Gauge fields with respect to $d=(3+1)$ in the Kaluza-Klein theories and in the spin-charge-family theory

Abstract

It is shown that in the spin-charge-family theory, as well as in all the Kaluza-Klein like theories, vielbeins and spin connections manifest in $d=(3+1)$ space equivalent vector gauge fields, when space with $d\ge5$ manifests large enough symmetry. The authors demonstrate this equivalence in spaces with the symmetry of the metric tensor in the space out of $d=(3+1)$ - $g^{\sigma \tau} = \eta^{\sigma \tau} \,f^{2}$ - for any scalar function $f$ of the coordinates $x^{\sigma}$, where $x^{\sigma}$ denotes coordinates of space out of $d=(3+1)$. Also the connection between vielbeins and scalar gauge fields in $d=(3+1)$ (offering the explanation for the Higgs's scalar) is discussed.