Abstract
The important role of the operator γ5 in the physics of elementary particles and their interactions is considered basic in this paper where it is shown that it corresponds to the duality rotation in space-time and how the dual geometry can be used to consider new symmetries for the wave equations of particles and their interaction fields. The new set of symmetries is shown to exactly correspond to what is needed to find the known schemes of leptons and quarks and to be the origin of many of the properties which are found expermentally for these elementary particles.
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