a4119
| 19 |
|---|
带有自旋的粒子。
...庞加莱群描述了一个点状物体的相对论运动。同样，巴格曼群描述了非相对论运动。动量的分量作为纯粹的几何量出现。这是一种物理学的几何化。物理学家熟悉能量E和动量矢量p。但他们可能会对另外两个物体感到有些困惑：通过量f和自旋矢量l。动量分量的形式取决于坐标的选择。...每个动态群都有其动量空间及其在该空间上的共轭作用。如果我们先选择相对论世界（庞加莱群），而不是非相对论世界，我们就必须参考巴格曼群。有关计算细节，请参阅我的关于群论的课程。巴格曼群是伽利略群的一个非平凡扩展：(272)
正如读者可以看到的，这个群作用在一个五维空间上：
r：空间
t：时间
z：一个额外的变量。
...这些关于额外变量的问题将在后面讨论。在这个网站上，已经给出了庞加莱群共轭作用的完整计算。也可以推导出巴格曼群在其动量空间上的共轭作用计算。令人惊讶的是，非相对论世界中的计算比相对论世界中的计算稍微复杂一些。结果如下：(273)
物理学家识别出一些熟悉的物体，比如速度：(274)
和动能：(275)
mv是动量。相对于什么的速度？一个群改变了运动的参数，给一个粒子一个速度v和动能E。我们可以选择相反的解释，认为一个群是某种事物的特殊视角，比如一个粒子。如果我们考虑SO(3)群，矩阵a，这意味着“从另一个角度看到”。如果我们考虑O(3)群，矩阵a，这增加了从镜子中观察“事物”的可能性。
欧几里得群的平移矢量(276)
增加了“从别处看到”。
在动态群中，群中存在速度v意味着观察者在移动。时间平移e = Dt意味着观察者在一段时间后看到事物。平移矢量Dr和时间延迟Dt可以组合成一个时空平移矢量：(277)
看看公式，从巴格曼群中我们可以看到：
m' = m
无论从哪个角度看，质量保持不变。
让我们稍微简化一下这个“视角”，选择a = 1。
共轭作用变为：(278)
...共轭作用表示运动参数的变化。如果我们认为我们从一个稳定状态转移到非稳定状态，初始条件对应于：
E = 0（零能量）
p = 0（零动量，零速度）
“通过”f = 0
那么共轭作用给出：(279)
“考虑”必须按照其词源意义来理解。
一个执行官说：- 制作一份清单和记录。
...对事物的静态（v = 0）观察对应于欧几里得群。执行官在距离c处观察事物。他在事情发生时观察（Dt = 0）。最终，他从某个角度（a不同于1）观察。
...一位将军驾驶飞机飞越战场，是一种观察事物的移动视角（从以速度v飞行的飞机上）。...但一位将军在指挥部，观看几小时前由无人驾驶飞机（无人机）拍摄的电影，他说：- 考虑目标，它在一个小时前的样子（Dt不为零），从一个移动的观察点（v不为零）观察，从五千米的高度（c不为零），以速度v飞行，并从某个角度（a不同于1）拍摄照片。
...目标没有定义的速度、位置或方向，即使它被认为是一个“固定”的建筑物。一切都是相对的。甚至地球、太阳和我们的银河系都在太空中移动。
...地球的“北极”与太阳的“北极”相差23度，并且随时间变化（26,000年），这是由于春分点的进动。太阳指示的北极（其自身的旋转轴）与我们银河系（银河系）指示的北极不同，银河系有自己的旋转运动（相差90度）。甚至一个星系也以每小时三百英里的速度移动。相对于什么？相对于其他星系。这就是我们能说的全部。群对应于两种不同的视角。
...如果我认为物体是静止的，固定在空间和时间中，并且没有旋转运动，那么我能说的就是：
- 如果我远离距离c。
- 如果我以速度v观察事物。
- 如果来自这个事物的信息以时间延迟Dt到达我。
相对于我：
---> 物体的质量没有改变。
----> 我给物体一个动量mv，视为表观的。
-----> 物体获得一个“通过”f = m[c - v Dt]
-----> 它获得一个自旋（279b）
更明确地写出来：(280)
(281)
(282)
或者：(283)
可以将自旋矩阵l的三个独立分量视为一个矢量的分量：(283b)
...尽管我们的空间中没有定义矢量积，也就是说，我们没有给空间赋予左右方向，但我们仍可以将最后的表达式视为矢量积。(284)
...倒置的v表示矢量积。我们看到，给出共轭作用的公式最后一行对应于：(285)
l是一个矩阵，而不是矢量。但根据所选的符号，粗体字母可以表示矩阵或矢量。
这个矢量开始看起来对物理学家来说很熟悉：动量。
原始版本（英文）
a4119
| 19 |
|---|
Particles with spin.
...The Poincarés' group describes the relativistic movement of a point-like object. Similarly the Bargmann group describes the non-relativistic movement. The components of the moment arise as pure geometric quantities. It's a geometrization of physics . The physicists are familiar to the energy E and impulsion vector p. But they can be a little bit puzzled by the two other objects : the passage f and the spin vector l . The form of the momentum's components depends on the choice of coordinates. ...Each dynamic group owns its momentum space and coadjoint action on this momentum space. If, instead chosing at first the relativist world (Poincaré's group) we had chosen the non-relativistic world, we would have to refer to the Bargmann's group. For computational detail see my lectures on groups. The Bargmann group is a non-trivial extension of the Galileo's group : (272)
As the reader can see, this group acts on a five dimensional space :
**r **: space
t : time z : an additional variable.
...These questions of additional variable or additional variables will be treated further. In this site full calculation of the coadjoint action of the Poincaré group has been given above. One could also derive the calculation of the coadjoint action of the Bargmann's group on its momentum. Paradoxically, the calculation in the non-relativist world is somewhat more complicated than the one in relavistice world. The result is the following : (273)
The physicist identifies somes familiar objects, like the velocity : (274)
and kinetic energy : (275)
m v is the impulsion. Velocity with respect to what ? A group changes the parameters of the movement, gives to a particle a velocity v and a kinetic energy E . We can choose the opposite interpretation, and consider that a group is a peculiar point of view on something, on a particle. If we consider the group SO(3), the matrixes a, it means "seen along another angle of view". If we consider the group O(3), matrixes a, it adds the possibility to observe the "thing" through a looking glass.
The translation vector (276)
of the Euclid's group adds "seen from elsewhere".
In dynamic groups, the presence of a velocity v in the group means that the observer moves. The time-translation e = Dt means that the observer sees the thing after some delay. The translation vector Dr and the time delay** **Dt can be put together in a space-time translation vector : (277)
Look at the formulas, from Bargmann's group, we see that :
m' = m
Whatever is the point of view, the mass in unchanged.
Let us simplify a little bit this "point of view", choosing a = 1.
The coadjoint action becomes : (278)
...The coadjoint action indicates the change in the movement's parameters. If we consider we pass from a steady situation to a non-steady situation, the inital conditions correspond to :
E = 0 ( zero energy )
**p **= 0 ( zero impulsion, zero velocity )
"passage" f = 0
The the coadjoint action gives : (279)
"To consider" must be read in its etymologic meaning.
A process server says : - Drawing up a survey cum inventory.
...A static (v = 0) vision of things, corresponds to the Euclid's group. The process server observes things at distance **c **. He observes facts at the time they happen (Dt = 0). Eventuallly he looks from a certain angle (a different from 1).
...A general, flying over a battle field, in a plane, is some sort of a process server, who observes things from a moving point of view (from a plane cruising at velocity v). ...But a general, in his headquarters, looking at a movie taken by a pilotless plane, a drone, some hours before, says : - Considering the target, as it was one hour before (non zero Dt), seen from a moving point of observation (non zero v) , from an altitude of five thousand feet (non zero c), cruising at velocity v and taking picture through a certain angle (a being different from 1).
...A target has no defined velocity, or position, or orientation, even if it is supposed to be a "fixed" building. Everything is relative. Even the Earth, the Sun, our galaxy move over space.
...The "north pole" of the Earth is different from the "north pole of the Sun", from 23°, and it changes in time ( 26,000 years), due to equinox precession. The north indicated by the sun ( its own rotation axis ) is not the north indicated by our galaxy, the milky way, which has its own spining movement ( 90°'s gap ). Even a galaxy moves, at a trhree hundred miles per hour. With respect to what ? To the other ones. That's all we can say. The group corresponds to two different points of view.
...If I consider that the object is steady, fixed in space and time, and owns no spining movement, all that I can say is :
- If I move off at distance c . - If I observe the thing, when cruising at velocity v . - If the information, coming from this thing, joins me with a time-delay Dt.
*With respect to me *:
---> The mass of the object is not modified.
----> I give the object an impulsion mv, considered as an apparent one. -----> The object gets a "passage" **f **= m [ c - v Dt ] ----> It gets a spin (279b)
Write it in a more explicit way : (280)
(281)
(282)
or : (283)
One may consider the three independent components of the spin matrix l as the component of a vector : (283b)
...Although the vectorial product has not been defined in our space, i.e : we did not give space a right-left orientation, we can consider the last expression as a vectorial product. (284)
...The reversed v indicates the vectorial product. We see that the last line of the formulas giving the coadjoint action corresponds to : (285)
l is a matrix, not a vector. But, according to the chosen notation, bold letters indicates indifferently a matrix or a vector.
This vector begins to look like something familiar to the physicist : the kinetic momentum .