论文标题
八元，痕量动力学和非交通性几何形状：自发量子重力统一的案例
Octonions, trace dynamics and non-commutative geometry: a case for unification in spontaneous quantum gravity
论文作者
论文摘要
我们最近在基于痕量动力学理论的基础上，在普朗克量表上提出了一种新的矩阵动力学。这是一种拉格朗日动力学，其中矩阵的自由度是由格拉曼（Grassmann）数字制成的，而拉格朗日（Lagrangian）是矩阵多项式的痕迹。由格拉曼代数的均匀元素制成的矩阵被称为玻色子，由奇数元素制成的矩阵被称为费米金：他们一起描述了一个“ aikyon”。在本文中，我们在此矩阵动力学中提供了旋转角动量的基本定义，并引入了玻色子（费米子）构型变量偶联物（fermion）的旋转。然后，我们表明，在低于Planck量表的能量下，矩阵动力学减少为量子理论，费米子具有半固定旋转（在普朗克常数的倍数中），并且玻色子具有积分的自旋。我们还表明，这种自旋的定义与对相对论量子力学中自旋的常规理解相符。因此，我们获得了自旋统计连接的基本证明。然后，我们激励为什么八元离子空间是Aikyon演变的自然空间。在这个空间中的一组自动形态是具有14个发电机的杰出谎言组$ G_2 $ [它们可以代表十二个矢量玻色子和两个重力的自由度吗？ ]。 Aikyon也类似于一个封闭的字符串，在文献中建议10-D字符串理论可以表示为8-D Octonionic空间中的2-D字符串。从Cohl Furey和其他工作的工作中，众所周知，由四个分区代数制成的Dixon代数[实数，复数，四元和八元]可以描述标准模型的对称性。在本文中，我们概述了迪克森代数自然出现的工作中，并可能导致与标准模型的重力统一。
We have recently proposed a new matrix dynamics at the Planck scale, building on the theory of trace dynamics. 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'. 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 fourteen generators [could they stand for the twelve 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.
