換群包含無限多個變換,由連續實數來標訂,
1det U
Groups that can be parameterized by continuous variables are called Lie groups.
)( kU
kkTiU 1)(0
0
kk
UiT
It is natural to assign the variable to zero when there is no transformation.
generators
kkTiU exp)(
These generators form a linear space.
Group elements can be expressed as the exponent of their linear combinations.
For Lie groups, communicators of generators are linear combinations of generator.
kkijji TiCTT ,
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!
1 UUUU
1det U
kkTiU exp)(
TT
0tr T
For SU(N), generators are N by N traceless hermitian matrices.
kkTiU exp
kk TiU trexpdet
TT 0tr T
For SU(N), generators are N by N traceless hermitian matrices.
For SU(2)
There are 3 independent generators.
There are N2-1 independent generators.
We can choose the Pauli Matrices:
10
01
0
0
01
10321
i
i
2i
iJ
For generators:
kijkji JiJJ ,
SU(2) algebra structure:
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!
21 iJJJ
JJJ 3, 32, JJJ
J could raise (lower) the eigenvalues of
3J
mmmJ 3
mJmmJmJmJJmJJmJJ 1, 333
mJ is a eigenstate of J3 with a eigenvalue
1m
We can change the base of the Lie algebra:
We can organize a representation by eigenstates of J3.
SU(2) Representations
mmmJ 3
In every rep, there must be a eigenstate with the highest J3
eigenvalue j ,0 jJ
From this state, we can continue lower the eigenvalue by J-:
1 jjjJ
121 kjkjkkjJ
until the lowest eigenvalue j - l 0 ljJ
2
lj
in general
021 ljlThe coefficient must vanish:
A representation can be denoted by j.
2
lj jjjjm ,1,1,
The rep is dim 12 j
121 kjkjkkjJFrom
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.
For every l and therefore every j, there is one and only one representation.
m are the basis of the rep.
Doublet
2
1j
2
1m
2
1m2D rep
b
aA state in the rep:
2i
iJ
10
01
0
0
01
10321
i
i
Two basis vectors
0
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!
1j 1m 1m3D rep
Triplet
3 basis vectors0m
Triplets of SU(2) is actually equivalent to vectors in O(3).
There is only one 3D rep.
2,
22121 iWW
WiWW
W
zmyix
myix
m
0,2
1,2
1
n
p
p
n
除了不帶電的 Pion ,還有兩種帶電的 Pion ,質量非常接近: 0
當 p,n 互換, Pion 也要互換:
帶電 Pion 的存在正是”為了”維護這個 p,n 互換的對稱
0 np
n
p
p
n
Isospin SU(2)
Élie Joseph Cartan 1869-1951
Cartan proved there are only finite numbers of forms of Lie algebras
428,7,6 ,,),2(),(),( FGENSpNSONSU
u d s 三個風味的夸克的互換對稱:
想像 u,d,s 三個物體性質相近,彼此可以互換:
1
UU
ihg
fed
cba
U SU(3)
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:
commute
We can use their eigenstates to organize a representation.
Generators are divided into two groups.
The remaining 6 generators form 3 sets of lowering and raising generators
U could raise (lower) the eigenvalues (t3,y) of
YT ,3 by 1,1
Fortune telling diagrams?
這是核力遵守 SU(3) 對稱的證據
將性質質量相近的粒子列表
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!
(8) Octet
自旋 3/2 的重子, (10) decuplet
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