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...One considers a particle. If we move away to a distance c and observe it while moving at velocity v, everything suggests that, conversely, the particle orbits around the observer at velocity v and distance d (radial distance r).
(286)
We now need to explain what the "passage" f = m [ c - v Dt ] represents.
It simply vanishes when c = v D t, i.e. when we relate the velocity v to the combined space translation c and time translation DDt.
(287)
Let us now return to Poincaré's momentum, expressed in a coordinate system where the passage f is zero:
(288)
...A particle corresponds to a specific choice of components of the momentum, which depend on the chosen coordinate system. There always exists a particular coordinate system in which the passage f becomes zero, and in which the momentum vector p can be reduced to a single component (for example, motion along z).
(289)
Thus, the object described by the Poincaré group corresponds to:
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An energy E
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A momentum p
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A proper spin l
...Spin is mass multiplied by length and by velocity. Its dimension is therefore M L² T⁻¹. This is the same dimension as that of the Planck constant.
(289b)
...Geometric quantization, developed by J.M. Souriau (See Structure des Systèmes Dynamiques, Dunod 1983, or Structure of Dynamical Systems, Birkhäuser Ed, 1997), shows that proper spin must be equal to:
(289b)
multiplied by n/2, where n is an integer. The proper spin s equals unity for the photon, and equals 1/2 for the proton, neutron, electron, neutrinos, and their corresponding antiparticles.
The photon.
...We obtain two distinct photons, possessing different helicities, right and left, even if they travel in the same direction with the same energy.
(290)
The energy E and momentum p of a photon are not independent quantities:
(291) E = h n
which gives:
(292)
...Beyond these characteristics (energy, trajectory, helicity), the photon possesses no further properties. In particular, it has no "charge." In other words, we can consider all its charges to be zero. Thus, the photon is identical to its antiparticle (since + zero = - zero).