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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 of the 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 possible 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 movements of a same particle.
...We are not going to deal with many-particles systems, like atoms, molecules. We will focus 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. They are P-symmetrical.
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One owns a right helicity .
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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 mass is mb . Even if their trajectories can be identical in space-time we consider they are different movements of two distinct 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 ?