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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 mean that the number of the 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 matrixes form the
groups SO(3) which a sub-group of the group O(3) composed of all the orthogonal
matrixes. We have :
(324)
Recall the difference with :
(325)
(325b)
are the most general
orthogonal matrixes, whose determinants obey :
(326)
End of this parenthesis.
The next group of square
matrixes (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 matrixes 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.
Index Dynamic Groups Theory