Brownian Motion in the Hilbert Space of Quantum States along with the Ricci Flow and Stochastically Emergent Einstein-Hilbert Action: Formulating a Well-Defined Feynman Path-Integral Measure for Quantum Fields in the Presence of Gravity

TL;DR

This paper links quantum field theory in curved spacetime with Ricci flow and Brownian motion, deriving a well-defined path-integral measure that incorporates gravity and suggests gravity's thermodynamic origin.

Contribution

It introduces a novel approach connecting Ricci flow and stochastic processes to formulate a consistent quantum gravity path integral measure.

Findings

Wiener measure contains Einstein-Hilbert action at first order

Emergence of Einstein-Hilbert action is geometric, not matter-dependent

Derived explicit formula for the cosmological constant

Abstract

In this paper, we aim to interpret the background gravitational effects appearing in quantum field theory on curved space-time by studying the Brownian motion of quantum states along with the Hamilton-Perelman Ricci flow. It has been shown that the Wiener measure automatically contains the Einstein-Hilbert action and the path-integral formulation of the scalar quantum field theory on curved space-time at the first order of local approximations. This provides a well-defined formulation of the path-integral measure for quantum field theory in the presence of gravity. However, we establish that the emergence of Einstein-Hilbert action is independent of the matter field interactions and is a merely entropic/geometric effect stemming from the nature of the Ricci flow of the universe geometry. We also extract an explicit formula for the cosmological constant in terms of the Ricci flow and the Hamilton theorem for 3-manifolds. Then, we discuss the cosmological features of the FLRW solution in the LambdaCDM Model via the derived equations of the Ricci flow. We also argue the correlation between our formulations and the entropic aspects of gravity. Finally, we provide some theoretical evidence that proves the second law of thermodynamics is the basic source of gravity and probably a more fundamental concept.