Abstract
We present a unified geometric model of Standard Model fields within a compact three-complex-dimensional
Clifford Algebra fiber over 4D spacetime.
Fiber Algebra
We ascribe the fiber a Hermitian metric h m n ˉ h_{m\bar{n}} h m n ˉ
α m α n + α n α m = 0 α m α n † + α n † α m = h m n ˉ , m , n ∈ 1 , 2 , 3 \begin{equation}
\begin{aligned}
\alpha_m \alpha_n + \alpha_n \alpha_m &= 0 \\
\alpha_m \alpha^{\dagger}_n + \alpha^{\dagger}_n \alpha_m &= h_{m\bar{n}}, \quad m, n \in {1,2,3}
\end{aligned}
\end{equation} α m α n + α n α m α m α n † + α n † α m = 0 = h m n ˉ , m , n ∈ 1 , 2 , 3 Rotors in the fiber are rotations between the different α \alpha α U ( 3 ) U(3) U ( 3 ) U ( 3 ) U(3) U ( 3 )
λ 1 ≡ 1 2 ( [ α 2 , α 3 † ] + [ α 3 , α 2 † ] ) λ 2 ≡ i 2 ( [ α 2 , α 3 † ] − [ α 3 , α 2 † ] ) λ 3 ≡ 1 2 ( [ α 3 , α 3 † ] − [ α 2 , α 2 † ] ) λ 4 ≡ 1 2 ( [ α 1 , α 3 † ] + [ α 3 , α 1 † ] ) λ 5 ≡ i 2 ( [ α 1 , α 3 † ] − [ α 3 , α 1 † ] ) λ 6 ≡ 1 2 ( [ α 1 , α 2 † ] + [ α 2 , α 1 † ] ) λ 7 ≡ i 2 ( [ α 1 , α 2 † ] − [ α 2 , α 1 † ] ) λ 8 ≡ 1 2 3 ( [ α 3 , α 3 † ] + [ α 2 , α 2 † ] − 2 [ α 1 , α 1 † ] ) Y R ≡ 1 3 ( [ α 3 , α 3 † ] + [ α 2 , α 2 † ] + [ α 1 , α 1 † ] ) \begin{equation}
\begin{aligned}
\lambda_1 &\equiv \tfrac{1}{2} \left( [\alpha_2, \alpha^{\dagger}_3] + [\alpha_3, \alpha^{\dagger}_2] \right) \\
\lambda_2 &\equiv \tfrac{i}{2} \left( [\alpha_2, \alpha^{\dagger}_3] - [\alpha_3, \alpha^{\dagger}_2] \right) \\
\lambda_3 &\equiv \tfrac{1}{2} \left( [\alpha_3, \alpha^{\dagger}_3] - [\alpha_2, \alpha^{\dagger}_2] \right) \\
\lambda_4 &\equiv \tfrac{1}{2} \left( [\alpha_1, \alpha^{\dagger}_3] + [\alpha_3, \alpha^{\dagger}_1] \right) \\
\lambda_5 &\equiv \tfrac{i}{2} \left( [\alpha_1, \alpha^{\dagger}_3] - [\alpha_3, \alpha^{\dagger}_1] \right) \\
\lambda_6 &\equiv \tfrac{1}{2} \left( [\alpha_1, \alpha^{\dagger}_2] + [\alpha_2, \alpha^{\dagger}_1] \right) \\
\lambda_7 &\equiv \tfrac{i}{2} \left( [\alpha_1, \alpha^{\dagger}_2] - [\alpha_2, \alpha^{\dagger}_1] \right) \\
\lambda_8 &\equiv \tfrac{1}{2\sqrt{3}} \left( [\alpha_3, \alpha^{\dagger}_3] + [\alpha_2, \alpha^{\dagger}_2] - 2[\alpha_1, \alpha^{\dagger}_1] \right) \\
Y_R &\equiv \tfrac{1}{3} \left( [\alpha_3, \alpha^{\dagger}_3] + [\alpha_2, \alpha^{\dagger}_2] + [\alpha_1, \alpha^{\dagger}_1] \right)
\end{aligned}
\end{equation} λ 1 λ 2 λ 3 λ 4 λ 5 λ 6 λ 7 λ 8 Y R ≡ 2 1 ( [ α 2 , α 3 † ] + [ α 3 , α 2 † ] ) ≡ 2 i ( [ α 2 , α 3 † ] − [ α 3 , α 2 † ] ) ≡ 2 1 ( [ α 3 , α 3 † ] − [ α 2 , α 2 † ] ) ≡ 2 1 ( [ α 1 , α 3 † ] + [ α 3 , α 1 † ] ) ≡ 2 i ( [ α 1 , α 3 † ] − [ α 3 , α 1 † ] ) ≡ 2 1 ( [ α 1 , α 2 † ] + [ α 2 , α 1 † ] ) ≡ 2 i ( [ α 1 , α 2 † ] − [ α 2 , α 1 † ] ) ≡ 2 3 1 ( [ α 3 , α 3 † ] + [ α 2 , α 2 † ] − 2 [ α 1 , α 1 † ] ) ≡ 3 1 ( [ α 3 , α 3 † ] + [ α 2 , α 2 † ] + [ α 1 , α 1 † ] ) Spacetime Algebra
γ μ γ ν + γ ν γ μ = 2 g μ ν , μ , ν ∈ 0 , 1 , 2 , 3 \begin{equation}
\begin{aligned}
\gamma_{\mu} \gamma_{\nu} + \gamma_{\nu} \gamma_{\mu} &= 2g_{\mu\nu}, \quad \mu, \nu \in {0, 1,2,3}
\end{aligned}
\end{equation} γ μ γ ν + γ ν γ μ = 2 g μν , μ , ν ∈ 0 , 1 , 2 , 3 
fusis