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

TL;DR

This paper develops a mathematically rigorous reformulation of the Feynman path-integral measure for quantum fields using a fractal norm in the Hilbert space of quantum states, potentially advancing quantum gravity research.

Contribution

It introduces the Wiener fractal measure based on a fractal norm, addressing limitations of the classic Wiener measure in infinite-dimensional Hilbert spaces for quantum field theory.

Findings

Defined the Wiener fractal measure for quantum states

Reformulated Feynman path-integral measure with fractal geometry

Provided a framework for quantum gravity applications

Abstract

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 In this research, we focus on the fact that the classic Wiener measure is no longer applicable in infinite-dimensional Hilbert spaces due to fundamental differences between displacements in low and extremely high dimensions.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. ...