量子态希尔伯特空间中的布朗运动与随机涌现的洛伦兹对称性：从维纳过程到构建相对论量子场费曼路径积分测度的分形几何方法
Brownian Motion in the Hilbert Space of Quantum States and the Stochastically Emergent Lorentz Symmetry: A Fractal Geometric Approach from Wiener Process to Formulating Feynman's Path-Integral Measure for Relativistic Quantum Fields
AI总结：
本文通过分形范数修正无限维希尔伯特空间中的维纳过程，构造维纳分形测度，从而为相对论量子场提供数学上良定义的费曼路径积分测度。
摘要：
本文旨在通过研究量子态无限维希尔伯特空间中的维纳随机过程，为量子场提供一种一致、有限值且在数学上良定义的费曼路径积分测度重构。这一重构无疑将在数学良定义框架下表述量子引力方面发挥关键作用。事实上，本研究与以往任何关于费曼路径积分与维纳随机过程之间关系的研究都有根本不同。在本研究中，我们关注这样一个事实：由于低维与极高维中位移之间的根本差异，经典维纳测度在无限维希尔伯特空间中不再适用。因此，受分形函数在 Wilsonian 重整化方法中作用启发，我们构造了一种解析范数，以恰当刻画紧致平坦流形上量子态希尔伯特空间中的布朗运动。这一范数，即所谓分形范数，将较粗糙的函数，或物理上能量较高的量子态，推向希尔伯特空间中更远的点，直至作为最粗糙对象的分形函数被移至无穷远。将维纳随机过程与分形范数结合实施，得到一种修正形式的维纳测度，称为维纳分形测度，它是量子场费曼路径积分表述的一个良定义测度。
英文摘要：
This paper aims to provide a consistent, finite-valued, and mathematically well-defined reformulation of the Feynman path-integral measure for quantum fields obtained by studying the Wiener stochastic process in the infinite-dimensional Hilbert space of quantum states. This reformulation will undoubtedly have a crucial role in formulating quantum gravity within a mathematically well-defined framework. In fact, the present study is fundamentally different from any previous research on the relationship between the Feynman path-integral and the Wiener stochastic process. In this research, we focus on the fact that the classic Wiener measure is no longer applicable in infinite-dimensional Hilbert spaces due to fundamental differences between displacements in low and extremely high dimensions. Thus, an analytic norm motivated by the role of the fractal functions in the Wilsonian renormalization approach is worked out to properly characterize Brownian motion in the Hilbert space of quantum states on a compact flat manifold. This norm, the so-called fractal norm, pushes the rougher functions, or physically the quantum states with higher energies, to the farther points of the Hilbert space until the fractal functions as the roughest ones are moved to infinity. Implementing the Wiener stochastic process with the fractal norm, results in a modified form of the Wiener measure called the Wiener fractal measure, which is a well-defined measure for the Feynman path-integral formulation of quantum fields. ...
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