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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 ofthe complete, four components Galileo
group :
(163)
which rules P, Tand
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 possble
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 ovements of
a same particle.
...We
are not going to deal with many-particles systems, like atoms, molecules.
We will focuss 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.
-
A proton and a neutron
can cruise along the same path, at same velocity. -
Two particles can follow
the same path, at v = c but one can be a photon and the other a neutrino. -
As we will see later,
two photons following the same path, in the same direction, at the velocity
of light, can be different. The are P-symmetrical. -
One owns a right helicity
. -
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 mas is
mb . Even if their trajectories can be identical in space-time we consider
they are different movements of two distincts 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 ?
Index Dynamic Groups Theory