"Fermion-Boson Duality and Quantum Field Theory Using an Extended Dirac Equation" — Supplementary Mathematica Files (256×256 Γ/Ω verification & e-QED scattering notes)
Abstract
This record provides supplementary materials for the project “Fermion-Boson Duality and Quantum Field Theory Using an Extended Dirac Equation.”Using Wolfram Mathematica, we reproduce and verify the anticommutation relations and basic properties of the 256×256 two-index gamma matrices Γμν\Gamma_{\mu\nu}Γμν and the bosonic counterparts Ωμν\Omega_{\mu\nu}Ωμν. We also include minimal tree-level notes for representative e-QED scattering processes (Compton, e+e− → μ+μ−e^+e^-\!\to\!\mu^+\mu^-e+e−→μ+μ−, Møller, Bhabha). In the flat limit the results coincide with standard QED.
Full text
Print ["------------(*γmatrix*)-------------------------"]; γ [0] = {{1, 0, 0, 0},{0, 1, 0, 0},{0, 0, -1, 0},{0, 0, 0, -1}}; γ [1]=I* {{0, 0, 0, 1},{0, 0, 1, 0},{0, 1, 0, 0},{1, 0, 0, 0}}; γ [2]=I* {{0, 0, 0, -I},{0, 0, I, 0},{0, -I, 0, 0},{I, 0, 0, 0}}; γ [3]=I* {{0, 0, 1, 0},{0, 0, 0, -1},{1, 0, 0, 0},{0, -1, 0, 0}}; Print ["γ0=", MatrixForm[γ[0]]]; Print ["γ1=", MatrixForm[γ[1]]]; Print ["γ2=", MatrixForm[γ[2]]]; Print ["γ3=", MatrixForm[γ[3]]]; Print ["--------------(*Anticommutation relation of γmatrix*)-----------------------"]; For[kh =0, kh ≤3, kh++, For[ks1 =0, ks1 ≤3, ks1++, yf =γ[kh].γ[ks1]+γ[ks1].γ[kh]; Print["γ", kh, "*γ", ks1, "+γ", ks1, "*γ", kh, "=", MatrixForm[yf]]; ]]; Print ["------------(*ωmatrix*)-------------------------"]; ω [0]=(γ[0] + γ[3]) / 2; ω [1]=(γ[1] + γ[1]) / 2; ω [2]=(γ[2] + γ[2]) / 2; ω [3]=(γ[3] + γ[0]) / 2; Print ["ω0=γ0+γ3=", MatrixForm[ω[0]]]; Print ["ω1=γ1+γ1=", MatrixForm[ω[1]]]; Print ["ω2=γ2+γ2=", MatrixForm[ω[2]]]; Print ["ω3=γ3+γ0=", MatrixForm[ω[3]]]; Print ["--------------(*Anticommutation relation of ωmatrix*)-----------------------"]; For[kh =0, kh ≤3, kh++, For[ks1 =0, ks1 ≤3, ks1++, yf =ω[kh].ω[ks1]+ω[ks1].ω[kh];
Print["ω", kh, "*ω", ks1, "+ω", ks1, "*ω", kh, "=", MatrixForm[yf]]; ]]; Print ["--------------(*Invariant using ωmatrix*)-----------------------"]; s1 =ω[0]*A0 +ω[1]*A1 +ω[2]*A2 +ω[3]*A3; y =s1.s1; Print ["(ω0*A0+ω1*A1+ω2*A2+ω3*A3)*(ω0*A0+ω1*A1+ω2*A2+ω3*A3)=", Simplify[y〚1〛]〚1〛]; ------------(*γmatrix*)------------------------- γ 0= 1 0 0 0 0 1 0 0 00-1 0 000-1 γ 1= 0 0 0 0 0 0 00 0 0 0 0 γ 2= 0 0 0 1 0 0 -1 0 0 1 0 0 -1 0 0 0 γ 3= 0 0 0 000- 000 0-0 0 --------------(*Anticommutation relation of γmatrix*)----------------------- γ 0*γ0+γ0*γ0= 2 0 0 0 0 2 0 0 0 0 2 0 0 0 0 2 γ 0*γ1+γ1*γ0= 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 γ 0*γ2+γ2*γ0= 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 γ 0*γ3+γ3*γ0= 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 γ 1*γ0+γ0*γ1= 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 γ 1*γ1+γ1*γ1= - 2 0 0 0 0-2 0 0 0 0 -2 0 0 0 0 -2 2 01_omega_matrix_properties_check_Ver1.nb
γ 1*γ2+γ2*γ1= 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 γ 1*γ3+γ3*γ1= 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 γ 2*γ0+γ0*γ2= 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 γ 2*γ1+γ1*γ2= 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 γ 2*γ2+γ2*γ2= -2 0 0 0 0-2 0 0 0 0 -2 0 0 0 0 -2 γ 2*γ3+γ3*γ2= 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 γ 3*γ0+γ0*γ3= 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 γ 3*γ1+γ1*γ3= 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 γ 3*γ2+γ2*γ3= 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 γ 3*γ3+γ3*γ3= -2 0 0 0 0-2 0 0 0 0 -2 0 0 0 0 -2 ------------(*ωmatrix*)------------------------- ω 0=γ0+γ3= 1 20 20 01 20- 2 20-1 20 0- 2 0-1 2 ω 1=γ1+γ1= 0 0 0 0 0 0 00 0 0 0 0 ω 2=γ2+γ2= 0 0 0 1 0 0 -1 0 0 1 0 0 -1 0 0 0 01_omega_matrix_properties_check_Ver1.nb 3
ω 3=γ3+γ0= 1 20 20 01 20- 2 20-1 20 0- 2 0-1 2 --------------(*Anticommutation relation of ωmatrix*)----------------------- ω 0*ω0+ω0*ω0= 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ω 0*ω1+ω1*ω0= 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ω 0*ω2+ω2*ω0= 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ω 0*ω3+ω3*ω0= 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ω 1*ω0+ω0*ω1= 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ω 1*ω1+ω1*ω1= - 2 0 0 0 0-2 0 0 0 0 -2 0 0 0 0 -2 ω 1*ω2+ω2*ω1= 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ω 1*ω3+ω3*ω1= 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ω 2*ω0+ω0*ω2= 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ω 2*ω1+ω1*ω2= 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ω 2*ω2+ω2*ω2= - 2 0 0 0 0-2 0 0 0 0 -2 0 0 0 0 -2 ω 2*ω3+ω3*ω2= 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 4 01_omega_matrix_properties_check_Ver1.nb
ω 3*ω0+ω0*ω3= 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ω 3*ω1+ω1*ω3= 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ω 3*ω2+ω2*ω3= 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ω 3*ω3+ω3*ω3= 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 --------------(*Invariant using ωmatrix*)----------------------- (ω0*A0+ω1*A1+ω2*A2+ω3*A3)*(ω0*A0+ω1*A1+ω2*A2+ω3*A3)=-A1 2 -A2 2 01_omega_matrix_properties_check_Ver1.nb 5