This article presents the description of the internal spaces of fermion and boson fields in d-dimensional spaces, with the odd and even “basis vectors” which are the superposition of odd and even products of operators ${\gamma }^a$. While the Clifford odd “basis vectors” manifest properties of fermion fields, appearing in families, the Clifford even “basis vectors” demonstrate properties of the corresponding gauge fields. In $d\ge (13+1)$ the corresponding creation operators manifest in $d=(3+1)$ the properties of all the observed quarks and leptons, with the families included, and of their gauge boson fields, with the scalar fields included, making several predictions. The properties of the creation and annihilation operators for fermion and boson fields are illustrated on the case $d=(5+1)$, when $SO(5,1)$ demonstrates the symmetry of $SU\left(3\right)\times U\left(1\right)$.