Quantum Repulsion and Cosmic Evolution: A Unified Theory from Calabi-Yau Manifolds to Macroscopic Dynamics

Abstract Objective Based on the fundamental hypothesis that space consists of discrete Calabi-Yau manifolds¹ and that compactification itself contributes a non-zero vacuum energy (termed "quantum repulsion"), this paper attempts to construct a unified dynamical framework spanning from microscopic transitions to macroscopic cosmic evolution. The goal is to reconcile the incompatibility between General Relativity and Quantum Field Theory and to explore the dual role of quantum repulsion as both a "global stabilizer" and a "local structure source." Methods We introduce a scalar field action with noise, dynamizing the quantum noise originating from the zero-point energy of compactified spaces. We clarify that the geometric phase field Φ serves as the order parameter for the collective behavior of a large ensemble of manifolds, while the noise ξ arises from statistical fluctuations in microscopic landscape transitions (in this paper, 'landscape' specifically refers to a concrete topological configuration of a Calabi-Yau manifold, while 'landscape' as a whole denotes the ensemble of all possible configurations; both correspond to the English word "landscape" in the literature). A scale parameter η, defined as the ratio of Compton wavelength to Schwarzschild radius, characterizes the competition between quantum repulsion and gravitational collapse. Using Noether's theorem, we derive a covariant expression for instantaneous power. Combining Quantum Field Theory in Curved Spacetime, the Fluctuation-Dissipation Theorem, and the Holographic Principle, we calculate the ensemble average and fluctuation behavior of this formula near extreme black hole horizons and cosmic horizons, respectively. Starting from landscape transitions of discrete geometric units, we attempt to derive the statistical properties of the noise ξ. Furthermore, by employing the influence functional method, we elucidate the deep connection between macroscopic stochastic dynamics and microscopic unitary evolution, revealing a two-layer statistical structure of the noise: two-point correlations characterize thermalization information, while higher-order correlations carry recoverable memory. This provides a microscopic foundation for understanding information transfer across cosmic generations. Results The model yields four solutions: inflation, dark energy, black holes, and cosmic contraction. It demonstrates a mathematical duality between the cosmic contraction phase and black hole horizons (established via Wick rotation). At the extreme black hole end, the power formula is consistent with Hawking radiation power under the quasi-static approximation and predicts critical divergent behavior at the end of evaporation, requiring a Planck-scale cutoff. At the cosmic horizon end, the formula derives the energy fluctuation scale in the heat death limit based on the Holographic Principle. We propose an operational definition of time as the relaxation process outside the event horizon. We attempt to derive the two-point correlation of noise as white noise from the independent randomness of microscopic transitions; its higher-order correlations may serve as ensemble parameters carrying historical information from previous cosmic generations, embodying the Holographic Principle. Limitations The discrete mechanism at the Planck scale requires further verification by a theory of Quantum Gravity. The strict relationship between higher-order correlation functions of noise and holographic entropy awaits future research. The image of cosmic reshaping in Chapter 8 is a conceptual framework; its rigorous realization requires a deepening of Quantum Gravity theory. Conclusion The universal power formula unifies the dynamical role of quantum repulsion into a concise expression. Quantum repulsion is both the dynamical basis for the overall stability of the manifold matrix and the common source of relaxation (time) and landscape transitions (particle seeds). This paper attempts to reveal that the Heisenberg Uncertainty Principle in this framework may not be an independent axiom but rather a statistical result of the compactification of discrete geometric structures, potentially more fundamental in terms of explanatory hierarchy. The two-point and higher-order correlations of noise characterize short-term fluctuations and long-term memory, respectively, providing a possible unified framework for understanding black hole information, cosmic origins, and matter structure. We propose a conceptual framework of an eternal cyclic universe where each cycle involves "reshaping" rather than "restarting." The noise correlation structure may constrain the range of physical constants in new universes, offering a possible picture for the unification of causality and creativity. Keywords: Quantum Repulsion; Calabi-Yau Manifolds; Extreme Black Holes; Cosmic Horizon; Hawking Radiation; Fluctuation-Dissipation Theorem; Holographic Principle; Higher-Order Correlations; Superstring Theory; Cosmic Reshaping ¹ In this paper, "discrete" specifically means that the topological types of Calabi-Yau manifolds take discrete values, which is compatible with the continuity of macroscopic spacetime. See Section 2.1 for details. 摘要 目的 基于空间由分立卡拉比-丘流形¹⁾构成,且紧致化本身贡献非零真空能(我们称之为量子排斥)的根本假设,本文尝试构建一个从微观跃迁到宏观宇宙演化的统一动力学框架,以期协调广义相对论与量子场论的不兼容难题,并探索量子排斥作为“整体稳定器”与“局部结构源”的双重角色。 方法 引入带噪声的标量场作用量,将源于紧致化空间零点能的量子噪声动力学化。阐明几何相场Φ作为大量流形集体行为的序参量,噪声ξ源于微观景致跃迁(本文中“景致”特指卡拉比-丘流形的具体拓扑构型,“景观”指所有可能构型的整体集合,二者在英文中均对应 landscape)的统计涨落。定义尺度参数η为康普顿波长与史瓦西半径之比,表征量子排斥与引力塌缩的竞争。通过Noether定理导出瞬时功率的协变表达式。结合弯曲时空量子场论、涨落-耗散定理和全息原理,分别计算该公式在极端黑洞视界附近和宇宙视界附近的系综平均与涨落行为。从离散几何单元的景致跃迁出发,尝试推导噪声ξ的统计性质。进一步,通过引入影响泛函方法,阐明宏观随机动力学与微观幺正演化之间的深层联系,并揭示噪声的两层统计结构——两点关联刻画热化信息,高阶关联承载可恢复记忆——为理解信息在宇宙世代间的传递提供微观基础。 结果 模型导出暴胀、暗能量、黑洞和宇宙收缩四个解,证明宇宙收缩相与黑洞视界存在数学对偶性(通过Wick转动建立)。在极端黑洞端,功率公式在准静态近似下与霍金辐射功率自洽,并预言蒸发末期临界发散行为,需普朗克尺度截断;在宇宙视界端,公式基于全息原理导出热寂极限下的能量涨落标度。提出时间的操作定义——事件视界外部的弛豫过程。尝试从微观跃迁的独立随机性推导噪声的两点关联为白噪声形式;其高阶关联可能作为系综参数承载前宇宙世代的历史信息,体现全息原理。 局限 普朗克尺度下的离散机制需量子引力理论进一步验证,噪声高阶关联函数与全息熵的严格关系有待后续研究。第8章的宇宙重塑图像属于概念框架,其严格实现需量子引力理论深化。 结论 普适功率公式将量子排斥的动力学作用统一于一个简洁表达。量子排斥既是流形矩阵整体稳定的动力学基础,又是弛豫(时间)与景致跃迁(粒子种子)的共同源头。本文尝试揭示海森堡不确定性原理在此框架中可能并非独立公理,而是离散几何结构紧致化的统计结果,在解释层级上或许更为基础。噪声的两点关联与高阶关联分别刻画短期涨落与长期记忆,为理解黑洞信息、宇宙起源及物质结构提供了一个可能的统一框架。提出宇宙永恒循环且每次“重塑”而非“重启”的概念框架,噪声关联结构可能约束新宇宙物理常数的取值范围,为因果性与创造性的统一提供一种可能图景。 关键词:量子排斥;卡拉比-丘流形;极端黑洞;宇宙视界;霍金辐射;涨落-耗散定理;全息原理;高阶关联;超弦理论,宇宙重塑 ¹⁾ 本文所称“分立”,特指卡拉比-丘流形的拓扑类型取离散值,与宏观时空的连续性兼容。详见第2.1节。