Quantum Gravity Without Metric Quantization: From Hidden Variables to Hidden Spacetime Curvatures

Abstract: Bohmian mechanics offers a deterministic alternative to conventional quantum theory through well-defined particle trajectories. While successful in nonrelativistic contexts, its extension to curved spacetime-and hence quantum gravity-remains unresolved. Here, we develop a covariant extension of Bohmian mechanics in curved spacetime that removes the need for metric quantization. From a Lagrangian formulation, we derive a generalized guidance equation in which Bohmian trajectories generate hidden curvature, replacing metric superposition with a statistical ensemble constrained by Heisenberg uncertainty, offering a novel perspective on quantum gravity. Consequently, in our approach, measuring the gravitational potential at a point unveils a pre-existing trajectory and its associated curvature-a departure from the observer-centric paradigm of standard quantum mechanics-providing an alternative in which gravitational effects emerge from deterministic quantum trajectories rather than wavefunction collapse. Numerical simulations in Robertson-Walker and cigar soliton spacetimes reveal that while quantum interference is curvature-sensitive, Zitterbewegung remains invariant, distinguishing fundamental quantum effects. Moreover, deviations from the Born rule in inhomogeneous spacetimes are observed and suggest gravity-induced quantum non-equilibrium. This new approach has far-reaching implications for the role of determinism and potential observational signatures of quantum non-equilibrium in cosmology.