Quantum Gravity, Constant Negative Curvatures, and Black Holes

Abstract

For purposes of quantization, classical gravity is normally expressed by canonical variables, namely the metric $g_{ab}(x)$ and the momentum ${\pi }^{cd}(x)$. Canonical quantization requires a proper promotion of these classical variables to quantum operators, which, according to Dirac, the favored operators should be those arising from classical variables that formed Cartesian coordinates; sadly, in this case, that is not possible. However, an affine quantization features promoting the metric $g_{ab}(x)$ and the momentric ${\pi }_d^c(x)[\equiv {\pi }^{ce}(x)g_{de}(x)]$ to operators. Instead of these classical variables belonging to a constant zero curvature space (i.e., instead of a flat space), they belong to a space of constant negative curvatures. This feature may even have its appearance in black holes, which could strongly point toward an affine quantization approach to quantize gravity.

9 pages: affine quantization; quantum gravity; constant fixed curvatures; black holes; acknowledgement added; minor correction