## The Dirac equation and a non-chiral electroweak theory in six dimensional spacetime from a locally gauged SO(3,3) symmetry group

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Dartora, C. A.

Cabrera, G. G.

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A toy model for the electroweak interactions(without chirality) is proposed
in a six dimensional spacetime with 3 timelike and 3 spacelike coordinates. The
spacetime interval $ds^2=dx_\mu dx^\mu$ is left invariant under the symmetry
group SO(3,3). We obtain the six-dimensional version of the Dirac gamma
matrices, $\Gamma_\mu$, and write down a Dirac-like lagrangian density, ${\cal
L}=i \bar{\psi} \Gamma ^\mu \nabla_\mu \psi$. The spinor $\psi$ is decomposed
into two Dirac spinors, $\psi_1$ and $\psi_2$, which we interpret as the
electron and neutrino fields, respectively. In six-dimensional spacetime the
electron and neutrino fields are then merged in a natural manner. The SO(3,3)
Lorentz symmetry group must be locally broken to the observable SO(1,3) Lorentz
group, with only one observable time component, $t_z$. The $t_z$-axis may not
be the same at all points of the spacetime and the effect of breaking the
SO(3,3) spacetime symmetry group locally to an SO(1,3) Lorentz group is
perceived by the observers as the existence of the gauge fields. The origin of
mass may be attributed to the remaining two hidden timelike dimensions. We
interpret the origin of mass and gauge interactions as a consequence of extra
time dimensions, without the need of the so-called Higgs mechanism for the
generation of mass. Further, we are able to give a geometric meaning to the
electromagnetic and non-abelian gauge symmetries.

##### Keywords

High Energy Physics - Theory