  In this manuscript, we will discuss the construction of covariant derivative
operator in quantum gravity. We will find it is more perceptive alternative 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 conventional Palatini and metric-affine formalisms, where the actions are
linear in the scalar curvature with metric and affine connections being the the
independent variables, are not much suitable to construct a source-free theory
of gravity with general affine 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 affine connections and
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 Einstein's equation. This field brings us beyond
strict local Minkowski geometries. These scalar fields can be useful to explain
inflation and dark energy.
