A trial to understand the supersymmetry relations through extension of the second quantized fermion and boson fields, either to strings or to odd dimensional spaces

Abstract

The article studies the extension of the internal spaces of fermion and boson second quantized fields, described by the superposition of odd (for fermions) and even (for bosons) products of the operators $\gamma^ {a}$, to strings and odd dimensional spaces.\\ For any symmetry $SO(d-1,1)$ of the internal spaces, it is the number of fermion fields (they appear in families and have their Hermitian conjugated partners in a separate group) equal to the number of boson fields (they appear in two orthogonal groups), manifesting a kind of supersymmetry, which differs from the usual supersymmetry.\\ The article searches for the supersymmetry arising from extending the ``basis vectors'' of second quantized fermion and boson fields described in $d=2(2n+1)$ (in particular $d=(13+1)$) either to strings or to odd-dimensional spaces ($d=2(2n+1)+1$).