a4123

Recall the components of
the Poincaré's group :

E : energy

p : impulsion

f : passage

l : spin matrix.

To be close of Souriau's text, let us call

• Ln
  the element of the neutral component Ln of the complete Lorentz group L.

• Ls
  the element of the the one which inverses space.

• Lt the
  element of the the one which inverses time.

• Lst
  the element of the the one which inverses both space and time.

C being the space-time
translation vector, we get the following components of the Poincaré's
group :

gp
( Ln , C) element of the neutral component Gpn.

gp
( Ls , C) element of the component Gps
, which inverses space.

gp
( Lt , C) element of the Gpt , which inverses
time.

gp
( Lst , C) element of the Gpst , which
inverses the two.

The coadjoint action
is :
(313)

P is the four-vector
:
(314)

We have four characteristic
matrixes :
(315)

with l = ± 1 et m = ± 1 .

Ln
= L ( l = 1 ; m = 1)

Ls
= L ( l = - 1 ; m = 1)

Lt =
L ( l = 1 ; m = - 1)

Lst
= L ( l = - 1 ; m = - 1)

(316)

(317)

(318)

We are interested in
C = 0
(319)

whence l' = l
et f' = l m
f

and :
(320)

gp
( Ln , C) : I E --> E ; p --> p
; f ---> f ; l ----> l

gp
( Ls , C) : I E --> E ; p --> - p
; f ---> - f ; l ----> l

gp (
Lt , C) : I E --> - E ; p --> p
; f ---> - f ; l ----> l

gp
( Lst , C) : I E --> - E ; p --> - p
; f ---> f ; l ----> l

The inversions do not modify the spin matrix l .

Oppositely, T-inversion
and energy inversion are synonym

E --->
-E

( we could call it a "E-symmetry"
)

cThe spin s, as the modulus of the spin vector
s, is only a number, unchanged by the action of all the components
of the groups, whatever they are orthochron or antichron. .
...The rest energy of a particle is mc2.
As we see the mass-inversion goes with the time-inversion. But space-inversion
does not change the energy, nor the mass.

Souriau calls the two
first connex components of the complete Poincaré's group :

Gpn , Gps

the orthochron components
( Gpn is the neutral component ).

And the two others :

Gpt , Gpst

the antichron components.
It arises the problem
of negative masses. Do they exist ? If yes, what about the collision between
particles with opposite masses and energies :

• mc2 and

• mc2

...Notice
it does not correspond to the so-called "annihilation" of a pair particle
anti-particle. From the collision of these last we get radiative energy, photons.
The result of the collision of a positive energy and a negative energy particles
should be much more puzzling, for it should be zero : nothing.

...What's
Nature, what are particles ? In this approach we start from a given group
: the Poincaré's group. Then we build the action of this group on its
momentum space. This momentum space is made of points. Each point is a movement
of one of the geometrical objects which compose space associated to the group.

...In
the following we will show that the Poincaré's group is unable to take
in charge all the characteristics of the particles.

...The
dimension of the Poincaré's group is 10.

Then the dimension of
the momentum is also ten.
(321)

J = { E
, p , f , l }

If we choose a system of coordinates linked to the particle f = 0 .

To sum up, the only charactristics
whick arise naturally from the Poincaré group, as ure geoemtrical quantities,
are :

For a zero mass particle:

• Its Energy
• Its spin and helicity

For non zero mass particle :

• Its rest mass
• Its spin.

The other characteristics
:

• Electric charge

• Baryonic charge

• Leptonic charge

• Muonic charge

• Tauonic charge

• Gyromagnetic factor

and the fact that this particle could correspond to matter or anti-matters'
world are not "contained" in the Poincarés' group. We will enrich the
group further to deal with.

At the present time the
group does not "build" particles and anti-particles. But, if completed by
his two sub-sets ( the two orthochron components plus the two antiochron components
) it "builds" as distinct species, postive and negative energy particles.

...If
the complete Poincaré's group "rules" the universe, then positive and
negative energy may cohabit, so that meet and give full annihilation phenomenon.
If the universe would be filled by 50 % of positive energy particles and 50
of negative energy particles there would we be a great risk that all the universe
would self-annihilate so that nothing would stay :

• No positive energy
  particles.

• No negative energy
  particles

• No positive energy
  photon

• No negative energy
  photon.

Nothing. Nothing at all. What a drag !

...As
suggested by Souriau, God, in his infinite wisdom, would have created only
positive energy particles and positive energy photons. Similarly his angels
would forbide the use of antichron components of the Poincaré's group,
that would be kept severely jailed somewhere.

...We
will think about another possibility in a future section.

Index Dynamic Groups Theory