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Recall the components of
the Poincaré group:
E: energy
p: momentum
f: passage
l: spin matrix.
To stay close to 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 one which reverses space. -
Lt the
element of the one which reverses time. -
Lst
the element of the one which reverses both space and time.
C being the space-time
translation vector, we get the following components of the Poincaré
group:
gp
( Ln , C) element of the neutral component Gpn.
gp
( Ls , C) element of the component Gps
, which reverses space.
gp
( Lt , C) element of the Gpt , which reverses
time.
gp
( Lst , C) element of the Gpst , which
reverses the two.
The coadjoint action
is:
(313)
P is the four-vector
:
(314)
We have four characteristic
matrices:
(315)
with l = ± 1 and 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
and 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 synonymous
E --->
-E
( we could call it an "E-symmetry"
)
The 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 connected components of the complete Poincaré 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 particle
anti-particle pair. From the collision of these last we get radiative energy, photons.
The result of the collision of a positive energy and a negative energy particle
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 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 group is unable to take
charge of all the characteristics of the particles.
...The
dimension of the Poincaré 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 characteristics
which arise naturally from the Poincaré group, as pure geometrical 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-matter's
world are not "contained" in the Poincaré 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
its two sub-sets ( the two orthochron components plus the two antichron components
) it "builds" as distinct species, positive and negative energy particles.
...If
the complete Poincaré 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 forbid the use of antichron components of the Poincaré group,
that would be kept severely jailed somewhere.
...We
will think about another possibility in a future section.
