Intrinsic Regularization via Curved Momentum Space: A Geometric Solution to Divergences in Quantum Field Theory

TL;DR

This paper introduces a geometric regularization method for quantum field theory that uses curved momentum space to naturally suppress divergences while preserving symmetries, offering a conceptually satisfying alternative to traditional techniques.

Contribution

It proposes a self-consistent, geometry-based regularization framework that inherently ensures convergence without external regulators or symmetry breaking.

Findings

Successfully constructs a measure that guarantees finiteness of key QFT integrals.

Maintains Lorentz invariance and other fundamental symmetries.

Extends seamlessly from Riemannian to Minkowski space.

Abstract

The problem of UV divergences in QFT has long been a fundamental challenge. Standard regularization techniques modify high-energy behavior to ensure well-defined integrals. However, these approaches often introduce unphysical parameters, rely on arbitrary prescriptions, or break fundamental symmetries, making them mathematically effective but conceptually unsatisfactory. We propose a novel and self-consistent approach in which UV regularization emerges naturally from the curved geometry of momentum space. Through curved momentum space, imposed by a geodesic metric, we construct an integral measure that inherently suppresses high-energy divergences while preserving fundamental symmetries, including full Lorentz invariance. This framework is self-sufficient, i.e. requires no external regulators. It retains equations of motion and is fully compatibility with standard field theory Our approach guarantees the weakest possible suppression necessary for convergence, avoiding excessive modifications to quantum behavior, still achieving convergence. While formulated in Riemannian Geometry, we show seamless extension to Minkowski space, maintainining regularization properties in relativistic QFT. This offers an alternative to ad hoc renormalization, providing an intrinsic and mathematically well-motivated suppression mechanism, purely rooted in curved geometry of momentum space. We rigorously construct the measure-theoretic framework and demonstrate its effectiveness by proof of finiteness for key QFT integrals. Beyond resolving divergences, this work suggests broader applications in spectral geometry, effective field theory, and potential extensions to quantum gravity.