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伽利略群（时空定向和完整群）。
我们可以为这个群起不同的名字。
GGSOTO（时空定向伽利略群）
或：GSG（特殊伽利略群）。
或者简单地说：SG(3,1)：特殊伽利略群。
3个空间维度，1个时间维度。回想一下，我们曾将PT群的作用表示如下：
（158）
然后我们转向一个时空定向的群。我们可以类似地写出这种群的作用：
（159）
这是一个更精细群的子群：
（160）
“时空定向的伽利略群”。其中：
（161）
相应的动作是：
（162）
这是一个单连通群。它是伽利略完整群的一个子群，该群有四个连通分支：
（163）
它规定了P、T和PT对称性：
（164）
同时也提出了反时序物体的问题（如我们将在后面所做的，但基于相对论）。
运动。
四维几何物体是“动态全息图”。在四维结构中，我们可以在连续的时间点上进行切片。每个切片是一个三维物体，由点(xi, yi, zi)组成。更简单的是考虑一个在时空中的点状物体的运动。这样，所考虑的时空结构就变成了一条轨迹，即一个运动。
...我们决定将物理中的粒子视为点的运动。它们可能是“质点”，也可能是点状能量（光子、中微子）。
...我们可以考虑所有可能的粒子的所有可能运动，并将它们包含在一个
（165）
运动空间中。
...
在时空内，我们可以确定光子、质子、中子、中微子、反质子等的所有可能轨迹。我们考虑了无数可能的位置、速度和其他参数，这些参数将在以后被发现。在这无数轨迹中，有属于某一特定粒子的轨迹：例如一个电子。其他轨迹对应于光子。它们是不同的。它们形成了两个不同的家族，两种
不同的运动种类。我们试图找出如何对粒子进行分类。然后我们试图找出如何定义运动种类。
我们将使用类似于欧几里得的方法。核心问题是：
哪些“物体”属于同一类？
...答案是：那些可以通过属于一个称为这些物体的规则性的子群的元素作用而相互叠加的物体。
...在欧几里得世界中，你不能将一个球体变成一个立方体，反之亦然。它们属于不同的种类。不存在一个子群可以将球体变成立方体，反之亦然。
...同样地，在某个待定义的群中，不存在一个属于某个特定子群的元素，可以将光子的运动转化为电子的运动。它们本质上是不同的；它们属于不同的种类。
如果群中存在一个元素，其作用可以将一个运动转化为另一个运动，那么这些运动就属于同一类。它们是同一粒子的两种不同运动。
...我们不会涉及多粒子系统，如原子或分子。我们将专注于自由粒子的分析，在空旷的时空中移动。在移动过程中，一些参数会被保留（质量、能量等）。
但仅仅检查粒子的时空轨迹不足以识别它并将其归入一个明确的种类。
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质子和中子可能以相同的速度沿着相同的轨迹运动。
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两个粒子可能以速度v = c沿相同的轨迹运动，但其中一个可能是光子，另一个可能是中微子。
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如我们将在后面看到的，两个沿相同方向以光速运动的光子可能不同。它们是P对称的。
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一个具有右旋性。
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另一个具有左旋性。
这对应于光的偏振。它们属于不同的种类吗？这取决于所选择的群。
一种种类是相对于给定的群而言的。
动量。
...一个运动是一种特殊的选择，是动量空间中的一个点。考虑只在质量上有所不同的运动种类。我们取两个种类。质量为ma的粒子不能转化为质量为mb的粒子。即使它们的轨迹在时空中可能相同，我们仍将它们视为属于两个不同种类的运动，即：
两种不同的运动种类。（166）
动量是一组参数：J = {J1, J2, J3, ..., Jn}，其中一个是能量：J1 = E。
另外三个是（J2 = px, J3 = py, J4 = pz），构成了动量矢量p，这些都是物理学家熟悉的量。
...这些量可能表现为纯粹的几何量，直接与所选的群相关。你将看到，构成动量的量的数目等于群的维度。
...那么，我们将要玩的游戏规则是什么？
原始版本（英文）
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**The Galileo groups **( space-time oriented and complete group ).
We can suggest different names for this group.
GGSOTO ( Galileo space oriented and time oriented )
or : GSG ( Special Galileo group ).
Or, simply : SG(3,1) : Special Galileo group.
3 dimensions of space, 1 for time. Remember we wrote the action of the PT group as follows :
(158)
Then we shifted to some space and time oriented group. We could similarly write the action of such a group :
(159)
This is a sub-group of a more refined one :
(160)
The "space and time oriented Galileo's group". With :
(161)
The corresponding action is :
(162)
This is a one component ( connex ) group. It is a sub-group ofthe complete, four components Galileo group :
(163)
which rules P, T and PT symmetries :
(164)
and arises the problem of antichron objects too ( as it will be done further, but on relativistic grounds ).
Movements.
4d-geometrical objects are "animated holograms". In the 4d structure we can make cuts, at successive times. Each cut is a 3d object, made of (xi,yi,zi)
points. It is simpler if we consider a point-like object moving in space-time. Then the considered space-time structure becomes a paths, a movement.
...We decide to assimilate the particles of physics to movements of points. Either they will be some "mass-points", or punctual energy (photons, neutrinos).
...We can consider all the possible movements of all the possible particles and include them in a
(165)
space of movements.
...In space time we can find all possible paths of photons, proton, neutrons, neutrinos, anti-proton, an son on. We consider an infinite number of possible positions, velocities, and other parameters, to be discovered. Among this infinity of paths are the paths refering to a given particle : an electron, for example. Other paths refers to photon. They are different. They form two distinct families, two
distinct species of movements.
We search how to classify particles. Then we search how to define movements' species.
We will use a method similar to Euclid's. The central question is :
What "objects" belong to the same species ?
...Answer : those that can be put one on the other through the action of elements of a group which belong to some sub-group called the regularity of such objects.
...In Euclid's world you cannot transform a sphere into a cube, and vice-versa. They belong to distinct species. There is no sub-group which makes possble to transform spheres into cubes, and vice-versa.
...Similarly, in some group, to be defined, there is no element, belonging to some sub-group, which makes possible to transform the movement of a photon into the movement of an electron. They are basically different, they belong to distinct species.
If there is an element of the group whose action transforms a movement into another movement, then these movements refer to a same species. They are two different ovements of a same particle.
...We are not going to deal with many-particles systems, like atoms, molecules. We will focuss on free particle analysis, cruising in an empty space. Then, during this cruise, a certain number of parameters are conserved (mass, energy, others...)
But the simple examination of the space-time path of a particle is not enough to identify it and put it into a defined species.
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A proton and a neutron can cruise along the same path, at same velocity.
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Two particles can follow the same path, at v = c but one can be a photon and the other a neutrino.
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As we will see later, two photons following the same path, in the same direction, at the velocity of light, can be different. The are P-symmetrical.
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One owns a right helicity .
-
The other a left helicity .
This correspond to polarization of light. Do they belong to distinct species ? Depend the group we choose.
A species is relative to a given group.
The momentum.
...A movement is a peculiar choice, a point in the **momentum space **. Consider movements of species whose only difference is mass. We take two species. A particle whose mass is ma cannot be converted into a particle whose mas is mb . Even if their trajectories can be identical in space-time we consider they are different movements of two distincts species or :
two distinct species of movements. (166)
The momentum is a set of parameters : **J **= { J1 , J2 , J3 , ........, Jn } One is Energy J1 = E .
Three others : ( J2 = px , J3 = py , J4 = pz )
form the impulsion vector p , all quantities which are familiar to physicists.
...These quantities can rise as pure geometric quantities, directly linked to the chosen group. You will see further that the number of quantities which forms the momentum is equal to the dimension of the group.
...Then what's the rules of the game we are going to play to ?