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 $\phi^4$ system for sound propagation in Fermi sea is applied and feasibility of numerical simulation is discussed.