论文标题
痕量动力学和分裂代数：朝向量子重力和统一
Trace dynamics and division algebras: towards quantum gravity and unification
论文作者
论文摘要
我们最近在普朗克量表上提出了一个痕量动力学的拉格朗日，以统一重力，杨米尔斯田地和费米子。动力学变量通过奇数（费米子）和偶数（玻色症）格拉曼矩阵描述。进化发生在康涅斯时间。在远低于普朗克量表的能量下，痕量动力学减少了量子场理论。在本文中，我们解释说，对旋转的正确理解要求我们在8-D八元离子空间中提出理论。属于最小的特殊谎言组$ g_2 $的八元代数的自动形态，取代时空的差异和内部仪表变换，使它们以通用的统一折叠。在其他研究人员对分区代数的早期工作的基础上，我们提出了Planck量表的Lorentz-Weak统一，对称组是八分代中四分之一的稳定器组。这是$ g_2 $的两个最大子组之一，另一个是$ su（3）$，是元素八元组的元素保存组。后一个组与$ u（1）_ {em} $相结合，描述了电彩对称性，如Furey前面所示。我们预测一个新的无质旋转一个玻色子（Lorentz Boson），应该在实验中寻找。我们的Lagrangian正确地描述了三个Fermion世代，该世代通过了$ g_2 $的三个副本，这些副本嵌入了特殊的Lie Group $ f_4 $中。这是四个基本相互作用的统一组，它也恰好是杰出约旦代数的自动形态群体。引力被证明是一种新兴的经典现象。而在普朗克量表上，有一个由洛伦兹玻色子介导的洛伦兹对称性的量化版本。我们认为，在子倾斜量表上，八元痕迹动力学的自我伴侣部分与弦理论在11个维度上具有关系。
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 subgroups 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 electro-colour 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. Whereas 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 eleven dimensions.
