How does Clifford algebra show the way to the second quantized fermions with unified spins, charges and families, and with vector and scalar gauge fields beyond the standard model

Abstract

Fifty years ago the standard model offered an elegant new step towards understanding elementary fermion and boson fields, making several assumptions, suggested by experiments. The assumptions are still waiting for an explanation. There are many proposals in the literature for the next step. The spin-charge-family theory of one of us (N.S.M.B.) is offering the explanation for not only all by the standard model assumed properties of quarks and leptons and antiquarks and antileptons, with the families included, of the vectors gauge fields, of the Higgs scalar and Yukawa couplings, but also for the second quantization postulates of Dirac and for cosmological observations, like there are the appearance of the dark matter, of matter–antimatter asymmetry, making several predictions. This theory proposes a simple starting action in $d\ge (13+1)$-dimensional space with fermions interacting with the gravity only (the vielbeins and the two kinds of the spin connection fields), what manifests in $d=(3+1)$ as the vector and scalar gauge fields, and uses the odd Clifford algebra to describe the internal space of fermions, what enables that the creation and annihilation operators for fermions fulfil the anticommutation relations for the second quantized fields without Dirac’s postulates: Fermions single particle states already anticommute. We present in this review article a short overview of the spin-charge-family theory, illustrating shortly on the toy model the breaks of the starting symmetries in $d=(13+1)$-dimensional space, which are triggered either by scalar fields — the vielbeins with the space index belonging to $d>(3+1)$ — or by the condensate of the two right handed neutrinos, with the family quantum number not belonging to the observed families. We compare properties and predictions of this theory with the properties and predictions of $SO\left(10\right)$ unifying theories.