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...One considers a particle. If we move off to distance c and observe it, while cruising at velocity v, everything looks as if, conversely, the particle were orbiting around the observer, at velocity v and distance d (radial distance r).
(286)
Now we have to explain what the "passage" f = m [ c - v Dt ] is.
It simply vanishes when c = v D t, i.e. when we link the velocity v to the combined space-translation c and time translation DDt.
(287)
Let us return to Poincaré's momentum, written in a system of coordinates where the passage f is zero:
(288)
...A particle as a peculiar choice of the momentum's components, which depend on the chosen system of coordinates. There is always a peculiar system of coordinates in which the passage f becomes zero, and in which the impulsion vector p may reduce to a single component (for example z-movement).
(289)
Then the object described by the Poincaré group corresponds to:
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An energy E
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An impulsion p
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A proper spin l
...A spin is a mass multiplied by a length and by a velocity. Then its dimension is M L2 T-1. It is the dimension of the Planck constant
(289b)
...The geometric quantification, developed by J.M.Souriau (See Structure des Systèmes Dynamiques, Dunod 1983, or Structure of Dynamical Systems, Birkhauser Ed, 1997) shows that proper spin must be equal to:
(289b)
multiplied by n/2, n being an integer. The proper spin s is unity for photon and is equal to 1/2 for proton, neutron, electron and neutrinos and the corresponding antiparticles.
.
The photon.
...We get two distinct photons, which own distinct helicities, right and left, even if they cruise along the same direction, with the same energy.
(290)
The Energy E and the impulsion p of a photon are not independent quantities:
(291)
E = h n
which gives:
(292)
...Besides these characteristics (energy, path, helicity) the photon owns no more ones. In particular it owns no "charge". In other terms we can consider that all its charges are zero. So that the photon is identical to its anti-particle (for + zero = - zero).
