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The special Galileo's group.
...The reader will find this extension in Souriau's book: Structure of Dynamical Systems, Birkhasuer 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. Compute the number of parameters which the Galileo's group depends on. We start from the 3d rotation matrix:
(322)
It's an orthogonal matrix:
(323)
these matrices form the groups SO(3) which a sub-group 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's angles. So that the dimension of the group is ten.
Using the notations:
(328)
we get:
(329)
Associated to the space time vector:
(330)
so that the corresponding action of the Spacial Galileo's group is:
(331)
...Given the Special Galileo's group, it is possible to compute the action of the group on its momentume space. The calculation will not be given here. The the reader can find it in my lectures of groups, available.
Let us give the result:
(332)
We recognize the momentum p and the energy E. The momentum is composed by:
(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 by three independent components lx , ly , lz , forming the "spin vector" ).
The trivial extension of the Special Galileo's group.
The next matrices form a new group.
(334)
It introduces a new component f, a scalar, the "phasis" ( connected to 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 ).
Here again, one can compute 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 one more scalar m and we identify it to the mass. We see that the Special Galileo's 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 arbitrarly, and which does not interact with the other components of the momentum.
