p n Flavor symmetry p,n 在原子核中、在核力作用時性質相近 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 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 a u + b d u d d c u + d d 古典 量子 u-d 互換對稱

This is quite general. 1 1’ 2 2’ 1 2 3 N 古典 N N’ is a set of orthonormal bases. 量子 must be a set of orthonormal bases. There is a unitary operator U connecting the two bases

a + b c + d SU(2) 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 d u d d u d u-d 互換對稱 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.

It is natural to assign the variable to zero when there is no transformation. generators 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. 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!

For SU(N), generators are N by N traceless hermitian matrices.

For SU(2) There are 3 independent generators. We can choose the Pauli Matrices: For SU(N), generators are N by N traceless hermitian matrices. There are N2-1 independent generators.

For generators: 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！

We can change the base of the Lie algebra: could raise (lower) the eigenvalues of is a eigenstate of J3 with a eigenvalue

SU(2) Representations We can organize a representation by eigenstates of J3. In every rep, there must be a eigenstate with the highest J3 eigenvalue j , From this state, we can continue lower the eigenvalue by J-: in general until the lowest eigenvalue j - l The coefficient must vanish: A representation can be denoted by j.

For every l and therefore every j, there is one and only one representation. are the basis of the rep. The rep is dim From 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 2D rep Two basis vectors A state in the rep:

3 basis vectors 3D rep Triplet 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.

除了不帶電的Pion，還有兩種帶電的Pion，質量非常接近： 帶電Pion的存在正是”為了”維護這個p,n互換的對稱 n p p n n p p n 當p,n互換，Pion也要互換： Isospin SU(2)

Cartan proved there are only finite numbers of forms of Lie algebras Élie Joseph Cartan 1869-1951

SU(3) 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 s 三個風味的夸克的互換對稱： 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 could raise (lower) the eigenvalues (t3,y) of by

Fortune telling diagrams? 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