# On Gauge Theories and Covariant Derivatives in Metric Spaces

## TL;DR

This paper explores the use of more general affine connections in quantum gravity, revealing new scalar fields that could explain dark energy and inflation, and challenging traditional metric-compatible approaches.

## Contribution

It introduces a formalism for affine connections beyond metric compatibility in quantum gravity, with potential implications for cosmology and the structure of spacetime.

## Key findings

- General affine connections introduce two massless scalar fields.
- One scalar field contributes a negative stress-tensor in Einstein's equations.
- This approach suggests new avenues for explaining dark energy and inflation.

## Abstract

In this manuscript, we will discuss the construction of covariant derivative operator in quantum gravity. We will find it is more perceptive to use affine connections more general than metric compatible connections in quantum gravity. We will demonstrate this using the canonical quantization procedure. This is valid irrespective of the presence and nature of sources. The Palatini and metric-affine formalisms, where metric and affine connections are the independent variables, are not sufficient to construct a source-free theory of gravity with affine connections more general than the metric compatible Levi-Civita connections. This is also valid for many minimally coupled interacting theories where sources only couple with metric by using the Levi-Civita connections exclusively. We will discuss potential formalism of affine connections to introduce affine connections more general than metric compatible connections in gravity. We will also discuss possible extensions of the actions for this purpose. General affine connections introduce new fields in gravity besides metric. In this article, we will consider a simple potential formalism with symmetric Ricci tensor. Corresponding affine connections introduce two massless scalar fields. One of these fields contributes a stress-tensor with opposite sign to the sources of Einstein's equation when we state the equation using the Levi-Civita connections. This means we have a massless scalar field with negative stress-tensor in the familiar Einstein equation. These scalar fields can be useful to explain dark energy and inflation. These fields bring us beyond strict local Minkowski geometries.