Quantum Geometric Perspective on the Origin of Quantum-Conditioned Curvatures

Abstract The quantization of the gravitational field, which includes the metric field, has been investigated using various methods such as loop quantum gravity, quantum field theory, and string theory. Nevertheless, an alternative strategy to tackle the challenge of merging the fundamentally different theories of general relativity (GR) and quantum mechanics (QM) is through a quantum geometric approach. This particular approach entails extending QM to relativistic energies and finite gravitational fields, while also expanding the continuous Riemann to a discretized (quantized) Finsler–Hamilton geometry. By embracing this method, it may be feasible to bridge the gap between GR and QM or even achieve their unification. The resulting fundamental tensor appears to blend its original classical and quantum characteristics, effectively integrating quantum-mechanically induced revisions to the affine connections and spacetime curvatures. Our study primarily focuses on investigating the Ricci curvature tensor in the context of the Einstein–Gilbert–Straus metric. By employing both analytical and numerical methods, we have identified quantum-conditioned curvatures (QCC) that act as additional sources of gravitation. These QCC exhibit a fundamental difference from the traditional curvatures described by Einsteinian GR. While the Ricci curvatures are predominantly positive across most regions, the quantized Ricci curvatures display negativity. We conclude that the QCC (a) possess an intrinsic, essential, and real character, (b) should not be disregarded due to their significant magnitude, and (c) are fundamentally different from the curvatures found in classical GR. Moreover, we conclude that the proposed quantum geometric approach may offer an alternative mathematical framework for understanding the emergence of quantum gravity.