Covariant Matching of Quantum Mechanics and General Relativity within the Nonstandard Analysis Framework, and Curvature Regularization and Short-Range Repulsion at the Planck Scale under a Maximal Momentum Cutoff

@misc{junjie2026covariant,
  title = {{Covariant Matching of Quantum Mechanics and General Relativity within the Nonstandard Analysis Framework, and Curvature Regularization and Short-Range Repulsion at the Planck Scale under a Maximal Momentum Cutoff}},
  author = {Zhao Junjie and Natzin · Tlāloc ·Potrykus(joint codename) Ignatius·},
  howpublished = {Zenodo},
  year = {2026},
  month = oct,
  doi = {10.5281/zenodo.23647592},
  url = {https://zenodo.org/records/23647592},
  note = {CC BY 4.0},
}

TY  - UNPB
TI  - Covariant Matching of Quantum Mechanics and General Relativity within the Nonstandard Analysis Framework, and Curvature Regularization and Short-Range Repulsion at the Planck Scale under a Maximal Momentum Cutoff
AU  - Zhao Junjie
AU  - Natzin · Tlāloc ·Potrykus(joint codename) Ignatius·
PY  - 2026
DA  - 2026/10/25
PB  - Zenodo
DO  - 10.5281/zenodo.23647592
UR  - https://zenodo.org/records/23647592
AB  - Quantummechanics and general relativity rest on different mathematical frameworks: the former  on operator algebras over Hilbert space, the latter on covariant geometry over a continuous manifold.  However, how to incorporate the operator noncommutativity of quantum mechanics and spacetime  curvature into a single algebraic framework, without modifying general relativity, remains an open  problem. The two mathematical structures exhibit a well-recognized discrepancy: the canonical  commutation relation of quantum mechanics is defined on a fixed-background operator algebra, with  a constant on the right-hand side; the curvature commutator of general relativity is defined on the  covariant geometry of dynamical spacetime, with a tensor field on the right-hand side. An algebraic  framework that simultaneously accommodates both types of commutators is currently unclear.  The monad of nonstandard analysis provides a physically interpretable foothold for this obstacle.  A monad is a hyperreal neighbourhood of infinitesimal scale ε around a standard spacetime point;  physically, it corresponds to a mesoscopic fluid particle that appears as a point at macroscopic scales  but contains sufficiently many molecules at microscopic scales. Through the standard part map, the  structure of this infinitesimal neighbourhood can be rigorously projected back onto the continuous  standard theory.  On this infinitesimal neighbourhood, the most natural operation is translation. It is known that  momentum generates translations in quantum mechanics (Sakurai, Modern Quantum Mechanics);  in general relativity, covariant derivatives fail to commute in the presence of curvature, and the  discrepancy of parallel transport around a closed loop is measured by the Riemann curvature ten  sor (EPFL Lecture Notes). A difference-differential algebra is an algebraic structure simultaneously  equipped with derivations (differentiation) and automorphisms (translation) (Ritt–Kolchin–Cohn,  1950s; Cohn, A Difference-Differential Basis Theorem, 1970), in which the “difference” refers pre  cisely to the translation operator. In this paper, within the difference-differential algebra, an in  finitesimal translation operator on the monad is taken as the fundamental representational object,  and a commutator is constructed so that, under appropriate conditions, it corresponds respectively  to the canonical commutation relation of quantum mechanics and the Riemann curvature of general  relativity. This construction does not modify general relativity, nor does it reduce quantum me  chanics to an approximation of a standard theory. Upon taking the standard part, the two types of  commutators return respectively to the canonical commutation relation of quantum mechanics and  the Riemann curvature of general relativity.  When the framework is extended to the Planck scale, a maximal momentum cutoff renders the  Fermi degeneracy pressure of collapsing matter finite in the high-density limit, the central metric  component remains nonsingular, and the effective behaviour is a short-range repulsion; in the low  energy limit, the first-order corrections to the Higgs vacuum expectation value and to the gauge  boson masses vanish, consistent with current experimental precision.  量子力学与广义相对论建立在不同的数学框架之上:前者依赖希尔伯特空间上的算符代数,后者  依赖连续流形上的协变几何。然而,如何在不对广义相对论作任何修改的前提下,将量子力学的算符  非交换性与时空曲率纳入同一代数框架,仍是一个未解决的问题。两者的数学结构存在公认的差异:  量子力学的正则对易关系定义在固定背景的算符代数上,交换子右端是常数;广义相对论的曲率交换  子定义在动力学时空的协变几何上,交换子右端是张量场。一个同时容纳这两种交换子的代数框架,  目前尚不清楚。  非标准分析中的单子为上述障碍提供了一个物理上可理解的立足点。单子是围绕标准时空点、尺  度为无穷小量ε的超实数邻域,物理上对应一个从宏观上可视为点、微观上包含足够多分子的介观  流体微团。通过标准部分映射,这一无穷小邻域的结构可严格投影回连续的标准理论。  在单子这一无穷小邻域上,最自然的操作是平移。已知量子力学中动量生成平移(Sakurai,Modern  Quantum Mechanics);广义相对论中协变导数在曲率存在时不对易,平行移动绕闭合回路的差异由  黎曼曲率张量度量(EPFL Lecture Notes)。差分工代数是同时配备导子(微分)与自同构(平移)  的代数结构(Ritt–Kolchin–Cohn, 1950s; Cohn, A Difference-Differential Basis Theorem, 1970),其  “差分”正是指平移算子。本文在差分工代数中,以单子上的无穷小平移算符为表示基本对象,构造  一个交换子,使其在适当条件下分别对应量子力学的正则对易关系与广义相对论的黎曼曲率。该构造  不修改广义相对论,也不将量子力学降格为标准理论的近似。当取标准部分时,两种交换子分别回到  量子力学的正则对易关系与广义相对论的黎曼曲率。  将该框架推广到普朗克尺度后,最大动量截断使坍缩物质的费米简并压力在高密度极限下保持  有限,中心度规无奇点,等效为短程排斥;低能极限下,希格斯真空期望值与规范玻色子质量的一阶  修正为零,与当前实验精度一致。
ER  -