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Neutrinos.
... Considered as massless particles (which they are, until someone proves they have mass), neutrinos possess momentum matrices similar to those of photons. However, their spin is 1/2.
(293)
... Neutrinos travel at the speed of light. They have a quantized spin, different from that of the photon. We know there are three distinct types of neutrinos (electron, muon, tau). However, the Poincaré group does not allow this distinction to be expressed in terms of new geometric characteristics. To achieve this, we will be obliged to introduce charges into the momentum. For neutrinos, we have three distinct charges:
cL = lepton charge = ± 1
cm = muon charge = ± 1
cn = tau charge = ± 1
... Charge inversion corresponds to matter-antimatter duality (according to Dirac). It is called charge conjugation or C-symmetry.
Thus, each neutrino has its own antiparticle, corresponding to:
(295)
Particles with non-zero mass, possessing spin.
There is then no longer a direct link between energy and momentum:
(296)
Denoting m as the "rest mass," we can write:
(297) (297b)
Limit our classification to:
- Proton
- Electron
- Neutron
as well as their respective antiparticles.
... These particles possess various properties, called charges, which do not originate from the Poincaré group, unlike geometric attributes.
These charges are:
- Electric charge e = ± 1
- Baryonic charge cB = ± 1
- Leptonic charge cL = ± 1
- Muonic charge cm = ± 1
- Tauonic charge ct = ± 1
- Gyromagnetic coefficient v (positive or negative)
... Inverting all these quantities (charge conjugation or C-symmetry) corresponds to the matter-antimatter duality (according to Dirac). In summary:
(298)
(298b)
which can point in any direction. The magnetic moment, the spin vector s, and the gyromagnetic factor v are related by the following equation:
(299)
... Here, we use a bold letter to denote the spin vector, which can point in any direction in space. However, its magnitude is quantized. C-symmetry (charge conjugation) reverses the charges and the gyromagnetic factor v, but not the spin. Thus, it reverses the magnetic moment of the particles.