Quantum Theory, Gravity and Second Order Geometry

TL;DR

The paper argues that coupling quantum theory to gravity requires extending standard Riemannian geometry to a second order framework that includes both a line and an area element, yielding 18-dimensional tangent spaces in 4D. It develops the mathematical structure (second order tangent/cotangent spaces, covariant mappings, and a second order metric) and shows how to construct covariant quantum actions on curved spacetimes, with Lagrangians/Hamiltonians that depend on both first- and second-order velocity data. A key result is that higher-derivative terms can be present without Ostrogradsky ghosts due to order mixing, and the framework naturally generates a Pauli-DeWitt-type correction to the Hamiltonian while maintaining a sensible energy-momentum relation. The work also discusses generalizations, potential connections to generalized and non-commutative geometry, and outlines directions for extending to field theories and quantum gravity contexts, including the appearance of a fundamental area scale and links to broader quantum geometric frameworks.

Abstract

We argue that a consistent coupling of a quantum theory to gravity requires an extension of ordinary `first order' Riemannian geometry to second order Riemannian geometry, which incorporates both a line element and an area element. This extension results in a misalignment between the dimension of the manifold and the dimension of the tangent spaces. In particular, we find that for a 4-dimensional spacetime, tangent spaces become 18-dimensional. We then discuss the construction of physical theories within this framework, which involves the introduction of terms that are quadratic in derivatives in the action. On a flat spacetime, the quadratic sector is perpendicular to the first order sector and only affects the normalization of the path integral, whereas in a curved spacetime the quadratic sector couples to the first order sector. Moreover, we show that, despite the introduction of higher order derivatives, the Ostragradski instability can be avoided, due to an order mixing of the two sectors. Finally, we comment on extensions to higher order geometry and on relations with non-commutative and generalized geometry.