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The photon.
It is observed that two types of photons are obtained:
(126)
A "right" photon (right polarization) and a "left" photon (left polarization), differing by their helicity. Two photons traveling in the same direction (along OZ), at speed c, and having the same color (same energy E).
For a photon, energy E and momentum p are not independent.
(127)
E = hν
which gives us:
(128)
Beyond these characteristics (energy, direction of propagation, helicity), the photon has no other properties. It possesses no "charge," so to speak, "all its charges are zero," making it identical to its antiparticle (plus zero being identical to minus zero).
Neutrinos.
Considered as massless particles (which they are, to date), they have momentum matrices identical to those of photons, except that their spin is half:
(128b)
(129)
Neutrinos, traveling at speed c, possess an energy-momentum, spin, also quantized, though different from that of the photon. Neutrinos also have helicity. There are neutrinos with right polarization and neutrinos with left polarization.
But it is known that there are three different types of neutrinos, a fact not revealed by the Poincaré group, nor can it be revealed by it (we will need to modify it in what follows to geometrically account for the charges of different particles).
Thus, neutrinos come in three types:
- electronic
- muonic
- tauonic
meaning they can be assigned three types of charges:
e = electric = ±1 (±; the "unit charge").
cₘ = muonic charge = ±1
cₙ = tauonic charge = ±1
This reversal of the sign of charges is also called
charge conjugation or C-symmetry.
Therefore, the three types of neutrinos must be associated with their corresponding antineutrinos:
(130)
However, this distinction—both in quantum numbers, charges, and matter-antimatter duality—is not inherently encoded in the Poincaré group either.
Back to the sub-outline "Groups and Physics"