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