PT-Symmetric Quaternionic Spacetime: A Rigorous Framework Bridging General Relativity and Quantum Mechanics

The unification of general relativity (GR) and quantum mechanics (QM) remains an open challenge in fundamental physics. We propose a novel framework extending spacetime into a quaternionic manifold with parity-time (PT\mathcal{PT}PT) symmetry, where each spacetime coordinate is promoted to a quaternion zμ=xμ+\iiyμ+\jjvμ+\kkwμz^\mu = x^\mu + \ii y^\mu + \jj v^\mu + \kk w^\muzμ=xμ+\iiyμ+\jjvμ+\kkwμ, introducing three hidden imaginary dimensions. Utilizing a symmetrized derivative and a real-projected (Moore/Dieudonné-inspired) determinant, we construct an extended Einstein–Hilbert action where the imaginary components of the metric contribute an effective energy–momentum tensor. This modification leads to observable effects that can naturally account for dark-sector phenomena, including dark energy and dark matter.We demonstrate that this construction is globally consistent under hyperkähler/quaternionic-Kähler conditions, and that PT\mathcal{PT}PT-symmetry ensures physically measurable quantities remain real. A toy FLRW cosmological model with small perturbations (ϵ≲10−5\epsilon \lesssim 10^{-5}ϵ≲10−5) aligns with observed dark energy behavior. Constraints from gravitational-wave data (GW170817) impose a bound on deviations in gravitational-wave speed, while renormalization group (RG) flow suggests a natural suppression of imaginary components at high energies, ensuring compatibility with low-energy GR and QFT. Further, we explore a quaternionic Dirac operator that accommodates mirror states while preserving unitarity.This framework offers a new geometric approach to quantum gravity, providing testable predictions in cosmology, astrophysics, and high-energy physics. We discuss observational constraints, potential collider signatures, and future extensions involving K-theory and Hopf-algebra-based quantization.