Quantum Brownian Motion of Gauge Fields Along the Ricci Flow and the Emergence of Yang-Mills Theory in the Presence of Einstein-Hilbert Action: Toward a Coherent Path-Integral Measure for the Standard Model and Gravity

We propose a novel theoretical framework bridging quantum field theory andgeneral relativity, based on the Wiener stochastic process within the infinite-dimensionalHilbert space of quantum states. By constructing a rigorously defined path-integral measurefor quantum fields, we explore the geometry of Ehresmann connections on a principal fiberbundle with intrinsic fractal structure inspired by Weierstrass-like functions. We show thatbackground gravitational effects, conventionally described by the Einstein-Hilbert action,emerge naturally as an entropic consequence of the universe’s geometric evolution under theHamilton-Perelman Ricci flow. Through critical approximations, the stochastic dynamicsreproduce the Yang-Mills action, including a dynamically generated gauge-fixing term thataddresses the Gribov ambiguity via non-local mechanisms. Incorporating Grassmann-valuedspinor fields and applying the Lichnerowicz formula for the perturbed Dirac operator,we demonstrate a structural correspondence between the requirements for a well-definedWiener fractal measure and the fermionic and bosonic content of the Standard Model(including massive neutrinos) and N = 2 supersymmetric Yang-Mills theory. These resultssuggest that the fundamental forces and gravity may emerge from a unified underlyingfractal stochastic dynamics in a fluctuating spacetime manifold.Keywords: Weierstrass functions, fractals, Wiener stochastic process, Wiener measure,Brownian motion, Laplace-Beltrami operator, principal bundle, Ehresmann connection,Bochner Laplacian, Weitzenb¨ock formula, Lichnerowicz formula, N = 2 supersymmetricYang-Mills theory, Standard Model, Hamilton-Perelman Ricci flow, Einstein-Hilbert action.