自旋跟时空变换直接联系，是外部对称性。
但另一方面，自旋是整数还是半奇数却决定着粒子的统计。
一种“外部”的东西，如何会具有如此地“内部”性？

也许我的提法不对，只是觉得，是费米统计还是玻色统计，这是内禀的东西，
怎么又跟外部性质扯上关系了？
或许自旋还真有某种内部性？跟轨道角动量不一样？

我觉得粒子的统计性质是粒子集体行为的体现，而这种集体行为又是受空间拓补性质约束的。
所以自旋也是与空间有关系的。典型的例子有，一维空间中的粒子统计全都是玻色统计，
自旋都是整数。而二维空间中的粒子可能会出现任意分数统计，而不仅仅局限于玻色
统计和费米统计两种。只有在三维及以上的空间中才会仅有玻色统计与费米统计。

粒子恰好有玻色统计及费米统计是由Lorentz群的表示决定的，恰好有两种情况：参

The Connection Between Spin and Statistics
W. Pauli
Phys. Rev. 58, 716 (1940)

相应的，与薛定额方程相对应的相对论性波动方程也只有两种情况，Klein-Gordon方程与 Dirac方程。

<<相应的，与薛定额方程相对应的相对论性波动方程也只有两种情况
，Klein-Gordon方程与 Dirac方程>>
-----------------------------------------------------------

不止两种情况,还有任意自旋粒子的类Dirac方程.

二维空间中的任意分数统计源于 酉群 相关的规范变换，与时空变换群导出的结果本质上不同。
微观粒子的拓扑性质都是酉群相关的。

———————————————

Dirac方程源于Lorentz群的最低阶旋量表示，论坛上以前好像有类似讨论。
自然界中满足Dirac方程的粒子比其它自旋的Fermion更稳定。

内部对称性有些问题让我很迷惑. 比如 isotopic spin. 是不是说谈论这个对称性的时候就没有单独的中子态和质子态, 而只有一个中子和一个质子的结合态? 而 isospin 的三个荷都是这个结合态所带的荷?

自旋跟时空变换直接联系，是外部对称性。
但另一方面，自旋是整数还是半奇数却决定着粒子的统计。
一种“外部”的东西，如何会具有如此地“内部”性？

也许我的提法不对，只是觉得，是费米统计还是玻色统计，这是内禀的东西，
怎么又跟外部性质扯上关系了？
或许自旋还真有某种内部性？跟轨道角动量不一样？
======================================================

Rotational group (such as SO(3)) is a sub-group of the Lorentz group.

As a result, the irreducible representations of Lorentz group could be classified
according to their transformation properties under the rotational
group, also known as spin. The well-known representations are: trivial j=0
(scalar), j=1/2 ((1/2,0) and (0,1/2),fermion), and j=1((0,1) and
(1,0), vector). All of them are used by nature. It is also possible to
Rarita-Schwinger j=3/2 (1,1/2) and (1/2,1) (such as gravitino) and j=2
such as the graviton. Higher spin representations exist, but somewhat
less useful.

Spin is ``intrinsic'' in the sense that it is not possible change the
spin of a particle by kinematical method. A particle of particular
kind is born with its definite spin and states with different spin
will not mix. In comparison, we could
change spatial angular momentum by accelerate a particle. Beside this,
I don't know a meaningful way to talk about ``intrinsic''.

Spin statistics can be divide into two questions:



1) how is the spin of the particle connect to the statistics?

This is the so-called spin statistics connection, first point out by
Fierz.

It has everything to do with Lorentz invariance.

It is a long story with details. However, the basic story line is that
Lorentz invariance of the S-matrix requires that the commutator of
interaction Lagrangian to commute outside of the light-cone (a
somewhat similar argument leads to the same conclusion is
causality). Lorentz invariance again together with cluster
decomposition principle require us to construct it with relativistic
quantum field (Very crudely speaking, cluster decomposition requires
we use creation and annihilation operators with specific
momentum. So we are in general talking about many particles. Lorentz
covariance requires we consider a linear combination
of them, also known as fields.).

Putting these together leads to the conclusion that
quantum fields will either have to commute or anti-commute. The form
of the relativistic quantum fields are also constrained by
transformation laws under Lorentz.

These two constraints fixes that spin half integer must anti-commute
and spin integer must commute.

2) How many different statistics are possible? This has nothing to do
with Lorentz invariance.

Let's ask how we distinguish particles. One way to do it is the
following:

We start with a set of particles in a particular initial state, then
we permute particles. If the particles are indistinguishable, this at
most creates some complex phase in front of the state. We now test
this result by asking the amplitude this state evolve into some
particular final state. The physical amplitude is the sum of all
indistinguishable permutations. In the path integral, we have
integrate over all possible paths. These include all paths which can
be continuously deform into each other, and sum over all topologically
different paths, different term in this sum in principle could
contribute different phases.

In d>=3, all paths for the same permutation are equivalent. So there
is no second part (the sum over topologies). Therefore, there is only
permutation of initial state. The complex number must form a rep of
permutation group. Moreover, two consecutive permutation (product of
two complex numbers) should be the same as permute once with the
product permutation. There are only two reps of the permutation group
which has this property, trivial rep (1), and Z_2(-1, 1). Therefore,
for d>=3, there are only two statistics possible, bose-einstein or
fermi-dirac, independent of spin.

In d=2, there is a topology sum since there are inequivalent paths
(which circles some particle any number of times). Therefore, the
complex number does not have to for a rep of the permutation group (It
has a name called braid group). In
fact, it could be any thing. Therefore, we could have any kind of
statistics, the so called anyons.

内部对称性有些问题让我很迷惑. 比如 isotopic spin. 是不是说谈论这个对称性的时候就没有单独的中子态和质子态, 而只有一个中子和一个质子的结合态? 而 isospin 的三个荷都是这个结合态所带的荷?

It is the same as spin, in principle, or any generic two state system. proton and neutron form a doublet. It could be proton, neutron, or linear superposition of the two.

what are the three charges you are talking about?

The isospin here is the SU(2).

我说三个荷因为 dim SU(2)=3.

我说三个荷因为 dim SU(2)=3.
============================

So did I answer your question?

(p,n) is a doublet under SU(2). You can use them to form Noether currents
which has three component, which are three generators of SU(2).

States generically are not eigenstates of these generators. In this sense, they are not c-number charges.

They can only have two possible eigen-values.

yes. Thank you for explanation. Weinberg was not completely clear on shis issue. It seems that the 2-dim irreducible representation of SU(2) corresponds to isospin. How about the 3-dim irr. rep. of SU(2), and all other irr. rep.?

Spin is ``intrinsic'' in the sense that it is not possible change the
spin of a particle by kinematical method. A particle of particular
kind is born with its definite spin and states with different spin
will not mix. In comparison, we could
change spatial angular momentum by accelerate a particle. Beside this,
I don't know a meaningful way to talk about ``intrinsic''.
～～～～～～～～～～～～～～～～～～～

这就好像换一坐标系平行的惯性系观测同一体系的角动量时，
会发现轨道角动量会改变，但自旋角动量不会改变。
所以自旋的这一“内禀”性是即使在经典物理中也是存在的。
所以，我对常强调的以下说法很反感和不满：
1、电子自旋是内禀性质，是电子的内在属性；
2、自旋是量子效应，无经典对应（与轨道角动量比较）；

sage兄乃高人也，解释的很完整很有条理，赞