f4203 通过群在其动量空间上的共轭作用，实现物质和反物质的几何化。1：作为作用于10维空间的群的动量的额外标量分量的电荷。反物质的几何定义。(p3) 完整的庞加莱群是：
(31) Gp = Gn U Gs U Gt U Gst
中性部分Gn是第一个子群。正时序群[1]：
(32) Go = Gn U Gs
也是庞加莱群的一个子群。
反时序部分[1]：
(33) Gac = Gt U Gst 不是一个群。显然：
(34) Gp = Go U Gac
...如[1]中所述，Gac = Gt U Gst元素的存在可能会产生负质量粒子，如时间倒流的特殊物质运动。在他的书[1]中，J.M. Souriau提出了两种解决方案：
-
要么我们简单地决定负质量不能存在。
-
要么庞加莱群被限制在其正时序子群中。
(35) Go = Gn U Gs
2) 庞加莱群的中心扩张。 (36)
是通过正时序子群构建的庞加莱群的中心扩张。相应的动作是：(37)
z是一个额外的维度，第五个维度。群的维度变为11，动量获得相应的额外分量：
(38) Jpe = { c , M , P } = { c , Jp }
共轭作用给出：(39)
...这个第11个分量c的物理意义从未被清楚地理解。通过他的几何量化方法，J.M. Souriau表明自旋必须被量子化[1]。选择一个使该分量为零的坐标系，并且只考虑z方向的运动，动量矩阵Jp变为：
(40)
其中E是能量，p是动量矢量的模，s是自旋。
光子对应于：
(41)
具有两种不同的螺旋度：右旋和左旋（偏振）。
中微子对应于：
(42)
同样具有两种不同的螺旋度。
像质子、电子、中子这样的非零质量粒子对应于：
(43)
其中：(44)
(45))
...通过Kostant-Kirilov-Souriau方法，从扩展的庞加莱群（36）可以推导出[1]的相对论量子克莱因-戈登方程。同样[1]，非相对论的Bargmann群（1960年）给出了非相对论的薛定谔方程。
那么反物质呢？
...在之前的著作[2]中，J.M. Souriau发展了五维的广义相对论，将额外的维度z加入时空（x，y，z，t）中。
...然后在参考文献[2]，第七章，第413页，他将第五维的反转（z→-z）与电荷共轭（或电荷反转，或C对称性）相联系，将物质转化为反物质。
原始版本（英文）
f4203 Geometrization of matter and antimatter through coadjoint action of a group on its momentum space. 1 : Charges as additional scalar components of the momentum of a group acting on a 10d-space. Geometrical definition of antimatter. (p3) The complete Poincaré group is :
(31) Gp = Gn U Gs U Gt U Gst
The neutral component Gn is the first sub-group. The orthochron group [1] :
(32) Go = Gn U Gs
is also a sub-group of the Poincaré group.
The antichron part of the group [1] :
(33) Gac = Gt U Gst is not a group. Obvioulsy :
(34) Gp = Go U Gac
...As pointed out in [1] the presence of the elements of Gac = Gt U Gst may produce negative mass particles, as peculiar movements of matter, runing backward in time. In his book [1] J.M.Souriau suggests two solution :
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Either one simply decides that negative masses cannot exist.
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Either the Poincaré group is limited to its orthochron subgroup.
(35) Go = Gn U Gs
2) The central extension of the Poincaré group. (36)
is the central extension of the Poincaré group, built from the orhochron sub-group. The corresponding action is : (37)
z is an additional dimension, a fifth dimension. The dimension of the group becomes 11 and the momentum gets a corresponding extra component :
(38) Jpe = { c , M , P } = { c , Jp }
The coadjoint action gives : (39)
...The physical meaning of this 11th component c was neven clearly undestood. Through his geometric quantification method, J.M.Souriau shows than the spin must be quanticized [1]. Choosing a coordinate system in which the passage becomes zero, and considering only z-motions, the Jp the momentum matrix becomes :
(40)
where E is the energy, p the modulus of the vector impulsion and s the spin.
Photons correspond to
(41)
with two distinct helicities : right and left (polarization).
Neutrinos correspond to :
(42)
with also two distinct helicities.
Non zero mass particles like proton, electron, neutron, correspond to :
(43)
with : (44)
(45))
...From the extended Poincaré group (36), through the Kostant-Kirilov-Souriau method one can derive [1] the relativistic quantum Klein-Gordon equation. Similarly [1] the non-relativist Bargmann group (1960) gives the non-relativistic Schödinger equation.
What about antimatter ?
...In a former book [2] J.M. Souriau developped general relativity in five dimensions, adding an extra dimension z to space-time ( x , y , z , t )
...Then, reference [2], Chater VII , page 413, he identifies the inversion of the fifth dimension ( z ---> - z ) to the charge conjugation ( or charge inversion, or C-symmetry ) transforming matter into anti-matter.