p n Flavor symmetry p,n 在原子核中、在核力作用時性質相近

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1 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！

2 實用主義者！ 這個世界是部分美麗的！

3 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。

4 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

5 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

6 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.

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8 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.

9 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!

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

11 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.

12 For generators: SU(2) algebra structure: This is just the commutation relation for angular momenta.

13 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！

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

15 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.

16 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.

17 Doublet 2D rep Two basis vectors A state in the rep:

18 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.

19 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)

20 Cartan proved there are only finite numbers of forms of Lie algebras
Élie Joseph Cartan

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22 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)

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24 Generators are divided into two groups.
commute We can use their eigenstates to organize a representation.

25 The remaining 6 generators form 3 sets of lowering and raising generators
could raise (lower) the eigenvalues (t3,y) of by

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30 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

31 自旋3/2的重子，(10) decuplet

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