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  4. "Three Fermion Generations from Octonions" HEPHY-PUB-98/07
  6. *arXiv:hep-ph/9810316*
  8. March 1998
  10. [**Gauge Unification with Octonions**]{}
  12. [**S. Meljanac and D. Meljanac**]{} [^1]
  14. [*Institute of Physics,University of Sarajevo,Agenzija for Polytechnic*]{}
  16. [**Abstract**]{}
  18. A group of 8 fermions is the smallest number of quarks and leptons that can be accommodated within the framework of the group theory and contains three generations. In this work, we explore the possibility of unifying the weak and electromagnetic interactions by constructing a nonassociative group of octonions and using it to formulate the Standard Model. We show that, with the help of the real octonions, one can take into account the standard model bosons and fermions. The real octonions also enable us to consider quarks and leptons as linear representations of the algebra. We also present the results for electric charge.
  20. Introduction
  21. ============
  23. The Standard Model (SM) has successfully accommodated all observed data on elementary particles. However, this theory is still far from a perfect description of the physics of the subatomic world and the need for a unification of the different interactions.
  25. A popular direction for unification is the incorporation of higher-dimensional space-time into field theories. This can be achieved by introducing an extra dimension, but an alternative way is to use an appropriate extension of the symmetry groups. Group-theoretical approach to the standard model is being investigated by several authors, but in a general case the number of the generations and the role of fermions in the group must be investigated. The nonassociative groups are the groups of quaternions and octonions. The first approach has been made by J.P. Gazeau and co-workers \[1-3\] where the role of the quaternion group in the SM has been investigated and this model has been extended to the octonions \[4\].
  27. This paper is organized as follows. In Section 2 we review the properties of the octonions. Section 3 is devoted to the unification of the weak and electromagnetic interactions by considering the octonions as the base ring. A group of eight fermions is constructed in Section 4. We also make use of the complex octonions to describe the quarks and leptons. In Section 5 the electric charge of the leptons is obtained. We conclude in Section 6.
  29. Octonions
  30. =========
  32. The octonions were introduced in the early years of the 20th century \[5\]. The space of all $8 \times 8$ octonionic matrices $\mathcal{O}$ is constructed as a real vector space $V$ with the non-associative product $[,]$ and the neutral element $\bf{0}$.
  34. The product $[,]$ is defined by the following property $$\begin{aligned}
  35. \nonumber
  36. \label{a.1}
  37. [u,v] &=& uv + vu \\
  38. \label{a.2}
  39. [u+v,x] &=& [u,x] + [v,x] \\
  40. \label{a.3}
  41. [u,x+y] &=& [u,x] + [u,y] \end{aligned}$$ It can be checked that any octonionic matrix can be expressed as a linear combination of $\bf{i}, \bf{j}, \bf{k}$ and $\bf{0}$, where $$\begin{aligned}
  42. \label{a.4}
  43. \bf{i} &=& \frac{1}{\sqrt{2}}(\bf{e_1} + \bf{e_2}) \\
  44. \label{a.5}
  45. \bf{j} &=& \frac{1}{\sqrt{2}}(\bf{e_1} - \bf{e_2}) \\
  46. \label{a.6}
  47. \bf{k} &=& \bf{0} \end{aligned}$$ and $\bf{e_1}$, $\bf{e_2}$ are the basic unit vectors $$\begin{aligned}
  48. \label{a.7}
  49. \bf{e_1} &=& \left( \begin{array}{c} 1 \\ 0 \end{array} \right) \\
  50. \label{a.8}
  51. \bf{e_2} &=& \left( \begin{array}{c} 0 \\ 1 \end{array} \right) \end{aligned}$$
  53. In analogy with the quaternions \[6\] there are four main operations with the octonions:
  55. 1\. (\[a.1\]) Multiplication of an octonionic matrix $A$ by a real scalar $\alpha$ $$\label{a.9}
  56. \alpha A = \alpha [u_1, \ldots, u_8] = [\alpha u_1, \ldots, \alpha u_8]$$
  58. 2\. (\[a.2\]) Left division of an octonionic matrix $A$ by a non-zero octonionic scalar $\alpha$ $$\label{a.10}
  59. A / \alpha = \frac{A [u_1, \ldots, u_8]}{\alpha [u_1, \ldots, u_8]}$$
  61. 3\. (\[a.3\]) Right division of an octonionic matrix $A$ by a non-zero octonionic scalar $\alpha$ $$\label{a.11}
  62. \alpha A = A / \alpha = \frac{[u_1, \ldots, u_8]}{\alpha [u_1, \ldots, u_8]}A$$
  64. 4\. (\[a.4\]) The generalized product of an octonionic matrix $A$ with a non-zero octonionic vector $u$ $$\label{a.12}
  65. A \otimes u = A u = [u_1, \ldots, u_8] \; A$$
  67. The quaternion and octonionic division operations (\[a.10\]) and (\[a.11\]) can be summarized by the following relations: $$\label{a.13}
  68. A = \alpha A + u \; \beta = \frac{\alpha A + u}{\alpha + \beta}$$ $$\label{a.14}
  69. \alpha A + u = \frac{\alpha A + u}{\alpha}$$ $$\label{a.15}
  70. \frac{\alpha A + u}{\alpha + \beta} = \frac{\alpha A + u}{\alpha}$$ where $\alpha$ and $\beta$ are non-zero octonionic scalars.
  72. Quaternions
  73. ===========
  75. The structure of the quaternions can be described in terms of the three basic operations (\[a.1\])-(\[a.3\]). The four generators $\bf{i}, \bf{j}, \bf{k}, \bf{0}$ satisfy the following multiplication rules $$\begin{aligned}
  76. \label{b.1}
  77. \bf{i} \bf{j} &=& \bf{k} \\
  78. \label{b.2}
  79. \bf{j} \bf{k} &=& \bf{i} \\
  80. \label{b.3}
  81. \bf{k} \bf{i} &=& \bf{0} \\
  82. \label{b.4}
  83. \bf{0} \bf{0} &=& \bf{0} \end{aligned}$$ and the quaternion conjugate operation is defined by $$\label{b.5}
  84. A^\dagger = \frac{1}{2}[A,\bf{0}] = \frac{1}{2}(A-\bf{0}(A))$$ It can be shown that the quaternions satisfy the associative law.
  86. To obtain the real octonions we define the inverse operation $$\label{b.6}
  87. A^{-1} = \frac{1}{2}(A + \bf{0}(A))$$ It can be proved that the inverse operation is unique and $AA^{-1} = A^{-1}A = 1$. The set of all non-zero octonionic matrices can be defined by the following relation $$\label{b.7}
  88. \mathcal{O} = \{\alpha_0 + \alpha_1 \bf{i} + \alpha_2 \bf{j} + \alpha_3 \bf{k} + \alpha_4 \bf{0} \; | \; \alpha_i \in \mathcal{R} \}$$ where $\mathcal{R}$ is the set of all real numbers.
  90. Octonions
  91. =========
  93. To form the real octonions we need to take the
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