Abstract We have recently proposed a Lagrangian in trace dynamics at the Planck scale, for unification of gravitation, Yang–Mills fields, and fermions. Dynamical variables are described by odd-grade (fermionic) and even-grade (bosonic) Grassmann matrices. Evolution takes place in Connes time. At energies much lower than Planck scale, trace dynamics reduces to quantum field theory. In the present paper, we explain that the correct understanding of spin requires us to formulate the theory in 8-D octonionic space. The automorphisms of the octonion algebra, which belong to the smallest exceptional Lie group G 2, replace space-time diffeomorphisms and internal gauge transformations, bringing them under a common unified fold. Building on earlier work by other researchers on division algebras, we propose the Lorentz-weak unification at the Planck scale, the symmetry group being the stabiliser group of the quaternions inside the octonions. This is one of the two maximal sub-groups of G 2, the other one being SU(3), the element preserver group of octonions. This latter group, coupled with U(1) em , describes the electrocolour symmetry, as shown earlier by Furey. We predict a new massless spin one boson (the ‘Lorentz’ boson) which should be looked for in experiments. Our Lagrangian correctly describes three fermion generations, through three copies of the group G 2, embedded in the exceptional Lie group F 4. This is the unification group for the four fundamental interactions, and it also happens to be the automorphism group of the exceptional Jordan algebra. Gravitation is shown to be an emergent classical phenomenon. Although at the Planck scale, there is present a quantised version of the Lorentz symmetry, mediated by the Lorentz boson, we argue that at sub-Planck scales, the self-adjoint part of the octonionic trace dynamics bears a relationship with string theory in 11 dimensions.

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迹动力学和除法代数：走向量子引力和统一

摘要 我们最近在普朗克尺度的迹动力学中提出了拉格朗日函数，用于统一引力、杨-米尔斯场和费米子。动力学变量由奇数级（费米子）和偶数级（玻色子）格拉斯曼矩阵描述。进化发生在康纳时代。在远低于普朗克标度的能量下，迹动力学归结为量子场论。在本文中，我们解释了正确理解自旋需要我们在 8 维八元空间中制定理论。八元代数的自同构属于最小的例外李群 G 2 ，取代了时空微分同胚和内部规范变换，使它们处于一个共同的统一折叠之下。基于其他研究人员在除法代数方面的早期工作，我们提出了普朗克尺度的洛伦兹弱统一，对称群是八元数内四元数的稳定群。这是 G 2 的两个最大子群之一，另一个是 SU(3)，八元数的元素保护群。后一组与 U(1) em 相结合，描述了电色对称性，如 Furey 先前所示。我们预测了一种新的无质量自旋单玻色子（“洛伦兹”玻色子），应该在实验中寻找它。我们的拉格朗日量正确地描述了三个费米子代，通过群 G 2 的三个副本，嵌入例外李群 F 4。这是四个基本相互作用的统一群，它也恰好是例外乔丹的自同构群代数。引力被证明是一种新兴的经典现象。虽然在普朗克尺度上，