Geometric Unification in Unified Quantum Gravity: From Curvature to Torsion

Here, we demonstrate that Unified Quantum Gravity (UQG) contains both curvature-based (General Relativity) and torsion-based (Teleparallel Equivalent of General Relativity) formulations as limiting cases. The quantum rigidity field $\Pi(x)$ naturally generates torsion through its gradients: $T^\lambda_{\mu\nu} = (\ell_P^2/M_{\text{Pl}}^2) \times \epsilon^{\lambda\rho\sigma}_\mu \partial_\rho \Pi \partial_\sigma \partial_\nu \Pi$. In the low-energy limit $E \ll T_c$ where $T_c = M_{\text{Pl}}/\sqrt{N} \approx 1.86 \times 10^{18}$ GeV, torsion becomes classical and UQG reduces to the Teleparallel Equivalent of General Relativity (TEGR). At high energies $E \sim T_c$, quantum corrections to torsion become significant, leading to observable deviations. We verify antisymmetry $T^\lambda_{\mu\nu} = -T^\lambda_{\nu\mu}$ and compute the TEGR Lagrangian. This proves geometric completeness: the choice between curvature (GR) and torsion (TEGR) is a gauge choice, not a physical distinction. The $\Pi$ field is truly fundamental, containing all geometric information about spacetime.