Quantum Theory, Gravity and Second Order Geometry

TL;DR

This paper proposes extending Riemannian geometry to second order to consistently couple quantum theory with gravity, leading to higher-dimensional tangent spaces and new physical theory constructions that avoid instabilities.

Contribution

It introduces second order Riemannian geometry for quantum gravity coupling, revealing higher-dimensional tangent spaces and new stable higher-derivative theories.

Findings

Tangent spaces become 18-dimensional in 4D spacetime.

Quadratic derivative terms are perpendicular to first order terms on flat spacetime.

Higher order derivatives do not lead to Ostrogradsky instability due to order mixing.

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.