This paper aims to provide a consistent, finite-valued, and mathematically well-defined reformulation of Feynman’s 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, this study is fundamentally different from any previous research on the relationship between Feynman’s 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 In this research, we focus on the fact that the classic Wiener measure is no longer applicable in infinite-dimensional Hilbert spaces due to fundamental differences between displacements in low and extremely high dimensions.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 (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 Feynman’s path-integral formulation of quantum fields. Wiener fractal measure has a complicated formula of non-local terms but produces the Klein–Gordon action at the first order of approximation. Using complex integrals to compensate for the removal of non-local terms appearing in higher orders of approximation, the Wiener fractal measure turns into a complex measure and generates Feynman’s path-integral formulation of scalar quantum fields. This brings us to the main objective of this study. Finally, some various significant aspects of quantum field theory (such as renormalizability, RG flow, Wick rotation, regularization, etc.) are revisited by means of the analytical aspects of the Wiener fractal measure.

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获得量子场的费曼路径积分测度，提供一致的、有限值的、数学上明确定义的重新表述。这种重新表述无疑对于在数学上明确定义的框架内表述量子引力具有至关重要的作用。事实上，这项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的不同。在本研究中，我们关注的事实是，由于低维和极高维位移之间的根本差异，经典维纳测度不再适用于无限维希尔伯特空间。因此，由分形函数在威尔逊重整化方法中的作用驱动的分析范数被制定出来，以正确表征紧致平面流形上量子态希尔伯特空间中的布朗运动。这个范数，即所谓的分形范数，将较粗糙的函数（物理上具有较高能量的量子态）推向希尔伯特空间的更远的点，直到分形函数作为最粗糙的函数移动到无穷大。使用分形范数实现维纳随机过程，会产生维纳测度的修改形式，称为维纳分形测度，它是费曼量子场路径积分公式的明确定义的测度。维纳分形测度具有复杂的非局部项公式，但会产生一阶近似的克莱因-戈登作用。 使用复积分来补偿在高阶近似中出现的非局部项的去除，维纳分形测度变成复测度并生成标量量子场的费曼路径积分公式。这给我们带来了这项研究的主要目标。最后，通过维纳分形测度的分析方面重新审视了量子场论的一些重要方面（例如可重整性、RG流、威克旋转、正则化等）。