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Particles and anti-particles'
zoos.
...Particles are species,
but there are also peculiar movements and peculiar species in the momentum
space. We can build the following two zoos :
(362)
From these two zoos we can write the corresponding moments :
(363)
Jpe
= { q , cB , cL , cm , ct ,
v , Jp }
Jj
= { 0 , 0 , 0 , 0 , 0 , 0 , Jp } : photon
Jp
= { 1 , 1 , 0 , 0 , 0 , vp , Jp } : proton
Jn
= { 0 , 1 , 0 , 0 , 0 , vn , Jp } : neutron
Je
= { -1 , 0 , 1 , 0 , 0 , ve , Jp } : electron
Jne
= { 0 , 0 , 1 , 0 , 0 , vne , Jp } : electronic
neutrino
Jnm
= { 0 , 0 , 0 , 1 , 0 , vnm , Jp } : m neutrino
Jnt
= { 0 , 0 , 0 , 0 , 1 , vnt , Jp } : t neutrino
...Doing this, we have created
a priori these two different zoos : species of matter and species of anti-matter.
We have no group's action which make possible to transform a particle into an
antiparticle.
All that is based on the following dynamic group :
(364)
What is the momentum
?
...Remember
: when building the Poincaré's group, we started from the Lorentz group
element L , which was axiomatically defined, using a "mirror" matrix
G :
(365
(366)
This being linked to a quadratic form : the Minkowski metric.
(367)
...A
Minkowski metric refers to an empty space. Our group decribes lonely particles,
not interacting systems of several particles. The movement of a particle is
a geodesic of Minkowski space : a straight line. If it is a zero mass particle
it corresponds to a "zero-length" geodesic, but it is not a wrong image to
figure the movements of particles as straight lines in space time.
(365b)
...The
set of points composing the momentum space represents all the possible movements
of all possible species of particles. A group's action (coadjoint action),
based on a given element g of the dynamic group G changes a movement
into another movement.
(366b)
...On
the above figure we see how an element of the group makes possible to transform
a given movement of an electron into another movement of this same species.
But, through coadjoint action and elements of the group we could not transform
the movement of an electron into the movement of a neutron, or of a photon.
The movement space is divided into sub-sets, each refering to all possible
movements of a given species.
...We
have seen above that the complete Poincaré group gives negative energy
particles. Then, if now we do not refuse to deal with, we must consider two
distinct sub-spaces :
(367b)
