Quantum Theory, Gravity and Higher Order Geometry
Folkert Kuipers
TL;DR
The paper argues that quantum non-differentiability and classical gravity's differentiability are fundamentally incompatible and proposes higher order geometry, especially second order geometry, as a framework to accommodate Hölder-regular, non-differentiable paths such as those appearing in path integrals. It develops the mathematical structure of rough paths and the It group to extend the tangent space to include second order velocity data, introduces a generalized second order metric, and formulates a toy worldline model—including nonrelativistic and relativistic cases—that yields modified geodesic equations and a complex diffusion-like wave equation. The results show how a covariant, second order description can recover standard physics in appropriate limits (flat spacetime, first order sector) while enabling a gravity-quantum coupling that avoids certain covariant anomalies and hints at a deformed Lorentz symmetry at high energies. The outlook outlines extending the formalism to full field theories, exploring connections to string theory, generalized and non-commutative geometries, and other quantum gravity frameworks, with potential implications for renormalizability and ghost-free higher derivative gravity. The work thus provides a concrete mathematical route to unify quantum and gravitational phenomena within a higher order geometric setting.
Abstract
The fact that quantum theory is non-differentiable, while general relativity is built on the assumption of differentiability sources an incompatibility between quantum theory and gravity. Higher order geometry addresses this issue directly by extending differential geometry, such that it can be applied to theories that are non-differentiable, but have a certain degree of Hölder regularity. As this includes the path integral formulation of quantum theory, it provides a natural mathematical framework for describing the interplay between gravity and quantum theory. In this article, we review the motivation for and the basic features of this framework and point towards future developments.