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The special Galileo's group.
...The reader will find this extension in Souriau's book : Structure of Dynamical Systems, Birkhäuser Ed. 1997 and, in French, Structure des Systèmes Dynamiques, Ed. Dunod 1973.
...A group can be extended. It means that the number of parameters it depends on will be increased. Calculate the number of parameters on which the Galileo group depends. We start from the 3D rotation matrix :
(322)
It is an orthogonal matrix :
(323)
These matrices form the group SO(3), which is a subgroup of the group O(3) composed of all the orthogonal matrices. We have :
(324)
Recall the difference with :
(325) (325b)
are the most general orthogonal matrices, whose determinants obey :
(326)
End of this parenthesis.
The next group of square matrices (5,5) will be called the special Galileo group :
(327)
The rotation matrix depends on three free parameters, the Euler angles. Thus, the dimension of the group is ten.
Using the notations :
(328)
we obtain :
(329)
Associated with the space-time vector :
(330)
so that the corresponding action of the special Galileo group is :
(331)
...Given the special Galileo group, it is possible to calculate the action of the group on its momentum space. This calculation will not be given here. The reader can find it in my lectures on groups, available.
Let us give the result :
(332)
We recognize the momentum **p **and the energy E. The momentum is composed of :
(333) JSG = { E , p , f , **l **}
...Ten scalar quantities. Ten dimensions for the group. We still have the passage vector **f **and the antisymmetric spin matrix **l **(composed of three independent components lx , ly , lz , forming the "spin vector").
The trivial extension of the special Galileo group.
The following matrices form a new group.
(334)
It introduces a new component f, a scalar, the "phasis" (connected to the quantum world). The dimension of the group becomes 10 + 1 = 11
This new group acts on a five-dimensional space :
(335)
z is an "additional dimension". It was first introduced by the Polish Kaluza, in 1921, then by J.M. Souriau, in 1964 (Géométrie et relativité Hermann Editeur, not translated in English).
Once again, one can calculate the corresponding coadjoint action of the group on its momentum space. We find this :
(336)
The momentum becomes :
(337) JTESG = { m , E , p , **f **, **l **}
...We have an additional scalar m and we identify it to the mass. We see that the special Galileo group, acting on space-time, brings the energy, but not the mass, as a component of the momentum. At the present time (through trivial extension), our particle gets an additional attribute, which is identified to the mass, very arbitrarily, and which does not interact with the other components of the momentum.