Action Functional Incorporating $\pi$ for Quantum-Modified $f(Q)$ Gravity

Abstract

We propose a novel action functional that integrates the mathematical constant $\pi$ to connect modified $f(Q)$ gravity with quantum-mechanical phase dynamics. This construction utilizes complex-analytic structures, including poles and essential singularities, alongside angle-amplitude relationships where $\pi$ arises naturally. The action is formulated with mathematical rigor, featuring key equations in dedicated environments. We demonstrate how this functional captures quantum-gravitational effects, such as by imposing periodic phase identifications modulo $2\pi$, akin to quantum interference conditions, and how non-metricity in $f(Q)$ gravity couples to these quantum phases. The significance of complex singularities is clarified by associating pole-type singularities with discrete spectra or stable states, and essential singularities with non-perturbative or chaotic regimes. Drawing inspiration from a recent discrete-action graph model featuring layered graph structures and emergent $SU(3)/U(1)$ symmetries, we highlight how discrete time steps of $\hbar$ can be modeled effectively through a $\pi$-periodic phase functional in the continuum limit. Finally, we propose observational strategies, including precision interferometry and cosmological surveys, to provide falsifiable tests for this quantum-modified $f(Q)$ gravity framework.

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