论文标题
标准模型的Octonion内部空间代数
Octonion Internal Space Algebra for the Standard Model
论文作者
论文摘要
该论文在寻找粒子物理的标准模型（SM）寻找适当的内部空间代数方面的最新进展。作为起点，涉及左乘法的运算符的克利福德代数。杰出的复杂结构扮演着一个核心角色，该结构实现了八元的分裂$ {\ mathbb o} = {\ mathbb c} \ oplus {\ mathbb c}^3 $反映了Lepton-Quark对称性。 $ c \ ell_6（\ subset c \ ell_8 \ subset c \ ell_ {10}）$体积形式，$ c \ ell_6（$ c \ ell_6（$ c \ ell_6）生成了这种复杂的结构。 （4）\ times spin（6） / {\ mathbb z} _2 $。而$ spin（10）$不变卷$ω_{10} =γ_1...γ_{10} $是众所周知的，可以将$ c \ ell_ {10} $的零旋转器拆分为左和右Chiral（semi）纺纱器，$ {\ cal p} = \ frac12（\ frac12（1-12）$ nised at on thate inim as inim at thate thate thate the thate thate the thate thate the thate ate at the thate at thate at thate at thate at thate at thate at the atuim \ textit {粒子子空间}（歼灭反粒子）。标准模型量规组是保留无菌中微子（用Fock真空识别）的$ G_ {PS} $的子组。然后将$ \ mathbb {z} _2 $ - 加工的内部空间代数$ \ mathcal {a} $包含在投影张量产品中：$ \ MATHCAL {a} \ subset \ subset \ subset \ mathcal {p} p} C \ ell_6^0 \ Mathcal {p} $。 Higgs字段作为第一个因素的SuperConnection的标量术语（奇数零件的一个元素）的标量项。 $ c \ ell_ {10} $的投影仅涉及第二个因子的偶数$ c \ ell_6^0 $，这一事实可以保证颜色对称性保持不间断。作为一个应用程序，我们将希格斯的比率$ \ frac {m_h} {m_w} $与$ w $ -boson群众相对于{\ it理论} weinberg anger的余弦而言。
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, $ω_6 = γ_1 \cdots γ_6$, 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 $ω_{10}=γ_1 ... γ_{10}$ is known to split the Dirac spinors of $C\ell_{10}$ into left and right chiral (semi)spinors, ${\cal P} = \frac12 (1 - iω_6)$ 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.
