spin group spin(n)

[Caravel]

指的是用

x-> RxR^-1

来代替SO(n)的直接旋转。spinor是其中d=2的不可约表象。

说明旋转可以裁成两半。

[TheMatrix]

指的是用

x-> RxR^-1

来代替SO(n)的直接旋转。spinor是其中d=2的不可约表象。

说明旋转可以裁成两半。

d是什么？

[Caravel]

d是什么？

representation是说等价nxn 的矩阵，这些矩阵作用的vector的dimension等于n，n=2就是spinor

[TheMatrix]

representation是说等价nxn 的矩阵，这些矩阵作用的vector的dimension等于n，n=2就是spinor

spin group的二维representation？

[Caravel]

spin group的二维representation？

https://math.stackexchange.com/question ... ms-under-t

这个answer总结了一下，我觉得比较契合我心中的定义。

奇怪的是，这么一个常用的概念，竟然很难找到一个consistent的定义，看的数学定义越多越糊涂。

[TheMatrix]

https://math.stackexchange.com/question ... ms-under-t

这个answer总结了一下，我觉得比较契合我心中的定义。

奇怪的是，这么一个常用的概念，竟然很难找到一个consistent的定义，看的数学定义越多越糊涂。

这个回答确实很好。也解了我的疑惑。

这种容易发生混淆的概念，一般出现在数学物理之中 - 知识量大，研究的人背景各不相同，常用语不同。正确的知识掌握在少数人的头脑中，大量的言论都是不准确的，初学者越看越混淆，以盲引盲。不过情况好于经济金融领域，至少没有人故意散布误导的言论。

[Caravel]

这个回答确实很好。也解了我的疑惑。

这种容易发生混淆的概念，一般出现在数学物理之中 - 知识量大，研究的人背景各不相同，常用语不同。正确的知识掌握在少数人的头脑中，大量的言论都是不准确的，初学者越看越混淆，以盲引盲。不过情况好于经济金融领域，至少没有人故意散布误导的言论。

wiki的历史部分也很有有趣，讲了spinor这个概念的演化

Spinors were first applied to mathematical physics by Wolfgang Pauli in 1927, when he introduced his spin matrices.[14] The following year, Paul Dirac discovered the fully relativistic theory of electron spin by showing the connection between spinors and the Lorentz group.[15] By the 1930s, Dirac, Piet Hein and others at the Niels Bohr Institute (then known as the Institute for Theoretical Physics of the University of Copenhagen) created toys such as Tangloids to teach and model the calculus of spinors.
Spinor spaces were represented as left ideals of a matrix algebra in 1930, by Gustave Juvett[16] and by Fritz Sauter.[17][18] More specifically, instead of representing spinors as complex-valued 2D column vectors as Pauli had done, they represented them as complex-valued 2 × 2 matrices in which only the elements of the left column are non-zero. In this manner the spinor space became a minimal left ideal in Mat(2, 
C
{\displaystyle \mathbb {C} }).[r][20]
In 1947 Marcel Riesz constructed spinor spaces as elements of a minimal left ideal of Clifford algebras. In 1966/1967, David Hestenes[21][22] replaced spinor spaces by the even subalgebra Cℓ01,3(
R
{\displaystyle \mathbb {R} }) of the spacetime algebra Cℓ1,3(
R
{\displaystyle \mathbb {R} }).[18][20] As of the 1980s, the theoretical physics group at Birkbeck College around David Bohm and Basil Hiley has been developing algebraic approaches to quantum theory that build on Sauter and Riesz' identification of spinors with minimal left ideals.

[FoxMe]

Clifford algebra的主要用处似乎就是这里：

旋转群Spin(n) = {R: RR' = 1, R属于Clifford group(即Clifford algebra中那些使得RxR^-1还在线性空间V内的元素)}

可以证明，R能写成最多n个线性空间V内的矢量之积。

初略地说，它是SO(n)的double cover. 很显然R和- R的旋转是相同的。

例子：Spin(3)就是unit quaternion，计算机图形学中用它来代替SO(3)，有些好处。

指的是用

x-> RxR^-1

来代替SO(n)的直接旋转。spinor是其中d=2的不可约表象。

说明旋转可以裁成两半。

[FoxMe]

spinor中文翻译是啥？

wiki的历史部分也很有有趣，讲了spinor这个概念的演化

Spinors were first applied to mathematical physics by Wolfgang Pauli in 1927, when he introduced his spin matrices.[14] The following year, Paul Dirac discovered the fully relativistic theory of electron spin by showing the connection between spinors and the Lorentz group.[15] By the 1930s, Dirac, Piet Hein and others at the Niels Bohr Institute (then known as the Institute for Theoretical Physics of the University of Copenhagen) created toys such as Tangloids to teach and model the calculus of spinors.
Spinor spaces were represented as left ideals of a matrix algebra in 1930, by Gustave Juvett[16] and by Fritz Sauter.[17][18] More specifically, instead of representing spinors as complex-valued 2D column vectors as Pauli had done, they represented them as complex-valued 2 × 2 matrices in which only the elements of the left column are non-zero. In this manner the spinor space became a minimal left ideal in Mat(2, 
C
{\displaystyle \mathbb {C} }).[r][20]
In 1947 Marcel Riesz constructed spinor spaces as elements of a minimal left ideal of Clifford algebras. In 1966/1967, David Hestenes[21][22] replaced spinor spaces by the even subalgebra Cℓ01,3(
R
{\displaystyle \mathbb {R} }) of the spacetime algebra Cℓ1,3(
R
{\displaystyle \mathbb {R} }).[18][20] As of the 1980s, the theoretical physics group at Birkbeck College around David Bohm and Basil Hiley has been developing algebraic approaches to quantum theory that build on Sauter and Riesz' identification of spinors with minimal left ideals.

[TheMatrix]

wiki的历史部分也很有有趣，讲了spinor这个概念的演化

Spinors were first applied to mathematical physics by Wolfgang Pauli in 1927, when he introduced his spin matrices.[14] The following year, Paul Dirac discovered the fully relativistic theory of electron spin by showing the connection between spinors and the Lorentz group.[15] By the 1930s, Dirac, Piet Hein and others at the Niels Bohr Institute (then known as the Institute for Theoretical Physics of the University of Copenhagen) created toys such as Tangloids to teach and model the calculus of spinors.
Spinor spaces were represented as left ideals of a matrix algebra in 1930, by Gustave Juvett[16] and by Fritz Sauter.[17][18] More specifically, instead of representing spinors as complex-valued 2D column vectors as Pauli had done, they represented them as complex-valued 2 × 2 matrices in which only the elements of the left column are non-zero. In this manner the spinor space became a minimal left ideal in Mat(2, 
C
{\displaystyle \mathbb {C} }).[r][20]
In 1947 Marcel Riesz constructed spinor spaces as elements of a minimal left ideal of Clifford algebras. In 1966/1967, David Hestenes[21][22] replaced spinor spaces by the even subalgebra Cℓ01,3(
R
{\displaystyle \mathbb {R} }) of the spacetime algebra Cℓ1,3(
R
{\displaystyle \mathbb {R} }).[18][20] As of the 1980s, the theoretical physics group at Birkbeck College around David Bohm and Basil Hiley has been developing algebraic approaches to quantum theory that build on Sauter and Riesz' identification of spinors with minimal left ideals.

Wiki没有把Spinor这个概念说清楚。

你给的那个stackexchange回答说清楚了。

一个Spinor是spin group的representation space（一个vector space）中的一个元素。

有时候名词的混用会给人造成长期的困惑，尤其是对于不在团体之中的人。

[TheMatrix]

Clifford algebra的主要用处似乎就是这里：

旋转群Spin(n) = {R: RR' = 1, R属于Clifford group(即Clifford algebra中那些使得RxR^-1还在线性空间V内的元素)}

可以证明，R能写成最多n个线性空间V内的矢量之积。

Spin(n)，也就是Spin group的定义方法，和SO(n)的定义方法相差挺大的：

Spin(n)的定义是：
给定一个V，一个V的orthonormal basis，可以构造V的Clifford algebra。
Spin(n)是Clifford algebra的一个子集：其元素R，
1，在Clifford algebra中可逆：RR^-1=1。
2，作用在V上结果还在V中：x∈V --> RxR^-1∈V。
3，norm等于1。（norm在V中的定义是清楚的，在Clifford algebra中要扩展一下）。

SO(n)的定义是：
给定一个V，一个V的orthonormal basis。
SO(n)是：V上的可逆线性变换，保长度，保符号（basis的排列方式）。

所以这两个的定义差别还是很大的。Spin(n)是SO(n)的double cover，也就它们基本相同，Spin(n)比SO(n)只差一点点，这一点还是惊讶的。因为感觉Spin(n)比SO(n)大得多 - 因为Clifford algebra的维度太大了。

[FoxMe]

所以我说是杀鸡用牛刀

Spin(n)，也就是Spin group的定义方法，和SO(n)的定义方法相差挺大的：

Spin(n)的定义是：
给定一个V，一个V的orthonormal basis，可以构造V的Clifford algebra。
Spin(n)是Clifford algebra的一个子集：其元素R，
1，在Clifford algebra中可逆：RR^-1=1。
2，作用在V上结果还在V中：x∈V --> RxR^-1∈V。
3，norm等于1。（norm在V中的定义是清楚的，在Clifford algebra中要扩展一下）。

SO(n)的定义是：
给定一个V，一个V的orthonormal basis。
SO(n)是：V上的可逆线性变换，保长度，保符号（basis的排列方式）。

所以这两个的定义差别还是很大的。Spin(n)是SO(n)的double cover，也就它们基本相同，Spin(n)比SO(n)只差一点点，这一点还是惊讶的。因为感觉Spin(n)比SO(n)大得多 - 因为Clifford algebra的维度太大了。

[TheMatrix]

所以我说是杀鸡用牛刀

Clifford algebra除了能spin，就没别的用途了？

[FoxMe]

有人也叫spinor group,确实容易混淆

https://encyclopediaofmath.org/wiki/Spinor_group

Wiki没有把Spinor这个概念说清楚。

你给的那个stackexchange回答说清楚了。

一个Spinor是spin group的representation space（一个vector space）中的一个元素。

有时候名词的混用会给人造成长期的困惑，尤其是对于不在团体之中的人。

[FoxMe]

物理里好像可以用来构造泡利矩阵，狄拉克矩阵等.

看过Clifford algebra的矩阵表示，忘了

Clifford algebra除了能spin，就没别的用途了？

[Caravel]

Spin(n)，也就是Spin group的定义方法，和SO(n)的定义方法相差挺大的：

Spin(n)的定义是：
给定一个V，一个V的orthonormal basis，可以构造V的Clifford algebra。
Spin(n)是Clifford algebra的一个子集：其元素R，
1，在Clifford algebra中可逆：RR^-1=1。
2，作用在V上结果还在V中：x∈V --> RxR^-1∈V。
3，norm等于1。（norm在V中的定义是清楚的，在Clifford algebra中要扩展一下）。

SO(n)的定义是：
给定一个V，一个V的orthonormal basis。
SO(n)是：V上的可逆线性变换，保长度，保符号（basis的排列方式）。

所以这两个的定义差别还是很大的。Spin(n)是SO(n)的double cover，也就它们基本相同，Spin(n)比SO(n)只差一点点，这一点还是惊讶的。因为感觉Spin(n)比SO(n)大得多 - 因为Clifford algebra的维度太大了。

Spin(n) 是 CL(n) even grade element span的的，似乎比可逆要强一些

[TheMatrix]

Spin(n) 是 CL(n) even grade element span的的，似乎比可逆要强一些

Spin(n)是个群。CL(n) even grade element是一个子代数，一个线性空间。怎么span呢？应该是exponential map。

[Caravel]

物理里好像可以用来构造泡利矩阵，狄拉克矩阵等.

看过Clifford algebra的矩阵表示，忘了

物理里面的思路很简单，物理里面确定一个态需要做实验，stern gerlach实验出来两个结果。如果自旋势态只有两个本征态，一个|+>, 一个|->，还有它们的线性叠加。这就是二分量vector的来源。

把这个和旋转联系起来，要用 sigma_xyz Pauli 算符作用上去， <x| sigma_x | x> 得到矢量。

[Caravel]

Spin(n)是个群。CL(n) even grade element是一个子代数，一个线性空间。怎么span呢？应该是exponential map。

即使类似 1+ ex ey 这样的元素。

[Caravel]

现在知道了为什么物理里面，工程里面到处都是矩阵，这就是representation theory，