Abstract We have recently proposed a new matrix dynamics at the Planck scale, building on the theory of trace dynamics and Connes noncommutative geometry program. This is a Lagrangian dynamics in which the matrix degrees of freedom are made from Grassmann numbers, and the Lagrangian is trace of a matrix polynomial. Matrices made from even grade elements of the Grassmann algebra are called bosonic, and those made from odd grade elements are called fermionic—together they describe an ‘aikyon’. The Lagrangian of the theory is invariant under global unitary transformations and describes gravity and Yang–Mills fields coupled to fermions. In the present article, we provide a basic definition of spin angular momentum in this matrix dynamics and introduce a bosonic(fermionic) configuration variable conjugate to the spin of a boson(fermion). We then show that at energies below Planck scale, where the matrix dynamics reduces to quantum theory, fermions have half-integer spin (in multiples of Planck’s constant), and bosons have integral spin. We also show that this definition of spin agrees with the conventional understanding of spin in relativistic quantum mechanics. Consequently, we obtain an elementary proof for the spin-statistics connection. We then motivate why an octonionic space is the natural space in which an aikyon evolves. The group of automorphisms in this space is the exceptional Lie group G 2 which has 14 generators [could they stand for the 12 vector bosons and two degrees of freedom of the graviton?]. The aikyon also resembles a closed string, and it has been suggested in the literature that 10-D string theory can be represented as a 2-D string in the 8-D octonionic space. From the work of Cohl Furey and others it is known that the Dixon algebra made from the four division algebras [real numbers, complex numbers, quaternions and octonions] can possibly describe the symmetries of the standard model. In the present paper we outline how in our work the Dixon algebra arises naturally and could lead to a unification of gravity with the standard model. From this matrix dynamics, local quantum field theory arises as a low energy limit of this Planck scale dynamics of aikyons, and classical general relativity arises as a consequence of spontaneous localisation of a large number of entangled aikyons. We propose that classical curved space–time and Yang–Mills fields arise from an effective gauging which results from the collection of symmetry groups of the spontaneously localised fermions. Our work suggests that we live in an eight-dimensional octonionic universe, four of these dimensions constitute space–time and the other four constitute the octonionic internal directions on which the standard model forces live.

中文翻译：

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八元数、迹动力学和非对易几何——自发量子引力统一的一个案例

摘要 我们最近提出了一种新的普朗克矩阵动力学，建立在迹动力学理论和 Connes 非对易几何程序的基础上。这是一个拉格朗日动力学，其中矩阵的自由度由格拉斯曼数构成，而拉格朗日是矩阵多项式的迹。由格拉斯曼代数的偶数级元素构成的矩阵称为玻色子，由奇数级元素构成的矩阵称为费米子——它们一起描述了一个“aikyon”。该理论的拉格朗日量在全局幺正变换下是不变的，并描述了与费米子耦合的引力场和杨-米尔斯场。在本文中，我们提供了该矩阵动力学中自旋角动量的基本定义，并介绍了与玻色子（费米子）的自旋共轭的玻色子（费米子）配置变量。然后我们证明，在低于普朗克标度的能量下，矩阵动力学简化为量子理论，费米子具有半整数自旋（普朗克常数的倍数），而玻色子具有积分自旋。我们还表明，自旋的这种定义与相对论量子力学中自旋的传统理解一致。因此，我们获得了自旋统计连接的基本证明。然后我们激发了为什么八元空间是 aikyon 进化的自然空间。这个空间中的自同构群是特殊的李群 G 2 ，它有 14 个发生器 [它们可以代表 12 个矢量玻色子和引力子的两个自由度吗？]。aikyon 也像一根封闭的弦，并且在文献中已经提出，10-D 弦理论可以表示为 8-D 八元空间中的 2-D 弦。从 Cohl Furey 等人的工作中得知，由四除法代数 [实数、复数、四元数和八元数] 构成的狄克逊代数可能可以描述标准模型的对称性。在本文中，我们概述了狄克逊代数如何在我们的工作中自然产生，并可能导致重力与标准模型的统一。从这个矩阵动力学中，局域量子场论作为 aikyons 的这种普朗克尺度动力学的低能量极限出现，而经典广义相对论作为大量纠缠 aikyons 的自发局域化的结果出现。我们提出，经典的弯曲时空和杨米尔斯场源于自发局域费米子的对称群的收集所产生的有效测量。我们的工作表明我们生活在一个八维的八维宇宙中，其中四个维度构成了时空，另外四个构成了标准模型力赖以生存的八维内部方向。