Title: On Gauge Invariance and Covariant Derivatives in Metric Spaces,
Abstract: In this manuscript, we will discuss the construction of covariant derivative
operator in quantum gravity. We will find it is appropriate 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 standard Palatini
formalism, where metric and affine connections are the independent variables,
is 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 minimally coupled interacting theories where sources
only couple with metric by using the metric compatible Levi-Civita connections
exclusively. We will discuss a potential formalism and possible extensions of
the action to introduce nonmetricity in these cases. This is also required to
construct a general interacting quantum theory with dynamical general affine
connections. We will have to use a modified Ricci tensor to state Einstein's
equation in the Palatini formalism. General affine connections can be described
by a third rank tensor with one contravariant and two covariant indices.
Antisymmetric part of this tensor in the lower indices gives torsion with a
half factor. In the Palatini formalism or its generalizations considered here,
symmetric part of this tensor in the lower indices is finite when torsion is
finite. This part can give a massless scalar field in a potential formalism. We
will have to extend the local conservation laws when we use general affine
connections. General affine connections can become significant to solve
cosmological problems.