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A geometrical definition
of anti-matter.
...As
mentioned by Souriau in 1964 in "Géometry and Relativité", Editions
Hermann, chapter VII "La Relativité à Cinq Dimensions" ( the
five dimensional relativity ), page 413, "the inversion of the fifth dimension
corresponds to the charge conjugation".
...It
is true if the anti-matter corresponds to Dirac's definition. Let us give
an a priori geometric definition of anti-matter. We can figure space with
dimensions :
(368)
This can be figured schematically
as follows, with fibered space-time :
(369)
...We
decide that matter's movements correspond to positive z i 's values and anti-matter's
movements to negative ones, which corresponds to :
(370)
It is easy to modify
the group in order to integrate this in it.
(371)
This becomes a four-components
group ( l = ± 1 ) x 2 ( the extended orthochron group owns two connex
components).
The component ( l = +1
) is a sub-group.
...Clearly,
the ( l = - 1 ) elements change the signs of the additional variables. We
decide that it corresponds to matter anti-matter duality, on pure geometric
grounds.
Let :
(380)
Then we can write, in a more compact way :
(381)
l = 1 corresponds
to the orthochron sub-group.
(382)
Introduce what we will call a : " l-commuter " :
(383)
It belongs to the second
component. But any element of this second component can be written :
(384)
go
= glc x go
being an element of the
orthochron component of the group.
Schematically :
(385)
Left : the movement space,
with two half-spaces, corresponding to
(z i >
0) movements ( matter )
and
(z i >
0) movements ( anti-matter )
Between the two the :
z i = 0 movements
( photons ).
...Right,
the four components group. All are orthochron. All movements correspond to
positive energy ( below, momentum space ).
Call the ( l = - 1 )
elements "anti-elements".
We have figured the l-commuter
anti-element.
...Normal
orthochron elements transform a momentum corresponding to a positive energy
movement J1+ into another positive energy movement J2+.
...But
anti-elements transform positive energy matter's movement into positive energy
anti-matter's movement ( J1+ -----> J3+
) in momentum space. The figurative point is in the quarter which corresponds
to anti-matter.
The corresponding paths
are figured in the evolution space
(385b)
The calculation of the
coadjoint action of the group
(386)
on its momentum gives
:
(387)
see :
J.P.Petit and P.Midy : "Geometrization of matter and anti-matter through
coadjoint action of a group on its momentum space. 2 : Geometrical description
of Dirac's anti-matter". Geometrical Physics B, 2 , 1998.
Index Dynamic Groups Theory