Do we understand the internal spaces of second quantized fermion and boson fields, with gravity included? Relation with strings theories

Abstract

The article proposes the description of internal spaces of fermion (quarks and leptons and antiquarks and antileptons) and boson (photons, weak bosons, gluons, gravitons and scalars) second quantized fields in a unique way if they all are massless. The internal spaces are described by ``basis vectors'', which are the superposition of odd (for fermions) and even (for bosons) products of the operators $\gamma^ {a}$. For an arbitrary 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 differ of the string supersymmetry. On the assumption that fermions and bosons are active (they have momenta different from zero) only in $d=(3+1)$ ordinary space-time, bosons present vectors if they carry the space index $\mu=(0,1,2,3)$, and present scalars if they carry the index $\sigma \ge 5$. The author discusses this theory's latest achievements, with a trial to understand whether the extension to strings or to odd-dimensional spaces can lead to a new kind of supersymmetry. This model, named {\it spin-charge-family} theory, manifests in a long series of papers on the phenomenological success of the theory in elementary particle physics and cosmology.