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A geometrical definition of anti-matter.
...As mentioned by Souriau in 1964 in "Geometry and Relativity", Editions Hermann, Chapter VII "Five-Dimensional Relativity", page 413, "the inversion of the fifth dimension corresponds to charge conjugation".
...This holds true if anti-matter corresponds to Dirac's definition. Let us provide an a priori geometric definition of anti-matter. We can represent space with dimensions:
(368)
This can be schematically depicted as follows, with fibered spacetime:
(369)
...We decide that matter's motions correspond to positive values of the zᵢ and anti-matter's motions to negative values, which corresponds to:
(370)
It is straightforward to modify the group in order to incorporate this feature.
(371)
This becomes a four-component group (l = ±1) × 2 (the extended orthochronous group has two connected components).
The component (l = +1) is a subgroup.
...Clearly, the elements with (l = -1) reverse the signs of the additional variables. We decide that these correspond to matter-antimatter duality, based purely on geometric grounds.
Let:
(380)
Then we can write, in a more compact form:
(381)
l = 1 corresponds to the orthochronous subgroup.
(382)
Introduce what we will call an: "l-commutator":
(383)
It belongs to the second component. However, any element of this second component can be written as:
(384) gₒ = gₗ꜀ × gₒ
where gₒ is an element of the orthochronous component of the group.
Schematically:
(385)
On the left: the space of motions, with two half-spaces corresponding to
(zᵢ > 0) motions (matter)
and
(zᵢ > 0) motions (antimatter)
Between them: motions with (zᵢ = 0) (photons).
...On the right: the four-component group. All are orthochronous. All motions correspond to positive energy (below, momentum space).
Call the elements with (l = -1) "anti-elements".
We have depicted the anti-element of the l-commutator.
...Standard orthochronous elements transform a momentum corresponding to a positive-energy motion J₁⁺ into another positive-energy motion J₂⁺.
...But anti-elements transform a positive-energy matter motion into a positive-energy antimatter motion (J₁⁺ → J₃⁺) in momentum space. The representative point lies in the quadrant corresponding to antimatter.
The corresponding paths are illustrated in the evolution space:
(385b)
The calculation of the coadjoint action of the group
(386)
on its momentum space yields:
(387)
see:
J.P. Petit and P. Midy: "Geometrization of matter and antimatter through the coadjoint action of a group on its momentum space. 2: Geometrical description of Dirac's antimatter". Geometrical Physics B, 2, 1998.