Title:Representations in Grassmann space and fermion degrees of freedom

Abstract:In Ref. [arXiv:1802.05554v3] one of the authors (N.S.M.B.) studies the second quantization of fermions with integer spin while describing the internal degrees of freedom of fermions in Grassmann space. In this contribution we study the representations in Grassmann space of the groups $SO(5,1)$, $SO(3,1)$, $SU(3) \times U(1)$, and $SO(4)$, which are of particular interest as the subgroups of the group $SO(13,1)$. The second quantized integer spin fermions, appearing in Grassmann space, not observed so far, could be an alternative choice to the half integer spin fermions, appearing in Clifford space. The spin-charge-family theory, using two kinds of Clifford operators --- $\gamma^a$ and $\tilde{\gamma}^a$ --- for the description of spins and charges (first) and family quantum numbers (second), offers the explanation for not only the appearance of families but also for all the properties of quarks and leptons, the gauge fields, scalar fields and others. In both cases the gauge fields in $d \ge(13+1)$ --- the spin connections $\omega_{ab \alpha}$ (of the two kinds in Clifford case and of one kind in Grassmann case) and the vielbeins $f^{\alpha}{}_{\alpha}$ --- determine in $d=(3+1)$ scalars, those with the space index $\alpha=(5,6,\cdots,d)$, and gauge fields, those with the space index $\alpha=(0,1,2,3)$. While states of the Lorentz group and all its subgroups (in any dimension) are in Clifford space in the fundamental representations of the groups, with the family degrees of freedom included, states in Grassmann space manifest with respect to the Lorentz group adjoint representations, allowing no families.