We have recently proposed a Lagrangian in trace dynamics to describe a possible unification of gravity, Yang–Mills fields, and fermions, at the Planck scale. This Lagrangian for the unified entity — called the aikyon — is invariant under global unitary transformations, and as a result possesses a novel conserved charge, known as the Adler–Millard charge. In this paper, we derive an eigenvalue equation, analogous to the time-independent Schrödinger equation, for the Hamiltonian of the theory. We show that in the emergent quantum theory, the energy eigenvalues of the aikyon are characterized in terms of a fundamental frequency times Planck’s constant. The eigenvalues of this equation can, in principle, determine the values of the parameters of the standard model. We also report a ground state, in this theory of spontaneous quantum gravity, which could characterize a non-singular initial epoch in quantum cosmology.

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迹动力学和自发量子引力的基态

我们最近提出了迹动力学中的拉格朗日算子，以描述普朗克尺度上重力、杨-米尔斯场和费米子的可能统一。这个统一实体的拉格朗日量 - 称为 aikyon - 在全局单一变换下是不变的，因此具有一种新的守恒电荷，称为 Adler-Millard 电荷。在本文中，我们为理论的哈密顿量推导出了一个特征值方程，类似于与时间无关的薛定谔方程。我们表明，在新兴的量子理论中，aikyon 的能量特征值以基频乘以普朗克常数为特征。该方程的特征值原则上可以确定标准模型的参数值。我们还报告了一个基态，在这个自发量子引力理论中，