On Gauge Invariance and Covariant Derivatives in Metric Spaces.
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.
Publisher URL: http://arxiv.org/abs/1702.02384
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