Crisis and Hope in Quantum Field Theory through Dimensional Analysis and Geometric Resummation

Abstract

Quantum field theory organizes interactions into calculable contributions, but the usefulness of a perturbative expansion depends on its physical scales and coefficients. This article examines the contrast between quantum electrodynamics and gravity through dimensional analysis, then considers whether reorganizing a divergent expansion can provide insight into unfamiliar gravitational behavior. The electromagnetic fine-structure constant illustrates the value of a dimensionless coupling without guaranteeing convergence. For gravity, the dimensionless parameter associated with a characteristic microscopic energy grows as the square of its ratio to the Planck energy, indicating where a low-energy expansion may lose control. A geometric-series model makes the distinction between convergence and analytic continuation explicit. Its continued expression changes sign across a pole, but that mathematical property does not establish gravitational repulsion. Connecting such a proposal to cosmology requires a physical observable and consistent equations for spacetime evolution. Established inflationary mechanisms provide examples of how specified dynamics can produce accelerated expansion. The article argues that the promise of perturbative breakdown lies in identifying the additional structure needed for a controlled description, while retaining dimensional analysis and mathematical reformulation as useful starting points for theoretical exploration.