Abstract 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. 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.

中文翻译：

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新矩阵动力学中自旋的基本定义

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。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。