Renormalization of $(2+1)D$ scalar Weyl spinors interactions on lattices using the Clifford groups

We consider symplectic quaternions instead of unitary spinors sitting on a lattice, and calculate the fixed point Wilson action on a finite $2D$ plane expanded by $u_{1}a{e}_1+u_{2}a{e}_2$ and on two $2D$ planes separated by $a e_1\wedge e_2$. Only the nearlest neighbor interactions are considered. Following Migdal and Kadanoff, we perform the renormalization of the Wilson action by making the lattice spacing $(\frac{1}{2}{)}^{h}a$ $(h=0,\cdots ,11)$, in order to simulate bosonic and solitonic phonon propagation in materials. Renormalization group method of Benfatto and Gallavotti for $(2+1)D$ scalar ${\varphi }^4$ system for sound propagation in Fermi sea is applied and feasibility of numerical simulation is discussed.