论文标题
布朗在量子状态的希尔伯特空间中的布朗运动和随机出现的洛伦兹对称性：从维纳过程到制定feynman的路径综合量的分形几何方法
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
论文作者
论文摘要
本文旨在为Feynman Path-Contemental措施提供一致的，有限的价值和数学定义明确定义的重新印度，以通过研究量子状态的无限二维Hilbert Space中的Wiener随机过程获得的量子场。毫无疑问，这种重新制定将在数学定义明确的框架内制定量子重力方面起着至关重要的作用。实际上，本研究与Feynman路径综合和Wiener随机过程之间关系的任何研究根本不同。在这项研究中，我们关注以下事实：经典的维纳措施不再适用于无限二维的希尔伯特空间，这是由于低维度和极高维度之间的基本差异。因此，制定了由分形功能在威尔逊重新归一化方法中的作用所激发的分析规范，以正确地表征量子状态在紧凑型平坦歧管上希尔伯特空间中的布朗运动。该规范是所谓的分形规范，将较粗糙的函数或具有较高能量的量子状态推向希尔伯特空间的较远点，直到分形功能随着最粗糙的作用移动到无限为止。通过分形规范实施维纳随机过程，导致了一种称为维纳尔分形度量的维纳仪的修改形式，这是Feynman Quynman路径综合配方的明确定义的度量。 ...
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 In this research, we focus on the fact that the classic Wiener measure is no longer applicable in infinite-dimensional Hilbert spaces due to fundamental differences between displacements in low and extremely high dimensions.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. ...
