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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 denote:
- Ln the element of the neutral component Ln of the full Lorentz group L.
- Ls the element that inverts space.
- Lt the element that inverts time.
- Lst the element that inverts both space and time.
Let C be the spacetime translation vector. Then the following components of the Poincaré group arise:
gp (Ln, C) — element of the neutral component Gpn.
gp (Ls, C) — element of the component Gps, which inverts space.
gp (Lt, C) — element of the component Gpt, which inverts time.
gp (Lst, C) — element of the component Gpst, which inverts both.
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)
so that 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 affect the spin matrix l.
On the contrary, T-inversion and energy inversion are synonymous.
E → –E (we could call this a "E-symmetry").
The spin s, like the magnitude of the spin vector s, is merely a number, unchanged by the action of any group component, whether orthochronous or antichronous. ... The rest energy of a particle is mc². As we see, mass inversion goes hand in hand with time inversion. But space inversion does not alter energy or mass.
Souriau calls the first two connected components of the full Poincaré group:
Gpn, Gps
the orthochronous components (Gpn being the neutral component).
The other two: Gpt, Gpst
are the antichronous components. This raises the issue of negative masses. Do they exist? If so, what happens during a collision between particles with opposite masses and energies:
+mc² and –mc²?
...Note that this does not correspond to the so-called "annihilation" of a particle-antiparticle pair. When such pairs collide, radiative energy—photons—are produced. The result of colliding a positive-energy particle with a negative-energy one would be far more troubling, as it should yield nothing at all.
...What is Nature? What are particles? In this approach, we begin with a given group: the Poincaré group. Then we construct the action of this group on its momentum space. This momentum space consists of points, each representing the motion of one of the geometric objects forming the space associated with the group.
...In what follows, we will show that the Poincaré group cannot account for all particle characteristics.
...The dimension of the Poincaré group is 10.
Thus, the dimension of momentum space is also ten. (321) J = {E, p, f, l}
If we choose a coordinate system tied to the particle, f = 0.
In summary, the only characteristics that naturally emerge from the Poincaré group as geometric quantities are:
For a massless particle:
- Its energy — its spin and helicity
For a massive particle:
- Its rest mass — its spin.
Other characteristics:
- Electric charge
- Baryonic charge
- Leptonic charge
- Muonic charge
- Tauonic charge
- Gyromagnetic factor
- Whether the particle belongs to the matter or antimatter world
are not "contained" in the Poincaré group. We will enrich the group later to handle these.
Currently, the group does not "construct" particles and antiparticles. But when completed by its two subsets (the two orthochronous components plus the two antichronous components), it "constructs" distinct species: positive-energy and negative-energy particles.
...If the full Poincaré group "governs" the universe, then positive and negative energies could coexist, so that their encounter would lead to complete annihilation. If the universe were half filled with positive-energy particles and half with negative-energy particles, there would be a great risk that the entire universe annihilated itself, leaving nothing behind:
- No positive-energy particles
- No negative-energy particles
- No positive-energy photons
- No negative-energy photons
Nothing. Absolutely nothing. What a disaster!
...As suggested by Souriau, in His infinite wisdom, God created only positive-energy particles and positive-energy photons. Likewise, His angels would forbid the use of antichronous components of the Poincaré group, which would be strictly confined somewhere.
...We will consider another possibility in a later section.