Octonion Internal Space Algebra for the Standard Model

The paper surveys recent progress in the search for an appropriate internal space algebra for the Standard Model (SM) of particle physics. As a starting point serve Clifford algebras involving operators of left multiplication by octonions. A central role is played by a distinguished complex structure which implements the splitting of the octonions ${\mathbb O} = {\mathbb C} \oplus {\mathbb C}^3$ reflecting the lepton-quark symmetry. Such a complex structure in $C\ell_{10}$ is generated by the $C\ell_6(\subset C\ell_8\subset C\ell_{10})$ volume form, $\omega_6 = \gamma_1 \cdots \gamma_6$, left invariant by the Pati-Salam subgroup of $Spin(10)$, $G_{\rm PS} = Spin (4) \times Spin (6) / {\mathbb Z}_2$. While the $Spin(10)$ invariant volume form $\omega_{10}=\gamma_1 ... \gamma_{10}$ is known to split the Dirac spinors of $C\ell_{10}$ into left and right chiral (semi)spinors, ${\cal P} = \frac12 (1 - i\omega_6)$ is interpreted as the projector on the 16-dimensional \textit{particle subspace} (annihilating the antiparticles). The standard model gauge group appears as the subgroup of $G_{PS}$ that preserves the sterile neutrino (identified with the Fock vacuum). The $\mathbb{Z}_2$-graded internal space algebra $\mathcal{A}$ is then included in the projected tensor product: $\mathcal{A}\subset \mathcal{P}C\ell_{10}\mathcal{P}=C\ell_4\otimes \mathcal{P} C\ell_6^0\mathcal{P}$. The Higgs field appears as the scalar term of a superconnection, an element of the odd part, $C\ell_4^1$, of the first factor. The fact that the projection of $C\ell_{10}$ only involves the even part $C\ell_6^0$ of the second factor guarantees that the colour symmetry remains unbroken. As an application we express the ratio $\frac{m_H}{m_W}$ of the Higgs to the $W$-boson masses in terms of the cosine of the {\it theoretical} Weinberg angle.