f4202 Geometrization of matter and antimatter through the coadjoint action of a group on its momentum space. 1 :
Charges as additional scalar components of the momentum of a group acting on a 10-dimensional space.
Geometrical definition of antimatter. (p2) – A third component :
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constructed from the component $L_t$ of the Lorentz group.
– and a fourth :
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constructed from the component $L_{st}$ of the Lorentz group.
A group acts on its momentum space [1]. Let $J_p$ denote the momentum space associated with the Poincaré group.
...Each particular element J$_p$ of $J_p$ corresponds to a particular movement of a relativistic mass point, described by this group. One can compute the coadjoint action of the group on the momentum [1].
The momentum is a set of 10 components (equal to the dimension of the group). These components are :
(16) J$_p$ = { $E$, $p_x$, $p_y$, $p_z$, $f_x$, $f_y$, $f_z$, $s_x$, $s_y$, $s_z$ } = { $E$, p, f, s }
$E$ is the energy.
p is the momentum vector :
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f is the passage vector [1].
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s is an antisymmetric (3,3) matrix, whose independent components are
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{ $s_x$, $s_y$, $s_z$ }
The momentum can be arranged in matrix form [1], with :
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and :
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Introduce the four-vector of momentum-energy :
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or :
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Then the coadjoint action of the Poincaré group can be written in matrix form :
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More explicitly :
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...It is interesting to study the effect of the different components of the complete Poincaré group on the components of its momentum space. One can focus on specific matrices :
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A is the Lorentz matrix associated.
The coadjoint action gives :
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where $I_4$ is the neutral component of the complete Poincaré group.
The corresponding coadjoint action is :
$E \mapsto E$ ; p $\mapsto$ p ; f $\mapsto$ f ; s $\mapsto$ s
— which reverses space. The corresponding coadjoint action is :
$E \mapsto E$ ; p $\mapsto$ –p ; f $\mapsto$ –f ; s $\mapsto$ s
— which reverses time. The corresponding coadjoint action is :
$E \mapsto$ –$E$ ; p $\mapsto$ p ; f $\mapsto$ –f ; s $\mapsto$ s
— which reverses both space and time. The corresponding coadjoint action is :
$E \mapsto$ –$E$ ; p $\mapsto$ –p ; f $\mapsto$ f ; s $\mapsto$ s
As pointed out by J.M. Souriau [1], the two components
\begin{pmatrix} E \ \mathbf{p} \end{pmatrix}
are accompanied by the inversion of the energy $E \mapsto$ –$E$, which implies the inversion of the mass $m \mapsto$ –$m$.
Define the following sets of matrices :
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